Energy Dissipation and Fluctuation-Response in Driven Quantum Langevin Dynamics: A Generating Function Approach

Energy dissipation in a nonequilibrium steady state is studied in driven quantum Langevin systems. We analyze a characteristic function that generates energy dissipation flow to thermal environment. This function gives a general formula for the average rate of energy dissipation using an autocorrelation function for the system variable. This leads to a general expression of the equality that connects the violation of the fluctuation-response relation to the rate of energy dissipation, the classical version of which was first studied by Harada and Sasa. Introduction. – Recent developments in nonequilibrium statistical mechanics have clarified fundamental aspects of nonequilibrium fluctuations of work, power flux, heat absorbed etc [1–6]. The fluctuation theorem (FT) is one of the most remarkable discoveries in nonequilibrium statistical mechanics. This theorem quantifies the probability of negative entropy, which can be important for short measurement times in small systems, and provides a precise statement of the second law of thermodynamics. The relation between the transient version of FT and the Jarzynski equality has been demonstrated and clarified [5]. Both the FT and the Jarzynski relation [4] have been tested experimentally in systems, such as micromechanically manipulated biomolecules [7, 8], colloids in time-dependent laser traps [9, 10], and optically driven single two-level systems [11]. Studying robust properties valid in far-from-equilibrium regime is obviously important for understanding the general structures of nonequilibrium statistical mechanics. Another important aspect of fluctuations is fluctuationresponse. In the linear response regime, a fluctuationdissipation relation (FDR) relates a response function with an auto correlation function for physical quantities at equilibrium [12–14]. However, the FDR is generically violated if the system is driven into a nonequilibrium state beyond this regime. Several recent studies considered extensions of FDR to far-from-equilibrium regime [15–17]. In Refs. [15, 16], the violation function was introduced to generalize the Einstein relation, and its validity was experimentally studied. Harada and Sasa considered the relationship between the degree of violation of FDR and the rate of energy dissipation in over-damped Langevin dynamics, and found an equality valid in a far-from-equilibrium regime [17]. The equality was first derived for the following γẋ(t) = F (x(t), t) + η(t) + εf(t), (1) where x are the coordinates, η is the white Gaussian noise satisfying 〈η(t)η(u)〉 = 2γkBTδ(t− u), and F (x(t), t) represents a force dependent on space and time. Let C(t) be the autocorrelation function for the velocity fluctuation without perturbation, ε = 0 C(t) = 〈[ẋ(t)− vs][ẋ(0)− vs]〉, (2) where 〈...〉 denotes an ensemble average over thermal noise. When the external perturbation is finite ε 6= 0, the response of the velocity obeys the linear response form [14]