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 
of a Brownian particle of mass m is expressed as the sum of a viscous force which is proportional to the particle's velocity 
(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 
which is the systematic interaction force due to the intramolecular and intermolecular interactions:
The diagram at right shows a phase portrait of the time evolution of the momentum, p = mv, vs. position, r of a harmonic oscillator. Deterministic motion would follow along the ellipsoidal trajectories which cannot cross each other without changing energy. The presence of a molecular fluid environment (represented by diffusion and damping terms) continually adds and removes kinetic energy from the system, causing an initial ensemble of stochastic oscillators (dotted circles) to spread out, eventually reaching thermal equilibrium.
Essentially similar equations govern random noise in other Brownian systems, such as thermal noise in an electrical resistor:
The Langevin equation is equivalent to the equation used in the theory of Ito diffusion. In the simplest case, the solution is an Ornstein–Uhlenbeck process.
Many interesting results can often be obtained without solving the Langevin equation, from the fluctuation dissipation theorem.
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).
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
- R. Friedrich, J. Peinke and Ch. Renner. How to Quantify Deterministic and Random Influences on the Statistics of the Foreign Exchange Market, Phys. Rev. Lett. 84, 5224 - 5227 (2000)
- L.C.G. Rogers and D. Williams. Diffusions, Markov Processes, and Martingales, Cambridge Mathematical Library, Cambridge University Press, Cambridge, reprint of 2nd (1994) edition, 2000.
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