Generalized Langevin Equations

A derivation is presented for a generalized Langevin equation of motion for a dynamical variable $\phi {(}\mathbf{R}\mathrm{(t),}\mathbf{P}\mathrm{(t)}{)}$ where R and P are the position and momentum of a single heavy particle in a bath of light particles. A detailed analysis is given for the conditions required for the validity of the equation. A stochastic Fokker–Planck equation is derived for the quantity $\mathrm{D(t)\hspace{0.17em}=\hspace{0.17em}\delta }{(}\mathbf{R}\mathrm{(t)\hspace{0.17em}-\hspace{0.17em}}\mathbf{R̂}{)}\delta {(}\mathbf{P}\mathrm{(t)\hspace{0.17em}-\hspace{0.17em}}\mathbf{P̂}{)}$ which may be used to obtain both the generalized Langevin equation for $\phi$ and a Fokker–Planck equation for the heavy particle distribution function. The correlation function for $\mathrm{D(t)}$ is computed and it is shown how this quantity may be used to obtain all other correlation functions of interest.