Entropy time asymmetry

Natural phenomena are striking us every day by the time asymmetry of their evolution. Various examples of this time asymmetry exist in physics, chemistry, biology, and the other natural sciences. This asymmetry manifests itself in the dissipation of energy due to friction, viscosity, heat conductivity, or electric resistivity, as well as in diffusion and chemical reactions. The second law of thermodynamics has provided a formulation of their time asymmetry in terms of the increase of the entropy. The aforementioned irreversible processes are fundamental for biological systems which are maintained out of equilibrium by their metabolic activity. 

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. 

This principle and the formula (101) show that entropy production results from a time asymmetry in the dynamical randomness in nonequilibrium steady states. 

It is most remarkable that the entropy production in a nonequilibrium steady state is directly related to the time asymmetry in the dynamical randomness of nonequilibrium fluctuations. The entropy production turns out to be the difference in the amounts of temporal disorder between the backward and forward paths or histories. In nonequilibrium steady states, the temporal disorder of the time reversals is larger than the temporal disorder h of the paths themselves. This is expressed by the principle of temporal ordering, according to which the typical paths are more ordered than their corresponding time reversals in nonequilibrium steady states. This principle is proved with nonequilibrium statistical mechanics and is a corollary of the second law of thermodynamics. Temporal ordering is possible out of equilibrium because of the increase of spatial disorder. There is thus no contradiction with Boltzmann s interpretation of the second law. Contrary to Boltzmann s interpretation, which deals with disorder in space at a fixed time, the principle of temporal ordering is concerned by order or disorder along the time axis, in the sequence of pictures of the nonequilibrium process filmed as a movie. The emphasis of the dynamical aspects is a recent trend that finds its roots in Shannon s information theory and modem dynamical systems theory. This can explain why we had to wait the last decade before these dynamical aspects of the second law were discovered. 

Such an attitude to equilibrium thermodynamics - the science which revealed irreversibility of the evolution of isolated systems and asymmetry of natural processes with respect to time - is related to some circumstances that require a thorough analysis. Here we will emphasize only one of them which is the most important for imderstanding further text. It lies in the fact that the most important notion of thermodynamics, i.e. equilibrium, became interpreted exclusively as the state of rest (absence of any forces and flows in the thermodynamic system) and equilibrium processes - as those identical to reversible ones. These one-sided interpretations ignored the Galileo principle of relativity, the third law of Newton and the Boltzmann probabilistic interpretations of entropy that allow dynamic interpretations of equilibria and irreversible interpretations of equilibrium processes. 

However, these two contributions are not enough to account for the full asymmetry in the Tafel plots, although they can account for a factor of ca. 40 times faster rates for the anodic process when compared with the cathodic ones. Further corrections are needed to account for the experimental data. An additional contribution may be due to a difference in entropy change for the cathodic and anodic processes. 

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