Information compressibility, entropy variation and approach to steady state in open systems

We introduce the concept of information compressibility, KI , which measures the relative change of number of available microstates of an open system in response to an energy variation. We then prove that at the time in which the system reaches a steady state, the second and third time derivatives of the information entropy are proportional to the corresponding time derivatives of the energy, the proportionality constant being KI . We argue that if two steady states with different but same-sign KI are dynamically connected in a non-adiabatic way it takes a longer time to reach the state with compressibility closer to zero than the reverse. We also show analytically that for a two-level system in contact with external baths, the information compressibility is inversely proportional to the temperature measured at any given time by a probe that is coupled to the system, and whose temperature is adjusted so that the system dynamics is minimally perturbed. This concept, that applies to both classical and quantum open systems, thus provides insight into the properties of non-equilibrium steady states. Introduction. – Dynamical systems are to some extent always in interaction with one or more external environments. The latter ones may exchange particles and/or energy with the system of interest. Under certain conditions this leads to a non-equilibrium steady state (NESS) of the system dynamics. This situation arises in various physical systems, from chemical and biological processes [1] to nanoscale systems [2]. The properties of NESSs, whether quantum or classical, have long been the subject of numerous studies but a comprehensive theoretical description is still lacking, especially for NESSs far from equilibrium [3]. One aspect, in particular, that has received much less attention is the approach to steady state, [4] and the entropy variation in such instance. The difficulty in describing properties of NESSs and the approach to steady state can be in part attributed to the lack of a general physical quantity that characterizes such states, especially far from thermodynamic equilibrium. Such a quantity needs to take into account the microscopic dynamics of the system and, at the same time, provide a global physical description that is, in principle, easy to access theoretically and/or experimentally. In this Letter we introduce such a quantity that we name information compressibility, KI . The latter measures the “easiness,” or “difficulty,” to vary the relative number of available microstates of an open system when its energy changes due to the interaction with the environment(s). This concept is the information counterpart of the similar concept for solids, liquids or gases, where the standard compressibility quantifies the relative change of the volume with respect to a pressure variation. The advantage of the information compressibility in characterizing non-equilibrium systems stems from the fact that it allows us to establish several results regarding the approach to steady state. We prove analytically that at the time in which the system reaches a steady state the second and third time derivatives of the information entropy are proportional to the corresponding time derivatives of the energy. In both cases, the proportionality constant is KI at that instant of time. The entropy production at that moment can be thus minimal or maximal according to the sign of the information compressibility and the concavity or convexity of the energy function. In addition, we can argue about the time it takes the system to reach a steady state with a given information compressibility from another steady state with different KI , when the two are connected non-adiabatically. We find that when the two compressibilities have the same sign it takes a longer time to reach the state with compressibility closer to zero than the reverse process. The reason for this be-