Statistical thermodynamics fluctuation theorems

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. 

Equation (3.12) is an identity that does not depend on the details of the kinetic reaction mechanism that is operating in a particular system [19], We [19] have shown that Equation (3.12) is intimately related to the Crooks fluctuation theorem [41] - an important result in non-equilibrium statistical thermodynamics - as well as to theories developed by Hill [87, 90], Ussing [201], and Hodgkin and Huxley [95],... 

Assume that a finite system is in contact with a heat bath at constant temperature and driven away from equilibrium by some external time-dependent force. Many nonequilibrium statistical analyses are available for the systems in the vicinity of equilibrium. The only exception is the fluctuation theorems, which are related to the entropy production and valid for systems far away from global equilibrium. The systems that are far from global equilibrium are stochastic in nature with varying spatial and timescales. The fluctuation theorem relates to the probability distributions of the time-averaged irreversible entropy production a. The theorem states that, in systems away from equilibrium over a finite time t, the ratio between the probability that CT takes on a value A and the probability that it takes the opposite value, —A, will be exponential in At. For nonequilibrium system in a finite time, the fluctuation theorem formulates that entropy will flow in a direction opposite to that dictated hy the second law of thermodynamics. Mathematically, the fluctuation theorem is expressed as ... 

Crooks stationary fluctuation theorem relates entropy production to the dynamical randomness of the stochastic processes. Therefore, it relates the statistics of fluctuations to the nonequilibrium thermodynamics through the entropy production estimations. The theorem predicts that entropy production will be positive as either the system size or the observation time increases and the probability of observing an entropy production opposite to that dictated by the second law of thermodynamics decreases exponentially. 

The derivation given above is not entirely satisfactory. Having been derived from theorems of classical mechanics, concerned for instance with the behaviour of pendulums in resisting media, it assumes that the particle is much larger than the molecules of the solvent, which is therefore treated as a continuum. In many solutions, however, the solute and solvent molecules are of comparable size, and this assumption cannot hold. Moreover, the force F on a molecule is subject to rapid fluctuations, which have been neglected in the derivation of Equation (3.38). Fortunately, they can be taken into account, in a more sophisticated treatment by the methods of non-equihbrium statistical thermodynamics, and the important relations (3.38) and (3.39) can then be rigorously derived. This derivation is beyond the scope of the present volume the reader is referred to monographs on diffusion [7]. 

Before we come to these models, we will first introduce a basic law of statistical thermodynamics which we require for the subsequent treatments and this is the fluctuation-dissipation theorem . We learned in the previous chapter that the relaxation times showing up in time- or frequency dependent response functions equal certain characteristic times of the molecular dynamics in thermal equilibrium. This is true in the range of linear responses, where interactions with applied fields are always weak compared to the internal interaction potentials and therefore leave the times of motion unchanged. The fluctuation-dissipation theorem concerns this situation and describes explicitly the relation between the microscopic dynamics in thermal equilibration and macroscopic response functions. 

Imagine that we select within a sample a subsystem contained in a volume Vj which is small but still macroscopic in the sense that statistical thermodynamics can be applied. If we could measure the properties of this subsystem we would observe time dependent fluctuations, for example in the shape of the volume, i.e. the local strain, the internal energy, the total dipole moment, or the local stress. The fluctuation-dissipation theorem relates these spontaneous, thermally driven fluctuations to the response functions of the system. We formulate the relationship for two cases of interest, the fluctuations of the dipole moments in a polar sample and the fluctuations of stress in a melt. 

Callen Herbert Bernard (1919-1990) US. phys., research on solid-state physics, thermodynamics and statistical mechanics, fluctuation-dissipation theorem (book Thermodynamics introduction to thermostatics 1960)... 

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. 

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