Langevin equation
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In statistical physics, a Langevin equation is a stochastic differential equation describing Brownian motion in a potential.
The first Langevin equations to be studied were those in which the potential is constant, so that the acceleration <math>\mathbf{a}<math> of a Brownian particle of mass <math>m<math> is expressed as the sum of a viscous force which is proportional to the particles velocity <math>\mathbf{v}<math> (Stokes' law), and 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:
- <math>m\mathbf{a} = m\frac{d\mathbf{v}}{dt} = - \beta \mathbf{v} + \eta(t).<math>
The main method of solution is by use of the Fokker-Planck equation, which provides a deterministic equation satisfied by the time dependant 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 Schrodinger equation by rescaling a few variables).