81 Entropy production described by Fokker-Planck equation 福克-普朗克方程與熵的產生

Today we are going to describe a relatively complex topic. The mathematics alone is quite easy but the hidden concepts are huge. Before we start, please review our previous posts to make sure you could understand what we are talking about:
71 Fundamentals--Langevin equation
72 Computer simulation of Langevin equation
73 The equivalence of Langevin equation and Fokker-Planck equation
74 The application of Fokker-Planck equation-I
75 The application of Fokker-Planck equation-II
If you don't even know the fundamental concepts of diffusion, we suggest the following episodes:
50 Fundamentals--Dirac delta function
12 How enzymes propel themselves-II (Fick's law)
13 How enzymes propel themselves-III (Microscopic basis of diffusion)
14 How enzymes propel themselves-IV (Langevin formulation of diffusion)
There are still other posts related to diffusion and we summarize them here to show how important this topic is.
24 蛋白質表現的隨機過程 (no English episode yet)
66 Gradient Sensing-I (computer simulation of diffusion)
67 Gradient Sensing-II (computer simulation of diffusion)
(computer simulation of 3D diffusion)

Tânia Tomé (2006). Entropy Production in Nonequilibrium Systems Described by a Fokker-Planck Equation. Braz. J. Phys.36(4a).
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Wow we have a long and dreadful introduction today. Before we dive into any mathematics, we have to make sure we know why Fokker-Planck equation is linked to entropy production. First we know that Fokker-Planck equation is equivalent to Langevin equation, which describes the evolution of a system subject to some dissipative forces. However, Fokker-Planck equation describe the evolution of the distribution associated with some dissipative processes. Since the dissipative processes are associated with entropy production, we should be able to describe the entropy production in terms of Fokker-Planck equation.

So now let's consider a system composed of n possibly interacting particles following a coupled system of Langevin equations:
The ζ term is the additive white noise. This means:
If the above 2 equations seem incomprehensible to you, please review our previous posts. This system of Langevin equations describe the evolution of each particles in the system, while there is another equation describing the evolution of the probability distribution, the associated Fokker-Planck equation:
Note that from the continuity equation, the evolution of probability distribution could be described in terms of sum of probability fluxes, J. So the above Fokker-Planck equation could be written as
If we divide this equation with P(x,t), we will get

According to Gibbs entropy formula, the entropy of the above system could be written as
With the aids of Fokker-Planck equation and integration by parts, the time evolution of entropy could be derived
The time evolution of the entropy of a system could be divided into 2 terms -- the entropy production and the entropy flow. The first term of the above equation could be either positive or negative while the second term is always positive. Therefore the first term could be identified as the entropy flow while the second term is entropy production rate.

If the system is a reversible system, the entropy production would eventually vanish and the system would reach equilibrium. However, if the system is irreversible and far from equilibrium, but it is in contact with a heat reservoir and is kept in a steady state, the energy dissipation rate, or the heat production rate would be

The above discussion might be a little bit confusing and abstract. I hope these would make sense to you guys after we move on to a concrete system of sensory adaptation in the following episodes. Stay tuned!