Langevin dynamics

Langevin dynamics describes a dynamical variable driven by deterministic forces and random noise. Underdamped Langevin dynamics retains inertia, whereas overdamped Langevin dynamics neglects it. At thermal equilibrium, the fluctuation-dissipation relation for a Langevin particle relates Gaussian white noise strength to drag and temperature.

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.

Independent-record averaging and increasing detector bandwidth change a measurement in different ways. Let $Y_1,\dots ,Y_N$ be independent measurements of zero-mean noise, made with the same fixed filter, and suppose each has finite variance ${\sigma }_B^2$. Their average satisfiesThe vanishing mean is an ensemble statement; the finite-record noise fluctuations only decrease as the number of independent records grows. For correlated records the variance is instead $N^{-2}{\sum }_{i,j}\mathrm{Cov}(Y_i,Y_j)$, so merely increasing the number of samples need not give the same reduction.

Bandwidth controls which fluctuations enter a record. With two-sided angular-frequency power spectral density $S(\omega )$ and an ideal low-pass filter retaining $|\omega |\le \Omega$, the filtered variance isFor flat noise spectrum $S(\omega )=S_0>0$ in that range, this is $S_0\Omega /\pi$, and averaging $N$ independent records gives $S_0\Omega /(\pi N)$. Increasing bandwidth exposes more of the physical fluctuations; independent averaging reduces their variance. Sampling faster from one correlated trajectory is not automatically equivalent to obtaining independent records.

For $m{\dot{v}}_i=-\zeta v_i+{\xi }_i$, classical equilibrium white noise satisfies $⟨{\xi }_i(t){\xi }_j(t^{\prime })⟩=2\zeta k_{B}T{\delta }_{ij}\delta (t-t^{\prime })$. The normalization produces the equipartition velocity variance $k_{B}T/m$.

For a particle of mass $m$ with linear drag $-\zeta v$, the velocity relaxation time is the exponential relaxation time ${\tau }_v=m/\zeta$. Under a constant applied force $F$ and zero-mean random forcing, the averaged Langevin equation is $md⟨v⟩/dt=-\zeta ⟨v⟩+F$. Solving givesThus the initial mean-velocity deviation decays by a factor $e^{-1}$ after time ${\tau }_v$. Since the force-per-velocity coefficient $\zeta$ has units $\mathrm{kg}{\mathrm{s}}^{-1}$, $m/\zeta$ has units of time. If the damping parameter is instead defined as a rate $\gamma =\zeta /m$, the same time is $1/\gamma$. For a sphere obeying Stokes's law, ${\tau }_v=m/(6\pi \eta R)$. The fluctuation-dissipation relation for a Langevin particle gives the same decay time for its equilibrium velocity covariance.