Fluctuation theorems

Felderhof B U 1980 Fluctuation theorems for dielectrics with periodic boundary conditions Physice A 101 275-82... 

The end effects have been neglected here, including in the expression for change in reservoir entropy, Eq. (178). This result says in essence that the probability of a positive increase in entropy is exponentially greater than the probability of a decrease in entropy during heat flow. In essence this is the thermodynamic gradient version of the fluctuation theorem that was first derived by Bochkov and Kuzovlev [60] and subsequently by Evans et al. [56, 57]. It should be stressed that these versions relied on an adiabatic trajectory, macrovariables, and mechanical work. The present derivation explicitly accounts for interactions with the reservoir during the thermodynamic (here) or mechanical (later) work,... 

Closely related to the fluctuation theorem is the work theorem. Consider the average of the exponential of the negative of the entropy change,... 

Fluctuation-dissipation theorem, transition state trajectory, white noise, 203—207 Fluctuation theorem, nonequilibrium thermodynamics, 6—7... 

Crooks, G. E., Entropy production fluctuation theorem and the nonequilibrium work relation for free energy differences, Phys. Rev. E 1999,60, 2721-2726... 

Jarzynski, C. Wojcik, D. K., Classical and quantum fluctuation theorems for heat exchange, Phys. Rev. Lett. 2004, 92, 230602... 

Collin, D. Ritort, F. Jarzynski, C. Smith, S.B. Tinoco, I. Bustamante, C. Verification of the Crooks fluctuation theorem and recovery of RNA folding free energies. Nature... 

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. 

Different expressions can be given to the fluctuation theorem. If we introduce the decay rates of the probabilities that the currents take some values as... 

Figure 19, Diagram illustrating the fluctuation theorem for the currents (140) showing how the decay rate at negative values of the currents equates the decay rate at positive value plus the irreversible value. Compare with Figs. 13 and 16.

A direct consequence of the fluctuation theorem for the currents (138) is that these coefficients obey remarkable relations [63-65], the first of which are... 

The world surrounding us is mostly out of equihbrium, equilibrium being just an idealization that requires specific conditions to be met in the laboratory. Even today we do not have a general theory about nonequilibrium macroscopic systems as we have for equilibrium ones. Onsager theory is probably the most successful attempt, albeit its domain of validity is restricted to the linear response regime. In small systems the situation seems to be the opposite. Over the past years, a set of theoretical results that go under the name of fluctuation theorems have been unveiled. These theorems make specific predictions about energy processes in small systems that can be scrutinized in the laboratory. 

Section 11 introduces two examples, one from physics and the other from biology, that are paradigms of nonequilibrium behavior. Section in covers most important aspects of fluctuation theorems, whereas Section IV presents applications of fluctuation theorems to physics and biology. Section V presents the discipline of path thermodynamics and briefly discusses large deviation functions. Section VI discusses the topic of glassy dynamics from the perspective of nonequilibrium fluctuations in small cooperatively rearranging regions. We conclude with a brief discussion of future perspectives. 

A physical insight on the meaning of the total dissipation S can be obtained by deriving the fluctuation theorem. We start by defining the reverse path T of a given path V. Let us consider the path E = Co Ci Cm corresponding... 

Equation (21) already has the form of a fluctuation theorem. However, in order to get a proper flucmation theorem we need to specify relations between probabilities for physically measurable observables rather than paths. From Eq. (21) it is straightforward to derive a fluctuation theorem for the total dissipation S. Let us take b C) = With this choice we get... 

This relation is called the Jarzynski equality (hereafter referred to as JE) and can be used to recover free energies from nonequilibrium simulations or experiments (see Section IV.B.2). The FT in Eq. (27) becomes the Crooks fluctuation theorem (hereafter referred to as GET) [45, 46] ... 

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