Generalization of the second law for a nonequilibrium initial state

Abstract

We generalize the second law of thermodynamics in its maximum work formulation for a nonequilibrium initial distribution. It is found that in an isothermal process, the Boltzmann relative entropy (H-function) is not just a Lyapunov function but also tells us the maximum work that may be gained from a nonequilibrium initial state. The generalized second law also gives a fundamental relation between work and information. It is valid even for a small Hamiltonian system not in contact with a heat reservoir but with an effective temperature determined by the isentropic condition. Our relation can be tested in the Szilard engine, which will be realized in the laboratory. The maximum work formulation of the second law of thermodynamics relates the work needed to move a system from one equilibrium state to another to the free energy difference between those states. It tells us that the work must be greater than or equal to the difference in free energies. In recent refinements of the second law, such as the fluctuation theorem[1] and the Jarzynski equality [2][3], the maximum work formulation in the sense of the average work is rigorously shown for an initial canonical distribution [2].In this letter we generalize the maximum work formulation of the second law to transitions between nonequilibrium states. Our generalization allows one to find the maximum work that can be gained from such a transition and the processes that realize it. We are able to consider isolated systems as well as those coupled to a heat reservoir and both adiabatic and isothermal transitions.The derivation is based on the Jarzynski equality modified for a nonequilibrium initial distribution. This leads to a relation between the work and the Boltzmann relative entropy with an effective temperature. The Boltzmann relative entropy, also known as the KullbackLeibler divergence, is always positive and gives a "distance" between the nonequilibrium initial distribution and the canonical distribution [4].For a finite Hamiltonian system without a heat reservoir, the effective temperature is determined by an isentropic condition. The maximum work is realized in two successive processes: an instantaneous stabilization of the nonequilibrium initial distribution and an isentropic process.When the system is coupled to a large heat reservoir, the effective temperature is the temperature of the heat reservoir. From the generalized second law, the max-

Thermodynamic cost of measurements

“…(4). This distribution is a non-equilibrium one and exploiting the relaxation of S to equilibrium allows one to extract a maximum average amount of work linked to the Kullback-Leibler distance or relative entropy of the non-equilibrium distribution p(x|y) and the equilibrium one p(x) [4,6]:…”

“…Recently, experimental and theoretical work have specified the relation between information and dissipated work in the thermodynamics of small systems [2,3,4,5,6,7,8,9]. In its traditional formulation, the second law of thermodynamics states that the average work W needed to change the state of a system in contact with a heat bath is bounded from below by the difference in free energy of the final and the initial states:…”

Clausius Inequality and Optimality of Quasistatic Transformations for Nonequilibrium Stationary States

“…This issue has been addressed by several authors in different contexts. For instance, following the basic papers [8][9][10], the case of Hamiltonian systems with finitely many degrees of freedom has been recently discussed in [11,12] while the case of Langevin dynamics is considered in [13].We here consider thermodynamic transformations for driven diffusive systems in the framework of the macroscopic fluctuation theory. With respect to the authors mentioned above, the main difference is that we deal with systems with infinitely many degrees of freedom and the spatial structure becomes relevant.…”

“…This issue has been addressed by several authors in different contexts. For instance, following the basic papers [8][9][10], the case of Hamiltonian systems with finitely many degrees of freedom has been recently discussed in [11,12] while the case of Langevin dynamics is considered in [13].…”

Optimizing Non-ergodic Feedback Engines

“…This protocol has been suggested in Refs. [9,13,14] and at first sight seems to be the most general procedure for a reversible preparation, since in a reversible process the system must be in equilibrium at any time, in particular, at the beginning and end of the process.…”