Langevin equation

In physics, Langevin equation (named after Paul Langevin) is a stochastic differential equation describing the time evolution of a subset of the degrees of freedom. These degrees of freedom typically are collective (macroscopic) variables changing only slowly in comparison to the other (microscopic) variables of the system. The fast (microscopic) variables are responsible for the stochastic nature of the Langevin equation. In physics, Langevin equation (named after Paul Langevin) is a stochastic differential equation describing the time evolution of a subset of the degrees of freedom. These degrees of freedom typically are collective (macroscopic) variables changing only slowly in comparison to the other (microscopic) variables of the system. The fast (microscopic) variables are responsible for the stochastic nature of the Langevin equation. The original Langevin equation describes Brownian motion, the apparently random movement of a particle in a fluid due to collisions with the molecules of the fluid, The degree of freedom of interest here is the position r {displaystyle mathbf {r} } of the particle, m {displaystyle m} denotes the particle's mass. The force acting on the particle is written as a sum of a viscous force proportional to the particle's velocity (Stokes' law), and a noise term η ( t ) {displaystyle {oldsymbol {eta }}left(t ight)} (the name given in physical contexts to terms in stochastic differential equations which are stochastic processes) representing the effect of the collisions with the molecules of the fluid. The force η ( t ) {displaystyle {oldsymbol {eta }}left(t ight)} has a Gaussian probability distribution with correlation function where k B {displaystyle k_{B}} is Boltzmann's constant, T {displaystyle T} is the temperature and η i ( t ) {displaystyle eta _{i}left(t ight)} is the i-th component of the vector η ( t ) {displaystyle {oldsymbol {eta }}left(t ight)} . The δ-function form of the correlations in time means that the force at a time t {displaystyle t} is assumed to be completely uncorrelated with it at any other time. This is an approximation; the actual random force has a nonzero correlation time corresponding to the collision time of the molecules. However, the Langevin equation is used to describe the motion of a 'macroscopic' particle at a much longer time scale, and in this limit the δ {displaystyle delta } -correlation and the Langevin equation become exact. Another prototypical feature of the Langevin equation is the occurrence of the damping coefficient λ {displaystyle lambda } in the correlation function of the random force, a fact also known as Einstein relation. Consider a free particle of mass m {displaystyle m} with equation of motion described by where v = d r / d t {displaystyle mathbf {v} =dmathbf {r} /dt} is the particle velocity, μ {displaystyle mu } is the particle mobility, and F ( t ) = m a ( t ) {displaystyle mathbf {F} (t)=mmathbf {a} (t)} is a rapidly fluctuating force whose time-average vanishes over a characteristic timescale t c {displaystyle t_{c}} of particle collisions, i.e. F ( t ) ¯ = 0 {displaystyle {overline {mathbf {F} (t)}}=0} . The general solution to the equation of motion is where τ = m μ {displaystyle au =mmu } is the relaxation time of the Brownian motion. As expected from the random nature of Brownian motion, the average drift velocity ⟨ v ( t ) ⟩ = v ( 0 ) e − t / τ {displaystyle langle mathbf {v} (t) angle =mathbf {v} (0)e^{-t/ au }} quickly decays to zero at t ≫ τ {displaystyle tgg au } . It can also be shown that the autocorrelation function of the particle velocity v {displaystyle mathbf {v} } is given by where we have used the property that the variables a ( t 1 ′ ) {displaystyle mathbf {a} (t_{1}')} and a ( t 2 ′ ) {displaystyle mathbf {a} (t_{2}')} become uncorrelated for time separations t 2 ′ − t 1 ′ ≫ t c {displaystyle t_{2}'-t_{1}'gg t_{c}} . Besides, the value of lim t → ∞ ⟨ v 2 ( t ) ⟩ = lim t → ∞ R v v ( t , t ) {displaystyle lim _{t o infty }langle v^{2}(t) angle =lim _{t o infty }R_{vv}(t,t)} is set to be equal to 3 k B T / m {displaystyle 3k_{B}T/m} such that it obeys the equipartition theorem. Note that if the system is initially at thermal equilibrium already with v 2 ( 0 ) = 3 k B T / m {displaystyle v^{2}(0)=3k_{B}T/m} , then ⟨ v 2 ( t ) ⟩ = 3 k B T / m {displaystyle langle v^{2}(t) angle =3k_{B}T/m} for all t {displaystyle t} , meaning that the system remains at equilibrium at all times.