Irreversibility of the Smoluchowski equation

3 Irreversibility of the Smoluchowski equation An important property of the Smoluchowski equation is that if C/( jc ) is independent of time and if there b no flux at the boundary, the distribution function P always approaches V equilibrium (Veq)  

Using eqn (3.21) and the integral by parts (the integral at the boundary being zero by the assumption), we have 

When the system is not in equilibrium, is larger than. s/[ I eq] and the difference —. s [ PeJ represents how far the system is away from the equilibrium state. We shall call sA the dynamical free energy. This quantity plays an important role in the subsequent discussions. 

The approach to the equilibrium state is also shown directly by the eigenfunction expansion of the distribution function. This is discussed in Appendix 3.1. More detailed discussion on the Smoluchowski equation is given in ref. 7. 

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