Fluctuation Relations for Diffusion Processes

Abstract

The paper presents a unified approach to different fluctuation relations for classical nonequilibrium dynamics described by diffusion processes. Such relations compare the statistics of fluctuations of the entropy production or work in the original process to the similar statistics in the time-reversed process. The origin of a variety of fluctuation relations is traced to the use of different time reversals. It is also shown how the application of the presented approach to the tangent process describing the joint evolution of infinitesimally close trajectories of the original process leads to a multiplicative extension of the fluctuation relations.

Equalities and Inequalities: Irreversibility and the Second Law of Thermodynamics at the Nanoscale

“…This result has been derived in various ways, using an assortment of equations of motion to model the microscopic dynamics [6,7,8,17,18,9,51,52,50,53,54,55,56,57,58,59,60,61,62,63], and has been confirmed experimentally. [64,65,66,67] In the following paragraph I will sketch how it can be obtained for the toy model of Fig.…”

Entropy production of a Brownian ellipsoid in the overdamped limit

“…It can be shown (see, e.g., [18] or the Supplementary Material of [19]) that the probability P [q(t), p(t)|q 0 , p 0 ] for a specific trajectory (q(t), p(t)) starting at (q 0 , p 0 ) at time t 0 and ending at the point (q 1 , p 1 ) at a later time t 1 is related to the probabilityP [q(t),p(t)|q …”

“…The entropy change of the particle is given as [38] ∆S over p = k B ln ρ(x 0 , n 0 , m 0 , t 0 ) − k B ln ρ(x(t), n(t), m(t), t) , (45) for a trajectory which starts at at a point (x 0 , n 0 , m 0 ) at time t 0 and is located at a point (x(t), n(t), m(t)) at a later time t. The entropy change in the environment can in principle be defined from the heat exchanged with the environment. However, due to the variations of temperature with position such an identification is subtle for the translational degrees of freedom [19,49] and thus the definition of entropy production in the environment is better based on path probability ratios [18]. It reads…”

Clausius Relation for Active Particles: What Can We Learn from Fluctuations

“…On the contrary if f nc is odd, for instance if the coarse-graining has delivered a force which is proportional to odd powers of the velocity of external bodies [14,26], or if magnetic fields are involved [25], the relation (10) does not hold anymore. In such cases, things seem to improve when the so-called conjugated dynamics is considered, by changing the sign of odd external non-conservative forces when computing the probability of inverse paths appearing in the denominator of Equation (6) [18,24,[26][27][28][29]: basically this amounts to change the parity of the force and get back the result in Equation (10). The problem of such an artificial prescription, however, is that the conjugated dynamics cannot be realized in experiments and therefore an empirical evaluation (i.e., without a detailed knowledge of the equation of motions) of the conjugated probability is not available, neither it is possible to experimentally observe the associated fluctuation relation.…”