Heat and fluctuations from order to chaos

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, 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 (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. * Review

Anomalies and absence of local equilibrium, and universality, in one-dimensional particles systems

“…[47] and Section 4 below. 5 5 In Ref. [47] it was observed that small temperature gradients allow the cold thermostat to behave as "thermodynamically" as the hot thermostat.…”

Fluctuation theorem and chaos

“…(2) the probability distribution µ has an "explicit" expression "similar to the equilibrium Gibbs distribution", [7]. (3) µ is concentrated on a 0 volume "attractor".…”

“…Conversely the analysis of the CH anf the fluctuation theorems has implications on the theory of turbulence, [29,7,14].…”

“…(xiii) The theory can be extended to quantum systems, [7], modeled by finite thermostats. Considering the system in Fig.1 let H be the operator on L 2 (C 3N 0 0 ), of symmetric or antisymmetric wave functions Ψ, H = −h 2 2 ∆ X 0 + U 0 (X 0 ) + j>0 (U 0j (X 0 , X j ) + U j (X j ) + K j ), parameterized by the configurations (X j ,Ẋ j ) j>0 of the particles in the thermostats (here −ihΨ(X 0 ) = (HΨ)(X 0 ),…”

“…If the CH is assumed the FT is expected to hold for this model, hence for the entropy creation rate, ε. See for more details [7]. …”