Fluctuation Theorems and Entropy Production with Odd-Parity Variables

Abstract

We show that the total entropy production in stochastic processes with odd-parity variables (under time reversal) is separated into three parts, only two of which satisfy the integral fluctuation theorems in general. One is the usual excess entropy production, which can appear only transiently and is called nonadiabatic. Another one is attributed solely to the breakage of detailed balance. The last part not satisfying the fluctuation theorem comes from the steady-state distribution asymmetry for odd-parity variables, which is activated in a non-transient manner. The latter two contributions combine together as the house-keeping (adiabatic) entropy production, whose positivity is not guaranteed except when the excess entropy production completely vanishes. where P r is the probability of a sequence r. As a corollary, the Jensen's inequality guarantees R ≥ 0. Consider r as a path or trajectory in state space, generated during a time interval by a stochastic dynamics. In case when its functional R [7] represents the total entropy production during the process, the FT has been derived for various nonequilibrium(NEQ) processes, and the thermodynamic 2nd law ∆S tot ≥ 0 automatically follows [3,4,8].More recently, Hatano and Sasa found that a part of the total entropy production (excess entropy), ∆S ex , also satisfies the FT, which represents the entropy production associated with transitions between steady states [9,10]. Later, Speck and Seifert showed that the remaining part (house-keeping entropy), ∆S hk , also satisfies the FT, which is required to maintain the NEQ steady state (NESS) [11,12]. In case of (quasi-static) reversible processes, the system stays at equilibrium almost always during the process, then the house-keeping entropy production vanishes, ∆S eq hk = 0. Most recently, Esposito et. al.[6] interpreted the house-keeping entropy as an adiabatic part and the excess entropy as a nonadiabatic part of the total entropy production, through a time-scale argument.Most of findings about the FTs so far hold only when all state variables have even parity under time reversal, such as position variables. A typical example is the driven Brownian motion in the over-damped limit. Including odd-parity variables, such as momentum, the mathematical description becomes more complicated in particular for NEQ processes. Recently, Spinney and Ford suggested a separation of the total entropy production into three terms for the stochastic system with odd-parity variables [13]. The excess entropy production can be cleanly separated out (in fact, exactly the same as in the case with even-parity variables only) and it satisfies the FT. However, the house-keeping part composes of two different terms and only one term satisfies the FT. Especially, the other term not satisfying the FT turns out to be transient, which seems inconsistent with the usual adiabatic feature of the house-keeping entropy. Thus, it was concluded that the physical interpretation of separated entropies is not as clear as in the even-variable only case (adiab...

Heat fluctuations and initial ensembles

“…We also define the time-reverse trajectory q † with q † (t) = q(τ − t) with q representing the mirrored trajectory with a parity for each state variable [37,38]. This trajectory starts at the mirrored state of the final state of the original trajectory: q † 0 = q τ .…”

Stochastic thermodynamics for linear kinetic equations

“…(12), the probability for having no collisions, determined by the rates k(v → v) and k(−v → −v) when we have a velocity v and −v , respectively, is the same in forward and backward trajectories. Without the parity property, this is no longer the case, and as mentioned in the introduction, the theory appears to become much more involved [12,13]. Hence the corresponding terms cancel out, and we have i s = s − e s as required.…”

Stochastic thermodynamics with odd controlling parameters

“…Spinney and Ford [42][43][44] recently investigated systems with odd dynamical variables (such as momentum that changes its sign under time-reversal operation) and found that the total entropy production can be separated into three parts, with two of them satisfying the integral fluctuation theorem. Lee et al [45,46] modified the separation rule for the total entropy production put forward in [42][43][44] and endowed each part of the total entropy production with clear physical origins.…”