Langevin Equation - Generic Langevin Equation

Generic Langevin Equation

There is a formal derivation of a generic Langevin equation from classical mechanics. This generic equation plays a central role in the theory of critical dynamics, and other areas of nonequilibrium statistical mechanics. The equation for Brownian motion above is a special case.

An essential condition of the derivation is a criterion dividing the degrees of freedom into the categories slow and fast. For example, local thermodynamic equilibrium in a liquid is reached within a few collision times. But it takes much longer for densities of conserved quantities like mass and energy to relax to equilibrium. Densities of conserved quantities, and in particular their long wavelength components, thus are slow variable candidates. Technically this division is realized with the Zwanzig projection operator, the essential tool in the derivation. The derivation is not completely rigorous because it relies on (plausible) assumptions akin to assumptions required elsewhere in basic statistical mechanics.

Let A={A_i} denote the slow variables. The generic Langevin equation then reads

The fluctuating force η_i(t) obeys a Gaussian probability distribution with correlation function

This implies the Onsager reciprocity relation λ_i,j=λ_j,i for the damping coefficients λ. The dependence dλ_i,j/dA_j of λ on A is negligible in most cases. The symbol =-ln(p₀) denotes the Hamiltonian of the system, where p₀(A) is the equlibribium probability distribution of the variables A. Finally, is the Poisson bracket of the slow variables A_i and A_j.

In the Brownian motion case one would have = p2/(2mk_BT), A={p} or A={x, p} and =δ_i,j. The equation of motion dx/dt=p/m for x is exact, there is no fluctuating force η_x and no damping coefficient λ_x,p.