Feature Article

21502 J. Phys. Chem. B 2005, 109, 21502-21515 FEATURE ARTICLE The Mesoscopic Dynamics of Thermodynamic Systems D. Reguera and J. M. Rubı´* Departament de Fı´sica Fonamental, Facultat de Fı´sica, UniVersitat de Barcelona, Martı´ i Franque`s, 1, 08028-Barcelona, Spain J. M. G. Vilar Computational Biology Center, Memorial Sloan-Kettering Cancer Center, 307 East 63rd Street, New York, New York 10021 ReceiVed: June 1, 2005; In Final Form: September 1, 2005 Concepts of everyday use such as energy, heat, and temperature have acquired a precise meaning after the development of thermodynamics. Thermodynamics provides the basis for understanding how heat and work are related and the general rules that the macroscopic properties of systems at equilibrium follow. Outside equilibrium and away from macroscopic regimes, most of those rules cannot be applied directly. Here we present recent developments that extend the applicability of thermodynamic concepts deep into mesoscopic and irreversible regimes. We show how the probabilistic interpretation of thermodynamics together with probability conservation laws can be used to obtain Fokker-Planck equations for the relevant degrees of freedom. This approach provides a systematic method to obtain the stochastic dynamics of a system directly from its equilibrium properties. A wide variety of situations can be studied in this way, including many that were thought to be out of reach of thermodynamic theories, such as nonlinear transport in the presence of potential barriers, activated processes, slow relaxation phenomena, and basic processes in biomolecules, such as translocation and stretching. 1. Introduction reminiscent of nonequilibrium thermodynamics, by which to study fluctuations in nonlinear systems. It is still a major challenge to understand how the wide variety of behaviors observed in everyday experience, such as the usual We show that the probabilistic interpretation of the density processes of living systems, arise from the relatively simple and together with conservation laws in phase space and positiveness small set of laws that rule the microscopic world. There are a of global entropy changes set the basis of a theory similar to few exceptions. Systems in the condition of equilibrium strictly nonequilibrium thermodynamics but of a much broader range follow the rules of thermodynamics.1 In such cases, the intricate of applicability. In particular, the fact that it is based on behavior of large numbers of molecules can completely be probabilities instead of densities allows us to consider meso- characterized by a few variables that describe general average scopic systems and their fluctuations. The situations that can properties. It is possible to extend thermodynamics to situations be studied with this formalism, named mesoscopic nonequilib- that are at local equilibrium. This is the domain of validity of rium thermodynamics (MNET), include, among others, slow nonequilibrium thermodynamics.2 Despite its generality, this relaxation processes, barrier crossing dynamics, chemical reac- theory has notorious limitations: it is applicable only to tions, entropic driving, and nonlinear transport. These processes macroscopic systems, for which fluctuations are not important, are in general nonlinear. From the methodological point of view, and it operates within the linear response domain. Whereas the given the equilibrium properties of a system, this theory provides linear approximation is valid for many transport processes, such a systematic and straightforward way to obtain its stochastic as heat conduction and mass diffusion, even in the presence of nonequilibrium dynamics in terms of Fokker-Planck equations. large gradients3,4 it is not appropriate for activated processes in To set grounds for the development of the formalism, we which the system immediately enters the nonlinear domain. discuss first the basic nonequilibrium thermodynamic concepts Small systems,5 such as single molecules in a thermal bath, in used in a local equilibrium description of the systems. We then which fluctuations can be even larger than the mean values, describe how MNET provides equilibrium statistical mechanics are beyond the scope of that theory. with a dynamics at the mesoscopic level. The key idea is to In this feature article, we present recent advances aimed at introduce statistical concepts in the framework of conservation obtaining a simple and comprehensive description of the laws of nonequilibrium thermodynamics. After introducing the dynamics of nonequilibrium systems at the mesoscopic scale. general ideas, we discuss applications to nonlinear transport These advances have provided not only a deeper understanding phenomena and activated processes that further illustrate the of the concept of local equilibrium but also a framework, usefulness of the approach. Applications of the use of the 10.1021/jp052904i CCC: $30.25 © 2005 American Chemical Society Published on Web 10/29/2005 Feature Article J. Phys. Chem. B, Vol. 109, No. 46, 2005 21503 David Reguera received his B.S. degree in Physics in 1998 and his Jose Vilar was born in Barcelona, Spain. He attended the University Ph.D. in 2001, both from the University of Barcelona, Spain. His thesis of Barcelona, where he received a B.S. in Physics in 1995 and a Ph.D. about nucleation phenomena was recognized by the “Ilya Prigogine in Physics in 1998. The subject of his graduate research was the Prize 2003” as the best European thesis in the field of thermodynamics. constructive role of non-equilibrium fluctuations in systems with After completion of his Ph.D., he moved to the Department of temporal and spatial dependence. After completion of his Ph.D., he Chemistry and Biochemistry at UCLA as a postdoctoral researcher in moved to the Department of Molecular Biology at Princeton University, the group of Prof. H. Reiss. In 2004, his research work on fluctuations where he redirected his research towards the biophysical basis of cellular in nanosystems was recognized by the “Amgen Award”. Since 2004, behavior, and then to The Rockefeller University. Since 2004, he is an he is a “Ramo´n y Cajal” Researcher at the University of Barcelona. Assistant Member of the Sloan-Kettering Institute for Cancer Research, His current research interest focuses on biophysics of viruses, nucleation where he heads the Laboratory for Integrative Biological Modeling. phenomena, and nonequilibrium statistical thermodynamics of nano- He is also an Assistant Professor of Physiology, Biophysics, and systems. Systems Biology at Weill Graduate School of Medical Sciences of Cornell University. His current research focuses on the mesoscopic behavior of gene regulatory and signal transduction networks in cells, with emphasis on the molecular mechanisms underlying the development and progression of cancer. thermodynamic variables defined in each subsystem of a conveniently partitioned system admit the same interpretation as in equilibrium. Second, the entropy production of any isolated system is always nonnegative. The theory attributes the devia- tions from equilibrium to the presence of unbalanced forces, such as electric fields or gradients, which give rise to fluxes, such as electric or heat currents. Forces and fluxes are in a relationship that is cause-effect compatible with the second law of thermodynamics and with the inherent symmetries, either macroscopic or microscopic. The dynamics follows from the local conservation laws for the thermodynamic field quantities, in which the fluxes are linear functions of the forces whose Miguel Rubı´ was born in Barcelona. He received the B. S. degree in coefficients, the Onsager coefficients, satisfy reciprocity rela- physics from the Universitat Auto`noma of Barcelona in 1975 and his tions. PhD degree in theoretical physics from the same university in 1979. In his research career, he has carried out theoretical investigation on The scheme of nonequilibrium thermodynamics as sketched nonequilibrium statistical physics and soft condensed matter. Rubı´ has previously has been used successfully to analyze irreversible been Sandoval Vallarta Professor of the UAM, Mexico and Onsager processes in systems of very different nature.6,7 To illustrate Professor of the university of Trondheim. In 2003, he was awarded explicitly its method, we will apply it to the simple case of the Onsager medal and an Alexander von Humboldt Prize for his contributions to nonequilibrium thermodynamics and to the theory of mass diffusion in one dimension. The first step is to compute stochastic processes. Miguel Rubı´, is Professor of condensed matter the entropy production. At equilibrium, changes in entropy S physics at the University of Barcelona. are given by the Gibbs equation concept of local equilibrium at the mesoscale are also discussed. T dS ) dE + p dV - µ dM (1) We treat in particular inertial effects in diffusion, the slow relaxation dynamics, and the translocation of a biomolecule. in which the thermodynamic extensive variables are the internal We conclude with a brief overview of other situations where energy E, the volume V, and the mass M of the system, and the this approach has been used and with a discussion of how far- intensive variables are the temperature T, the pressure p, and away-from-equilibrium situations can be recast into a local the chemical potential µ. All these quantities may in general equilibrium description. depend on time. For the sake of simplicity, we assume that the process takes place at constant temperature, energy, and

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