Three detailed fluctuation theorems

Autonomous modular quantum systems: contextual Jarzynski relations

For autonomous quantum systems with modular structure we demonstrate that the Jarzynski relation can be reinterpreted to apply even locally: For this purpose certain contexts have to be introduced by selecting the system of interest versus its environment. The respective energy exchange is then divided into heat and work based on a generalized definition of these notions. In this way we are able to identify functional parts of the environment as either heat or work sources, respectively. We investigate different combinations of these functional parts with respect to contextual Jarzynski relations. Our analytical results are confirmed by numerical investigations on small multipartite systems.

Irreversible thermodynamics in multiscale stochastic dynamical systems

This work extends the results of a recently developed theory of a rather complete thermodynamic formalism for discrete-state, continuous-time Markov processes with and without detailed balance. We investigate whether and in what way the thermodynamic structure is invariant in a multiscale stochastic system, that is, whether the relations between thermodynamic functions of state and process variables remain unchanged when the system is viewed at different time scales and resolutions. Our results show that the dynamics on a fast time scale contribute an entropic term to the internal energy function u(S)(x) for the slow dynamics. Based on the conditional free energy u(S)(x), we can then treat the slow dynamics as if the fast dynamics is nonexistent. Furthermore, we show that the free energy, which characterizes the spontaneous organization in a system without detailed balance, is invariant with or without the fast dynamics: The fast dynamics is assumed to reach stationarity instantaneously on the slow time scale; it has no effect on the system's free energy. The same cannot be said for the entropy and the internal energy, both of which contain the same contribution from the fast dynamics. We also investigate the consequences of time-scale separation in connection to the concepts of quasi-stationarity and steady adiabaticity introduced in the phenomenological steady-state thermodynamics.

Nonequilibrium thermodynamics and Nose-Hoover dynamics

We show that systems driven by an external force and described by Nose-Hoover dynamics allow for a consistent nonequilibrium thermodynamics description when the thermostatted variable is initially assumed in a state of canonical equilibrium. By treating the "real" variables as the system and the thermostatted variable as the reservoir, we establish the first and second law of thermodynamics. As for Hamiltonian systems, the entropy production can be expressed as a relative entropy measuring the system-reservoir correlations established during the dynamics.

Entropy production theorem for a charged particle in an electromagnetic field

In this work, it is shown that the detailed fluctuation theorem for the total entropy production of a charged particle in a two-dimensional harmonic trap under the action of an electromagnetic field is valid in two physical situations. The proof of the theorem is achieved if the particle is initially distributed with a canonical distribution at equilibrium with the thermal bath. The two examined cases are the following: in the first case, the charged particle in the harmonic trap is subjected to an arbitrary time-dependent electric field; in the second one, the minimum of the harmonic trap is arbitrarily dragged by such an electric field. The theoretical framework is developed within the context of stochastic thermodynamics and the Langevin dynamics for the charged particle.

Generalized detailed fluctuation theorem under nonequilibrium feedback control

It has been shown recently that the Jarzynski equality is generalized under nonequilibrium feedback control [T. Sagawa and M. Ueda, Phys. Rev. Lett. 104, 090602 (2010)]. The presence of feedback control in physical systems should modify both the Jarzynski equality and the detailed fluctuation theorem [K. H. Kim and H. Qian, Phys. Rev. E 75, 022102 (2007)]. However, the generalized Jarzynski equality under forward feedback control has been proved by considering that the physical systems under feedback control should locally satisfy the detailed fluctuation theorem. We use the same formalism and derive the generalized detailed fluctuation theorem for nonequilibrium driven systems under feedback control. We find that the feedback control in a physical system should preserve the detailed fluctuation theorem if the system has the same feedback information measure in forward and reverse directions.

Energy representation for nonequilibrium brownian-like systems: steady states and fluctuation relations

Stochastic dynamics in the energy representation is used as a method to represent nonequilibrium Brownian-like systems. It is shown that the equation of motion for the energy of such systems can be taken in the form of the Langevin equation with multiplicative noise. Properties of the steady states are examined by solving the Fokker-Planck equation for the energy distribution functions. The generalized integral fluctuation theorem is deduced for the systems characterized by the shifted probability flux operator. From this theorem, a number of entropy and fluctuation relations such as the Evans-Searles fluctuation theorem, the Hatano-Sasa identity, and the Jarzynski's equality are derived.

Unifying approach for fluctuation theorems from joint probability distributions

Any decomposition of the total trajectory entropy production for Markovian systems has a joint probability distribution satisfying a generalized detailed fluctuation theorem, when all the contributing terms are odd with respect to time reversal. The expression of the result does not bring into play dual probability distributions, hence easing potential applications. We show that several fluctuation theorems for perturbed nonequilibrium steady states are unified and arise as particular cases of this general result. In particular, we show that the joint probability distribution of the system and reservoir trajectory entropies satisfy a detailed fluctuation theorem valid for all times.

Three faces of the second law. II. Fokker-Planck formulation

The total entropy production is the sum of two contributions, the so-called adiabatic and nonadiabatic entropy productions, each of which is non-negative. We derive their explicit expressions for continuous Markovian processes, discuss their properties, and illustrate their behavior on two exactly solvable models.

Three faces of the second law. I. Master equation formulation

We propose a formulation of stochastic thermodynamics for systems subjected to nonequilibrium constraints (i.e. broken detailed balance at steady state) and furthermore driven by external time-dependent forces. A splitting of the second law occurs in this description leading to three second-law-like relations. The general results are illustrated on specific solvable models. The present paper uses a master equation based approach.

Physical origins of entropy production, free energy dissipation, and their mathematical representations

A unifying mathematical theory of nonequilibrium thermodynamics of stochastic systems in terms of master equations is presented. As generalizations of isothermal entropy and free energy, two functions of state play central roles: the Gibbs entropy S and the relative entropy F , which are related via the stationary distribution of the stochastic dynamics. S satisfies the fundamental entropy balance equation dS/dt = e p - h d/T with entropy production rate e p ≥ 0 and heat dissipation rate h d, while dF/dt = -f d ≤ 0. For closed systems that satisfy detailed balance: Te p(t)=f d(t). For open systems, one has Te p(t) = f d(t)+Q hk(t), where the housekeeping heat, Q hk ≥ 0, was first introduced in the phenomenological nonequilibrium steady-state thermodynamics put forward by Oono and Paniconi. Q hk represents the irreversible work done by the surrounding to the system that is kept away from reaching equilibrium. Hence, entropy production e p consists of free energy dissipation associated with spontaneous relaxation (i.e., self-organization), f d, and active energy pumping that sustains the open system Q hk. The amount of excess heat involved in the relaxation Q ex = h d - Q hk = f d -T(dS/dt). Two kinds of irreversibility, and the meaning of the arrow of time, emerge. Quasistationary processes, adiabaticity, and maximum principle for entropy are also generalized to nonequilibrium settings.