Generalized Langevin equation ($\dot{V}(t)=-{\int }_0^t\kappa (u)V(t-u)du+R(t)$)

A Langevin equation with a friction memory kernel, rather than only instantaneous friction. The equation couples the velocity to its earlier history through a causal convolution. Equilibrium noise and friction are related by a generalized fluctuation-dissipation relation. The name extends the Langevin equation associated with Paul Langevin.

For the equilibrium generalized Langevin equation with unit mass, the random force has zero mean and the displayed autocorrelation, where $\Theta$ is the equilibrium velocity variance. If the deterministic relaxation $g$ is stable, $g(0)=1$ and $g^{\prime }=-\kappa \ast g$, then the stationary velocity autocorrelation function is $\chi =\Theta g$, equivalentlyOne proof uses the power spectral density. The even force autocorrelation has spectrum $2\Theta \mathrm{Re}\tilde{\kappa }(i\omega )$. With transfer function $H(i\omega )=[i\omega +\tilde{\kappa }(i\omega ){]}^{-1}$, the velocity spectrum is $2\Theta |H{|}^2\mathrm{Re}\tilde{\kappa }=2\Theta \mathrm{Re}H$, the spectrum of the even extension of $\Theta g$. Stability removes initial transients with finite second moment. A valid force autocorrelation must have nonnegative power spectral density; an arbitrary proposed memory kernel does not automatically satisfy this condition.