1.1. Notations and statement of the problem. -The derivation of the laws of transport phenomenon from the underlying basic laws is one of the basic problems of non-equilibrium statistical mechanics. For classical systems the basic transport equation at the level of many-body theory is the Liouville equation (1) (partial deriv)_t ρ_t = -((partial deriv)_q ρ_t · (partial deriv)_p H -(partial deriv)_pρ_t · (partial deriv)_qH) = -{ρ_t,H}, which follows from Hamilton's equations of motion (2) q = (partial deriv)_p H, p = -(partial deriv)_q H, by taking the time derivative of a probability density ρ_t = ρ(q_t, P_t), calculated along a phase space trajectory (q_t,p_t) of a classical Hamiltonian system with Hamiltonian H. In (1) {·,·} denote the Poisson brackets. Replacing them with a commutator, we see that the relation between Liouville equation (1) and the usual Hamilton equation is the classical analogue of the relation between the (dual) Heisenberg equation and Schroedinger's equation.
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