In statistical physics, a Langevin equation is a stochastic differential equation describing Brownian motion in a potential. Statistical physics, one of the fundamental theories of physics, uses methods of statistics in solving physical problems. ... Albert Einstein, Paul Ehrenfest, Paul Langevin, Heike Kamerlingh Onnes, and Pierre Weiss at Ehrenfests home in Leiden Paul Langevin (January 23, 1872 â€“ December 19, 1946) was a prominent French physicist who developed Langevin dynamics and the Langevin equation. ... A stochastic differential equation (SDE) is a differential equation in which one or more of the terms is a stochastic process, thus resulting in a solution which is itself a stochastic process. ... Three different views of Brownian motion, with 32 steps, 256 steps, and 2048 steps denoted by progressively lighter colors. ...

The first Langevin equations to be studied were those in which the potential is constant, so that the acceleration a of a Brownian particle of mass m is expressed as the sum of a viscous force which is proportional to the particles velocity v (Stokes' law), a noise term (the name given in physical contexts to terms in stochastic differential equations which are stochastic processes) representing the effect of a continuous series of collisions with the atoms of the underlying fluid and F(x) which is the systematic interaction force due to the intramolecular and intermolecular interactions: In 1851, George Gabriel Stokes derived an expression for the frictional force exerted on spherical objects with very small Reynolds numbers (e. ... In the mathematics of probability, a stochastic process is a random function. ... Intramolecular describes a process or characteristic limited within the structure of a single, or each molecule; a property or phenomenon limited to the extent of a single, or each molecule. ... Intermolecular describes a process or characteristic that extends from one molecule to an adjacent one; a property or phenomenon that extends from one molecule to another. ...

Essentially similar equations govern other Brownian systems, such as thermal noise in an electrical resistor: Johnson-Nyquist noise (sometimes thermal noise, Johnson noise or Nyquist noise) is the noise generated by the equilibrium fluctuations of the electric current inside an electrical conductor, which happens without any applied voltage, due to the random thermal motion of the charge carriers (the electrons). ...

Many interesting results can often be obtained without solving the Langevin equation, from the fluctuation dissipation theorem. In statistical physics, the fluctuation dissipation theorem states that if a thermodynamic system responds linearly to an external perturbation, then the amount by which it responds is simply related to the fluctuation properties of the thermodynamic system. ...

The main method of solution if a solution is required is by use of the Fokker-Planck equation, which provides a deterministic equation satisfied by the time dependent probability density. Alternatively numerical solutions can be obtained by Monte Carlo simulation. Other techniques, such as path integration have also been used, drawing on the analogy between statistical physics and quantum mechanics (for example the Fokker-Planck equation can be transformed into the Schrödinger equation by rescaling a few variables). The Fokker-Planck equation (named after Adriaan Fokker and Max Planck; also known as the Kolmogorov Forward equation) describes the time evolution of the probability density function of position and velocity of a particle. ... Monte Carlo methods are a widely used class of computational algorithms for simulating the behavior of various physical and mathematical systems, and for other computations. ... This article or section is in need of attention from an expert on the subject. ... Fig. ... For a non-technical introduction to the topic, please see Introduction to quantum mechanics. ...

Further reading

• W. T. Coffey (Trinity College, Dublin, Ireland), Yu P. Kalmykov (Université de Perpignan, France) & J. T. Waldron (Trinity College, Dublin, Ireland), The Langevin Equation, With Applications to Stochastic Problems in Physics, Chemistry and Electrical Engineering (Second Edition), World Scientific Series in Contemporary Chemical Physics - Vol 14. (The First Edition is Vol 10)

• Reif, F. Fundamentals of Statistical and Thermal Physics, McGraw Hill New York, 1965. See section 15.5 Langevin Equation