Fluctuation theorems microscopic reversibility

The plan of this chapter is the following. Section II gives a summary of the phenomenology of irreversible processes and set up the stage for the results of nonequilibrium statistical mechanics to follow. In Section III, it is explained that time asymmetry is compatible with microreversibility. In Section IV, the concept of Pollicott-Ruelle resonance is presented and shown to break the time-reversal symmetry in the statistical description of the time evolution of nonequilibrium relaxation toward the state of thermodynamic equilibrium. This concept is applied in Section V to the construction of the hydrodynamic modes of diffusion at the microscopic level of description in the phase space of Newton s equations. This framework allows us to derive ab initio entropy production as shown in Section VI. In Section VII, the concept of Pollicott-Ruelle resonance is also used to obtain the different transport coefficients, as well as the rates of various kinetic processes in the framework of the escape-rate theory. The time asymmetry in the dynamical randomness of nonequilibrium systems and the fluctuation theorem for the currents are presented in Section VIII. Conclusions and perspectives in biology are discussed in Section IX. 

The transient fluctuation theorem is applied to the transient response of a system. It bridges the microscopic and macroscopic domains and links the time-reversible and irreversible description of processes. In transient fluctuations, the time averages are calculated from a zero time with the known initial distribution function until a finite time. The initial distribution function may be, for example, one of the equilibrium distribution functions of statistical mechanics. So, for arbitrary averaging times, the transient fluctuation theorems are exact. The transient fluctuation theorem describes how irreversible macroscopic behavior evolves from time-reversible microscopic dynamics as either the observation time or the system size increases. It also shows how the entropy production can be related to the forward and backward dynamical randomness of the trajectories or paths of systems as characterized by the entropies per unit time. 

Consider a system in thermal contact with a constant temperature heat bath and driven by a time-dependent process. Crooks fluctuation theorem (Crooks, 1999) is for stochastic microscopically reversible dynamics and given by... 

Perhaps the best starting point in a review of the nonequilibrium field, and certainly the work that most directly influenced the present theory, is Onsager s celebrated 1931 paper on the reciprocal relations [10]. This showed that the symmetry of the linear hydrodynamic transport matrix was a consequence of the time reversibility of Hamilton s equations of motion. This is an early example of the overlap between macroscopic thermodynamics and microscopic statistical mechanics. The consequences of time reversibility play an essential role in the present nonequilibrium theory, and in various fluctuation and work theorems to be discussed shortly. 

The fluctuation-dissipation theorem, of which this is one example, is referred to in the Appendix. Unlike (/), is finite for / < 0. In a non-rotating system in the absence of magnetic fields, the growth of thermal fluctuations, because of the perfect reversibility of the microscopic equations of motion, is the reverse of their decay. Thus... 

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