Langevin

Langevin refers primarily to Paul Langevin, a French physicist active in the early 20th century. He is best known for introducing a stochastic description of Brownian motion and for contributing to theories of radiation reaction, magnetism, and statistical mechanics. The term also appears in the Langevin function, associated with classical paramagnetism.

The Langevin equation is a stochastic differential equation that models the dynamics of a particle in a fluid subject to both systematic forces and random thermal fluctuations. A common form is m dv/dt = -γ v - dU/dx + ξ(t), where γ is a friction coefficient and ξ(t) represents a random force modeled as Gaussian white noise with ⟨ξ(t)⟩ = 0 and ⟨ξ(t)ξ(t′)⟩ = 2γkBT δ(t − t′). In the overdamped limit, where inertia is negligible, the equation reduces to γ dx/dt = -dU/dx + ξ(t). The Langevin equation connects to the Fokker–Planck equation and embodies the fluctuation–dissipation theorem, describing how thermal fluctuations and dissipation drive systems toward Boltzmann equilibrium.

Langevin dynamics is the use of this framework in molecular dynamics and computational physics to sample canonical ensembles and study systems such as polymers, liquids, and biomolecules. It provides a practical approach to incorporate thermal noise and damping in simulations.

The Langevin function, L(x) = coth(x) − 1/x, arises in classical paramagnetism to describe the average alignment of non-interacting magnetic dipoles in an external field. It appears in expressions for magnetization and magnetic susceptibility as a function of temperature and field strength.

Together, Langevin’s ideas have had lasting influence on stochastic processes, statistical mechanics, and computational methods across physics.

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