Typicality and the Role of the Lebesgue Measure in Statistical Mechanics

It is often claimed, or hoped, that some temporal asymmetries are explained by the thermodynamic asymmetry in time. Thermodynamics, the macroscopic physics of pressure, temperature, volume, and so on, describes many temporally asymmetric processes. Heat flows spontaneously from hot objects to cold objects (in closed systems), never the reverse. More generally, systems spontaneously move from non-equilibrium states to equilibrium states, never the reverse. Delving into the foundations of statistical mechanics, this chapter reviews the many open questions in that field as they relate to temporal asymmetry. Taking a stand on many of them, it tackles questions about the nature of probabilities, the role of boundary conditions, and even the nature and scope of statistical mechanics.

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THE HATSOPOULOS–GYFTOPOULOS RESOLUTION OF THE SCHRÖDINGER–PARK PARADOX ABOUT THE CONCEPT OF "STATE" IN QUANTUM STATISTICAL MECHANICS

Modern Physics Letters A ◽

10.1142/s0217732306021840 ◽

2006 ◽

Vol 21 (37) ◽

pp. 2799-2811 ◽

Cited By ~ 5

Author(s):

GIAN PAOLO BERETTA

Keyword(s):

Information Theory ◽

Quantum Theory ◽

Statistical Mechanics ◽

Quantum Statistical Mechanics ◽

Quantum Information Theory ◽

Individual State ◽

Fundamental Difficulty ◽

Quantum Statistical ◽

Theory Interpretation

A seldom recognized fundamental difficulty undermines the concept of individual "state" in the present formulations of quantum statistical mechanics (and in its quantum information theory interpretation as well). The difficulty is an unavoidable consequence of an almost forgotten corollary proved by Schrödinger in 1936 and perused by Park, Am. J. Phys.36, 211 (1968). To resolve it, we must either reject as unsound the concept of state, or else undertake a serious reformulation of quantum theory and the role of statistics. We restate the difficulty and discuss a possible resolution proposed in 1976 by Hatsopoulos and Gyftopoulos, Found. Phys.6, 15; 127; 439; 561 (1976).

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Statistical behaviour of the leaves of Riccati foliations

Ergodic Theory and Dynamical Systems ◽

10.1017/s0143385708001028 ◽

2009 ◽

Vol 30 (1) ◽

pp. 67-96 ◽

Cited By ~ 6

Author(s):

CH. BONATTI ◽

X. GÓMEZ-MONT ◽

R. VILA-FREYER

Keyword(s):

Lebesgue Measure ◽

Geodesic Flow ◽

Conformal Structure ◽

Holomorphic Foliation ◽

Horocycle Flow ◽

Linear Ordinary Differential Equations ◽

Positive Time ◽

Measure Class ◽

Dimension One

AbstractWe introduce the geodesic flow on the leaves of a holomorphic foliation with leaves of dimension one and hyperbolic, corresponding to the unique complete metric of curvature −1 compatible with its conformal structure. We do these for the foliations associated to Riccati equations, which are the projectivization of the solutions of linear ordinary differential equations over a finite Riemann surface of hyperbolic type S, and may be described by a representation ρ:π1(S)→GL(n,ℂ). We give conditions under which the foliated geodesic flow has a generic repeller–attractor statistical dynamics. That is, there are measures μ− and μ+ such that for almost any initial condition with respect to the Lebesgue measure class the statistical average of the foliated geodesic flow converges for negative time to μ− and for positive time to μ+ (i.e. μ+ is the unique Sinaï, Ruelle and Bowen (SRB)-measure and its basin has total Lebesgue measure). These measures are ergodic with respect to the foliated geodesic flow. These measures are also invariant under a foliated horocycle flow and they project to a harmonic measure for the Riccati foliation, which plays the role of an attractor for the statistical behaviour of the leaves of the foliation.

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