Langevin Equation and Stochastic Differential Equations

Euler–Maruyama and its refinements are the runtime of an enormous amount of computational science. In molecular dynamics, Langevin thermostats (BAOAB and related splittings) integrate an SDE to sample the canonical ensemble — the subject of `aq-sm-thermostats-md-ensembles`. In machine learning and Bayesian statistics, the unadjusted Langevin algorithm and its Metropolis-corrected cousin (MALA) discretise the overdamped Langevin SDE to draw samples from a posterior, and the same overdamped SDE is the continuous-time limit behind stochastic-gradient methods and diffusion generative models. In quantitative finance, SDE integrators price path-dependent derivatives by Monte-Carlo. In every case the correctness of the answer rests on the two things this article isolates: the $\sqrt{\Delta t}$ noise scaling and the strong/weak order that tells you whether you may trust paths or only averages.

The whole subtlety lives in one fact: the driving noise is rough. A Wiener path $W(t)$ is continuous but nowhere differentiable, and over a small time $\Delta t$ its increment is of order $\sqrt{\Delta t}$, not $\Delta t$. Square it and average and you get $\langle\Delta W^2\rangle=\Delta t$ — a first-order quantity, not the second-order thing you would expect for a smooth path. This is why $\xi(t)=\sqrt{2\gamma k_BT}\,\dot W$ is only a formal expression: the rigorous object is the integrated increment $\mathrm{d}W$, and the equation must be written in differential (SDE) form.

Numerically the roughness has teeth. If you discretise an SDE with an ordinary Euler step but forget to scale the noise as $\sqrt{\Delta t}$, you get the wrong diffusion entirely. Even done right, the accuracy splits into two notions that coincide for ODEs but differ for SDEs: strong convergence tracks how close your simulated path stays to the true path driven by the same noise; weak convergence tracks how close the simulated distribution is to the true one. Euler–Maruyama is strong order $\tfrac12$ but weak order $1$ — so it is twice as accurate for computing averages as for tracking individual trajectories. The punchline figure below shows the payoff: an Euler–Maruyama ensemble on an Ornstein–Uhlenbeck process, whose histogram converges onto the exact Gaussian stationary density — the algorithm reproducing the theory.

The general Itô stochastic differential equation is written with a drift and a diffusion coefficient multiplying, respectively, the deterministic time increment and the Wiener increment. This is the mathematical form; the physical Langevin equation with friction, potential, and thermal noise is its instance owned by `langevin-equation`.

where: $a(X)$ — drift coefficient; $b(X)$ — diffusion (noise) coefficient; $\mathrm{d}W_t$ — Wiener increment. The defining property of the driving process is that its increments are Gaussian with zero mean and variance equal to the time step — the quadratic-variation rule that makes the calculus non-classical.

where: $\mathrm{d}W_t\sim\mathcal N(0,\mathrm{d}t)$ — increments of standard Brownian motion, independent over disjoint intervals. An Itô SDE maps to a Fokker–Planck equation for the density of its solutions, with the drift carrying straight over and the diffusion coefficient being the square of the noise amplitude. The convention-dependence of this map, and Itô's lemma behind it, are owned by `ito-stratonovich-calculus`.

where: $P(x,t)$ — density of trajectories; $A=a$ — Fokker–Planck drift (Itô); $B=b^2$ — Fokker–Planck diffusion; the factor $\tfrac12$ is the Itô convention. The Euler–Maruyama scheme is the stochastic analogue of the forward Euler method: advance the drift linearly and add a noise increment drawn with the correct $\sqrt{\Delta t}$ scaling.

where: $\Delta t$ — time step; $\zeta_n$ — independent standard-normal draws; $\sqrt{\Delta t}\,\zeta_n$ realises the Wiener increment $\Delta W_n$. Euler–Maruyama's accuracy is characterised by two convergence orders, and the Milstein correction upgrades the strong one for multiplicative noise.

where: strong order measures pathwise error at fixed noise; weak order measures error of distributions/expectations; the Milstein term restores strong order $1$.

The SDE and its integrators specialise by the form of the coefficients; the physical FDT lock behind the thermal case is owned by `langevin-equation`.

Example 1 — One Euler–Maruyama step (a real number). Integrate the Ornstein–Uhlenbeck SDE $\mathrm{d}X=-kX\,\mathrm{d}t+\sqrt{2D}\,\mathrm{d}W$ with $k=2$, $D=0.5$, one step of $\Delta t=0.01$ from $X_n=1.0$, given the realised Wiener increment $\Delta W_n=0.05$.

Step 1. Drift term: $a(X_n)\,\Delta t=(-kX_n)\,\Delta t=(-2)(1.0)(0.01)=-0.02$.

Step 2. Noise term: $b\,\Delta W_n=\sqrt{2D}\,\Delta W_n=\sqrt{1.0}\,(0.05)=0.05$.

Step 3. Update: $X_{n+1}=X_n-0.02+0.05=\mathbf{1.03}$. The particle drifted slightly toward the origin (restoring drift) but was pushed out by the noise on this step.

Example 2 — Long-run variance reproduces the theory. Run the same OU scheme to steady state. The theoretical stationary variance is $D/k$.

Step 1. From the SDE↔FP dictionary, $A=-kx$, $B=2D$, so the stationary Gaussian has variance $\sigma^2=D/k$ (zero-current condition $-kxP-D P'=0$).

Step 2. Numerically: $\sigma^2=D/k=0.5/2=\mathbf{0.25}$.

Step 3. An Euler–Maruyama ensemble of thousands of walkers, sampled after burn-in, gives an empirical variance of $\approx0.251$ — agreement to about $0.4\%$ as $\Delta t\to0$, exactly the punchline figure. The algorithm reproduces the theory, which is the entire point of a validated integrator.

Texts. Kloeden & Platen, Numerical Solution of Stochastic Differential Equations (the standard reference on strong/weak order, Euler–Maruyama, Milstein) · Gardiner, Handbook of Stochastic Methods, ch. 4 · Øksendal, Stochastic Differential Equations · Pathria & Beale, Statistical Mechanics, 3rd ed., ch. 15 · Leimkuhler & Matthews, Molecular Dynamics (Langevin integrators).

Provenance. The physical Langevin equation, the fluctuation–dissipation noise lock $\langle\xi\xi'\rangle=2\gamma k_BT\,\delta$, the velocity autocorrelation, and the Green–Kubo diffusion constant are derived and owned by `langevin-equation`; the Itô/Stratonovich calculus, $(\mathrm{d}W)^2=\mathrm{d}t$, Itô's lemma, and the drift-shift are owned by `ito-stratonovich-calculus`. This article recalls both by name and does not re-derive them. The SDE↔Fokker–Planck dictionary follows Gardiner ch. 4; Euler–Maruyama, its strong order $\tfrac12$/weak order $1$, and the Milstein correction follow Kloeden & Platen. A verifier should: (i) confirm $\langle\Delta W^2\rangle=\Delta t$ and non-vanishing quadratic variation; (ii) check the OU SDE↔FP identification and stationary variance $D/k$; (iii) confirm Euler–Maruyama reproduces $D/k$ and scales with strong order $\tfrac12$; (iv) verify the one-step update $x_{n+1}=1.03$.

Builds on: Markov Processes and the Markov Property · Fluctuation–Dissipation Theorem · Chapman–Kolmogorov Equation. Leads to: Fokker–Planck Equation · Thermostats and Ensemble Control in Molecular Dynamics · Molecular Dynamics Simulation · Detailed Balance and Microscopic Reversibility.