(A) A simple example of the mechanical interpretation of heat theorem (B) Comments on Boltzmann’s statement: “aperiodic motions as periodic with infinite period”. (C) The heat theorem without details on the dynamics: Comments on the first paper in which the theory of the canonical ensemble appears. The discreetness assumption about the microscopic states is for the first time not only made very explicit but it is used to obtain in a completely new way that the equilibrium distribution is equivalently a canonical or a microcanonical one. (D) A further mechanical example realizing the heat theorem in the frame of Keplerian motion. (E) Gauss’ least constraint principle (as applied in thermostat models in recent literature). (F) Non smoothness of stable/unstable manifolds: A qualitative argument to understand why, even in analytic Anosov. systems they may be non smooth (G) Example of construction of Markov partitions (in general 2 dimensional Anosov systems). (H) Axiom C: definition. (I) Lyapunov’s pairing theory: Pairing theory in mechanical systems. (J) Fluid equations (pairing and ergodicity). Apparent pairing and equivalence of exponents in corresponding reversible and irreversible flows. (K) Fluctuation theorem and viscosity. Main question: is it meaningful to ask whether the fluctuation relation holds in irreversible evolutions? Then discussion about pairing of Lyapunov exponents, FT, and equivalence RNS and NS. (L) Reversibility and friction: Considers a regularization, which studies the regularized NS equations with cut-off \(R^{\frac{3}{4}}\) , together with the corresponding reversible equation. Model is inspired by the OK41 turbulence theory. (M) Reciprocity and fluctuation theorem: it is shown that Onsager’s reciprocity can be also obtained, and extended, in reversible systems verifying CH by applying the fluctuation patterns theorem. (N) Large deviations in SRB states: illustrates how a proof of the theorem is reduced (assuming Perron-Frobenius’ theorem extension of Ruelle) to a few bounds on quantities familiar in the theory of statistical mechanics and Markov processes. (O) An exact formula: An immediate consequence of the fluctuation theorem. (P) Transient FT: “transient fluctuation theorem”. It is extremely general and does not depend on any chaoticity assumption. Just reversibility and time reversal symmetry and the evolution of an initial distribution which is invariant under time reversal. It says nothing about the SRB distribution (which is singular with respect to the Liouville distribution).

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发表日期：

2025年

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Giovanni Gallavotti

摘要

(A) A simple example of the mechanical interpretation of heat theorem (B) Comments on Boltzmann’s statement: “aperiodic motions as periodic with infinite period”. (C) The heat theorem without details on the dynamics: Comments on the first paper in which the theory of the canonical ensemble appears. The discreetness assumption about the microscopic states is for the first time not only made very explicit but it is used to obtain in a completely new way that the equilibrium distribution is equivalently a canonical or a microcanonical one. (D) A further mechanical example realizing the heat theorem in the frame of Keplerian motion. (E) Gauss’ least constraint principle (as applied in thermostat models in recent literature). (F) Non smoothness of stable/unstable manifolds: A qualitative argument to understand why, even in analytic Anosov. systems they may be non smooth (G) Example of construction of Markov partitions (in general 2 dimensional Anosov systems). (H) Axiom C: definition. (I) Lyapunov’s pairing theory: Pairing theory in mechanical systems. (J) Fluid equations (pairing and ergodicity). Apparent pairing and equivalence of exponents in corresponding reversible and irreversible flows. (K) Fluctuation theorem and viscosity. Main question: is it meaningful to ask whether the fluctuation relation holds in irreversible evolutions? Then discussion about pairing of Lyapunov exponents, FT, and equivalence RNS and NS. (L) Reversibility and friction: Considers a regularization, which studies the regularized NS equations with cut-off \(R^{\frac{3}{4}}\) , together with the corresponding reversible equation. Model is inspired by the OK41 turbulence theory. (M) Reciprocity and fluctuation theorem: it is shown that Onsager’s reciprocity can be also obtained, and extended, in reversible systems verifying CH by applying the fluctuation patterns theorem. (N) Large deviations in SRB states: illustrates how a proof of the theorem is reduced (assuming Perron-Frobenius’ theorem extension of Ruelle) to a few bounds on quantities familiar in the theory of statistical mechanics and Markov processes. (O) An exact formula: An immediate consequence of the fluctuation theorem. (P) Transient FT: “transient fluctuation theorem”. It is extremely general and does not depend on any chaoticity assumption. Just reversibility and time reversal symmetry and the evolution of an initial distribution which is invariant under time reversal. It says nothing about the SRB distribution (which is singular with respect to the Liouville distribution).