A possible application of previous sections ideas, particularly on phase space discretization, leads to propose “the degree of irreversibility” of a process controlled by one or more parameters assigned to vary in time, while the system starts in a stationary state corresponding to initial parameters value and in time evolves, for instance, to a final stationary state. In particular initial, intermediary, final parameters values could be such that, if kept constant, would lead to an equilibrium state (a model of “thermodynamic process”). A definition of the rapidity of an evolution, or of the duration of a process, can be taken as a measure of its “irreversibility”, expected because, with Carnot, a reversible process is “infinitely slow”: several examples on well known processes are provided. The analysis in Sect. 5.2 relies on the remark of the existence of a cancellation in the formal expression of the SRB distribution. Applications of CH to fluctuations in reversible evolutions, Chap. 4, induces immediately to ask whether similar questions can be studied in the case of irreversible evolutions. Since reversible models are often studied in simulations while theory very often deals with irreversible models, it is natural to try to establish more formally a correspondence rule between reversible and irreversible models of the same evolution problem. A few conjectures are formulated that can be summarized in a general rule concerning a system described by two models, both fulfilling CH: one in which a parameter is a constant while in the other the parameter is replaced by a quantity designed to keep, say, dissipation rate constant at the average value occurring in the first system. A basic question, “equivalence problem”, is whether, or on which conditions, the statistics of the fluctuations of classes of observables of the two models are the same. Examples of conjectured equivalence are provided. An example is the incompressible Navier-Stokes fluid with UV cutoff parameter N on the harmonics of the velocity \(\textbf{u}\) ; consider the classic Navier-Stokes (NS) equation with viscosity \(\nu \) and the reversible Navier-Stokes (RNS) equation with viscosity replaced by a multiplier \(\alpha (\textbf{u})\) such that the enstrophy \(En=\int (\partial \textbf{u}(x))^2 dx\) is exactly constant. If at fixed viscosity the average enstrophy is \(En_N\) , then the statistics of the observables depending only on finitely many Fourier’s harmonics (“large scale harmonics”), evaluated following the two equations, is the same in the limit \(N\rightarrow \infty \) , i.e. at removal of the cut-off. The conjecture has been tested in dimension 2 and even in dimension 3 and results are commented. Other equivalence conjectures arise in mechanical systems exemplified also in problems on granular materials or for stochastic evolutions. And a proposal of employing the fluctuation theorem, to design a thermometer to measure absolute temperature of very small regions of a surface, is also discussed.

Conjectures and Suggested Applications

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发表日期：

2025年

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Giovanni Gallavotti

摘要

A possible application of previous sections ideas, particularly on phase space discretization, leads to propose “the degree of irreversibility” of a process controlled by one or more parameters assigned to vary in time, while the system starts in a stationary state corresponding to initial parameters value and in time evolves, for instance, to a final stationary state. In particular initial, intermediary, final parameters values could be such that, if kept constant, would lead to an equilibrium state (a model of “thermodynamic process”). A definition of the rapidity of an evolution, or of the duration of a process, can be taken as a measure of its “irreversibility”, expected because, with Carnot, a reversible process is “infinitely slow”: several examples on well known processes are provided. The analysis in Sect. 5.2 relies on the remark of the existence of a cancellation in the formal expression of the SRB distribution. Applications of CH to fluctuations in reversible evolutions, Chap. 4, induces immediately to ask whether similar questions can be studied in the case of irreversible evolutions. Since reversible models are often studied in simulations while theory very often deals with irreversible models, it is natural to try to establish more formally a correspondence rule between reversible and irreversible models of the same evolution problem. A few conjectures are formulated that can be summarized in a general rule concerning a system described by two models, both fulfilling CH: one in which a parameter is a constant while in the other the parameter is replaced by a quantity designed to keep, say, dissipation rate constant at the average value occurring in the first system. A basic question, “equivalence problem”, is whether, or on which conditions, the statistics of the fluctuations of classes of observables of the two models are the same. Examples of conjectured equivalence are provided. An example is the incompressible Navier-Stokes fluid with UV cutoff parameter N on the harmonics of the velocity \(\textbf{u}\) ; consider the classic Navier-Stokes (NS) equation with viscosity \(\nu \) and the reversible Navier-Stokes (RNS) equation with viscosity replaced by a multiplier \(\alpha (\textbf{u})\) such that the enstrophy \(En=\int (\partial \textbf{u}(x))^2 dx\) is exactly constant. If at fixed viscosity the average enstrophy is \(En_N\) , then the statistics of the observables depending only on finitely many Fourier’s harmonics (“large scale harmonics”), evaluated following the two equations, is the same in the limit \(N\rightarrow \infty \) , i.e. at removal of the cut-off. The conjecture has been tested in dimension 2 and even in dimension 3 and results are commented. Other equivalence conjectures arise in mechanical systems exemplified also in problems on granular materials or for stochastic evolutions. And a proposal of employing the fluctuation theorem, to design a thermometer to measure absolute temperature of very small regions of a surface, is also discussed.