Fluctuation Relation for Heat

Abstract

We present a fluctuation relation for heat dissipation in a nonequilibrium system. A nonequilibrium work is known to obey the fluctuation theorem in any time interval t. A heat, which differs from a work by an energy change, is shown to satisfy a modified fluctuation relation. Modification is brought by correlation between a heat and an energy change during nonequilibrium processes whose effect may not be negligible even in the t → ∞ limit. The fluctuation relation is derived for overdamped Langevin equation systems, and tested in a linear diffusion system. Fluctuations of thermodynamic quantities of nonequilibrium systems obey a universal relation referred to as fluctuation theorem (FT) [1][2][3][4][5][6][7][8][9]. Discovery of the FT leads to a great advance in nonequilibrium statistical mechanics. Based on the FT, one can generalize the fluctuation dissipation relation to nonequilibrium systems [10][11][12] and figure out fluctuations observed in experimental small-sized systems [13][14][15].The FT for a quantity R over a time interval t takes the form e −R = 1, where the average · is taken over a probability distribution for an initial state and over all time trajectories. Some quantities further satisfy the FT in the form P r (R)/P r (−R) = e R where P r (R) = δ(R − R) is a probability density function (PDF) for a nonequilibrium process andP r (R) for a corresponding reverse process. The latter is called the detailed FT and implies the former called the integral FT.Consider a system being in thermal equilibrium with a heat reservoir. We will set the temperature and the Boltzmann constant to unity. The system is driven into a nonequilibrium state if one adds a nonconservative force or applies a time-dependent perturbation. Then, there exist nonzero net flows of a nonequilibrium work W into the system and a heat dissipation Q into the reservoir. It is well established that the work W over a time interval t obeys the FT [2,6]. In addition, the total entropy change ∆S tot = ∆S sys + ∆S res with the system (reservoir) entropy S sys (S res ) satisfies the integral FT for an arbitrary initial state, and even the detailed FT for a steady state initial condition [7]. Thermodynamic quantities are measurable experimentally from time trajectories in classical systems [16], while their experimental measurability in quantum systems is still an open issue [17].Fluctuations of heat Q, or entropy production ∆S res = Q/T , has also been attracting much interest [18][19][20][21][22][23][24][25][26]. Note that a heat differs from a work by an energy change ∆E as Q = W − ∆E. When t becomes large, the system will reach a steady state with constant work and heat production rates on average. Hence one may expect the FT for heat in the large t limit where an energy change can be negligible (Q ≃ W ≫ ∆E). In fact, the FT for the heat production rate (Q/t) is derived formally in the t → ∞ limit [3,4]. On the other hand, some model studies demonstrate the FT for heat [25] or failure of the FT in the t → ∞ limit [19,23,24]. So, it is ...

Heat fluctuations for underdamped Langevin dynamics

“…This gives rise to exponential tails in the pdf of Q and is signaled by the presence of singularities in the corresponding characteristic function. Such temporal "boundary" effects that take place in systems with unbounded potentials are now well-documented, both theoretically [6][7][8][9][10][11][12][13][14][15][16][17] and experimentally [18][19][20].…”

Hidden entropy production by fast variables

“…In contrast to the previous works, our study focuses on the role of the slow and fast variable. Harmonic systems are analytically tractable [26][27][28][29][30][31]. We derive the formula for the entropy productions by the slow and fast variables as a function of the coupling constant, which will shed a light on the effect of the coarse graining.…”

Time Reversibility, Correlation Decay and the Steady State Fluctuation Relation for Dissipation

“…We note that this can then be used to write a fluctuation relation for joint probability distributions ofΛ 0,τ and H 0 (S τ Γ) − H 0 (Γ) considered for stochastic systems in [22][23][24]. In the deterministic systems studied here, these correlations will be significant, however limits under which these correlations become negligible are important to establish and will be considered in future work.…”