Entropy production in non-equilibrium systems: physical interpretation?

Question

I have been learning about entropy production in non-equilibrium systems as developed by Prigogine and others, especially in the context of chemical reactions. I now understand that from the first law of thermodynamics, one can derive an expression for entropy production in the form of force times flux. The force is the affinity and the flux the change in the extent of reaction: $ \frac{d_i S}{dt} = \left(\frac{A}{T}\right) \frac{d\xi}{dt} > 0$

Now in terms of the forward and reverse reaction rates $R_f$ and $R_r$, one can re-express the above expression as $ \frac{d_i S}{dt} = R(R_f - R_r) \ln{\frac{R_f}{R_n}}$.

My question is what this quantity physically represents or can be used for. In non-driven systems where the concentrations relax to an equilibrium value, the entropy production goes to zero as equilibrium is reached. However, if the system is driven, entropy may be continuously produced. What is the use quantifying the rate at which this happens in non-equilibrium systems?

Answer 1

The main postulate of equilibrium statistical physics is that in an isolated system all the microstates with the same values of energy are equally probable. One then defines the entropy as the logarithm of the number of these microstates $$S=\log\Omega,$$ which is consistent with the Shannon's definition of entropy $$S=-\sum_i p_i\log p_i,$$ where all the microstates are equally probable, $$p_i=\frac{1}{\Omega}.$$

Note also that the equiprobable distribution maximizes the Shannon's entropy. If we start with a different probability distribution vector, $\mathbf{p}$, the entropy will be lower and we expect it to increase till all the $p_i$ have the same value, i.e., till we reach a thermodynamically equilibrium state. Thus, the entropy production can be thought of as the speed at which the probability distribution among the microstates is equalizing itself. Any non-equilibrium process can be viewed as relaxation to thermodynamic equilibrium, although sometimes the time that it reaches equilibrium is taken to be infinitely long (e.g., for a stationary process).

Further, the entropy production characterizes only the production of entropy within the system, $dS_i/dt$ ($S_i$ is the internal system entropy). Driving the system would mean that its entropy is constantly being lowered by the external forces, at the rate that compensates the internal entropy production: $$\frac{dS_i}{dt} >0, \frac{dS_e}{dt} <0.$$ Note however, that Prigogine's minimum energy production principle addresses steady states. See this good discussion by Jaynes.