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In statistical physics
Statistical physics

Statistical physics is the area of physics that uses methods of probability theory and statistics, and particularly the Mathematics tools for dealing with large populations, in solving physical problems....
, a Langevin
Paul Langevin

Paul Langevin was a prominent France physicist who developed Langevin dynamics and the Langevin equation. He was one of the founders of the Comit? de vigilance des intellectuels antifascistes, an antifascist organization created in the wake of the February 6, 1934 far right riots....
 equation is a stochastic differential equation
Stochastic differential equation

A stochastic differential equation 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....
 describing Brownian motion
Brownian motion

Brownian motion is the seemingly random movement of particles suspended in a liquid or gas or the mathematical model used to describe such random movements, often called a particle theory....
 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 is expressed as the sum of a viscous force (which is proportional to the particles velocity) (Stokes' law
Stokes' law

In 1851, George Gabriel Stokes derived an expression, now known as Stokes' law, for the frictional force ? also called drag force ? exerted on sphere objects with very small Reynolds numbers in a continuous viscosity fluid....
), a noise term (the name given in physical contexts to terms in stochastic differential equations which are stochastic process
Stochastic process

A stochastic process, or sometimes random process, is the counterpart to a deterministic process in probability theory. Instead of dealing with only one possible 'reality' of how the process might evolve under time , in a stochastic or random process there is some indeterminacy in its future evolution described by probability distribu...
es) 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
Intramolecular

Intramolecular in chemistry describes a process or characteristic limited within the structure of a single molecule; a property or phenomenon limited to the extent of a single molecule....
 and intermolecular interactions:

Essentially similar equations govern 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
Ito diffusion

In mathematics — specifically, in stochastic processes — an Ito diffusion is a solution to a specific type of stochastic differential equation....
.

Many interesting results can often be obtained without solving the Langevin equation, from the fluctuation dissipation theorem
Fluctuation dissipation theorem

In statistical physics, the fluctuation dissipation theorem is a powerful tool for predicting the Non-equilibrium thermodynamics of a system ? such as the irreversibility dissipation of energy into heat ? from its reversible process fluctuations in thermodynamic equilibrium....
.

The main method of solution if a solution is required is by use of the Fokker-Planck equation
Fokker-Planck equation

The Fokker?Planck equation describes the time evolution of the probability density function of the position of a particle, and can be generalized to other observables as well....
, which provides a deterministic equation satisfied by the time dependent probability density.






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In statistical physics
Statistical physics

Statistical physics is the area of physics that uses methods of probability theory and statistics, and particularly the Mathematics tools for dealing with large populations, in solving physical problems....
, a Langevin
Paul Langevin

Paul Langevin was a prominent France physicist who developed Langevin dynamics and the Langevin equation. He was one of the founders of the Comit? de vigilance des intellectuels antifascistes, an antifascist organization created in the wake of the February 6, 1934 far right riots....
 equation is a stochastic differential equation
Stochastic differential equation

A stochastic differential equation 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....
 describing Brownian motion
Brownian motion

Brownian motion is the seemingly random movement of particles suspended in a liquid or gas or the mathematical model used to describe such random movements, often called a particle theory....
 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 is expressed as the sum of a viscous force (which is proportional to the particles velocity) (Stokes' law
Stokes' law

In 1851, George Gabriel Stokes derived an expression, now known as Stokes' law, for the frictional force ? also called drag force ? exerted on sphere objects with very small Reynolds numbers in a continuous viscosity fluid....
), a noise term (the name given in physical contexts to terms in stochastic differential equations which are stochastic process
Stochastic process

A stochastic process, or sometimes random process, is the counterpart to a deterministic process in probability theory. Instead of dealing with only one possible 'reality' of how the process might evolve under time , in a stochastic or random process there is some indeterminacy in its future evolution described by probability distribu...
es) 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
Intramolecular

Intramolecular in chemistry describes a process or characteristic limited within the structure of a single molecule; a property or phenomenon limited to the extent of a single molecule....
 and intermolecular interactions:

Essentially similar equations govern 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
Ito diffusion

In mathematics — specifically, in stochastic processes — an Ito diffusion is a solution to a specific type of stochastic differential equation....
.

Many interesting results can often be obtained without solving the Langevin equation, from the fluctuation dissipation theorem
Fluctuation dissipation theorem

In statistical physics, the fluctuation dissipation theorem is a powerful tool for predicting the Non-equilibrium thermodynamics of a system ? such as the irreversibility dissipation of energy into heat ? from its reversible process fluctuations in thermodynamic equilibrium....
.

The main method of solution if a solution is required is by use of the Fokker-Planck equation
Fokker-Planck equation

The Fokker?Planck equation describes the time evolution of the probability density function of the position of a particle, and can be generalized to other observables as well....
, which provides a deterministic equation satisfied by the time dependent probability density. Alternatively numerical solutions can be obtained by Monte Carlo
Monte Carlo method

Monte Carlo methods are a class of computational algorithms that rely on repeated random sampling to compute their results. Monte Carlo methods are often used when computer simulation physics and mathematics systems....
 simulation. Other techniques, such as path integration
Path integral formulation

The path integral formulation of quantum mechanics is a description of quantum theory which generalizes the action of classical mechanics. It replaces the classical notion of a single, unique trajectory for a system with a sum, or functional integral, over an infinity of possible trajectories to compute a probability amplitude....
 have also been used, drawing on the analogy between statistical physics and quantum mechanics
Quantum mechanics

Quantum mechanics is a set of principles underlying the most fundamental known description of all physical systems at the microscopic scale . Notable amongst these principles are both a dual wave-like and particle-like behavior of matter and radiation, and prediction of probabilities in situations where classical physics predicts certaintie...
 (for example the Fokker-Planck equation can be transformed into the Schrödinger equation
Schrödinger equation

In physics, especially quantum mechanics, the Schr?dinger equation is an equation that describes how the quantum state of a physical system changes in time....
 by rescaling a few variables).

Further reading


  • W. T. Coffey (Trinity College, Dublin
    Trinity College, Dublin

    Trinity College, Dublin , corporately designated as the Provost, Fellows and Scholars of the College of the Holy and Undivided Trinity of Queen Elizabeth near Dublin, was founded in 1592 by Queen Elizabeth I of England as the "mother of a university", and is the only constituent residential college of the University of Dublin....
    , Ireland), Yu P. Kalmykov (Université de Perpignan, France
    France

    France , officially the French Republic , is a country whose Metropolitan France is located in Western Europe and that also comprises various Overseas departments and territories of 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)