Non-equilibrium Statistical Mechanics

Abstract

The problem of transport, which is inherently statistical, becomes very subtle, if one wants to set up a formalism in which the starting point is the phase or Gibbs space. The key ingredient is of course that the density operator is now explicitly time dependent, in order to cover the changes that occur in time. In equilibrium statistical physics, the Liouville equation is

$$[H,\varrho ] = 0$$

(8.1)

and it is possible to find a density operator, namely ϱ = exp (−βH), which is an exact solution of this equation. For non-equilibrium statistical physics, the Liouville equation is

$$i\hbar \frac{{\partial \varrho }}{{\partial t}} = [H,\varrho ]$$

(8.2)

which is formally solved by

$$\varrho (t) = \exp \left( { - \frac{i}{\hbar }Ht} \right)\varrho (0)\exp \left( {\frac{i}{\hbar }Ht} \right).$$

(8.3)

However, this solution follows in a microscopic sense the time evolution, and as such it does not correspond to an entropy-increasing solution. Indeed

$$S(t) = - {{k}_{B}}Tr[\varrho (t)\ln \varrho (t)] = S(0).$$

(8.4)

As was emphasized in the introduction, transport phenomena are usually accompanied by dissipative effects, i.e. an increment of entropy should be one of the essential outcomes of a sensible non-equilibrium statistical theory of tranport.