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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.

Set $v=\dot{x}$, $\zeta =6\pi \mu a$, and $\tau =m/\zeta$. To keep dimensions and the thermal amplitude explicit, write the force covariance as $⟨F(t)F(s)⟩=A\delta (t-s)$. The printed unit coefficient is a choice of noise normalization, not a general dimensional thermal force. Assume the initial velocity is independent of the future white noise. The Underdamped Langevin dynamics has integrating-factor solutionTaking its expectation and using the delta covariance in the double integral givesHere the noise contribution is $A{m}^{-2}{\int }_0^t{e}^{-2(t-s)/\tau }ds$. The equipartition theorem requires the long-time value $⟨m{v}^2/2⟩=k_{B}T/2$. Consequently the fluctuation-dissipation relation for a Langevin particle isWith an initially thermal velocity, $⟨v_0⟩=0$ and $⟨v_0^2⟩=k_{B}T/m$, so the velocity variance is stationary. If one insists on the printed coefficient $A=1$ in dimensional units, the model has temperature fixed by $k_{B}T=1/(2\zeta )$ in those units; it cannot describe arbitrary temperature while retaining that coefficient.

Write $E(t)=\delta A/\delta x(t)$ for the functional derivative of the action. Under time reversal in classical mechanics, $\dot{x}$ changes sign whereas $E$ is even by assumption. Thus the forward and reversed Gaussian white noise histories associated with the same geometric path areThe Onsager--Machlup path probability therefore givesHere the path probabilities are densities conditional on their respective initial states; they are not separately normalized bridges conditioned on both endpoints. The noise-to-path Jacobian determinant must be treated with a consistent discretization: it cancels when it is invariant under time reversal in classical mechanics. This is the usual additive-noise Langevin dynamics convention, including a constant-mass Underdamped Langevin dynamics or the time-reversal invariance of a path Jacobian in midpoint overdamped Langevin dynamics. A completely arbitrary velocity-dependent Lagrangian would require specifying that measure rather than deducing its cancellation solely from the parity of $E$.

Equation (1) is Underdamped Langevin dynamics for a unit-mass particle in potential $U$, coupled to a heat bath of temperature $T$. The coefficient $\gamma >0$ is viscous friction, and the noise amplitude is fixed by the Fluctuation-dissipation theorem. Its Fokker-Planck equation isFor $\gamma \gg 1$, velocity relaxes rapidly, so formallyThus the overdamped Langevin dynamics isand its density obeys

Invariant distribution of an Itô diffusion, specialized to Underdamped Langevin dynamics, has densityThus $X$ has density proportional to $e^{-U(x)}$, and conditionally and marginally $P\sim N(0,\eta I_d)$. The Hamiltonian transport between $x$ and $p$ preserves this density, while the Ornstein-Uhlenbeck process in momentum has exactly that Gaussian invariant law.

The Euler-Maruyama method with step size $\delta$ and independent ${\xi }_k\sim N(0,I_d)$ is