Heat and Fluctuations from Order to Chaos

TL;DR

This paper explores the connection between thermodynamics, chaos theory, and statistical mechanics, emphasizing the Heat theorem, the Chaotic Hypothesis, and the Fluctuation Theorem, and discusses their implications for complex systems and nonequilibrium phenomena.

Contribution

It introduces a unified perspective linking equilibrium and nonequilibrium thermodynamics through chaos theory and the Fluctuation Theorem, extending the understanding of large fluctuations in complex systems.

Findings

The Heat theorem connects entropy to Hamiltonian mechanics.

The Chaotic Hypothesis leads to the Fluctuation Theorem for large fluctuations.

Implications for fluids and quantum systems are briefly discussed.

Abstract

The Heat theorem reveals the second law of equilibrium Thermodynamics (i.e.existence of Entropy) as a manifestation of a general property of Hamiltonian Mechanics and of the Ergodic Hypothesis, valid for 1 as well as $10^{23}$ degrees of freedom systems, {\it i.e.} for simple as well as very complex systems, and reflecting the Hamiltonian nature of the microscopic motion. In Nonequilibrium Thermodynamics theorems of comparable generality do not seem to be available. Yet it is possible to find general, model independent, properties valid even for simple chaotic systems ({\it i.e.} the hyperbolic ones), which acquire special interest for large systems: the Chaotic Hypothesis leads to the Fluctuation Theorem which provides general properties of certain very large fluctuations and reflects the time-reversal symmetry. Implications on Fluids and Quantum systems are briefly hinted. The physical meaning of the Chaotic Hypothesis, of SRB distributions and of the Fluctuation Theorem is discussed in the context of their interpretation and relevance in terms of Coarse Grained Partitions of phase space. This review is written taking some care that each section and appendix is readable either independently of the rest or with only few cross references.