Everlasting initial memory threshold for rare events in equilibration processes

Abstract

Conventional wisdom indicates that initial memory should decay away exponentially in time for general (noncritial) equilibration processes. In particular, time-integrated quantities such as heat are presumed to lose initial memory in a sufficiently long-time limit. However, we show that the large deviation function of time-integrated quantities may exhibit initial memory effect even in the infinite-time limit, if the system is initially prepared sufficiently far away from equilibrium. For a Brownian particle dynamics, as an example, we found a sharp finite threshold rigorously, beyond which the corresponding large deviation function contains everlasting initial memory. The physical origin for this phenomenon is explored with an intuitive argument and also from a toy model analysis. Hot coffee gets colder and iced coffee gets warmer at room temperatures. These phenomena are the examples of equilibration processes and can be generalized as the following situation; a system with initial temperature T s is in thermal contact with a heat bath with temperature T b . Then, the system gradually deviates from its initial state and approaches to the final equilibrium (EQ) state which is determined by the heat bath. Here, the initial distance from final equilibrium is parameterized by the temperature ratio β ≡ T b /T s . The relaxation process is usually exponentially fast, so the memory of the initial temperature will be lost for average values of most physical observables after a characteristic relaxation time. However, the initial memory can often survive in the tail part (rare-event region) of a probability distribution function (PDF) even in the long-time limit.What about time-integrated quantities such as heat, work, or entropy production, which are the key quantities for nonequilibrium (NEQ) fluctuation theorems [1][2][3][4][5]? These accumulated quantities are also affected by a finite transient period, but their average values increase (or decrease) linearly in time asymptotically in NEQ steady state. Therefore, in a sufficiently long-time limit, our conventional wisdom may lead us to expect that they will lose all initial memory (independent of β). Nevertheless, in this Letter, we show rigorously that this is false wisdom for time-integrated quantities and, in particular, corresponding large deviation functions depend strongly on the initial condition (β) even in the infinite-time limit. More surprisingly, there exists a sharp threshold for β −1 in general, only beyond which the initial memory lasts forever.In literatures, there have been some reports that initial conditions can affect the large deviation function in the long time limit [6][7][8][9]. For example, van Zon and Cohen [6] showed that heat transfer Q in a driven harmonic oscillator in contact with a heat bath violates the fluctuation theorem even in the long-time limit, starting initially from EQ. In contrast to work, heat is known to satisfy the fluctuation theorem, only starting with a uniform distribution (infinite-temperature initial state) [10]. ...

Stochastic efficiency of an isothermal work-to-work converter engine

“…(51), then, the integral can not approximated with the usual saddle-point solution, and one has to evaluate Eq. (39) carefully by taking into account of the singularities [20,21,56,57].…”

Modified saddle-point integral near a singularity for the large deviation function

“…Therefore, in the long-time limit, our conventional wisdom may lead us to expect that the PDF's of Q d and Q i will lose all initial memory, thus become independent of β. However, it has been noticed in various examples [23,24,25,28,37] that the effect of initial conditions can remain in the tail of the PDF's (rare-event region) even in the infinite-time limit.…”

“…However, there is another source for highly energetic particles generated by the heat bath, which is also exponentially rare. These two rare events compete each other and sometimes a sharp nontrivial threshold for the initial condition (β) appears with regard to the initial-condition dependence of the PDF tail shape or the LDF [37].…”

“…The calculation method is similar to that for the dissipated power. The generating function of the injected power is given by [37] …”

“…Therefore, we develop a saddle-point integration method to treat a non-Gaussian integration near this modified saddle point λ * (ε), especially when λ * (ε) asymptotically approaches the singular point. To illustrate our method explicitly, we revisited the equilibration process [37] as an example. In this case, the prefactor φ(λ) has a square-root singularity with a branch cut.…”