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David Ruelle

Publications and source records attributed to David Ruelle.

34 records · Page 2Linked to original sources
arXiv2006

Nonequilibrium statistical mechanics and entropy production in a classical infinite system of rotators

We analyze the dynamics of a simple but nontrivial classical Hamiltonian system of infinitely many coupled rotators. We assume that this infinite system is driven out of thermal equilibrium either because energy is injected by an external force (Case I), or because heat flows between two thermostats at different temperatures (Case II). We discuss several possible definitions of the entropy production associated with a finite or infinite region, or with a partition of the system into a finite number of pieces. We show that these definitions satisfy the expected bounds in terms of thermostat temperatures and energy flow.

cond-mat.stat-mech↗
arXiv2005

Analyticity of the Susceptibility Function for Unimodal Markovian Maps of the Interval

In a previous note [Ru] the susceptibility function was analyzed for some examples of maps of the interval. The purpose of the present note is to give a concise treatment of the general unimodal Markovian case (assuming $f$ real analytic). We hope that it will similarly be possible to analyze maps satisfying the Collet-Eckmann condition. Eventually, as explained in [Ru], application of a theorem of Whitney [Wh] should prove differentiability of the map $f\mapstoρ_f$ restricted to a suitable set.

math.DS↗
arXiv2004

Differentiation of SRB states for hyperbolic flows

Let the ${\cal C}^3$ vector field ${\cal X}+aX$ on $M$ define a flow $(f^t_a)$ with an Axiom A attractor $Λ_a$ depending continuously on $a\in(-ε,ε)$. Let $ρ_a$ be the SRB measure on $Λ_a$ for $(f^t_a)$. If $A\in{\cal C}^2(M)$, then $a\mapstoρ_a(A)$ is ${\cal C}^1$ on $(-ε,ε)$ and $dρ_a(A)/da$ is the limit when $ω\to0$ with ${\rm Im}ω>0$ of $$ \int_0^\infty e^{iωt}dt \intρ_a(dx) X(x)\cdot\nabla_x(A\circ f_a^t) $$

math.DS↗
arXiv2004

Differentiating the absolutely continuous invariant measure of an interval map f with respect to f

Let the map $f:[-1,1]\to[-1,1]$ have a.c.i.m. $ρ$ (absolutely continuous $f$-invariant measure with respect to Lebesgue). Let $δρ$ be the change of $ρ$ corresponding to a perturbation $X=δf\circ f^{-1}$ of $f$. Formally we have, for differentiable $A$, $$ δρ(A)=\sum_{n=0}^\infty\intρ(dx) X(x){d\over dx}A(f^nx) $$ but this expression does not converge in general. For $f$ real-analytic and Markovian in the sense of covering $(-1,1)$ $m$ times, and assuming an {\it analytic expanding} condition, we show that $$λ\mapstoΨ(λ)=\sum_{n=0}^\inftyλ^n \intρ(dx) X(x){d\over dx}A(f^nx) $$ is meromorphic in ${\bf C}$, and has no pole at $λ=1$. We can thus formally write $δρ(A)=Ψ(1)$.

math.DS↗
arXiv2003

Extending the definition of entropy to nonequilibrium steady states

We study the nonequilibrium statistical mechanics of a finite classical system subjected to nongradient forces $ξ$ and maintained at fixed kinetic energy (Hoover-Evans isokinetic thermostat). We assume that the microscopic dynamics is sufficiently chaotic (Gallavotti-Cohen chaotic hypothesis) and that there is a natural nonequilibrium steady state $ρ_ξ$. When $ξ$ is replaced by $ξ+δξ$ one can compute the change $δρ$ of $ρ_ξ$ (linear response) and define an entropy change $δS$ based on energy considerations. When $ξ$ is varied around a loop, the total change of $S$ need not vanish: outside of equilibrium the entropy has curvature. But at equilibrium (i.e. if $ξ$ is a gradient) we show that the curvature is zero, and that the entropy $S(ξ+δξ)$ near equilibrium is well defined to second order in $δξ$.

cond-mat.stat-mech↗
arXiv2001

Topics in quantum statistical mechanics and operator algebras

The language of operator algebras is of great help for the formulation of questions and answers in quantum statistical mechanics. In Chapter 1 we present a minimal mathematical introduction to operator algebras, with physical applications in mind. In Chapter 2 we study some questions related to the quantum statistical mechanics of spin systems, with particular attention to the time evolution of infinite systems. The basic reference for these two chapters is Bratteli-Robinson: Operator algebras and quantum statistical mechanics I, II. In Chapter 3 we discuss the nonequilibrium statistical mechanics of quantum spin systems, as it is currently being developped.

math-ph↗
arXiv2000

Grace-like polynomials

Results of somewhat mysterious nature are known on the location of zeros of certain polynomials associated with statistical mechanics (Lee-Yang circle theorem) and also with graph counting. In an attempt at clarifying the situation we introduce and discuss here a natural class of polynomials. Let $P(z_1,...,z_m,w_1,...,w_n)$ be separately of degree 1 in each of its $m+n$ arguments. We say that $P$ is a Grace-like polynomial if $P(z_1,...,w_n)\ne0$ whenever there is a circle in ${\bf C}$ separating $z_1,...,z_m$ from $w_1,...,w_n$. A number of properties and characterizations of these polynomials are obtained.

math-ph↗
arXiv2000

Entropy production in quantum spin systems

We consider a quantum spin system consisting of a finite subsystem connected to infinite reservoirs at different temperatures. In this setup we define nonequilibrium steady states and prove that the rate of entropy production in such states is nonnegative.

math-ph↗
arXiv2000

Absolutely singular dynamical foliations

We show that for the C^1-open set of partially hyperbolic diffeomorphisms constructed in (M. Shub and A. Wilkinson, "Pathological foliations and removable zero exponents," Invent. math. 139 (2000) 3, 495-508), Lebesgue measure on the 3-torus decomposes as atomic measure along the leaves of the central foliation.

math.DS↗
arXiv2000

Natural nonequilibrium states in quantum statistical mechanics

A quantum spin system is discussed, where a heat flow between infinite reservoirs takes place in a finite region. A time dependent force may also be acting. Our analysis is based on a simple technical assumption concerning the time evolution of infinite quantum spin systems. This assumption, physically natural but currently proved for few specific systems only, says that quantum information diffuses in space-time in such a way that the time integral of the commutator of local observables converges: $\int_{-\infty}^0dt ||[B,α^tA]||<\infty$. In this setup one can define a natural nonequilibrium state. In the time independent case, this nonequilibrium state retains some of the analyticity which characterizes KMS equilibrium states. A linear response formula is also obtained which remains true far from equilibrium. The formalism presented here does not cover situations where (for time independent forces) the time translation invariance and uniqueness of the natural nonequilibrium state are broken.

math-ph↗
arXiv1998

Smooth dynamics and new theoretical ideas in nonequilibrium statistical mechanics

This paper reviews various applications of the theory of smooth dynamical systems to conceptual problems of nonequilibrium statistical mechanics. We adopt a new point of view which has emerged progressively in recent years, and which takes seriously into account the chaotic character of the microscopic time evolution. The emphasis is on nonequilibrium steady states rather than the traditional approach to equilibrium point of view of Boltzmann. The nonequilibrium steady states, in presence of a Gaussian thermostat, are described by SRB measures. In terms of these one can prove the Gallavotti-Cohen fluctuation theorem. One can also prove a general linear response formula and study its consequences, which are not restricted to near equilibrium situations. Under suitable conditions the nonequilibrium steady states satisfy the pairing theorem of Dettmann and Morriss. The results just mentioned hold so far only for classical systems; they do not involve large size, i.e., they hold without a thermodynamic limit.

chao-dyn↗
arXiv1996

SRB states and nonequilibrium statistical mechanics close to equilibrium

Nonequilibrium statistical mechanics close to equilibrium is studied using SRB states and a formula for their derivatives with respect to parameters. We write general expressions for the thermodynamic fluxes (or currents) and the transport coefficients, generalizing previous results. In this framework we give a general proof of the Onsager reciprocity relations.

chao-dyn↗
arXiv1994

Dynamical zeta functions for maps of the interval

A dynamical zeta function $ζ$ and a transfer operator $\scr L$ are associated with a piecewise monotone map $f$ of the interval $[0,1]$ and a weight function $g$. The analytic properties of $ζ$ and the spectral properties of $\scr L$ are related by a theorem of Baladi and Keller under an assumption of ``generating partition''. It is shown here how to remove this assumption and, in particular, extend the theorem of Baladi and Keller to the case when $f$ has negative Schwarzian derivative.

math.DS↗