Anomalous Thermodynamics at the Microscale

Abstract

Particle motion at the microscale is an incessant tug-of-war between thermal fluctuations and applied forces on one side and the strong resistance exerted by fluid viscosity on the other. Friction is so strong that completely neglecting inertia--the overdamped approximation--gives an excellent effective description of the actual particle mechanics. In sharp contrast to this result, here we show that the overdamped approximation dramatically fails when thermodynamic quantities such as the entropy production in the environment are considered, in the presence of temperature gradients. In the limit of vanishingly small, yet finite, inertia, we find that the entropy production is dominated by a contribution that is anomalous, i.e., has no counterpart in the overdamped approximation. This phenomenon, which we call an entropic anomaly, is due to a symmetry breaking that occurs when moving to the small, finite inertia limit. Anomalous entropy production is traced back to futile phase-space cyclic trajectories displaying a fast downgradient sweep followed by a slow upgradient return to the original position.

Entropy production of a Brownian ellipsoid in the overdamped limit

“…Collecting all these pieces together, we rearrange the resulting stochastic integral into three parts [19], ∆S reg , ∆S time and ∆S anom , related to the regular entropy production, an entropy production due to time-changes of T , and a part of the entropy production related to spatial temperature variations, which we show to yield an anomalous contribution in the overdamped limit. The final result for ∆S env from (9) thus reads…”

“…To explicitly account for the observation that the system exhibits dynamics on different time-scales we apply the following multi-scale procedure. First, we introduce time variables θ and τ corresponding to the scales given by τ v and τ x (see (12) and (13)), and a variable ϑ for the intermediate scale [19,24,52],…”

“…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…”

“…Investigating this question, it has been shown in [19] that the overdamped approximation does not fully capture the entropy production rate if the thermal environment is inhomogeneous (but in equilibrium locally). Rather, there is a contribution to the entropy production in addition to those predicted from the overdamped approximation, which can not be obtained from the statistics of the overdamped trajectories.…”

“…The occurrence of the "anomalous entropy production" in the overdamped limit has been discovered in [19] for purely translational motion of a spherical Brownian particle through an inhomogeneous thermal environment, see also [23]. It has been further analyzed in [24] and shown to occur in general classes of stochastic systems, including discrete stochastic processes; a related analysis of discrete processes is presented in [25].…”