Steady state, nonequilibrium, entropy changes

The odd contribution to the nonequilibrium steady-state probability distribution is just the exponential of this entropy change. Hence the full nonequilibrium steady-state probability distribution is... 

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. 

In the equilibrium Second Law, the first entropy increases during spontaneous changes in structure, and when the structure stabilizes (i.e., change ceases), the first entropy is a maximum. This state is called the equilibrium state. Similarly, in the nonequilibrium Second Law, the second entropy increases during spontaneous changes in flux, and when the flux stabilizes, the second entropy is a maximum. This state is called the steady state. The present nonequilibrium Second Law has the potential to provide the same basis for the steady state that Clausius Second Law has provided for the equilibrium state. 

Of course, depending on the system, the optimum state identified by the second entropy may be the state with zero net transitions, which is just the equilibrium state. So in this sense the nonequilibrium Second Law encompasses Clausius Second Law. The real novelty of the nonequilibrium Second Law is not so much that it deals with the steady state but rather that it invokes the speed of time quantitatively. In this sense it is not restricted to steady-state problems, but can in principle be formulated to include transient and harmonic effects, where the thermodynamic or mechanical driving forces change with time. The concept of transitions in the present law is readily generalized to, for example, transitions between velocity macrostates, which would be called an acceleration, and spontaneous changes in such accelerations would be accompanied by an increase in the corresponding entropy. Even more generally it can be applied to a path of macrostates in time. 

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. 

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). 

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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