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--- abstract: | We study functions on topometric spaces which are both (metrically) Lipschitz and (topologically) continuous, using them in contexts where, in classical topology, ordinary continuous functions are used. 1. We define *normal* topometric spaces and characterise them by analogues of Urysohn’s Lemma and Tietze’s Extension Theorem. 2. We define *completely regular* topometric spaces and characterise them by the existence of a topometric Stone-Čech compactification. 3. For a compact topological space $X$, we characterise the subsets of $\cC(X)$ which can arise as the set of continuous $1$-Lipschitz functions with respect to a topometric structure on $X$. address: | Itaï <span style="font-variant:small-caps;">Ben Yaacov</span>\ Université Claude Bernard – Lyon 1\ Institut Camille Jordan, CNRS UMR 5208\ 43 boulevard du 11 novembre 1918\ 69622 Villeurbanne Cedex\ France author: - 'Itaï <span style="font-variant:small-caps;">Ben Yaacov</span>' bibliography: - 'begnac.bib' title: Lipschitz functions on topometric spaces --- [^1] $Id: LipTM.tex 1161 2010-10-07 16:19:06Z begnac $ [^2] Introduction {#introduction .unnumbered} ============ Topometric spaces are spaces equipped both with a metric and a topology, *which need not agree*. To be precise, \[dfn:TopoMetric\] A *topometric space* is a triplet $(X,\sT,d)$, where $\sT$ is a topology and $d$ a metric on $X$, satisfying: 1. The distance function $d\colon X^2 \to [0,\infty]$ is lower semi-continuous in the topology. 2. The metric refines the topology. We follow the convention that unless explicitly qualified, the vocabulary of general topology (compact, continuous, etc.) refers to the topological structure, while the vocabulary of metric spaces (Lipschitz function, etc.) refers to the metric structure. Excluded from this convention are separation axioms: we assimilate the lower semi-continuity of the distance function to the Hausdorff separation axiom, and stronger axioms, such as normality and complete regularity, will be defined for topometric spaces below. Compact topometric spaces were first defined in [@BenYaacov-Usvyatsov:CFO] as a formalism for various global and local type spaces arising in the context of continuous first order logic. General topometric spaces (i.e., non compact) were defined studied further from an abstract point of view in [@BenYaacov:TopometricSpacesAndPerturbations]. Further examples include types spaces for unbounded logic (merely locally compact), perturbation structures on type spaces. A very different class of examples, very far from being compact or even locally compact in general, is formed by automorphism groups of (metric) structures, as well as Polish groups or (completely) metrisable ones. In addition, there are two classes of examples which recur throughout the paper, arising from the embedding of the categories of (Hausdorff) topological spaces and of metric spaces in the category of topometric spaces. By a *maximal* topometric space we mean one equipped with the discrete $0/1$ distance, which can be identified, for all (or most) intents and purposes, with its underlying pure topological structure. Similarly, a *minimal* topometric space is one in which the metric and topology agree, which may be identified with its underlying metric structure. These sometimes serve as first sanity checks (e.g., when we define a normal topometric space we must check that a maximal one is normal if and only if it is normal as a pure topological space, and that minimal ones are always normal). The aim of this paper is to study some basic properties of the class of (topologically) continuous and (metrically) Lipschitz functions on a topometric space. These are naturally linked with separation axioms. For example, existence results such as Urysohn’s Lemma and Tietze’s Extension Theorem are tied with normality, discussed in , while the Stone-Čech compactification (defined in terms of a universal property with respect to continuous Lipschitz functions) is related to complete regularity, as discussed in . To conclude, characterises the bare minimum that the set of Lipschitz functions needs to satisfy. Lipschitz functions on an ordinary metric spaces, and algebras thereof, are extensively studied in Weaver [@Weaver:LipschitzAlgebras]. This is some natural resemblance between our object of study here and that of Weaver, with the increased complexity due to the additional topological structure. Th reader may wish to compare, for example, our version of Tietze’s Extension Theorem () with [@Weaver:LipschitzAlgebras Theorem 1.5.6] (as well as with the classical version of Tietze’s Theorem, see Munkres [@Munkres:Topology]). Normal topometric spaces and Urysohn–Tietze results {#sec:Normal} =================================================== For two topometric spaces $X$ and $Y$ we define $\cC_{\cL(1)}(X,Y)$ to be the set of all continuous $1$-Lipschitz functions from $X$ to $Y$. An important special case is $\cC_{\cL(1)}(X) = \cC_{\cL(1)}(X,\bC)$, where $\bC$ is equipped with the standard metric and topology (i.e., with the standard minimal topometric structure), which codes information both about the topology and about the metric structure of $X$. In the present paper we seek conditions under which $\cC_{\cL(1)}(X)$ codes the entire topometric structure, as well as analogues of classical results related to separation axioms, in which $\cC(X)$ would be replaced with $\cC_{\cL(1)}(X)$. As discussed in [@BenYaacov:TopometricSpacesAndPerturbations], we consider the lower semi-continuity of the distance function to be a topometric version of the Hausdorff separation axiom, so we may expect other classical separation axioms to take a different form in the topometric setting. We start with normality. Let $X$ be a topometric space. We say that a closed set $F \subseteq X$ has *closed metric neighbourhoods* if for every $r > 0$ the set $\overline B(F,r) = \{x \in X\colon d(x,F) \leq r\}$ is closed in $X$. We say that $X$ *admits closed metric neighbourhoods* if all closed subsets of $X$ do. It was shown in [@BenYaacov:TopometricSpacesAndPerturbations] that compact sets always have closed metric neighbourhoods, so a compact topometric space admits closed metric neighbourhoods. Indeed, the first definition of a *compact* topometric space in [@BenYaacov-Usvyatsov:CFO] was given in terms of closed metric neighbourhoods. While this property seems too strong to be part of the definition of a non compact topometric space, it will play a crucial role in this section. A *normal topometric space* is a topometric space $X$ satisfying: 1. Every two closed subset $F,G \subseteq X$ with positive distance $d(F,G) > 0$ can be separated by disjoint open sets. 2. The space $X$ admits closed metric neighbourhoods. One checks that a maximal topometric space $X$ (i.e., equipped with the discrete $0/1$ distance) is normal if and only if it is so as a topological space. Similarly, a minimal topometric space (i.e., equipped with the metric topology) is always normal. Also, every compact topometric space is normal (since it admits closed metric neighbourhoods and the underlying topological space is normal). We contend that our definition of a normal topometric space is the correct topometric analogue of the classical notion of a normal topological space. This will be supported by analogues of Urysohn’s Lemma and of Tietze’s Extension Theorem. The technical core of the proofs (and indeed, the only place where the definition of a normal topometric space is used) lies in the following Definition and Lemma. Let $X$ be a topometric space, $c > 0$ a constant, $S \subseteq \bR$ and $\Xi_S = \{(F_\alpha,G_\alpha)\colon \alpha \in S\}$ a sequence of pairs of closed sets $F_\alpha,G_\alpha \subseteq X$. 1. We say that $\Xi_S$ is an *approximation of a strictly $c$-Lipschitz partial continuous function on $X$*, or simply a *partial $c$-Lipschitz approximation*, if $d(F_\alpha,G_\beta)c > \beta - \alpha$ for $\alpha < \beta$ in $S$. 2. It is a *(total) approximation* if in addition $F_\alpha \cup G_\alpha = X$ for all $\alpha \in S$ (so particular $G_\alpha^c \subseteq F_\alpha \subseteq G_\beta^c \subseteq F_\beta$ for $\alpha < \beta$). 3. We say that $\Xi_S$ is an approximation of a function $f\colon X \to \bR$ if $f\rest_{F_\alpha} \leq \alpha$ and $f\rest_{G_\alpha} \geq \alpha$ for $\alpha \in S$. If $f\colon X\to \bR$ is $c$-Lipschitz and $S \subseteq \bR$ then the sequence $\{(F_\alpha,G_\alpha)\colon \alpha \in S\}$ defined by $F_\alpha = \{x\colon f(x) \leq \alpha\}$, $G_\alpha = \{x\colon f(x) \geq \alpha\}$ is a $c'$-Lipschitz approximation $f$ for all $c' > c$. \[lem:PartialApproximationExtension\] Let $\{(F_\alpha,G_\alpha)\colon \alpha \in S\}$ be a finite partial $c$-Lipschitz approximation in a normal topometric space, and let $\beta \in S$. Then there are $F'_\beta,G'_\beta \subseteq X$ such that - $F'_\beta \supseteq F_\beta$, $G'_\beta \supseteq G_\beta$. - $F'_\beta \cup G'_\beta = X$. - Letting $F'_\alpha = F_\alpha$ and $G'_\alpha = G_\alpha$ for $\alpha \neq \beta$ then $\{(F'_\alpha,G'_\alpha)\colon \alpha \in S\}$ is a partial $c$-Lipschitz approximation. Since the partial approximation is finite it is also $c'$-Lipschitz for some $c' < c$. Define: $$\begin{gathered} K = \bigcup_{\alpha \in S, \alpha < \beta} \overline B(F_\alpha,(\beta-\alpha)/c'), \qquad L = \bigcup_{\alpha \in S, \alpha > \beta} \overline B(G_\alpha,(\alpha-\beta)/c'). \end{gathered}$$ By construction $d(K,L) > 0$ and both are closed as finite unions of closed sets. Since $X$ is normal we can find disjoint open sets $U \supseteq K$ and $V \supseteq L$. We claim that $F'_\beta = F_\beta \cup V^c$ and $G'_\beta = G_\beta \cup U^c$ will do. The first two items are trivially verified, so we only need to check the last one. So assume that $\alpha < \beta$. We already know by hypothesis that $d(F_\alpha,G_\beta)c > \beta - \alpha$. We also know by construction that $U \supseteq \overline B(F_\alpha,(\beta-\alpha)/c')$, whereby $d(F_\alpha,U^c)c > d(F_\alpha,U^c)c' \geq \beta - \alpha$. Thus $d(F_\alpha,G'_\beta)c > \beta - \alpha$. We show similarly that if $\beta < \alpha$ then $d(F'_\beta,G_\alpha)c > \alpha - \beta$, and we are done. \[lem:ApproximationExtension\] Let $X$ be a normal topometric space, $\Xi_S = \{(F_\alpha,G_\alpha)\colon \alpha \in S\}$ a finite $c$-Lipschitz approximation. Then for every $\beta \in \bR$ there is a $c$-Lipschitz approximation $\Xi'_{S\cup\{\beta\}} \supseteq \Xi_S$. We may assume that $\beta \notin S$, and let $F_\beta = G_\beta = \emptyset$. Then $\big\{ (F_\alpha,G_\alpha)\colon \alpha \in S\cup\{\beta\} \big\}$ is a partial $c$-Lipschitz approximation and (with the same $\beta$) we obtain the required approximation $\Xi'_{S\cup\{\beta\}} = \bigl\{ (F'_\alpha,G'_\alpha) \bigr\}_{\alpha \in S\cup\{\beta\}}$. \[prp:ContinuousLipschitzApproximation\] In a normal topometric space every finite approximation of a $c$-Lipschitz continuous function approximates such a function. Let $X$ be a normal topometric space, $\{(F_\alpha,G_\alpha)\colon \alpha \in S\}$ a finite $c$-Lipschitz approximation. Since $S$ is finite its convex hull is a compact interval $I \subseteq \bR$. Let $T \subseteq I$ be a countable dense subset containing $S$. By repeated applications of one can extend the given approximation into a $c$-Lipschitz approximation $\{(F_\alpha,G_\alpha)\colon \alpha \in T\}$. Letting $f(x) = \sup\{\alpha\in I\colon x \in G_\alpha\} = \inf\{\alpha \in I\colon x\in F_\alpha\}$ (here $\inf \emptyset = \sup I$ and $\sup \emptyset = \inf I$) one obtains a continuous, $c$-Lipschitz function $f\colon X \to I$ which is approximated by $\{(F_\alpha,G_\alpha)\colon \alpha \in S\}$. The topometric analogue of Urysohn’s Lemma is obtained as an easy corollary. \[cor:TopometricUrysohn\] Let $X$ be a normal topometric space, $F,G \subseteq X$ closed sets, $0 < r < d(F,G)$. Then there exists a $1$-Lipschitz continuous function $f\colon X\to [0,r]$ equal to $0$ on $F$ and to $r$ on $G$. Conversely, every topometric space in which this property holds is normal. Apply to $S = \{0,r\}$, $F_0 = F$, $G_r = G$, $G_0 = F_r = X$. Assume now that the first property holds in $X$. Then closed sets of positive distance can be separated by a $1$-Lipschitz continuous function, and therefore by open sets. Also, if $F \subseteq X$ is closed and $d(x,F) > r$ then we may separate $F$ and $x$ by a $1$-Lipschitz continuous function such that $f\rest_F = 0$ and $f(x) > r$. Then $\{y\colon f(y) \leq r\}$ is a closed set containing $\overline B(F,r)$ but not $x$. If follows that $\overline B(F,r)$ is closed. \[lem:FiniteApproximationExtension\] Let $X$ be a normal topometric space, $Y \subseteq X$ closed. Then for every finite $c$-Lipschitz approximation $\{(F_\alpha,G_\alpha)\colon \alpha \in S\}$ in $Y$ there is one $\{(F'_\alpha,G'_\alpha)\colon \alpha \in S\}$ in $X$ such that $F'_\alpha \supseteq F_\alpha$, $G'_\beta \supseteq G_\beta$. Observe that $\{(F_\alpha,G_\alpha)\colon \alpha \in S\}$ is a partial $c$-Lipschitz approximation on $X$, so we may apply to each $\alpha \in S$ and obtain the required approximation. Observe that the forced limit operator $\flim\colon [0,1]^\bN \to [0,1]$ defined in [@BenYaacov-Usvyatsov:CFO] is $1$-Lipschitz where $[0,1]^\bN$ is equipped with the supremum metric. \[thm:Tietze\] Let $X$ be a normal topometric space. Then for every $c < c'$ every continuous $c$-Lipschitz function $f\colon Y \to [0,1]$ on a closed subset $Y \subseteq X$ extends to a continuous $c'$-Lipschitz function $g\colon X \to [0,1]$. Moreover, for an arbitrary topometric space the following are equivalent: 1. $X$ is a normal topometric space. 2. Tietze’s Extension Theorem for topometric spaces (i.e., the statement above) holds in $X$. 3. The statement of holds in $X$. 4. Urysohn’s Lemma (the main assertion of ) holds in $X$. Let $Y \subseteq X$ be closed, $f\colon Y \to [0,1]$ be continuous and $c$-Lipschitz. For $\alpha \in [0,1]$ let $F_\alpha = f^{-1}([0,\alpha])$ and $G_\alpha = f^{-1}([\alpha,1])$. For $n \in \bN$ let $S_n = \{k2^{-n}\colon 0 \leq k \leq 2^n\}$, and $\Xi_n = \{(F_\alpha,G_\alpha)\colon \alpha \in S_n\}$. Then $\Xi_n$ is a $c'$-Lipschitz approximation on $Y$ for any $c' > 0$. By it admits an extension $\Xi_n' = \{(F'_{n,\alpha},G'_{n,\alpha})\colon \alpha \in S_n\}$ to $X$ (which may depend on $n$) which is $c'$-Lipschitz as well. By there exists a continuous $c'$-Lipschitz function $g_n\colon X \to[0,1]$ approximated by $\Xi_n'$, and let $g = \flim g_n$. Notice that if $y \in Y$ and $k2^{-n} \leq f(y) \leq (k+1)2^{-n}$ then $y \in F_{(k+1)2^{-n}} \cap G_{k2^{-n}} \subseteq F'_{n,(k+1)2^{-n}} \cap G'_{n,k2^{-n}}$, whereby $k2^{-n} \leq g_n(y) \leq (k+1)2^{-n}$ as well. Thus $|g_n\rest_Y-f| \leq 2^{-n}$ for all $n$ whereby $g\rest_Y = f$. Also, a forced limit of a family of continuous $c'$-Lipschitz functions is continuous and $c'$-Lipschitz. For the moreover part, we have seen that if $X$ is normal then (ii)-(iv) hold. Conversely, each of (ii) and (iii) clearly implies (iv), and by (iv) implies that $X$ is normal. This proof of Tiezte’s theorem is fairly different from other the author managed to find in the literature. Indeed none of the more common proofs seems to be capable of preserving the Lipschitz condition. Completely regular topometric spaces and Stone-Čech compactification {#sec:StoneCech} ==================================================================== Let $\{X_i\colon i \in I\}$ be a family of topometric spaces. We equip the set $\prod_{i \in I} X_i$ with the product topology and the supremum metric $d(\bar x,\bar y) = \sup\{d(x_i,y_i)\colon i \in I\}$. One verifies easily the result is indeed a topometric space which we call the *product topometric structure*. In particular we obtain large compact topometric spaces of the form $[0,\infty]^I$, and we claim that these are in some sense universal, meaning that every compact topometric space embeds in one of those. Similarly, every bounded compact topometric (i.e., of finite diameter) can be embedded in $[0,M]^I$, and up to re-scaling in $[0,1]^I$. In fact we shall show that every completely regular topometric space embeds in such a space, obtaining a Stone-Čech compactification. Say that a family of functions $\cF \subseteq \bC^X$ *separates points from closed sets* if for every closed set $F \subseteq X$ and $x \in X \setminus F$, there is a function $f \in \cF$ which is constant on $F$ and takes some different value at $x$. \[fct:TopologicalCompletelyRegularEmbedding\] Let $X$ be a Hausdorff topological space, $\cF \subseteq \cC(X)$ a family separating points from closed sets. Then the map $\theta\colon X \to \bC^\cF$ defined by $x \mapsto (f \mapsto f(x))$ is a topological embedding. This is fairly standard. First of all $\cF$ separates points so $\theta$ is injective. To see that $\theta$ is continuous, it is enough to consider a sub-basic open set $U = \pi_f^{-1}(V) \subseteq{ \bC^\cF}$, where $V \subseteq \bC$ is open and $\pi_f$ is the projection on the $f$th coördinate. Then $\theta^{-1}(U) = f^{-1}(V)$ is open. In order to show that $\theta$ is a homeomorphism with its image it will be enough to show that for $F \subseteq X$ closed and $x \notin F$ there is a closed set $F' \subseteq \bC^\cF$ such that $\theta(F) \subseteq F'$ and $\theta(x) \notin F'$. Since $\cF$ separates points from closed sets there is $f \in \cF$ such that $f\rest_F = t$ and $f(x) \neq t$. Then $F' = \{\bar y \in \bC^\cF\colon y_f = t\}$ will do. \[dfn:CompletelyRegular\] Let $X$ be a topometric space. Say that a family of functions $\cF \subseteq \cC_{\cL(1)}(X)$ is *sufficient* if 1. It separates points and closed sets. 2. For $x,y \in X$ we have $$\begin{gathered} d(x,y) = \sup \bigl\{ |f(x)-f(y)|\colon f \in \cF \bigr\}. \end{gathered}$$ (Clearly, $\geq$ always holds.) A topometric space $X$ is *completely regular* if $\cC_{\cL(1)}(X)$ is sufficient. This is clearly equivalent to $\cC_{\cL(1)}(X,\bR^+)$ being sufficient. In view of we may say that a topometric space $X$ is completely regular if $\cC_{\cL(1)}(X)$ captures both the topological structure and the metric structure of $X$. 1. Every normal topometric space is completely regular. 2. Every subspace of a completely regular space is completely regular. 3. Let $X$ be a maximal topometric space. Then it is topologically completely regular if and only if it is topometrically completely regular. The first item follows from , keeping in mind that since the metric of a topometric space $X$ refines its topology, if $F \subseteq X$ is closed and $x \notin F$ then $d(x,F) > 0$. For the second item, assume that $X$ is completely regular, $Y \subseteq X$. If $F \subseteq Y$ is closed then $F = Y\cap \overline F$, where $\overline F$ is the closure in $X$. Thus if $x \in Y \setminus F$ then $x \in X \setminus \overline F$, so there is a $1$-Lipschitz continuous function separating $\overline F$ from $x$, and its restriction to $Y$ is continuous and $1$-Lipschitz as well. The same argument works for witnessing distances. The last item follows from the fact that every function from a maximal topometric space to $[0,1]$ is $1$-Lipschitz. \[prp:CompletelyRegularEmbedding\] Let $X$ be a completely regular topometric space and let $\cF = \cC_{\cL(1)}(X,\bR^+)$. Then the map $\theta\colon X \to (\bR^+)^\cF$ from is a topometric embedding, i.e., an isometric homeomorphic embedding. Immediate from the definitions. Every completely regular topometric space $X$ (and thus in particular every normal or compact one) embeds in some power of $[0,\infty]$. If in addition $X$ is bounded, say of diameter $1$, then it embeds in a power of $[0,1]$. We just have to show the last part. Indeed let $\theta \colon X \to [0,\infty]^I$ be any embedding. Define $\theta' \colon X \to [0,\infty]^I$ by $\theta'(x)(i) = \theta(x)(i) - \inf\{\theta(y)(i)\colon y \in X\}$ (here $\infty-\infty = d(\infty,\infty) = 0$). Then $\theta'$ is an embedding as well, and $\bar 0 \in \theta(X)$. If $X$ is bounded of diameter $1$ then $\theta(X) \subseteq [0,1]^I$. A topometric space admits a compactification if and only if it is completely regular. If $X$ is completely regular then we can identify it with a subspace of $[0,\infty]^I$, and then its closure there is a compactification. Conversely, assume $X$ admits a compactification $\bar X$. Then $\bar X$ is completely regular, whereby so is $X$. \[thm:StonCechUniversalProperty\] Let $X$ be completely regular. Then it admits a compactification $\beta X$ satisfying the following universal property: Every $1$-Lipschitz continuous function $f\colon X \to [0,\infty]$ can be extended to such a function on $\beta X$ (and the extension is unique). Moreover, $\beta X$ is unique up to a unique isomorphism (i.e., isometric homeomorphism) and satisfies the same universal property with any compact topometric space $Y$ instead of $[0,\infty]$. Let $\cF = \cC_{\cL(1)}(X,\bR^+)$ and let $\theta\colon X \to (\bR^+)^\cF \subseteq [0,\infty]^\cF$ be as in . Identify $X$ with $\theta(X)$ and let $\beta X$ be its closure in $[0,\infty]^\cF$. For $f \in \cF$, let $\pi_f\colon [0,\infty]^\cF \to [0,\infty]$ be the projection on the $f$th coordinate. Then $\pi_f \circ \theta = f$, so $\pi_f\colon \beta X \to [0,\infty]$ is as required. Given $f \in \cC_{\cL(1)}(X,[0,\infty])$ and $n \in \bN$, the truncation $f \wedge n\colon X \to [0,n]$ belongs to $\cF$ and the sequence $\pi_{f \wedge n}$ is increasing, converging point-wise to some $g\colon \beta X \to [0,\infty]$. The collection of open subsets of $[0,\infty]$ which are either bounded or contain $\infty$ forms a base. For such an open set $U$ there is $n$ such that either $[n,\infty] \subseteq U$ or $U \subseteq [0,n]$, and in either case $g^{-1}(U) = (f \wedge n)^{-1}(U)$ is open. Thus $g$ is continuous. (Of course we could have also let $\cF = \cC_{\cL(1)}(X,[0,\infty])$ to begin with.) Now let $Y$ be any compact topometric space. Then $Y$ embeds in $[0,\infty]^J$ for some $J$. If $f \in \cC_{\cL(1)}(X,Y)$ then $\pi_j \circ f \in \cC_{\cL(1)}(X,[0,\infty])$ for $j \in J$ and thus extends to $g_j \in \cC_{\cL(1)}(\beta X,[0,\infty])$. Let $g = (g_j)\colon \beta X \to [0,\infty]^J$, so $g\rest_X = f$. Then $g(X) \subseteq Y$, $X$ is dense in $\beta X$ and $Y$ is closed in $[0,\infty]^J$, so $g(\beta X) \subseteq Y$ as required. The uniqueness of an object satisfying this universal property is now standard. In other words, for every compact $Y$ the restriction $\cC_{\cL(1)}(\beta X,Y) \to \cC_{\cL(1)}(X,Y)$ is bijective. The compactification $\beta X$, if it exists (i.e., if $X$ is complete regular) is called the *Stone-Čech compactification* of $X$. Automorphism groups of metric structures probably form the most natural class of examples of non (locally) compact topometric spaces. They are easily checked to be completely regular. Let $\cM$ be a metric structure and let $G = \Aut(\cM)$, equipped with the topology $\sT$ of point-wise convergence and with the distance $d_u$ of uniform convergence. Then $(G,\sT,d_u)$ is a completely regular topometric space. Similarly, if $(G,\sT)$ is any metrisable topological group, with left-invariant compatible distance $d_L$, and $d_u(f,g) = \sup_h d_L(fh,gh)$, then $(G\sT,d_u)$ is a completely regular topometric space. Since $d_u(f,g) = \sup_{a \in M} d(fa,ga)$, and for each $a$ the function $(f,g) \mapsto d(fa,ga)$ is continuous, $d_u$ is lower semi-continuous. Assume that $d_u(f,g) > r$. Then there exists $a \in M$ such that $d(fa,ga) > r$, and we may define $\theta(x) = d(fa,xa)$. Then $\theta$ is continuous and $1$-Lipschitz (by definition of point-wise and uniform convergence). In addition, $\theta(f) = 0$ and $\theta(g) > r$. Thus continuous $1$-Lipschitz functions witness distances, and it follows that $d_u$ is lower semi-continuous. Now let $U$ be a topological neighbourhood of $f$. Then there is a finite tuple $\bar a \in M^n$ and $\varepsilon > 0$ such that $U$ contains the set $$\begin{gathered} U_{\bar a,f\bar a,\varepsilon} = \{ h\colon d(h\bar a,f\bar a) < \varepsilon\}. \end{gathered}$$ Then the function $\rho(x) = d(f\bar a,x\bar a)$ separates $f$ from $G \setminus U$. A similar reasoning applies to the case of an abstract group (acting on itself on the left). In fact, when $G$ is completely metrisable then this case can be shown to be a special case of the first, and every metrisable group can be embedded in a completely metrisable one. Are automorphism groups of metric structures topometrically normal? In other words, do continuous $1$-Lipschitz functions witness distance between closed sets? Most topometric spaces one would encounter, such as compact ones (e.g., type spaces) or automorphism groups, are (metrically) complete. If $X$ is an incomplete topometric space then the metric structure carries obviously over to the completion $\hat X$, and it is legitimate to ask whether, or how, the topological structure carries there as well. Let us concentrate on the case where $X$ is completely regular. Let $X$ be a completely regular topometric space. We equip its completion $\hat X$ with the least topology such that for every $f \in \cC_{\cL(1)}(X)$, the unique $1$-Lipschitz extension of $f$ to $\hat f \colon \hat X \to \bC$ is continuous. In other words, we define it so that the restriction map $\cC_{\cL(1)}(\hat X) \to \cC_{\cL(1)}(X)$ is a bijection. Let $X$ be a completely regular topometric space. Then so is $\hat X$. The Stone-Čech compactification $\beta X$ is compact and therefore complete, and the canonical identification of $\hat X$ with a subset of $\beta X$ is homeomorphic. The topometric structure we put on $\hat X$ is clearly the strongest possible regular one, and it is natural to ask whether it is unique. For a positive result in this direction, let us consider the following two conditions on a topometric space $X$: - For every open set $U \subseteq X$ and $r > 0$, the open metric neighbourhood $B(U,r)$ is (topologically) open. - For every open set $U \subseteq X$ and $r > 0$ we have $\overline U^d \subseteq B(U,r)^\circ$. Clearly $(*)$ implies $(**)$. \[prp:ContinuousLipschitzDenseExtension\] Let $X$ be a completely regular topometric space in which condition $(**)$ holds, and let $X_0 \subseteq X$ be a metrically dense subspace. Then every $f \in \cC_{\cL(1)}(X_0)$ extends to $\hat f \in \cC_{\cL(1)}(X)$. Let $f \in \cC_{\cL(1)}(X_0)$. Then it extends uniquely to a $1$-Lipschitz function $\hat f\colon X \to \bC$, and all we need to show is that $\hat f$ is continuous at every $x \in X$. Assuming, as we may, that $\hat f(x) = 0$, let $U = \{y \in X_0\colon |f(y)| < \varepsilon\}$ for some $\varepsilon > 0$. Then $U \subseteq X_0$ is open, so of the form $V \cap X_0$ for some open $V \subseteq X$. Since $x \in \overline V^d$, by $(**)$ we have $x \in B(V,\varepsilon)^\circ$. Now let $w \in B(V,\varepsilon)$. Then there is $z \in V \cap B(w,\varepsilon)$, and for some $0 < \delta < \varepsilon$ we have $B(z,\delta) \subseteq V$. Since $X$ is dense, there is $y \in B(z,\delta) \cap X_0 \subseteq U$. Thus $|\hat f(y)| = |f(y)| < \varepsilon$, so $|\hat f(w)| < 3\varepsilon$, which is enough. Condition $(*)$ holds in every topometric space of the form $\prod [s_i,r_i]$. More generally, it holds in every minimal or maximal topometric space, and if it holds in each $X_i$ then it holds in $\prod X_i$. Similarly, if condition $(**)$ holds in each $X_i$ then it also holds in $\prod X_i$. Easy. Condition $(*)$ holds in every topometric group. In fact, while we usually require that the distance in a topometric group be biïnvariant, here it is enough that it be invariant on one side. Assume that the distance is left-invariant. Then one checks that $B(U,r) = \bigcup_{d(h,1) < r} Uh$. On the other hands, it is not difficult to construct even compact topometric spaces where the properties discussed in this section fail. In [@BenYaacov:Perturbations Example 3.11 & Theorem 3.15] an example was given somewhat indirectly of a compact topometric space in which condition $(*)$ fails (in the terminology used there, in which the perturbation distance was not *open* or even weakly so). We give a more explicit example in which fails (so in particular, so do $(**)$ and $(*)$). Let $X$ be the disjoint union of $[0,1]$ with $\bN$, where $[0,1]$ is equipped with the usual minimal structure (i.e., usual topology and distance), $\bN$ is equipped with the discrete topology and $0/1$ distance (which is curiously both maximal and minimal). The distance between any point of $[0,1]$ and of $\bN$ is one, and $0$ (hereafter always referring to $0 \in [0,1]$ and not to $0 \in \bN$) is the limit of $\bN$. Thus $X$ is a compact topometric space, which can be naturally viewed as a subspace of $[0,1]^\bN$ by sending $t \in [0,1]$ to $(t,0,0,\ldots)$, and sending $n \in \bN$ to the sequence $(0,0,\ldots,0,1,1,\ldots)$ consisting of $n$ initial zeroes. Let $U = (0,1) \subseteq X$ and $0 < r < 1$. Then $\overline U^d = [0,1] = B(U,r)$, while every neighbourhood of $0$ must contains members of $\bN$, so $(**)$ fails. Now let $X_0 = (0,1) \cup \bN$. Then $X_0$ is metrically dense in $X$, and the function $\bone_{(0,1)}$ is continuous and $1$-Lipschitz on $X_0$, but its $1$-Lipschitz extension to $X$ fails to be continuous at $0$, failing . The topometric structure defined earlier on $\hat X_0$ differs from that on $X$ only in that $0$ is no longer an accumulation point of $\bN$. An abstract characterisation of the set of (continuous) $1$-Lipschitz functions {#sec:Abstract1Lipschitz} =============================================================================== It is a classical fact that for a compact space $X$, $\cC(X)$ is a commutative unital $C^*$-algebra, and that conversely, every such algebra is of the form $\cC(X)$ for a compact $X$ which is moreover unique up to a unique homeomorphism. Since a compact topometric space is completely regular, the distance is captured by the subset $\cC_{\cL(1)}(X) \subseteq \cC(X)$. Here we ask the opposite question, namely, given commutative unital $C^*$-algebra, which we may already consider to be of the form $\cC(X)$ for some compact space $X$, which subsets of the algebra can be of the form $\cC_{\cL(1)}(X)$ for some topometric structure on $X$. \[dfn:L1Set\] Let $X$ be a compact topological space. We say that a set $A \subseteq \cC(X)$ is an *$\cL(1)$-set* if 1. It is convex, closed under multiplication by scalars $\alpha \in \bC$, $|\alpha| \leq 1$ and under taking the absolute value. 2. It separates points in $X$. 3. $\bC \subseteq A$. 4. If $f \notin A$ then there are two points $x,y \in X$ and some $\varepsilon > 0$ such that for all $g \in \cC(X)$, if $|f(x) - g(x)|, |f(y) - g(y)| < \varepsilon$ then $g \notin A$ as well. \[lem:L1Set\] Let $X$ be a compact topological space and $A \subseteq \cC(X)$ an $\cL(1)$-set. Then $A$ is closed in the topology of point-wise convergence, separates points from closed sets and $A + \bC = A$. That $A$ is closed in point-wise convergence follows directly from the last condition of . Now let $f \in A$ and $\alpha \in \bC$. For $0 < \lambda < 1$ we have $\lambda f + (1-\lambda) \frac{\alpha}{1-\lambda} \in A$, and since this converges uniformly to $f + \alpha$ when $\lambda \to 1$ we have $f + \alpha \in \bC$. Now let $x \in X$ disjoint from a closed set $F$. For $y \in F$ there is $f_y \in A$ such that $f_y(x) \neq f_y(y)$. Translating by a constant and taking the absolute value we may assume that $f_y \geq 0$, $f_y(x) = 0$ and $f_y(y) > 0$. By compactness there is a finite family $\{y_i\}_{i<k}$ such that for all $y \in F$ there is $i < k$ for which $f_{y_i}(y) > \half f_{y_i}(y_i)$. Letting $f = \frac{1}{k} \sum f_{y_i} \in A$ we have $f(x) = 0$ and $f(y) \geq r > 0$ for all $y \in F$. Modulo conditions (i)–(iii), condition (iv) of is equivalent to 1. If $f \notin A$ then there are two points $x,y \in X$ and some $\varepsilon > 0$ such that for all $g \in \cC(X)$, if $|f(x) - g(x) - f(y) + g(y)| < \varepsilon$ then $g \notin A$ as well. Indeed, (iv$'$) clearly implies (vi). For the other direction we already know that $A$ is translation invariant, so we may always assume that $f(x) = g(x)$, in which case (iv) and (iv$'$) are the same. In pure $C^*$-algebraic terms, we can express $\cC(X \times X)$ as the $C^*$ tensor product $\cC(X) \otimes \cC(X)$, and define $\delta\colon \cC(X) \to \cC(X) \otimes \cC(X)$ by $\delta f = f \otimes 1 - 1 \otimes f$, i.e., $\delta f(x,y) = f(x) - f(y)$. Since a point in $X \times X$ corresponds to a maximal ideal in $ \cC(X) \otimes \cC(X)$, we obtain that (vi) is further equivalent to 1. If $f \notin A$ then there exists $\varepsilon > 0$ such that the family of all $\varepsilon \dotminus |\delta f - \delta g|$, in the sense of continuous functional calculus, as $g$ varies over $A$, generates a proper ideal in $\cC(X) \otimes \cC(X)$. \[thm:L1Set\] Let $X$ be a compact topological space, $A \subseteq \cC(X)$. Then the following are equivalent: 1. The set $A$ is an $\cL(1)$-set. 2. There is a topometric structure $(X,d)$ on $X$ such that $A = \cL_{\cL(1)}(X)$. In this case the metric $d$ is unique and can be recovered by $$\begin{gathered} \label{eq:DistanceFrom1Lipschitz} d(x,y) = \sup_{f \in A} |f(x)-f(y)|. \end{gathered}$$ Bottom to top is easy, and follows from Urysohn’s Lemma for normal topometric spaces and the fact that a compact topometric space is normal. Assume therefore that $A$ is an $\cL(1)$-set, and let us define $d$ by . Clearly $d$ is a pseudo-distance, and is lower semi-continuous being the supremum of continuous functions. Since $A$ separates points from closed sets, $d$ refines the topology, and in particular is a distance (rather than a pseudo-distance). Thus $(X,d)$ is a topometric space, and we view it henceforth as such. It is then immediate from the construction that $A \subseteq \cC_{\cL(1)}(X)$. Finally, assume that $f \notin A$, and let $x,y \in X$ and $\varepsilon > 0$ be such that if $|f(x)-g(x)-f(y)+g(y)| < \varepsilon$ then $g \notin A$. Since $A$ is closed under multiplication by complex scalar of absolute value $\leq 1$, this is only possible if $|f(x) - f(y)| \geq |g(x) - g(y)| + \varepsilon$ for all $g \in A$. It follows that $|f(x) - f(y)| \geq d(x,y) + \varepsilon$, so $f \notin \cC_{\cL(1)}(X)$, as desired. This is quite different from [@Weaver:LipschitzAlgebras Theorem 4.3.2], which still seems to be the most closely analogous result therein. [^1]: Author supported by ANR chaire d’excellence junior THEMODMET (ANR-06-CEXC-007) and by the Institut Universitaire de France. [^2]: *Revision* *of*

[**Non-Hermitian Oscillator and $\cal{R}$- deformed Heisenberg Algebra**]{} [*R.Roychoudhury[[^1] and B. Roy[[^2]]{}\ Physics & Applied Mathematics Unit\ Indian Statistical Institute\ Kolkata 700108, India.]{}*]{}\ [*P.P.Dube[[^3]\ Garalgacha Surabala Vidyamandir\ West Bengal, India]{}*]{} [**Abstract**]{} A non-Hermitian generalized oscillator model, generally known as the Swanson model, has been studied in the framework of $\cal{R}$-deformed Heisenberg algebra. The non-Hermitian Hamiltonian is diagonalized by generalized Bogoliubov transformation. A set of deformed creation annihilation operators is introduced whose algebra shows that the transformed Hamiltonian has conformal symmetry. The spectrum is obtained using algebraic technique. The superconformal structure of the system is also worked out in detail. An anomaly related to the spectrum of the Hermitian counterpart of the non-Hermitian Hamiltonian with generalized ladder operators is shown to occur and is discussed in position dependent mass scenario. **Introduction** ================ Recently there has been a surge of interest in deformed Heisenberg algebra containing the reflection operator $\cal{R}$ (R-deformed Heisenberg algebra (RDHA) \[1\]. It appears in the context of Wigner quantization schemes \[2\], initiated in 1950 by Wigner \[3\] generalizing the bosonic commutation relations. The single mode algebra that appeared in the work of Wigner implicitly, was further generalized for the field systems, and led to the concept of parabosons and parafermions \[4\]. In this sense it has some properties of universality \[5\]. Hamiltonians with reflection operators have most notably arisen in the context of quantum many-body integrable systems of Calogero-Sutherland type \[6\] and their generalizations with internal degrees of freedom \[7\]. In ref\[8\], a hidden nonlinear supersymmetry was revealed in purely parabosonic harmonic oscillator systems (and in related Calogero models with exchange interaction) with the help of RDHA, and a problem of the quantum anomaly related to nonlinear supersymmetry was identified. This was actually studied later and resulted in the discovery of the hidden supersymmetry in quantum mechanical systems with a local Hamiltonian \[9\]. The RDHA was applied for the universal description of bosons, fermions, anyons and supersymmetry in (2+1) dimensions. In particular, the non unitary finite and infinite dimensional representation of the RDHA algebra have been used to obtain extended supermultiplets of anyons, or of bosons and fermions\[10\]. Deformed Heisenberg algebra has also been used to construct $N = 2$ supersymmetric quantum mechanics \[11\] and the associated exactly solvable models \[12\]. Symmetry algebras containing reflection operator in the representations of the generators of the algebra have been examined in the context of quantum oscillators \[13\]. Also, mirror symmetry plays an important role in the design of spin chains for quantum information transport \[14\]. Recently RDHA algebra has been employed for bosonization of supersymmetric quantum mechanics \[15\] and for describing anyons in (2+1) \[16\] and (1+1) dimensions \[17\]. The findings of many interesting physical and mathematical properties associated with these studies has prompted us to consider a non-Hermitian generalized oscillator model, widely known as the Swanson model \[18\] within the framework of RDHA algebra. Though many features of this particular Hamiltonian have been studied \[19\], to the best of our knowledge, this has not been reported so far in the literature.\ The non-Hermitian generalized oscillator Hamiltonian is given by $$\hat{H}_S = \omega (\hat{a}^{\dagger}\hat{a} + \frac{1}{2}) + \alpha \hat{a}^2 + \beta {\hat{a}^{\dagger 2}}, ~~~~~~~~~\omega,\alpha,\beta \in \mathbf{R}$$ where $\hat{a}$, $\hat{a^{\dagger}}$ are bosonic harmonic oscillator annihilation and creation operators satisfying usual commutation relation $[\hat{a},\hat{a}^{\dagger}] = 1$. If $\alpha \neq \beta$, $H_S$ is non-Hermitian $\cal{PT}$-symmetric (or equivalently ${\cal{P}}$-pseudo-Hermitian, $\cal{P}$ being the parity operator) \[20\], possesses real spectrum given by $E_n = (n + \frac{1}{2})\Omega, n = 0, 1, 2, \cdots$ where $\Omega^2 = \omega^2 - 4 \alpha \beta$ so long as $\omega > \alpha + \beta$ and the eigenfunctions can be derived from those of the harmonic oscillator \[18\].\ In the present work we shall study the Hamiltonian (1) with $a$, $a^{\dagger}$ satisfying the (anti)commutation relations of the RDHA algebra given by $$\begin{array}{lcl} [\hat{a}, {\hat{a}}^{\dagger}] &=& 1 + \nu\mathcal{R}\\ \{\mathcal{R}, \hat{a}\} &=& \{\mathcal{R}, {\hat{a}}^{\dagger}\} = 0\\ {\mathcal{R}}^2 &=& 1~~\mathcal{R}^{\dagger} = \mathcal{R}^{-1} = \mathcal{R}\\ \end{array}$$ where $\nu \in \mathbf {R}$ is a deformation parameter and $\mathcal{R}$ is the reflection operator, $\mathcal{R} = (-1)^{\mathcal{N}} = exp(i \pi \mathcal{N})$. The number operator \[10\] $$\mathcal{N} = \frac{1}{2}\{\hat{a}^{\dagger}, \hat{a}\} - \frac{1}{2}(\nu + 1),~~~[\mathcal{N}, \hat{a}^{\dagger}] = \hat{a}^{\dagger}~~~[\mathcal{N}, \hat{a}] = -\hat{a}$$ defines the Fock space $$\mathcal{F} = \{|n>, n = 0, 1, 2, \cdots\},~~~ \mathcal{N}|n> = n|n>$$ where $|n> = C_n(a^{\dagger})^n |0>,~~ n = 0, 1, \cdots ,~~ \hat{a}|0> = 0,~~ <0|0> = 1$, $$C_n = ([n]_{\nu}!)^{-\frac{1}{2}},~~[0]_{\nu}! = 1,~~ [n]_{\nu}! = \prod_{i = 1}^n [l]_{\nu},~~ n\geq 1,~~ [l]_{\nu} = l +\frac{1}{2}(1 - (-1)^l)\nu$$ For $\nu > -1$, the RDHA algebra has infinite dimensional, parabosonic type unitary irreducible representations; for $\nu = -(2r+1),~r = 1, 2, \cdots$ it has $(2r+1)$-dimensional, non unitary parafermionic-type representations; for $\nu < -1,~~\nu \neq -(2r+1)$, it has infinite dimensional non-unitary representations. In what follows we have not considered any particular representation for the $\mathcal{R}$ deformed Heisenberg algebra. But it is important to mention that if one takes especially the non-unitary finite and infinite dimensional representations as was done in \[10\] then this will have an effect on the generalized Bogoliubov transformation (6) as well as on the spectrum of ${\hat{H}}_S'$ discussed in section 4. The objective of the present work is two-fold: i) First, to analyse the conformal and superconformal structure associated with the diagonalized version of the non-Hermitian Hamiltonian (1) ii) Second, to obtain the equivalent Hermitian counterpart $\tilde{h}$ \[21\] of (1) (with generalized ladder operators) by a similarity transformation \[22\] and show how the generalized quantum rule (2) in this case gives rise to an anomaly related to the spectrum of $\tilde{h}$. The reason for this apparent anomaly is attributed to the fact that the domain of $x$ and the domain of the variable used in the coordinate transformation to obtain the conformal Hamiltonian are not necessarily the same. Our findings will be illustrated in the framework of coordinate dependent mass models. **Non-Hermitian Oscillator Hamiltonian and Generalized Bogoliubov Transformation** ================================================================================== In this section we shall just outline the essential results as the derivation is practically the same as has been done by Swanson \[18\]. The two new operators $\hat{c}$, $\hat{d}$ are introduced by means of a generalized Bogoliubov transformation $$\begin{array}{lcl} \hat{c} &=& \displaystyle \frac{l_1}{(l_1 l_4 - l_2 l_3)} {\hat{a}}^{\dagger} - \frac{l_3}{(l_1 l_4 - l_2 l_3)} \hat{a}\\ \hat{d} &=& \displaystyle \frac{l_4}{(l_1 l_4 - l_2 l_3)} \hat{a} - \frac{l_2}{(l_1 l_4 - l_2 l_3)} {\hat{a}}^{\dagger} \\ \end{array}$$ where the coefficients $l_j = 1,2,3,4$ are taken to be complex numbers. It reduces to the standard Bogoliubov transformation when $l_4 = l_1^*$ and $l_3 = l_2^*$. Equation (6) ) is written in matrix form as $$\left( \begin{array}{c} \hat{d} \\ \hat{c} \\ \end{array} \right) = \frac{1}{(l_1 l_4 - l_2 l_3)}\left( \begin{array}{cc} l_4 & -l_2 \\ -l_3 & l_1 \\ \end{array} \right) \left( \begin{array}{c} \hat{a} \\ \hat{a}^{\dagger} \\ \end{array} \right)$$ Then the commutation relation (2) gives $$[\hat{d},\hat{c}] = \frac{1}{(l_1 l_4 - l_2 l_3)}[\hat{a}, \hat{a}^{\dagger}] = \frac{1}{(l_1 l_4 - l_2 l_3)}(1 + \nu \mathcal{R})$$ Inversion of the matrix in (7) gives $$\begin{array}{lcl} \hat{a} &=& l_1 \hat{d} + l_2 \hat{c}\\ \hat{a}^{\dagger} &=& l_3 \hat{d} + l_4 \hat{c}\\ \end{array}$$ Substituting (9) into (1) yields $$\hat{H}_S = \Lambda \hat{c} \hat{d} + \tilde{\alpha} \hat{c}^2 + \tilde{\beta} \hat{d}^2 + \epsilon + \frac{1}{2} \omega$$ where $$\begin{array}{lcl} \Lambda &=& \omega(l_4 \l_1 + \l_3 \l_2) + 2 \alpha l_2 l_1 + 2 \beta l_4 l_3\\ \tilde{\alpha} &=& \omega l_2 l_4 + l_2^2 \alpha + l_4^2 \beta\\ \tilde{\beta} &=& \omega l_3 l_1 + l_1^2 \alpha + l_3^2 \beta\\ \epsilon &=& \frac{1}{(l_1 l_4 - l_2 l_3)}(1 + \nu \hat{R}) (\omega l_3 l_2 + \alpha l_1 l_2 + \beta l_3 l_4)\\ \end{array}$$ The coefficients $l_j$ s are so chosen that $$\tilde{\alpha} = \tilde{\beta} = 0$$ are satisfied. If $\alpha = \beta = 0$ in (1), then (9) can be chosen to reduce to $\hat{c} = \frac{l_1}{l_1 l_4 - l_2 l_3}\hat{a}^{\dagger}$ and $\hat{d} = \frac{l_4}{l_1 l_4 - l_2 l_3}\hat{a}$. This yields the boundary conditions $$\lim_{\alpha, \beta \rightarrow 0}\frac{l_{2,3}}{(l_1 l_4 - l_2 l_3)} = 0$$ and $$\lim_{\alpha, \beta \rightarrow 0} \frac{l_{1,4}}{\sqrt{(l_1 l_4 - l_2 l_3)}} = \lim_{\alpha, \beta \rightarrow 0}\frac{1}{(l_1 l_4 - l_2 l_3)}(1 + \nu \mathcal{R})$$ Now the task is to find the solutions to (11) and (8) consistent with the limits (13) and (14). In the subsequent calculations, the value of $(l_1 l_4 - l_2 l_3)$ has been set to unity. Writing the constraints of (12) as two quadratic equations under the assumptions $l_1 \neq 0$ and $l_4 \neq 0$ and choosing appropriate signs consistent with the limits (13) and (14) while taking the square roots, gives \[18\] $$\begin{array}{lcl} l_1 l_4 &=& \displaystyle \frac{2 \alpha \beta}{(4 \alpha \beta - \omega^2 + \omega \sqrt{\omega^2 - 4 \alpha \beta})} \\ \l_2 l_3 &=& \displaystyle \frac{\omega - \sqrt{\omega^2 - 4 \alpha \beta}}{2\sqrt{\omega^2 - 4 \alpha \beta}} \\ l_1 \l_2 &=& - \displaystyle \frac{\beta}{\sqrt{\omega^2 - 4 \alpha \beta}} \\ l_3 l_4 &=& \displaystyle -\frac{\alpha}{\sqrt{\omega^2 - 4 \alpha \beta}} \end{array}$$ Therefore $\Lambda$ and $\epsilon$ in (11) are $$\begin{array}{lcl} \Lambda &=& \sqrt{\omega^2 - 4 \alpha \beta} \\ \epsilon &=& \frac{1}{2}(\Lambda - \omega)(1 + \nu \hat{R})\\ \end{array}$$ It is worth noting that in (16) $\epsilon$ is a reflection dependent operator. The transformed Hamiltonian (10) then reads $$\hat{H}_S = \sqrt{\omega^2 - 4 \alpha \beta} \hat{c} \hat{d} + \frac{1}{2}\sqrt{\omega^2 - 4 \alpha \beta} (1 + \nu \hat{R}) - \frac{1}{2}\omega \nu \hat{R}$$ In the coordinate representation \[23\], $\mathcal{R}$ is realized by the parity operator $\cal{P}$: $${\cal{P}} |x> = |-x> \Rightarrow \{{\cal{P}}, x\} = \{{\cal{P}}, p_x\} = 0 , ~~~~{{\cal{P}}}^{\dagger} = {{\cal{P}}}^{-1} = {\cal{P}}~~~{\cal{P}}^2 = 1 \label{e4}$$ whereas the deformed ladder operators can be realized in the form \[1,16\] $$\begin{array}{lcl} {\hat{c}} &=& \frac{1}{\sqrt{2}}(-x + ip_x) \\ {\hat{d}} &=& \frac{1}{\sqrt{2}}(-x - ip_x)) \end{array}$$ where $$p_x = -i(\frac{d}{dx} - \frac{\nu}{2x} \cal{P})$$ Substitution of (19) into (17) gives $${\hat{H}}_S' = \frac{1}{2} \{p^2 + x^2 + \frac{1}{4x^2}(\nu^2 - 2\nu\cal{P})\}$$ where ${\hat{H}}_S' = \displaystyle \frac{{\hat{H}}_S - \frac{1}{2}\omega \nu \cal{P}}{\sqrt{\omega^2 - 4 \alpha \beta}},~~(\omega^2 \neq 4\alpha \beta)$ **Conformal structure of ${\hat{H}}_S'$** ========================================= Let us recall that the conformal group $O(2,1)$ is spanned by three generators $H$, the Hamiltonian, $D$, the dilatation generator, and $K$, the conformal generator. These generators form together the algebra \[24\] $$[H,D] = iH~~~~[K,D] = -iK~~~~[H,K] = 2iD$$ If we define $$R = \frac{1}{2}(K + H)~~~~~~~~~~~~~~~~~~~~S = \frac{1}{2}(K - H)$$ then $R$, $D$ and $H$ satisfy the $O(2,1)$ algebra $$[D,R] = iS~~~~~~~~~~[S,R] = -iD~~~~~~~~~~~~~~[S,D] = -iR$$ Evidently, (23) corresponds to just a change of basis of the Lie algebra $O(2,1)$. The subgroup generated by $R$ is compact (rotation group in a plane) while those generated by $S$ and $D$ are not, being of the boost type \[24\]. Let us take the following representations for $H$, $K$, $D$ $$\begin{array}{lcl} H &=& \displaystyle \frac{1}{2} \{p^2 + \frac{\nu^2 - 2 \nu \cal{P}}{4 x^2}\}\\ K &=& \displaystyle \frac{x^2}{2}\\ D &=& \displaystyle -\frac{1}{4} (x p_x + p_x x) \end{array}$$ where $p = -i \frac{d}{dx}$ and $p_x$ is given by (20). Then it is easy to verify that $2R = {\hat{H}}_S'$, $S$ and $D$ satisfy (24), i.e. ${\hat{H}}_S'$ has the conformal symmetry. **Spectrum of ${\hat{H}}_S'$** ============================== Let us introduce the ladder operators $$L_{\pm} = S \pm iD$$ These play the role of ladder operators, since these, together with $R$, satisfy the commutation relations of $SL(2,R)$ algebra \[24\] $$[R, L_{\pm}] = \pm L_{\pm}~~~~~~~~~~~~~~~~~~~~~~~[L_+, L_-] = -2R$$ At this point it is worth noticing that $\hat{c}^2$, $\hat{d}^2$ and $\hat{c}\hat{d}$ given in equation (22), satisfy the following commutation relations $$[\hat{c}^2, \hat{c}\hat{d}] = -2 \hat{c}^2~~~~~~~~[\hat{d}^2, \hat{c}\hat{d}] = 2\hat{d}^2~~~~~~~~~~[\hat{c}^2,\hat{d}^2] = -4\hat{c}\hat{d} - 2(1+ \nu \cal{P})$$ These can be identified with the generators $L_-$, $L_+$ and $R$ of $SL(2,R)$ algebra in the following way $$\hat{c}^2 = -2 L_+~~~~~~~~~~~\hat{d}^2 = -2 L_-~~~~~~~~~~~\hat{c}\hat{d} = 2R - \frac{(1+\nu\cal{P})}{2}$$ In terms of $H$, $K$ and $D$ \[24\] $$\begin{array}{lcl} L_+ &=& \frac{1}{2}K-H+2iD)\\ L_- &=& \frac{1}{2}(K-H-2iD)\\ R &=& \frac{1}{2}(H+K) \end{array}$$ and the Casimir operator is given by $$\begin{array}{lcl} J^2 &=& R^2 + R - L_- L_+\\ &=&\displaystyle \frac{HK+KH}{2} - D^2\\ &=& \displaystyle \frac{g}{4} - \frac{3}{16} \end{array}$$ where $$g = \frac{\nu^2 - 2 \nu {\mathcal{P}}}{4}$$ Putting $J^2 = r_0 (r_0 - 1)$, one finds $$\begin{array}{lcl} r_0 &=& \frac{1}{2}(1 + \sqrt{g + \frac{1}{4}})\\ &=& \frac{1}{4}(\nu + 3) ~~~ \mbox {if}~~ {\cal{P}} = -1\\ &=& \frac{1}{4}(\nu + 1) ~~~ \mbox {if}~~ {\cal{P}} = +1 \end{array}$$ The reason for taking the positive sign before the square root in (33) arises from the conditions to be satisfied by the wavefunction of the quantum mechanical system. In this case the required conditions are the vanishing of both the lowest eigenfunction and its first derivative as $x \rightarrow 0$ \[24\]. This requires $r_0 > \frac{3}{4}$ which is possible for the choice of positive sign. Now we write the eigenvalue equation as $${\hat{H}}_S'|n> = \epsilon_n |n>$$ where $|n>$ are the eigenstates and $|0>$ labels the vacuum state.\ From the commutation relations (27), it follows that \[24\] $$L_{\pm} |n, r_0> = C_{\pm} (n, r_0) |n \pm 1, r_0>$$ implying that successive eigenvalues differ by unity. Using (31), one gets $$\begin{array}{lcl} |C_{\pm}(n, r_0)|^2 &=& \epsilon_n (\epsilon_n \pm 1) - J^2\\ &=& \epsilon_n (\epsilon_n \pm 1) - r_0(r_0 -1) \geq 0 \end{array}$$ so that   $\epsilon_n \geq r_0~,~\epsilon_n \leq -r_0$. Here we consider $\epsilon_n$ to be positive and hence $\epsilon_n > \frac{3}{4}$ (since $r_0 > \frac{3}{4}$), to obtain positive eigenvalues given by $$\epsilon_n = r_0 + n~~,~~~~~n = 0,1,2, \cdots$$ where $r_0$ is given by (33). **Superconformal structure associated with ${\hat{H}}_S'$** =========================================================== To study the supersymmetric version of conformal quantum mechanics of section 3, we shall make use of the general construction by Witten \[25\]. The supersymmetric generalization of ${\hat{H}}_S'$ is \[26\] $$\begin{array}{lcl} \cal{H} &=& \frac{1}{2} \{ Q_c, Q_c^{\dagger}\}\\ &=& \frac{1}{2} \{p_x^2 + W^2 - BW^{\prime}\} \end{array}$$ where the grading operator $B = [\psi^{\dagger}, \psi] = \sigma_3$, $\sigma_3$ being the third component of Pauli spin matrices \[34\]. $$\begin{array}{lcl} Q_c &=& (-i p_x + \frac{\sqrt{g}}{x}) \psi^{\dagger}\\ {Q_c}^{\dagger} &=& \psi(i p_x + \frac{\sqrt{g}}{x})\\ W(x) &=& \displaystyle \frac{-1 + \sqrt{1 + 2\nu^2 + 4(1+\nu^2)(g \pm \sqrt{g})}}{2x} - \displaystyle \frac{(\nu \pm 2\nu \sqrt{g})}{2x} ~ \mbox {for}~ B = \pm 1, \mathcal{P} = 1\\ &=& \displaystyle \frac{-1 + \sqrt{1 + 2\nu^2 + 4(1+\nu^2)(g \pm \sqrt{g})}}{2x} + \displaystyle \frac{(\nu \pm 2\nu \sqrt{g})}{2x} ~ \mbox {for}~ B = \pm 1, \mathcal{P} = -1 \end{array}$$ In (38), $W(x)$ is the superpotential generating the conformal supersymmetric quantum mechanics and is obtained by solving a Riccati equation. For $W(x)$ to be real $g > 0$ giving rise to the constraint $(\nu \mp 1)^2 > 1$ (corresponding to $\mathcal{P} = \pm 1$) which is consistent with the condition $r_0 > \frac{3}{4}$ (see equation (33)). The spinor operators $\psi$ and $\psi^{\dagger}$ in (39) obey the anticommutation relation $$\{\psi, \psi^{\dagger}\} = 1$$ Introduction of an extra pair of spinor operators $S$ and $S^{\dagger}$ as $$\begin{array}{lcl} S^{\dagger} &=& \psi x\\ S &=& x \psi^{\dagger}\\ \end{array}$$ endows the system with a richer algebraic structure, namely the superconformal algebra \[1,27\] which is given by $$\begin{array}{lcl} \frac{1}{2} \{ Q_c, Q_c^{\dagger}\} &=& \cal{H}\\ \frac{1}{2}\{S, S^{\dagger}\} &=& K\\ \frac{1}{2}\{Q_c, S^{\dagger}\} &=& -iD - \frac{1}{2}B(1+\nu {\cal{P}}) + \sqrt{g}\\ \frac{1}{2}\{Q_c^{\dagger}, S\} &=& +iD - \frac{1}{2}B(1+\nu {\cal{P}}) + \sqrt{g}\\ \end{array}$$ All other anticommutators like $\{Q_c, Q_c\}, \{Q_c, S\}$ vanish. Explicitly $$\begin{array}{lcl} \cal{H} &=& \displaystyle \frac{1}{2} (\mathbf{I} + p_x^2 + \frac{\mathbf{I} g + \sqrt{g} B (1-\nu {\cal{P}})}{x^2})\\ K &=& \displaystyle \frac{1}{2} x^2\\ D &=& \displaystyle \frac{1}{2} (p_x x + x p_x)\\ B &=& [\psi, \psi^{\dagger}] \end{array}$$ The fermionic raising and lowering operators are defined as $$\begin{array}{lcl} M &=& Q_c - S\\ M^{\dagger} &=& Q_c^{\dagger} - S^{\dagger}\\ N &=& Q_c + S\\ N^{\dagger} &=& Q_c^{\dagger} + S^{\dagger} \end{array}$$ which obey the anticommutator algebra $$\begin{array}{lcl} \frac{1}{2} \{M , M^{\dagger}\} &=& 2 {\cal{H}}_0 + \frac{1}{2}(1 + \nu {\cal{P}})B - \sqrt{g} \equiv {\cal{H}}_1\\ \frac{1}{2} \{N , N^{\dagger}\} &=& 2 {\cal{H}}_0 - \frac{1}{2}(1 + \nu {\cal{P}})B + \sqrt{g} \equiv {\cal{H}}_2 \\ -\frac{1}{4}\{M , N^{\dagger}\} &=& {\cal{L}}_-\\ -\frac{1}{4} \{M^{\dagger}, N\} &=& {\cal{L}}_+ \end{array}$$ where\ ${\cal{H}}_0 = \frac{1}{2}({\cal{H}} + K)$ is the supersymmetric generalization of the generator of compact rotation $2R = \hat{H}_S'$,\ each of ${\cal{H}}_1$ and ${\cal{H}}_2$ are supersymmetric Hamiltonians and\ ${\cal{L}}_{\pm}$ are the supersymmetric generalizations of the ladder operators defined in (30).\ The superconformal algebra (42) closes since \[33\] $$[B,N^{\dagger}] = - N^{\dagger},~~~[K, N^{\dagger}] = - S^{\dagger}$$ $$[\mathcal{H}, N^{\dagger}] = -Q_c^{\dagger} = -N^{\dagger} + S^{\dagger}$$ $$[B,S] = S, ~~~ [K,S] = 0$$ $$[\mathcal{H},S] = Q_{c} = N - S$$ In terms of the superpotentials $W_1(x)$ and $W_2(x)$ corresponding to ${\cal{H}}_1$ and ${\cal{H}}_2$ respectively, $M$, $M^{\dagger}$, $N$ and $N^{\dagger}$ are written as \[26\] $$\begin{array}{lcl} M &=& (-i p_x + W_1(x))\psi^{\dagger}\\ M^{\dagger} &=& \psi(i p_x + W_1(x))\\ N &=& \psi (i p_x + W_2(x))\\ N^{\dagger} &=& (-i p_x + W_2(x))\psi^{\dagger} \end{array}$$ The superpotentials $W_1(x)$, associated with ${\mathcal{H}_1}$ is given by \[28\] $$W_1(x) = - \frac{u_0'(x)}{u_0(x)}$$ where $u_0(x)$ (taking $\mathcal{P} = -1)$ is $$u_0(x) = A_0 e^{-\frac{x^2}{2}} x^{\frac{1}{2}(\nu + 2 + 2\sqrt{g})} ~ _1F_1(1, \frac{(\nu + 3 + 2\sqrt{g})}{2}, x^2)$$ $A_0$ being a constant, and $_1F_1 (a,b,z)$ is the confluent Hypergeometric function \[29\]. Similarly $W_2(x)$ can be obtained. The case $\mathcal{P} = +1$ can be treated analogously. **Reduction of the Hermitian counterpart of non-Hermitian generalized Oscillator into the conformal Hamiltonian $\hat{H}_S'$** ============================================================================================================================== In this section our aim is to obtain the equivalent Hermitian counterpart of the Hamiltonian (1) (with generalized ladder operators) by a similarity transformation and study the associated spectrum. The reason for taking generalized ladder operators giving rise to a generalized quantum condition (2) is that it allows access to those physical systems that are underlined by a coordinate dependent mass \[30\] whereby one can determine the isospectrality/ nonisospectrality of the Hermitian Hamiltonian with the Hamiltonian $\mathcal{H}_S^{\prime}$. The genralized ladder operators $\hat{a}, {\hat{a}}^{\dagger}$ are taken as $$\hat {a} = A(x)\frac{d}{dx} + B(x) + f(x){\cal{P}}~~~~{\tilde{a}}^{\dagger} = -A(x)\frac{d}{dx} + B(x) - A^{\prime} - f(x)\cal{P}$$ where $f(x)$ is a function to be determined later and prime denotes differentiation with respect to $x$. It is assumed that $A(-x) = A(x), B(-x) = - B(x), f(-x) = -f(x)$. It is to be noted that both for $\mathcal{P} = +1$ and $-1$, $B(x)$ and $f(x)$ should have the same parity. In this case $$[\hat{a}, \hat{a}^{\dagger}] = 2AB' -4Bf{\cal{P}} + 2A^{\prime}f{\cal{P}} -A A''$$ Substitution of (53) into (1) and removing the first derivative term of the resulting Hamiltonian with the help of a similarity transformation gives the equivalent Hermitian Hamiltonian $\tilde{h}$ \[21\] and is given by $$\tilde{h} = \tilde{{\rho}}_{(\alpha, \beta)} {H_S} \tilde{{\rho}}_{(\alpha, \beta)}^{-1} = -\frac{d}{dx} A^2 \frac{d}{dx} + V_{eff}(x)$$ where $$\tilde{\rho}(\alpha, \beta) = A(x)^{\frac{\alpha - \beta}{2}} exp(-(\alpha - \beta) \int^x\frac{B(x')}{A(x')}dx')$$ $$\begin{array}{lcl} V_{eff}(x) &=& \frac{1}{2}(\alpha + \beta)A A'' + [\frac{\alpha + \beta}{2} + \frac{(\alpha - \beta)^2}{4}]A'^2 - [1 + 2(\alpha + \beta) + 2(\alpha - \beta)^2]A'B\\ && +[1 + 2(\alpha + \beta) + (\alpha - \beta)^2]B^2 - (\alpha + \beta + 1)AB' + \frac{1}{2}(\alpha + \beta +1) + f^2 \\ && \mp Af' \mp (\alpha + \beta +1)A'f \pm 2(\alpha + \beta + 1) f B ~~~~\mbox{for} \mathcal{P} = \pm 1\\ \end{array}$$ taking $\omega - \alpha - \beta = 1$. It is important to mention here that $\tilde{\rho}(\alpha, \beta)$ should be well defined on $\mathbf {R}$ \[21\] so that the eigenfunctions of $\tilde{h}$ are normalizable. In fact this is consistent with the results obtained in section 6.1. There is a one to one correspondence between the energy eigenvalues of $\tilde{h}$ and $H_S$. Also, if $\psi_n(x)$ are the wave functions of the equivalent Hermitian Hamiltonian $\tilde{h}$ then the wave functions of the Hamiltonian $H_S$ are given by $(\tilde{\rho}_{(\alpha, \beta)})^{-1}\psi_n(x)$. If $\hat{a}$ and ${\hat{a}}^{\dagger}$ satisfy the generalized quantum rule $$[\hat{a}, {\hat{a}}^{\dagger}] = 1 + \nu \cal{P}$$ then comparison of (54) with (58) gives $$\begin{array}{lcl} z(x) &=& \displaystyle \int^x \frac{dx'}{A(x')}\\ B(x) &=& \displaystyle -\frac{z''}{2z'^2} + \frac{z}{2}\\ f(x) &=& \displaystyle \frac{k_1}{z}\\ \end{array}$$ where $k_1 = - \frac{\nu}{2}$ and for simplicity the integration constant is taken to be zero. Consequently, the Hamiltonian $\tilde{h}$ given in equation (55) becomes $$\begin{array}{lcl} \tilde{h} &=& -\frac{d}{dx} A^2 \frac{d}{dx} + V_{eff}(x)\\ V_{eff}(x) &=& \frac{z'''}{2z'^3} - \frac{5}{4}\frac{z''^2}{z'^4} + {\tilde{\omega}}^2 z^2 \pm (1+\alpha+\beta)k_1 + \frac{k_1(k_1\pm 1)}{z^2}~~~\mbox{for} \mathcal{P} = \pm 1 \end{array}$$ where $ {\tilde{\omega}}^2 = \frac{(\alpha - \beta)^2 + 2(\alpha + \beta) + 1}{4}$. For the change of variable (59) we have $A' = \frac{\dot{A}}{A}$,  $A'' = \frac{\ddot{A}}{A^2} - \frac{\dot{A}^2}{A^3}$ where ’dot’ denotes derivative with respect to $z$. Consequently the eigenvalue equation for the Hamiltonian $\tilde{h}$ given in (60), reduces to $$-\frac{d^2\phi}{dz^2} + [\tilde{\omega}^2 z^2 + \frac{\nu(\nu + 2)}{4z^2}] \phi(z) = [E - \frac{(1+\alpha + \beta)\nu}{2}]\phi(z)$$ where $\mathcal{P}$ has been taken to be equal to $-1$. Though in the subsequent calculations we have taken $\mathcal{P} = -1$, similar analysis can be made for $\mathcal{P} = 1$ also. It is easily seen that equation (61) is the Schr$\ddot{o}$dinger equation for the Hamiltonian $\mathcal{H}_S'$ given in (21) for $\mathcal{P} = -1$. Equation (61) can be transformed into the Kummer’s equation \[29\] $$y\frac{d^2\chi}{dy^2} + \frac{d\chi}{dy}[\frac{(1-\nu)}{2} - y] + \chi(y)[\frac{E'}{4\tilde{\omega}} - \frac{1}{4} + \frac{\nu}{4}] = 0$$ where $E' = E - \frac{(1 + \alpha + \beta)\nu}{2}$, by the transformations $$y = \tilde{\omega} z^2~~~~~~~~~~~~~~~\phi(y) = y^{-\frac{c_1}{4}} e^{-\frac{y}{2}} \chi(y)$$ Therefore the general solutions of equation (61) are given by $$\begin{array}{lcl} \phi_e(z) = N_e (\tilde{\omega}z^2)^{\frac{1}{2}+ \frac{\nu}{4}} e^{-\frac{\tilde{\omega} z^2}{2}} _1{F}_1 (\frac{1-\nu}{4} - \frac{E'}{4\tilde{\omega}}, \frac{1-\nu}{2}, \tilde{\omega} z^2)\\ \phi_o(z) = N_o (\tilde{\omega} z^2)^{\frac{1}{2} + \frac{\nu}{4}} e^{-\frac{\tilde{\omega} z^2}{2}} _1{F}_1 (\frac{\nu+3}{4} - \frac{E'}{4\tilde{\omega}}, \frac{\nu+3}{2}, \tilde{\omega} z^2)\\ \end{array}$$ where $\phi_e$ and $\phi_o$ denote respectively even and odd parity solutions and $N_e$, $N_o$ are normalisation constants. The eigenfunctions of the Hamiltonian $\tilde{h}$ are given by $$\psi(x) \sim A(z)^{-\frac{1}{2}} \phi(z)$$ where $z(x)$ is given by equation (59). Breaking of Isospectrality -------------------------- In what follows we shall take the odd parity solution in (64) corresponding to $\mathcal{P} = -1$. Similar results can be obtained for $\mathcal{P} = 1$ as well. To obtain the eigenvalues of equation (61) one has to analyze the behaviour of the odd parity eigenfunction in (64). If the domain of the argument $\tilde{\omega}{z^2}$ is unbounded then $\phi_o(z)$ will not in general, square integrable because of the asymptotic behaviour of the confluent Hypergeometric function \[29\], $$_1{F}_1(a,b,y) = \frac{\Gamma(b)}{\Gamma(a)} e^y y^{a-b}[1 + O(y^{-1})]$$ Therefore to make the eigenfunctions square integrable one must take $a = -m (m = 0,1,2,3 \cdots)$ in which case $_1{F}_1(a,b,y)$ reduces to a polynomial. Correspondingly $$E_{2m + 1} = \frac{(1 + \alpha + \beta) \nu}{2} + 2\tilde{\omega} (2m + \frac{(\nu+3)}{2})$$ Hence in this case the Hamiltonian corresponding to the eigenvalue equation (61) has the spectrum of the Hamiltonian $\mathcal{H}_S'$ subject to the constant $\frac{(1 + \alpha + \beta) \nu}{2}$\ But if the domain of $\tilde{\omega}z^2$ is finite then the odd parity eigenfunctions in (64) must vanish at the end points of the domain of $z$ and the eigenvalues will be given by the zeroes of the confluent hypergeometric functions when the arguments attain the end points. The first approximation of the m’th $(m = 1,2 \cdots)$ positive zero $X_0$ of $_1{F}_1(a,b,y)$ is given by \[29\] $$X_0 = \frac{\pi^2(m + \frac{b}{2} - \frac{3}{4})^2}{2b - 4a}[1 + O(\frac{1}{\frac{b}{2} - a)^2})], ~~~m = 1,2, \cdots$$ Hence from equation (64) we obtain, to the leading order (i.e. when $\frac{1}{(\frac{b}{2} - a)^2}$ is small in (68) which corresponds to large $\frac{E'}{4 \tilde{\omega}}$) $$E_n = \frac{(1 + \alpha + \beta) \nu}{2} + \frac{\tilde{\omega}\pi^2}{4{z_\pm}^2}[n+1+\frac{\nu}{2}]^2~~~~n = 1, 3, 5, \cdots$$ where $z_\pm$ are the end points of the argument of the confluent hypergeometric function. Hence in this the Hamiltonian corresponding to the eigenvalue equation (61) is not isospectral to the Hamiltonian $\mathcal{H}_S'$. Connection with coordinate dependent mass models ================================================ As mentioned earlier the Swanson Hamiltonian with generalized ladder operators enables one to connect it to those physical systems which are endowed with coordinate dependent mass by choosing $A(x) = m(x)^{-\frac{1}{2}}$ which is a strictly positive function \[22\]. In this case the Hamiltonian (55) reduces to the coordinate dependent mass Hamiltonians \[30\] for which the corresponding Schr$\ddot{o}$dinger equation reads $$(-\frac{d}{dx}\frac{1}{m(x)}\frac{d}{dx} + V_{eff}(x))\psi(x) = E \psi(x)$$ Example 1. Unbroken Isospectrality\ Let us consider the mass function $m(x) = e^x$. This mass function is used in studying the transport properties of semiconductors \[31\]. Correspondingly $z(x) = 2 e^{\frac{x}{2}}$ which belongs to $(0, \infty)$ as $x \in (-\infty, \infty)$. This choice of mass function gives rise to the effective potential $$V_{eff}(x) = - \frac{e^{-x}}{16} [4k_1^2 + 4k_1 -3] + 4 e^x \tilde{\omega} - k_1(1 + \alpha + \beta)$$ In this case, the eigenvalues of the Hamiltonian corresponding to the effective potential (71) is given by (67) i.e. the Hamiltonian is isospectral to the Hamiltonian (21)with $\mathcal {P} = -1$.\ Example 2. Broken Isospectrality\ In this case the mass function $m(x) = sech^2(x)$ which depicts the solitonic profile \[32\] is taken. Correspondingly $z(x) = \tan^{-1}(\sinh x)$. Clearly $z(x) \rightarrow z_\pm (=\pm \frac{\pi}{2})$ as $x \rightarrow \pm \infty$. The effective potential is $$V_{eff}(x) = \frac{1}{4} - \frac{3}{4}\cosh^2 x + \tilde{\omega}^2 [\tan^{-1}(\sinh x)]^2 + \frac{k_1(k_1 + 1)}{[\tan^{-1}(\sinh x)]^2} - (1 + \alpha + \beta) k_1$$ The eigenvalues of the Hamiltonian $\tilde{h}$ corresponding to the effective potential (72) is given by (69) putting $z_{\pm} = \pm \frac{\pi}{2}$ $$E_n = \frac{(1 + \alpha + \beta) \nu}{2} + \tilde{\omega}[n+1+\frac{\nu}{2}]^2~~~~n = 1, 3, 5, \cdots$$ Therefore in this case the Hamiltonian is nonisospectral to the Hamiltonian (21) with $\mathcal {P} = -1$. Conclusion ========== A generalized Bogoliubov transformation is used to diagonalize the non-Hermitian oscillator Hamiltonian. Within the framework of $\cal{R}$ deformed Heisenberg algebra the diagonalized Hamiltonian is shown to possess the conformal symmetry. The supersymmetric generalization of the above Hamiltonian and the associated superpotential have been constructed. 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--- abstract: 'A review of some recent results and ideas about the expected behaviour of large chaotic systems and fluids.' address: 'Fisica, Universitá di Roma, “La Sapienza”, 00185 Roma, Italia' author: - Giovanni Gallavotti title: 'Chaotic dynamics, fluctuations, nonequilibrium ensembles.' --- [*Keywords: Chaos, Statistical Mechanics, Nonequilibrium ensembles*]{} [Ergodic hypothesis]{} Giving up a detailed description of microscopic motion led to a statistical theory of macroscopic systems and to a deep understanding of their equilibrium properties. At the same time a far less successful (even for steady states theory) approach to nonequilibrium systems began. It is clear today, as it was already to Boltzmann and many others, that some of the assumptions and guiding ideas used in building up the theory were not really necessary or, at least, could be greatly weakened or just avoided. A typical example is the [*ergodic hypothesis*]{}. It is interesting for us a very short history of it (as I see it). Early in his works Boltzmann started publishing the first versions of the [*heat theorem*]{}. The theorem says that one can define in terms of time averages of total or kinetic energy, of density, and of average momentum transfer to the container walls, quantities that one could call [*specific internal energy*]{} $u$, [*temperature*]{} $T$, [*specific volume*]{} $v$, [*pressure*]{} $p$ and when two of them varied, say the specific energy and volume by $du$ and $dv$, they verify: $$\frac{du+pdv}T={\rm\ exact\ }\label{1.1}$$ At the beginning this was discussed in very special cases (like free gases). But about fifteen years later Helmoltz noted in a series of four ponderous papers that for a class of very special systems, that he called [*monocyclic*]{}, in which all motions were periodic and in a sense nondegenerate, one could give appropriate names, familiar in macroscopic thermodynamics, to various mechanical averages and then check that they verified the relations that would be expected between thermodynamic quantities with the same name. Helmoltz’ assumptions about monocyclicity are very strong and I do not see them satisfied other than in confined one dimensional Hamiltonian systems. Here is an example of Helmotlz’ reasoning (as reported by Boltzmann). Consider a $1$–dimensional system with potential $\varphi(x)$ such that $|\varphi'(x)|>0$ for $|x|>0$, $\varphi''(0)>0$ and $\varphi(x)\,\, {\vtop{\ialign{#\crcr\rightarrowfill\crcr \noalign{\kern0pt\nointerlineskip}\hglue3.pt${\scriptstyle {x\to\infty}}$\hglue3.pt\crcr}}}\,\,+\infty$ (in other words a $1$–dimensional system in a confining potential). There is only one motion per energy value (up to a shift of the initial datum along its trajectory) and all motions are periodic so that the system is [*monocyclic*]{}. We suppose that the potential $\varphi(x)$ depends on a parameter $V$. One defines [*state*]{} a motion with given energy $E$ and given $V$. And: A state is parameterized by $U,V$ and if such parameters change by $dU, dV$ respectively we define: $$dL=-p dV,\qquad dQ=dU+p dV\label{1.1a}$$ then: [*Theorem*]{} (Helmoltz): [*the differential $({dU+pdV})/{T}$ is exact.*]{} In fact let: $$\begin{aligned} S=&2\log \int_{x_-(U,V)}^{x_+(U,V)} \sqrt{K(x;U,V)}dx=\nonumber\\ =&2\log \int_{x_-(U,V)}^{x_+(U,V)} \sqrt{U-\varphi(x)}dx\label{1.2}\end{aligned}$$ ($\frac12S$ is the logarithm of the action), so that: $$dS=\frac{\int (dU-\partial_V\varphi(x) dV) \frac{dx}{\sqrt{K}}}{ \int K\frac{dx}{\sqrt{K}}}\label{1.3}$$ and, noting that $\frac{dx}{\sqrt K} =\sqrt{\frac2m} dt$, we see that the time averages are given by integrating with respect to $\frac{dx}{\sqrt K}$ and dividing by the integral of $\frac{1}{\sqrt K}$. We find therefore: $$dS=\frac{dU+p dV}{T}\label{1.4}$$ Boltzmann saw that this was not a simple coincidence: his interesting (and healthy) view of the continuum (which he probably never really considered more than a convenient artifact, useful for computing quantities describing a discrete world) led him to think that in some sense [*monocyclicity was not a strong assumption*]{}. Motions tend to recurr (and they do in systems with a discrete phase space) and in this light monocyclicity would simply mean that, waiting long enough, the system would come back to its initial state. Thus its motion would be monocyclic and one could try to apply Helmoltz’ ideas (in turn based on his own previous work) and perhaps deduce the heat theorem in great generality. The nondegeneracy of monocyclic systems becomes the condition that for each energy there is just one cycle and [*the motion visits successively all*]{} (discrete) [*phase space points*]{}. Taking this viewpoint one had the possibility of checking that in all mechanical systems one could define quantities that one could name with “thermodynamic names” and that would verify properties coinciding with those that Thermodynamics would predict for quantities with the same name. He then considered the two body problem, showing that the thermodynamic analogies of Helmoltz could be extended to systems which were degenerate, but still with all motions periodic. This led to somewhat obscure considerations that seemed to play an important role for him, given the importance he gave them. They certainly do not help in encouraging reading his work: the breakthrough paper of 1884 [@[B84]], starts with associating quantities with a thermodynamic name to Saturn rings (regarded as rigid rotating rings!) and checking that they verify the right relations (like the second principle, see Eq.(\[1.1\])). In general one can call [*monocyclic*]{} a system with the property that there is a curve $\ell\to x(\ell)$, parameterized by its curvilinear abscissa $\ell$, varying in an interval $0< \ell< L(E)$, closed and such that $ x(\ell)$ covers all the positions compatible with the given energy $E$. Let $ x= x(\ell)$ be the parametric equations so that the energy conservation can be written: $$\frac12 m\dot\ell^2+\varphi(x(\ell))=E \label{1.6}$$ then if we suppose that the potential energy $\varphi$ depends on a parameter $V$ and if $T$ is the average kinetic energy, $p=-\langle{\partial_V \varphi}\rangle$ it is, for some $S$: $$dS=\frac{dE+pdV}{T},\qquad p=-\langle{\partial_V \varphi}\rangle,\quad T=\langle{K}\rangle\label{1.7}$$ where $\langle \cdot\rangle$ denotes time average. A typical case to which the above can be applied is the case in which the whole space of configurations is covered by the projection of a single periodic motion and the whole energy surface consists of just one periodic orbit, or at least only the phase space points that are on such orbit are observable. Such systems provide, therefore, natural models of thermodynamic behaviour. Noting that a chaotic system like a gas in a container of volume $V$, which can be regarded as a parameter on which the potential $\varphi$ (which [*includes*]{} interaction with the container walls) depends, will verify “for practical purposes” the above property, we see that we should be able to find a quantity $p$ such that $dE+p dV$ admits the average kinetic energy as an integrating factor. On the other hand the distribution generated on the surface of constant energy by the time averages over the trajectory should be an invariant distribution and therefore a natural candidate for it is the uniform distribution, [*Liouville’s distribution*]{}, on the surface of constant energy. The only one if we accept the viewpoint, problably Boltzmann’s, that phase space is discrete and motion on the energy surface is a monocyclic permutation of its finitely many cells (ergodic hypothesis). It follows that if $\mu$ is the Liouville distribution and $T$ is the average kinetic energy with respect to $\mu$ then there should exist a function $p$ such that $T^{-1}$ is the integrating factor of $dE+p dV$. Boltzmann shows that this is the case and, in fact, $p$ is the average momentum transfer to the walls per unit time and unit surface, [*ie*]{} it is the [*physical*]{} pressure. Clearly this is not a proof that the equilibria are described by the microcanonical ensemble. However it shows that for most systems, independently of the number of degrees of freedom, one can define a [*mechanical model of thermodynamics*]{}. Thermodynamic relations are [*very general*]{} and simple consequences of the structure of the equations of motion. They hold for small and large systems, from $1$ degree of freedom to $10^{23}$ degrees. The above arguments, based on a discrete view of phase space, suggest that they hold in some approximate sense (as we have no idea on the precise nature of the discrete phase space). But they may hold [*exactly*]{} even for small systems, if suitably formulated: for instance in the 1884 paper[@[B84]] Boltzmann shows that in the [*canonical ensemble*]{} the relation Eq.(\[1.1\]) ([*ie*]{} the second principle) holds [*without corrections*]{} even if the system is small. Thus the ergodic hypothesis does help in finding out why there are mechanical “models” of thermodynamics: they are ubiquitous in small and large systems. But such relations are of interest in large systems and not really in small ones. For large systems any theory claiming to rest on the ergodic hypothesis may seem bound to fail because if it is true that a system is ergodic, it is also true that the time the system takes to go through one of its cycles is simply too long to be of any interest and relevance: this was pointed out very clearly by Boltzmann[@[B96]] and earlier by Thomson. The reason we observe approach to equilibrium over time scales far shorter than the recurrence times is due to the property that the microcanonical ensemble is such that [*on most of phase space the actual values of the observables, whose averages yield the pressure and temperature and the few remaining thermodynamic quantities, assume the same value*]{}[@[L]]. This implies that such value coincides with the average and therefore verifies the [*heat theorem*]{} if $p$ is the pressure (defined as the average momentum transfer to the walls per unit time and unit surface). The ergodic hypothesis loses it importance and fundamental nature and it appears simply as a tool used in understanding that some of the relations that we call “macroscopic laws” hold in some form for [*all*]{} systems, whether small or large. [The Chaotic hypothesis]{} A natural question is whether something similar to the above development can be achieved in systems out of equilibrium. Here I am not thinking of systems evolving in time: rather I refer to properties of systems that reach a stationary state under the influence of external non conservative forces acting on them. For instance I think of an electric circuit in which a current flows (stationarily) under the influence of an electromotive field. Or of a metal bar with two different temperatures fixed at the extremes. Or of a Navier–Stokes fluid in a Couette flow. The first two systems, regarded as microscopic systems ([*ie*]{} as mechanical systems of particles), do certainly have a very chaotic microscopic motions even in absence of external driving (while macroscopically they are in a stationary state and nothing happens, besides a continuous, sometimes desired, heat transfer from the system to the surroundings). The third system also behaves, as a macroscopic system, very chaotically at least when the Reynolds number is large. Can one do something similar to what Boltzmann did? The first problem is that the situation is quite different: there is no established nonequilibrium Thermodynamics to guide us. The great progresses of the theory of stationary nonequilibrium that took place in the past century (I mean the XX), at least the ones that are unanimously recognized as such, only concern properties of [*incipient*]{} non equlibrium: [*ie*]{} transport properties at vanishing external fields (I think here of Onsager’s reciprocity and its quantitative form given by Green–Kubo’s transport theory). So it is by no means clear that there is any general non equilibrium thermodynamics. Nevertheless in 1973 a first suggestion that a general theory might be possible for non equilibrium systems in stationary and chaotic states was made by Ruelle in talks and eventually written down in papers[@[R1]]. The proposal is very ambitious as it suggests a [*general and essentially unrestricted*]{} answer to which should be the ensemble that describes stationary states of a system, [*whether in equilibrium or not*]{}. The ergodic hypothesis led Boltzmann to the general theory of ensembles (as acknowledged by Gibbs, whose work has been perhaps the main channel through which the allegedly obscure works of Boltzmann reached us): besides giving the second law, Eq.(\[1.1\]), it also prescribed the microcanonical ensemble for describing equilibrium statistics. The reasoning of Ruelle was that from the theory of simple chaotic systems one knew that such systems, for the simple fact that they are chaotic, will reach a “unique” stationary state. Therefore simply assuming chaoticity would be tantamount to assuming that there is a uniquely defined ensemble which should be used to compute the statistical properties of a system out of equilibrium. Therefore one is, in a very theoretical way, in a position to inquire whether such unique ensemble has [*universal properties*]{} valid for small and large systems alike: of course we cannot expect too many of them to hold. In fact in equilibrium theory the only one I know is precisely the heat theorem, besides a few general (related) inqualities ([*eg positivity of the specific heat or of compressibility*]{}). The theorem leads, indirectly as we have seen, to the microcanonical ensemble and then, after one century of work, to a rather satisfactory theory of phenomena like phase transitions, phase coexistence, universality. In the end the role of the ergodic hypothesis emerges, at least in my view, as greatly enhanced: and the idea of Ruelle seems to be its natural (and I feel unique) extension out of equilibrium. Of course this would suffer from the same objections that are continuously raised about the ergodic hypothesis: namely “there is the time scale problem”. To such objections I do not see why the answer given by Boltzmann should not apply [*unchanged*]{}: large systems have the extra property that the interesting observables take the same value in the whole (or virtually whole) phase space. [*Therefore they verify any relation that is true no matter whether the system is large or small*]{}: such relations (whose very existence is, in fact, surprising) might be of no interest whatsoever in small systems (like in the above mentioned Boltzmann’s rigid Saturn ring, or in his other similar example of the Moon regarded as a rigid ring rotating about the Earth). Ruelle’s proposal was formulated in the case of fluid mechanics: but it is so clearly more general that the reason why it was not explicitly proposed for statistical systems is probably due to the fact that, as a principle, it required some “check” if formulated for Statistical MEchanics: as originally stated and without any further check it would have been analogous, in my view, to the ergodic hypothesis without the heat theorem (or other consequences drawn from the theory of statistical ensembles). Evidence for the non trivial applicability of the hypothesis built up quite rapidly and it was repeatedly hinted in various papers dealing with numerical experiments, mostly on very small particle systems ($<100$ to give an indication)[@[H]]. In attempting at underdstanding one such experiment[@[ECM2]] the following “formal” interpretation of the Ruelle’s priciple was formulated[@[GC]] for statistical mechanics (as well as for fluid mechanics, replacing “many particles system” with “turbulent fluid”)) in the form: [*Chaotic hypothesis: A many particle system in a stationary state can be regarded as a transitive Anosov system [*(see below)*]{} for the purpose of computing the macroscopic properties of the system.*]{} The hypothesis was made first in the context of reversible systems (which were the subject of the experimental work that we were attempting to explain theoretically). The assumption that the system is Anosov (see below) has to be interpreted when the system has an attractor strictly smaller than the available phase space ([*ie*]{} not dense in it), as saying that the attractor itself can be regarded as a smooth Anosov systems (see below). The latter interpretation [*rules out*]{} fractal attractors and, to include them, it could be replaced by changing “Anosov” into “Axiom A”: but I prefer to wait if there is real need of such an extension. It is certainly an essential extension for small systems, but it is not clear to me how relevant could fractality be when the system has $10^{23}$ particles). A transitive Anosov system is a [*smooth*]{} system with a dense orbit (the latter condition is to exclude trivial cases, like when the system consists of two chaotic but noninteracting subsystems) and such that around every point $x$ one can set up a local coordinate system that a) [*depends continuously on $x$ and is covariant*]{} ([*ie*]{} it follows $x$ in its evolution) and b) is [*hyperbolic*]{} ([*ie*]{} transversally to the phase space velocity of any chosen point $x$ the motion of nearby points looks, when seen from the coordinate frame covariant with $x$, as a hyperbolic motion near a fixed point. This means that on (each) plane transversal to the phase space velocity of $x$ there will be a “stable coordinate surface”, the [*stable manifold*]{} through $x$, whose points trajectories get close to the trajectory of $x$ at exponential speed as the time tends to $+\infty$ and an “unstable coordinate surface”, the [*unstable manifold*]{}, whose trajectories get close to the trajectory of $x$ at exponential speed as the time tends to $-\infty$. The direction parallel to the velocity can be regarded as a [*neutral*]{} direction where, in the average, no expansion or contraction occurs. Anosov systems are the [*paradigm*]{} of chaotic systems: they are the analogues of the harmonic oscillators for ordered motions. Their simple but surprising and deep properties are by and large very well understood; particularly in the discrete time cases that we consider below. Unfortunately they are not as well known as they should among physicists, who seem confused by the language in which they are usually presented: however it is a fact that such a remarkable mathematical object has been introduced by mathematicians and the physicists must therefore make an effort at understanding the new notion and its physical significance. In particular, if a system is Anosov: [*for all*]{} observables $F$ ([*ie*]{} continuous functions on phase space) and for almost all initial data $x$ the time average of $F$ exists and can be computed by a phase space integral with respect to a distribution $\mu$ uniquely determined on phase space ${\cal F}$: $$\lim_{T\to\infty} \frac1T \int_0^T F(S_t x) \,dt=\int_{\cal F} F(y)\,\mu(dy)\label{2.1}$$ [*“almost all”*]{} means apart from a set of zero volume in phase space. The distribution is called the SRB distribution: it was proven to exist by Sinai[@[SRB]] for Anosov systems and the result was extended to the much more general Axiom A attractors by Ruelle and Bowen[@[R1]]. [*Natural distributions*]{} were, independently, discussed and shown to exist[@[LY]] for other (related and simpler) dynamical systems. Clearly the chaotic hypothesis solves [*in general*]{} ([*ie*]{} for systems that can be regarded as “chaotic”) the problem of determining which is the ensemble to use to study the statistics of stationary systems in or out of equilibrium (it clearly implies the ergodic hypothesis in equilibrium), in the same sense in which the ergodic hypothesis solves the equilibrium case. Therefore the first problem with such an hypothesis is that it will be very hard to prove it in a mathematical sense: the same can be said about the ergodic hypothesis which is not only unproved for most cases, but it will remain such, in systems of statistical mechanical interest, for long if not forever, aside from some very special cases (like the hard core gas). The chaotic hypothesis might turn out to be false in interesting cases, like the ergodic hypothesis which does not hold for the simplest systems studied in statistical mechanics, like the free gas, the harmonic chain and the black body radiation. Worse: it is [*known*]{} to be false for trivial reasons in some systems in equilibrium (like the hard core gas): simply because the Anosov definition requires smoothness of the evolution and systems with collisions are not smooth systems (in the sense that the trajectories are not differentiable as functions of the initial data). However, interestingly enough, the case of hard core systems is perhaps the system closest to an Anosov system that can be thought of and that is also of statistical mechanical relevance. To an extent that there seem to be no known properties that such system does not share with an Anosov system. Aside from the trivial fact that it is not a smooth system, the hard core system behaves, for Statistical Mechanics purposes, [*as if it was a Anosov system*]{}. Hence it is the prototype system to study in looking for applications of the chaotic hypothesis. The [*problem*]{} that remains is whether the chaotic hypothesis has any power to tell us something about nonequilibrium statistical mechanics. This is the real, deep, question for anyone who is willing to consider it. [*One consequence*]{} is the ergodic hypothesis, hence the heat theorem: but this is [*too little*]{} even though it is a very important property for a theory with the ambition of being a [*general*]{} extension of the theory of equilibrium ensembles. I conclude this section with a comment useful in the following. As is well known by who has ever attempted a numerical (or real) experiment, one often does not observe systems in continuous time: but rather one records the state of the system at times when some event that is considered intereresting or characteristic happens. Calling such events “[*timing events*]{}” the system then appears as having a phase space of dimension one unit lower: because the set of timing events has to be thought of as a surface in phase space transversal to the phase space velocity of the trajectories $t\to S_t x$. If $x$ is a timing event and $\vartheta(x)$ is the time that one has to wait until the next timing event happens, the time evolution becomes a map $x\to Sx\equiv S_{\vartheta(x)}x$ of $x$ into the following timing event. For instance one could record the configuration of a system of hard balls every time that a collision takes place, and $S$ will map a collision configuration into the next one. The chaotic hypothesis can be formulated for such “Poincarè’s sections” of the continuous time flow in exaclty the same way: and this is in fact a simpler notion as there will be no “[*neutral direction*]{}” and the covariant local system of coordinates will be simply based on a stable and an unstable manifold through every point $x$. [*In the following section we take the point of view that time evolution has been discretized in the above sense*]{} ([*ie*]{} via a Poincarè’s section on a surface of timing events): this simplifies a discussion, but in a minor way[@[Ge]]. [Fluctuation theorem for reversibly dissipating systems.]{} The key to find applications is that the apparently inconsequential hypothesis that the system is Anosov provides us not only with an existence theorem of the SRB distribution $\mu$ but [*also*]{} with an explicit expression for it. How explicit? as we shall see not too far from what we are used to in equilibrium statistical mechanics ([*eg*]{} $\mu=e^{-\beta H}$): where apparently unmanageable expressions and hopeless integrals have important and beautiful applications in spite of their obvious non computability. The expression is the following: there is a partition of phase space into cells $E_1,E_2,\ldots$ which in a sense that I do not specify here[@[G1]] is “[*covariant*]{}” with respect to time evolution and to the other symmetries of the system (if any: think of parity or time reversal) such that the average value of an observable can be computed as: $$\langle{F}\rangle=\int_{\cal F} F(y)\,\mu(dy) =\frac{\sum_{E_i}\Lambda_{u,T}^{-1}(x_i) F(x_i)} {\sum_{E_i} \Lambda_{u,T}^{-1}(x_i)}\label{3.1}$$ where $x_i\in E_i$ is a point suitably chosen in $E_i$ (quite, but not completely, arbitrarily for technical, trivial, reasons[@[GC]; @[G1]]) and $\Lambda_{u,T}(x)$ is the expansion of a surface element lying on the unstable manifold of $S_{-\frac12T} x$ and mapped by $S_T$ into a surface element around $S_{\frac12T}x$. Of course Eq.(\[3.1\]) requires that the cells be so small that $F$ has neglegible variations inside them: if this is not the case then one simply has to [*refine*]{} the partiction into smaller cells, until they become so small that $F$ is a constant inside them (for practical purposes). This can be done simply by applying the time evolution map and its inverse to the partition that we already imagine to have, but which has large cells, and then intersecting the elements of the new partitions obtained to get a finer partition. The hyperbolicity of the evolution implies that the partition into cells can be made as fine as desired. Another reason why we need small cells is to insure that the weights themselves do not depend too much on which point $x_i$ is chosen to evaluate them: the precise condition is somewhat delicate[@[boundary]]. An example of an application of the above formula is obtained by studying the phase space volume contraction rate $\sigma(x)$: this is defined as the logarithm of the Jacobian determinant $\Lambda(x)$ of the time evolution map (recall that we are now considering a discrete time evolution $S$, as explained at the end of the preceding section). Suppose that we ask for the fluctuations of the average of the “dimensionless contraction” $\sigma(x)/\sigma_+$ where $\sigma_+$ is the (infinite) time average $\sigma_+=\int \sigma(y)\,\mu(dy)$, that is assumed strictly positive (it could be zero, for instance in a equilibrium system where the evolution is Hamiltonian and conserves volume in phase space; [*but*]{} it cannot[@[R2]] be $<0$). The positivity of the time average of $\sigma$ can be taken as the very definition of “dissipative” motions. This is the quantity: $p=\frac1{\tau\sigma_+}\sum_{k=- \frac12\tau}^{\frac12\tau} \sigma(S^k x)$. It will have a probability distribution, in the stationary state, that we write $\pi_\tau(p)$. We now compare $\pi_\tau(p)$ to $\pi_\tau(-p)$, which is clearly a ratio of probabilities of two events one of which will have an extremely small probability (the expected value of $p$ being $1$). [*Suppose that the system is time reversible*]{}: [*ie*]{} that there is an isometry of phase space $I$ that anticommutes with the evolution: $IS=S^{-1}I$. Then: $$\frac{\pi_\tau(p)}{\pi_\tau(-p)}=\frac {\sum_{E_i; p}\Lambda_{u,T}^{-1}(x_i)} {\sum_{E_i; -p}\Lambda_{u,T}^{-1}(x_i)}\label{3.2}$$ where the sum in the numerator extends over the cells $E_i$ in which the total dimensionless volume contraction rate is $p$ anf the sum in the denominator over those with contraction rate $-p$. Here we take $T=\tau$ for the purpose of a partial illustration: this is [*not*]{} allowed and in a sense it is the [*only*]{} difficulty in the discussion. But taking $T=\tau$ conveys some of the main ideas. If this “interchange of limits” is done then one simply remarks that the sum in the denominator of Eq.(\[3.2\]) can be performed over the same cells as that in the numerator, provided we evaluate the weight in the denominator at the point $I x_i$, [*ie*]{} provided we use in the denominator the weight $\Lambda_{u,T}^{-1}(Ix_i)$: this is so because time reversal maps a cell in which the dimensionless rate of volume contraction is $p$ into one in which it is $-p$ and viceversa. But time reversal also interchanges expasion and contraction so that $\Lambda_{u,T}^{-1}(Ix_i)=\Lambda_{s,T}(x_i)$, if the contraction rate along the stable manifold $\Lambda_{s,T}$ is defined in the same way as $\Lambda_{u,T}$ by exchanging stable and unstable manifolds. This means that the ratio between corresponding terms is now $\Lambda_{u,T}^{-1}(x_i)\Lambda_{s,T}^{-1}(x_i)$. Since the latter quantity is essentially the [*total contraction rate*]{} up to a factor bounded independently of the value of $T$ (because the angle between the stable and unstable manifolds is bounded away from zero by the continuity property of Anosov systems) it follows that the ratio Eq.(\[3.2\]), in this (rather uncontrolled) approximation $T=\tau$, is $\tau p \sigma_+$ [*ie*]{} simply the contraction rate which has the [*same value for all cells considered*]{}, by construction. Conclusion: $$\lim_{\tau\to\infty}\frac1{\tau\,p\,\sigma_+} \log\frac{\pi_\tau(p)}{\pi_\tau(-p)}=1 \label{3.3}$$ which is the [*fluctuation theorem*]{} if $\pi_\tau$ are evaluated with respect to the SRB distribution of the system. The above “proof” is missing a key point: namely the interchange of limits. Fixing $\tau=T$ means that we are not computing the probabilities in the SRB distribution but, at best, in some approximation of it. In experimental tests one need the theorem to hold when the limits are taken in the proper order ([*ie*]{} first $T\to\infty$ and [*then*]{} $\tau\to\infty$). The latter theoretical aspects have been discussed in the original papers[@[GC]], where it is shown that the limit is approached as $\tau^{-1}$; and formal proofs are also available[@[G2]; @[R3]]. That this is not a fine point of rigor can be seen from the fact that if one disregards it then other proofs of the “same” result [*but*]{} with $\tau=T$ become possible. In other words the result has a “tendency” to be general[@[MR]; @[K]] but it can be proved in the right form of Eq(\[3.3\]) only under strong chaoticity assumptions. It is very interesting that in weaker forms a result closely related to the fluctuation theorem can be obtained for [*completely different*]{} dynamical systems [*ie*]{} for stochastic evolutions[@[K]]. It is possible that for the stochastic evolution the result could be extended to become a closer analogue of the above, solving the mentioned problem of the interchange of limits: one would, in fact, think that the noise makes the system as chaotic as one may possibly hope. The result Eq.(\[3.3\]), has to be tested because in all applications we do not know whether the system is Anosov and to what extent it can be assumed such. And its verification provides a form of test of the chaotic hypothesis. Other equivalent formulations of the fluctuation theorem are in terms of the “free energy” of the observable $p$: $\zeta(p)=\lim_{\tau\to\infty} \frac1\tau\log \pi_\tau(p)$; it becomes: $$\frac{\zeta(p)-\zeta(-p)}{p\sigma_+}=1 \label{3.4}$$ which says that the odd part of $\zeta(p)$ is linear in $p$ with a [*determined and parameter free*]{}, slope: note that without reversibility one could only expect that $\zeta(p)$ had a quadratic maximum at $p=1$ ([*central limit theorem*]{} for the observable $\sigma(x)$) which stays quadratic as long as $|p-1|=O(\frac1{\sqrt{{\tau}}})$. The fluctuation theorem instead gives informations concerning huge deviations $|p-1|=O(2)$! it is a [*large deviation theorem*]{}. The main interest, so far, of the above theorem is that it has shown that Ruelle’s principle has some power of prediction. In fact the result has been checked in various small systems[@[BGG]]. The first of which was its experimental discovery[@[ECM2]] [*preceding*]{} the chaotic hypothesis and fluctuation theorem formulations. It is also interesting because of its [*universal validity*]{}: it is system independent (provided reversible), hence it is a general law that should be satisfied if the chaotic hypotesis is the correct mathematical translation of our intuitive notion of chaos, and Anosov systems catch it fully. The question whether the above results can also be obtained from the chaotic hypothesis formulated in terms of the continuous time flow on phase space (rather than for a map between timing events, see the last comments in the previous section) would leave us unhappy if it did not have a positive answer: it does have a positive answer[@[Ge]]. [Onsager’s reciprocity and Green–Kubo’s formula.]{} The fluctuation theorem degenerates in the limit in which $\sigma_+$ tends to zero, [*ie*]{} when the external forces vanish and dissipation disappears (and the stationary state becomes the equilibrium state). Since the theorem deals with systems that are time reversible [*at and outside*]{} equilibrium Onsager’s hypotheses are certainly verified and the system should obey reciprocal response relations at vanishing forcing. This led to the idea that there might be a connection between fluctuation theorem and Onsager’s reciprocity and also to the related (stronger) Green–Kubo’s formula. This is in fact true: if we define the [*microscopic thermodynamic flux*]{} $j(x)$ associated with the [*thermodynamic force*]{} $E$ that generates it, [*ie*]{} the parameter that measures the strength of the forcing (which makes the system not Hamiltonian), via the relation: $$j(x)=\frac{\partial\sigma(x)}{\partial E}\label{4.1}$$ (not necessarily at $E=0$) then in \[G2\] a heuristic proof shows that the limit as $E\to0$ of the fluctuation theorem becomes simply (in the continuous time case) a property of the average, or “macroscopic”, [*flux*]{} $J=\langle{j}\rangle_{\mu_E}$: $$\frac{\partial J}{\partial E}\big|_{E=0}=\frac12 \int_{-\infty}^{\infty} \langle{j(S_tx)j(x)}\rangle_{\mu_E}\Big|_{E=0} \,dt\label{4.2}$$ where $\langle{\cdot}\rangle_{\mu_E}$ denotes average in the stationary state $\mu_E$ ([*ie*]{} the SRB distribution which, at $E=0$, is simply the microcanonical ensemble). If there are several fields $E_1,E_2,\ldots$ acting on the system we can define several thermodynamic fluxes $j_k(x){\buildrel def\over =} \partial_{E_k}\sigma(x)$ and their averages $\langle{j_k}\rangle_\mu$: a simple extension of the fluctuation theorem[@[G3]] is shown to reduce, in the limit in which all forces $E_k$ vanish, to: $$\begin{aligned} L_{hk}{\,{\buildrel def\over=}\,} \frac{\partial J_{h}}{\partial E_k}\big|_{E=0}=\kern1cm{}\nonumber\end{aligned}$$ $$=\frac12 \int_{-\infty}^{\infty} \langle{j_h(S_tx)j_k(x)}\rangle_{E=0} \,dt=L_{kh}\label{4.3}$$ therefore we see that the fluctuation theorem can be regarded as [*an extension to non zero forcing*]{} of Onsager’s reciprocity and, actually, of Green–Kubo’s formula. Certainly assuming reversibility in a system out of equilibrium can be disturbing: therefore one can inquire if there is a more general connection between the chaotic hypothesis and Onsager’s reciprocity and Green–Kubo’s formula. This is indeed the case and provides us with a [*second application*]{} of the chaotic hypothesis valid, however, only in zero field. It can be shown that the relations Eq.(\[4.3\]) follow from the sole assumption that at $E=0$ the system is time reversible and that it verifies the chaotic hypothesis at $E=0$: at $E\ne0$ it can be, as in Onsager’s theory, not reversible[@[GR]]. It is not difficult to see, technically, how the fluctuation theorem, in the limit in which the driving forces tend to $0$, formally yields Green–Kubo’s formula. We consider time evolution in continuous time and simply note that Eq.(\[3.3\]) implies that, for all $E$ (for which the system is chaotic): $$\lim_{\tau\to\+\infty} \frac1\tau \log \langle e^{I_E}\rangle_{\mu_E}=0\label{4.4}$$ where $I_E\,{\buildrel def \over =}\,\int \sigma(S_tx) dt$ with $\sigma(x)$ being the divergence of the equations of motion ([*ie*]{} the phase space contraction rate, in the case of continuous time). This remark[@[Bo]] (that says that essentialy $\langle e^{I_E}\rangle_{\mu_E}\equiv1$ or more precisely it is not too far from $1$ so that Eq.(\[4.4\]) holds) can be used to simplify the analysis in [@[G3]] as follows. Differentiating both sides with respect to $E$, not worrying about interchanging derivatives and limits and the like, one finds that the second derivative with respect to $E$ is a sum of six terms. Supposing that for $E=0$ the system is Hamiltonian and (hence) $I_0\equiv 0$, the six terms are, when evaluated at $E=0$: $$\begin{aligned} &\frac1\tau\langle \partial^2_E I_E\rangle_{\mu_E}|_{E=0} %,\nonumber\\ & -\frac1\tau\langle (\partial_E I_E)^2\rangle_{\mu_E}|_{E=0}+\nonumber\\ &+\frac1\tau\int \partial_E I_E(x) \partial\mu_E(x)|_{E=0}+\nonumber\\ &-\frac1\tau\Big(\langle (\partial_E I_E)^2\rangle_{\mu_E}\cdot \int 1\,\partial_E \mu_E\Big)|_{E=0}+\label{4.5}\\ &\frac1\tau\int \partial_E I_E(x) \partial\mu_E(x)|_{E=0} +\frac1\tau\int 1\cdot\partial_E^2\mu_E|_{E=0}\nonumber\end{aligned}$$ and we see that the fourth and sixth terms vanish being derivatives of $\int \mu_E(dx)\equiv 1$, the first vanishes (by integration by parts) because $I_E$ is a divergence and $\mu_0$ is the Liouville distribution (by the assumption that the system is Hamiltonian at $E=0$ and chaotic). Hence we are left with: $$\Big(-\frac1\tau\langle (\partial_E I_E)^2\rangle_{\mu_E}+\frac2\tau \int \partial_E I_E(x) \partial_E \mu_E(x)\Big)_{E=0}=0\label{(4.6)}$$ where the second term is, since the distribution $\mu_E$ is stationary, $2\tau^{-1} \partial_E ( \langle \partial_E I_E\rangle_{\mu_E})|_{E=0}\equiv 2\partial_E J_E|_{E=0}$; and the first term tends to $\int_{-\infty}^{+\infty} \langle j(S_t x) j(x)\rangle_{E=0} dt$ as $\tau\to\infty$. Hence we get Green–Kubo’s formula in the case of only one forcing paprameter. The argument should be extended to the case in which $E$ is a vector describing the strength of various driving forces acting on the system[@[G3]]: but one needs a generalization of Eq([4.4]{}). The latter is a consequence of the fluctuation theorem, but the theorem had to be extended[@[G3]] to derive also Green–Kubo’s formula (hence reciprocity) when there were several independent forces acting on the system.. The above analysis is unsatisfactory because we interchange limits and derivatives quite freely and we even take derivatives ot $\mu_E$, which seems to require some imagination as $\mu_E$ is concentrated on a set of zero volume. On the other hand, under the strong hypotheses which we suppose to be (that the system is Anosov), we should not need extra assumptions. Indeed the above mentioned non heuristic analysis[@[GR]] is based on the study on the differentiability of SRB distributions with respect to parameters[@[Rdiff]]. A [*third application*]{} of the chaotic hypothesis, still limited to reversible systems, is the following: consider the probability that certain observables $O_1,O_2,\ldots$ are measured during a time interval $[-\frac11\tau,\frac12\tau]$ during which the system evolves between the point $S_{-\frac12\tau}x$ and $S_{\frac12\tau}x$. And suppose that we see the [*path*]{} or [*pattern*]{} $\omega$ given by $t\to O_1(S_tx),O_2(S_tx),\ldots$. Assuming, for simplicity, that $O_j$ are [*even*]{} under time reversal the “time reversed” pattern $I\omega$ will be $t\to O_1(S_{-t}x), O_2(S_{-t}x)$ and it will be clearly [*very unlikely*]{}. Suppose that we look at the relative probabilities of various patterns [*conditioned*]{} to an average (over the time interval $[-\frac11\tau,\frac12\tau]$) dimensionless volume contraction rate $p$. Then one can prove[@[G4]], under the chaotic hypothesis, that the relative probabilities of patterns in presence of rate $p$ is [*the same as that of the time reversed patterns in presence of rate*]{} $-p$. Since the contraction rate of volume in phase space can be interpreted as [*entropy creation rate*]{}, as suggested for instance by the above use, Eq.(\[1.3\]) of the phase space contraction to define the thermodynamic fluxes, as “conjugate” observables to the external thermodynamic forces, the latter statement has some interest as it can be read as saying that “it costs no extra effort to realize events normally regarded as impossible once one succeeds in the enterprise of reversing the sign of entropy creation rate”[@[G4]]. The interpretation of phase space contraction rate as [*entropy creation rate*]{} meets opposition, fierce at times: however it seems to me a very reasonable proposal for a concept that we should not forget has not yet received a universally accepted definition and therefore its definition should at least be considered as an open problem. [Reversible versus irreversible dissipation. Nonequilibrium ensembles?]{} A system driven out of equilibrium can reach a stationary state (and not steam out of sight) only if enough dissipation is present. This means that any mechanical model of a system reaching a stationary state out of equilibrium [*must*]{} be a model with non conservative equations of motion in which forces representing the action of the thermostats, that keep the system from heating up, are present. Thus a generic model of a system stationarily driven out of equilibrium will be obtained by adding to Hamilton’s equations (corresponding to the non driven system) other terms representing forces due to the thermostat action. Here one should avoid attributing a fundamental role to special assumptions about such forces. One has to realize that there is [*no privileged*]{} thermostat. One can consider many of them and they simply describe various ways to take out energy from the system. Thus one can use stochastic thermostats, and there are many types considered in the literature; or one can consider deterministic thermostats and, among them, reversible ones or irreversible ones. Each thermostat requires its own theory. However the same system may behave in the same way under the action of different thermostatting mechanisms: if the only action we make on a gas tube is to keep the extremes temperatures fixed by taking in or out heat from them the difference may be irrelevant, at least in the limit in which the tube becomes long enough and as far as what happens in the middle of it is concerned. But of course the form of the stationary state may be very different in the various cases, even when we think that the differences are only minor boundary effects. For instance, in the case of the gas tube, if our model is of deterministic dissipation we expect that the SRB state be concentrated on a set of [*zero phase space volume*]{} (because phase space will in the average contract, when $\sigma_+>0$, so that any stationary state has to be concentrated on a set of zero volume, [*which however could still be dense*]{} and ususally will be). While if the model is stochastic then the stationary state will be described by a [*density*]{} on phase space. Nothing could seem more different. Nevertheless it might be still true that in the limit of an infinite tube the two models give the same result: in the same sense as the canonical and microcanonical ensembles describe the same state even though the microcanonical ensemble is supported on the energy surface, which has zero volume if measured by using the canonical ensemble (which is given by a density over the whole available phase space). Therefore we see that out of equilibrium we have in fact [*much more freedom to define equivalent ensembles*]{}. Not only we have (very likely) the same freedom that we have in equilibrium (like fixing the total energy or not, or fixing the number of particles or not, passing from microcanonical to canonical to grand canonical [*etc*]{}) but [*we can also change the equations of motion and obtain different stationary states, [*ie*]{} different SRB distributions, which will however become the same in the thermodynamic limit*]{}. Being able to prove the mathematical equivalence of two thermostats will amount at proving their physical equivalence. This again will be a difficult task, in any concrete case. What I find fascinating is that the above remarks provide us with the possibility that a [*reversible thermostat can be equivalent in the thermodynamic limit to an irreversible one*]{}. I conclude by reformulating a conjecture, that I have already stated many times in talks and in writings[@[G5]], which clarifies the latter statement. Consider the following two models describing a system of hard balls in a periodic (large) box in which there is a lattice of obstacles that forbid collisionless paths (by their arrangement and size): the laws of motion will be Newton’s laws (elastic collisions with the obstacles as well as between particles) plus a constant force $E$ along the $x$–axis [*plus a thermostatting force*]{}. In the first model the thermostatting force is simply a constant times the momentum of the particles: it acts on the $i$-tth particle as $-\nu p_i$ if $\nu$ is a “friction” constant. Another model is a force proportional to the momentum but via a proportionality factor that is not constant and depends on the system configuration: it has the form $-\alpha(x)p_i$ with $\alpha(x)=E\cdot\sum_i p_i/\sum_i p_i^2$. The first model is essentially the model used by Drude in his theory of conduction in metals. The second model has been used very often in recent years for theoretical studies and has thus acquired a respected status and a special importance: it was among the first models used in the experiments and theoretical ideas that led to the connection between Ruelle’s ideas for turbulent motion in fluids and nonequilibrium statistical mechanics[@[H]]. I think that the importance of such works cannot be underestimated: without them the recent theoretical developments would have been simply unthinkable, in spite of the fact that [*a posteriori*]{} they seem quite independent and one could claim (unreasonably in my view) that everything could have been done much earlier. Furthermore the second model can be seen as derived from Gauss’ least constraint principle. It keeps the total (kinetic) energy exactly constant over time (taking in and out energy, as needed) and is called [*Gaussian thermostat*]{}. [*Unlike the first model it is reversible*]{}, with time reversal being the usual velocity inversion. Thus the above theory and results based on the chaotic hypothesis apply. My conjecture was (and is) that: 1\) compute the average energy per particle that the system has in the constant friction case and call it ${\cal E}(\nu)$ calling also $\mu_\nu$ the corresponding SRB distribution. 2\) call $\tilde \mu_{\cal E}$ the SRB distribution for the Gaussian thermostat system when the total (kinetic) energy is fixed to the value ${\cal E}$ 3\) then $\mu_\nu=\tilde \mu_{{\cal E}(\nu)}$ [*in the thermodynamic limit*]{} (in which the box size tends to become infinitely large but with the number of particles and the total energy correspondingly growing so that one keeps the density and the energy density constant) and for [*local*]{} observables, [*ie*]{} for observables that depend only on the particles of the system localized in a fixed finite region of the container. This means that the equality takes place in the usual sense of the theory of ensembles[@[R4]]. This opens the way to several speculations as it shows that the reversibility assumption might be not so strong after all. And results for reversible systems may carry through to irreversible ones. I have attempted to extend the above ideas also to cases of turbulent motions but I can only give here references[@[G5]; @[G6]]. [*Acknowledgements: this work has been supported also by Rutgers University and by the European Network \# ERBCHRXCT940460 on: “Stability and Universality in Classical Mechanics". I am indebted to F. Bonetto for several enlightening discussions[@[Bo]].*]{} Boltzmann, L.: [*Über die Eigenshaften monzyklischer und anderer damit verwandter Systeme*]{}, in “Wissenshafltliche Abhandlungen”, ed. F.P. Hasenöhrl, vol. III, Chelsea, New York, 1968, (reprint). Boltzmann, L.: [*Entgegnung auf die wärmetheoretischen Betrachtungen des Hrn. E. Zermelo*]{}, engl. transl.: S. Brush, “Kinetic Theory”, [vol. 2]{}, 218, Pergamon Press. Boltzmann, L.: [*Zu Hrn. Zermelo’s Abhandlung “Ueber die mechanische Erklärung irreversibler Vorgänge”*]{}, engl. trans. in S. Brush, “Kinetic Theory”, [**2**]{}, 238. Boltzmann, L.: [*Lectures on gas theory*]{}, english edition annotated by S. Brush, University of California Press, Berkeley, 1964. Lebowitz, J.L.: [*Boltzmann’s entropy and time’s arrow*]{}, Physics Today, 32–38, 1993. And [*Microscopic Reversibility and Macroscopic Behavior: Physical Explanations and Mathematical Derivations*]{}, in [*25 years of nonequilibrium statisticakl mechanics*]{}, ed. J. Brey, J. Marro, J. Rubi, M. San Miguel, [*Lecture Notes in Physics*]{}, Springer, 1995. Ruelle, D.: [*A measure associated with Axiom A attractors*]{}, American Journal of Mathematics, [**98**]{}, 619–654, 1976. Ruelle, D.: [*Measures describing a turbulent flow*]{}, Annals of the New York Academy of Sciences, [**357**]{}, 1–9, 1980. R.Bowen, D.Ruelle. [*Ergodic theory of Axiom A flows*]{}, Inventiones Math. [**2**]{},181-202(1975). See also Ruelle, D.: [*Chaotic motions and strange attractors*]{}, Lezioni Lincee, notes by S. Isola, Accademia Nazionale dei Lincei, Cambridge University Press, 1989. It is here impossible to account here, analytically, for the vast literature on the subject. The following papers are highlights and contain detailed references to related papers: Holian, B.L., Hoover, W.G., Posch. H.A.: [*Resolution of Loschmidt’s paradox: the origin of irreversible behavior in reversible atomistic dynamics*]{}, Physical Review Letters, [**59**]{}, 10–13, 1987. Evans, D.J., Morriss, G.P.: [*Statistical Mechanics of Nonequilibrium fluids*]{}, Academic Press, New York, 1990. Evans, D.J.,Cohen, E.G.D., Morriss, G.P.: [*Viscosity of a simple fluid from its maximal Lyapunov exponents*]{}, Physical Review, [**42A**]{}, 5990–5997, 1990. Dellago, C., Posch, H., Hoover, W.: [*Lyapunov instability in system of hard disks in equilibrium and non-equilibrium steady states*]{}, Physical Review, [**53E**]{}, 1485–1501, 1996. Evans, D.J.,Cohen, E.G.D., Morriss, G.P.: [*Probability of second law violations in shearing steady flows*]{}, Phys. Rev. Letters, [**71**]{}, 2401–2404, 1993. Gallavotti, G., Cohen, E.G.D.: [*Dynamical ensembles in nonequilibrium statistical mechanics*]{}, Physical Review Letters, [**74**]{}, 2694–2697, 1995. Gallavotti, G., Cohen, E.G.D.: [*Dynamical ensembles in stationary states*]{}, Journal of Statistical Physics, [**80**]{}, 931–970, 1995. Sinai, Y.G.: [*Gibbs measures in ergodic theory*]{}, Russian Math. Surveys, [**27**]{}, 21–69, 1972. Also: [*Introduction to ergodic theory*]{}, Princeton U. Press, 1977. Lasota, A., Yorke, J.A.: [*On the existence of invariant measures for piecewise monotonic transformations*]{}, Transactions American Mathematical Society, [**183**]{}, 481, 1973. Gentile, G: [*Large deviation rule for Anosov flows*]{}, mp$\_$arc@ math. utexas. edu, \#96–79, in print in Forum Mathematicum. Gallavotti, G. [*New methods in nonequilibrium gases and fluids*]{}, Maribor meeting lectures, june 1996– 5 july 1996, [mp$\_$arc@ math. utexas. edu]{} \#96-533, in print in Open Systems & Information Dynamics. And Gallavotti, G.: [*Topics in chaotic dynamics*]{}, Lectures at the Granada school 1994, Ed. P. Garrido, J. Marro, in Lect. Notes in Phys., Springer Verlag, [**448**]{}, p. 271–311, 1995. The reason can be easily understood by analogy with a statistical mechanics case. Imagine a $1$–dimensional nearest neighbour Ising spin system and that $\mu$ is the [*infinite volume*]{} zero field Gibbs state with coupling $J$. Then we can use a formula like Eq.(\[3.1\]) where $E_i$ are the spin configurations which have a specific form (one among the $2^T$ possible ones) in the box $[-\frac12T,\frac12T]$ and are arbitrary outside: clearly to get $\langle F\rangle|_\mu$ right the function $F$ has to depend only on few “centrally located spins”, which means that $F$ has to be essentially constant in each of the $E_i$’s; hence, as said, the $E_i$ have to be small enough, [*ie*]{} $T$ large enough. But we also need that the boundaries have no influence on the expectation values that we compute. A choice of $x_i\in E_i$ means selecting a spin configuration inside the box $[-\frac12T,\frac12T]$ [*and ouside it*]{}: hence the weight ([*ie*]{} $e^{-H}$ if $H$ is the energy) depends on the form of the spin configuration $x_i$ [*outside*]{} the box. This means that the relative weights of the possible configurations inside the box (corresponding to the possible choices of $x_i$) do depend on $x_i$ and change by order $O(J)$, [*not small as $T\to\infty$*]{}, by changing $x_i$ keeping it inside $E_i$ [*ie*]{} keeping the same spin configuration in the box. Therefore care has to be paid in the choice of $x_i$ so that the relative weights are “right”. However there are always “pathological” choices of $x_i$ for which something “goes wrong” for some $F$ unless some care is exercized in choosing $x_i$. Think of $F$ as being the value of the spin $\sigma_0$ located at the origin and of choosing $x_i$ so that the first spins out of the box, $\sigma_{\pm(\frac12T+1)}$, [*the same* ]{} as the one, $\sigma_0$, at the center of the box: this amounts, if such a $x_i$ is chosen, at computing the distribution of $\sigma_0$ in a model with an external field $2J$ acting on the spin at site $0$: this is obviously going to give the wrong expectation value for $F=\sigma_0$ although in the average we produce the same state as the correct one. In fact the only wrong averages concern those of spins located near the origin (note that the analogue of this this would be false in $2$–dimensions, in the phase transitions region). An example of a “correct” choice of $x_i$, in this analogy, would be simply to take $x_i$ with $\sigma_{\pm(\frac12T+1)}=\sigma_{\pm\frac12T}$ and arbitrary elsehwere outside the box (using that the range is nearest neighbour). Finally the analogy is not accidental: the theory of the SRB states is done precisely by showing that they can be thought of as Gibbs states for a one dimensional short range Ising model[@[SRB]; @[R1]; @[G1]]. Ruelle, D.: [*Positivity of entropy production in nonequilibrium statistical mechanics*]{}, Journal of Statistical Physics, [**85**]{}, 1–25, 1996. Gallavotti, G.: [*Reversible Anosov maps and large deviations*]{}, Mathematical Physics Electronic Journal, MPEJ, (http:// mpej.unige.ch), [**1**]{}, 1–12, 1995. Ruelle, D.: [*New theoretical ideas in nonequilibrium statistical mechanics*]{}, Lecture notes at Rutgers University, fall 1997. Morriss, G.P., Rondoni, L.: [*Applications of periodic orbit theory to $N$–particle systems*]{}, Journal of Statistical Physics, [**86**]{}, 991, 1997. Kurchan, J.: [*Fluctuation Theorem for stochastic dynamics*]{}, preprint [email protected], \#9709304. Bonetto, F., Gallavotti, G., Garrido, P.: [*Chaotic principle: an experimental test*]{}, Physica D, [**105**]{}, 226–252, 1997, Gallavotti, G.: [*Extension of Onsager’s’s reciprocity to large fields and the chaotic hypothesis*]{}, Physical Review Letters, [**77**]{}, 4334–4337, 1996. Gallavotti, G., Ruelle, D.: [*SRB states and nonequilibrium statistical mechanics close to equilibrium*]{}, Communications in Mathematical Physics, in print The Eq(\[3.4\]) as a rewriting the fluctuation theorem Eq(\[3.3\]) is remarkable and was pointed out by F. Bonetto. Ruelle, D.: [*Differentiation of SRB states*]{}, Communications in Mathematical Physics, $\clubsuit$, 1997. Gallavotti, G.: [*Fluctuation patterns and conditional reversibility in nonequilibrium systems*]{}, in print on Annales de l’ Institut H. Poincaré, chao-dyn@ xyz. lanl. gov \#9703007. Gallavotti, G.: [*New methods in nonequilibrium gases and fluids*]{}, Proceedings of the conference [*Let’s face chaos through nonlinear dynamics*]{}, U. of Maribor, 24 june– 5 july 1996, ed. M. Robnik. Ruelle, D.: [*Statistical Mechanics*]{}, Benjamin, 1969. Gallavotti, G.: [*Equivalence of dynamical ensembles and Navier Stokes equations*]{}, Physics Letters, [**223A**]{}, 91–95, 1996. And [*Dynamical ensembles equivalence in fluid mechanics*]{}, Physica D, [**105**]{}, 163–184, 1997. *Internet access: The author’s quoted preprints can be downloaded (latest revision) at:* http://chimera.roma1.infn.it in the Mathematical Physics Preprints page. *Author’s e-mail: [email protected]*

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