Irreversible process nonequilibrium steady states

MSN. 100. G. Nicolis and 1. Prigogine, Irreversible processes at nonequilibrium steady states and Lyapounov functions, Proc. Natl. Acad. Sci. USA 76, 6060-6061 (1979). 

Nonequilibrium Steady State (NESS). The system is driven by external forces (either time dependent or nonconservative) in a stationary nonequilibrium state, where its properties do not change with time. The steady state is an irreversible nonequilibrium process that cannot be described by the Boltzmann-Gibbs distribution, where the average heat that is dissipated by the system (equal to the entropy production of the bath) is positive. 

Figure 17.5 The time variation of the entropy production P = diS/dt— Y i FkJk for equilibrium and near equilibrium states, (a) For a lluctpation from the equilibrium state, the initial nonzero value of P decreases to its equilibrium value of zero, (b) In the Unear regime, a fluctuation from a nonequilibrium steady state can only increase the value of P above the stationary value Pgu irreversible processes drive P back to its minimum value Pst...

Therefore, the total entropy produced within the system must be discharged across the boundary at stationary state. For a system at stationary state, boundary conditions do not change with time. Consequently, a nonequilibrium stationary state is not possible for an isolated system for which deS/dt = 0. Also, a steady state cannot be maintained in an adiabatic system in which irreversible processes are occurring, since the entropy produced cannot be discharged, as an adiabatic system cannot exchange heat with its surroundings. In equilibrium, all the terms in Eq. (3.48) vanish because of the absence of both entropy flow across the system boundaries and entropy production due to irreversible processes, and we have dJS/dt = d dt = dS/dt = 0. 

The stability of transport and rate systems is studied either by nonequilibrium thermodynamics or by conventional rate theory. In the latter, the analysis is based on Poincare s variational equations and Lyapunov functions. We may investigate the stability of a steady state by analyzing the response of a reaction system to small disturbances around the stationary state variables. The disturbed quantities are replaced by linear combinations of their undisturbed stationary values. In nonequilibrium thermodynamics theory, the stability of stationary states is associated with Progogine s principle of minimum entropy production. Stable states are characterized by the lowest value of the entropy production in irreversible processes. The applicability of Prigogine s principle of minimum entropy production is restricted to stationary states close to global thermodynamic equilibrium. It is not applicable to the stability of continuous reaction systems involving stable and unstable steady states far from global equilibrium. The steady-state deviation of entropy production serves as a Lyapunov function. 

We thus arrive at an interesting conclusion regarding thermodynamics and process control. It is not the steady state irreversibility (inefficiency) that matters for control but the ability to alter the rate of total entropy production in response to the system s departure from steady state. We have previously indicated qualitatively how entropy is produced. To see how the rate of entropy production changes with the system s state, we need to perform a quantitative analysis. This requires a brief introduction to the subject of nonequilibrium thermodynamics (Callen, 1985 Haase, 1990). 

If the steady state concentrations of the components are shifted, but not too far from their equilibrium values, the interconnection between the fluxes and chemical forces (chemical affinities, in our case) should satisfy the well-known linear relationships that are usually postulated in the linear thermodynamics of irreversible processes [15-18]. We do not consider here the phenomenological equations of nonequilibrium thermodynamics. For details the reader can refer to numerous excellent monographs and review articles devoted to the applications of nonequilibrium thermodynamics in the description of chemical reactions and biological processes (see, for instance, [22-30]). In many cases, the conventional phenomenological approaches of linear and nonlinear nonequilibrium thermodynamics appear to be useful tools for the... 

In some cases a system undergoes an irreversible process in such a way that the nonequilibrium state of the system does not change during the process. We say that the system is in a steady state. We cannot calculate the entropy of the system since it is not in an equilibrium state, but we can conclude that its entropy does not change during the process. Any entropy changes occur in the surroundings. 

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