Hydrodynamics nonequilibrium thermodynamics, time

In references (Santamaria Holek, 2005 2009 2001), the Smoluchowski equation was obtained by calculating the evolution equations for the first moments of the distribution function. These equations constitute the hydrodynamic level of description and can be obtained through the Fokker-Planck equation. The time evolution of the moments include relaxation equations for the diffusion current and the pressure tensor, whose form permits to elucidate the existence of inertial (short-time) and diffusion (long-time) regimes. As already mentioned, in the diffusion regime the mesoscopic description is carried out by means of a Smoluchowski equation and the equations for the moments coincide with the differential equations of nonequilibrium thermodynamics. 

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 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 book is the outcome of decades of work. The senior author was a student of Theophile De Donder (1870-1957), the founder of the Brussels School of Thermodynamics. Contrary to the opinion prevalent at that time, De Donder considered that thermodynamics should not be limited to equilibrium situations. He created an active school in Belgium. But his approach remained isolated. Today the situation has drastically changed. There is a major effort going on in the study of nonequilibrium processes, be it in hydrodynamics, chemistry, optics or biology. The need for an extension of thermodynamics is now universally accepted. 

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