A question kept bothering me about the Non-Equilibrium Statistical mechanics, can somebody give a simple description of how one approaches this subject? Is there a exact formalism, as we have for Equilibrium Statistical Mechanics, or is it some kind of an approximation? I would also like to know the promising efforts in this field.
Answer
There exist an exact formalism to treat non equilibrium statistical mechanics. You start to write down the Hamiltonian for the N interacting particles. Then you introduce the distribution function in the phase space $f(r_1,r_2...r_n,p_1,p_2,...p_n,t)$.The time evolution of this distribution function is generated by the Hamiltonian and more precisely by the poisson brackets: ${x_i,p_i};{x_i,H};{p_i,H}$. The time evolution equation for f is named Liouvillian. However beautifull this formalism is, it is completly equivalent to solving the motion equation for the N particles, that is to say, it is useless. So on reduces by 2N-1 integrations over $x_i,p_i$ the problem to a 1 particle distribution function. The reduction is exact but one finds that $f_1$ is coupled to $f_{12}$; $f_{12}$ is coupled to $f_{123}$ etc. (BBGKY hierarchy). There are different methods to stop the expansion and the resulting equation for the 1 particle distribution function is named differently depending on the problem: Vlasov's equation, Laundau's equation, Balescu's equation, Fokker-Planck's equation or Boltzmann's equation.
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