![]() ![]() We will be concerned here with systems that are far from equilibrium, that is systems that are irreversible even in the stationary state. All results obtained below will be valid under these two types of boundary conditions. In this case the component of the force perpendicular to is assumed to vanish. For the reflecting boundary conditions the component of the current of probability J perpendicular to vanishes. In this case, the forces f i( x) must be periodic. For periodic boundary conditions both quantities are periodic. We will consider two types of boundary conditions: periodic and reflecting. The Fokker-Planck equation has to be solved inside a given region of the space spanned by the set of variables x i subject to a prescribed boundary condition which concerns the behavior of P( x,t) and J i( x,t) at the surface that delimits the region. Let us consider a set of n interacting particles that evolve in time according to the following coupled set of Langevin equations We will show that in this case P = / T, which is a fluctuation-dissipation type relation. In this study we will relate the entropy production P with the dissipated power occurring in nonequilibrium systems subject to nonconservative forces and in contact with a heat bath at a temperature T. The study of production of entropy in systems described by a Markovian process in continuous time but discrete configuration space or, in other words, described by a master equation, has been the object of several studies including the production of entropy in the majority vote model. ![]() When the force becomes conservative the production of entropy vanishes. As we shall see, if the forces are nonconservative the resulting entropy production is nonzero and positive. The irreversible character is determined by the type of force entering the Langevin and its associate Fokker-Planck equation. Systems described by a Fokker-Planck equation are systems that follow a Makovian process in continuous time and continuous configuration space. ![]() Here we are concerned with the definition and calculation of the entropy production in nonequilibrium systems described by a Fokker-Planck equation or in an equivalent description by a Langevin equation. In this case, the "fluid'' in phase space becomes compressible and the production of entropy may be related to the contraction of the phase space. Irreversible systems, on the other hand, are supposed to be described by non-Hamiltonian dynamics, that is, by dynamics coming from nonconservative forces. This property is a consequence of the incompressibility of the "fluid'' that represents the system in phase space. In deterministic Hamiltonian dynamics it is well known that the Gibbs entropy is invariant. In this sense it is interesting to see how one can define those quantities in systems that evolve in time according to specified dynamics. According to Gallavotti the problem of defining entropy in a system out of equilibrium has not been solved yet. The quantity F is defined as the flux from inside to outside the system, so that it will be positive in the nonequilibrium stationary state.Īlthough in equilibrium the entropy is a well defined quantity, in nonequilibrium systems the entropy as well as the production of entropy do not have a universal definition. In the stationary state the rate of change of the entropy vanishes so that F = P. The quantity P is positive definite whereas F can have either sign. Where P is the entropy production due to irreversible processes ocurring inside the system and F is the entropy flux from the system to the environment. The rate of change of the entropy S of a system can be properly decomposed into two contributions A measure of the distance from thermodynamic equilibrium can therefore be given by the production of entropy since this quantity vanishes in equilibrium. In the stationary state, irreversible systems are in a continuous process of production of entropy. Keywords: Entropy production Irreversible systems Fokker-Planck equation We have applied the results for a simple model for particles subjected to dissipative forces. We have also been able to obtain a fluctuation-dissipation type relation between the dissipated power, which was written as an ensemble average, and the production of entropy for the case of systems in contact with one heat bath. This result is similar to the one used by Lebowitz and Spohn for system following a Markovian process in discrete space. We have found that the entropy flux can be written as an ensemble average of an expression containing the force and its derivative. We have devised an expression for the entropy flux in the stationary state. We study the entropy production in nonequilibrium systems described by a Fokker-Planck equation. Instituto de Física, Universidade de São Paulo, Caixa postal 66318, 05315-970 São Paulo- SP, Brazil Entropy production in nonequilibrium systems described by a Fokker-Planck equation
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