Brownian motion as a prototype

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The original Langevin equation[1] 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   of the particle,   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   (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   has a Gaussian probability distribution with correlation function

 

where   is Boltzmann's constant,   is the temperature and   is the i-th component of the vector  . The δ-function form of the correlations in time means that the force at a time   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  -correlation and the Langevin equation become exact.

Another prototypical feature of the Langevin equation is the occurrence of the damping coefficient   in the correlation function of the random force, a fact also known as Einstein relation.

Trajectories of free Brownian particles

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Consider a free particle of mass   with equation of motion described by

 

where   is the particle velocity,   is the particle mobility, and   is a rapidly fluctuating force whose time-average vanishes over a characteristic timescale   of particle collisions, i.e.  . The general solution to the equation of motion is

 

where   is the relaxation time of the Brownian motion. As expected from the random nature of Brownian motion, the average drift velocity   quickly decays to zero at  . It can also be shown that the autocorrelation function of the particle velocity   is given by[2]

 
Simulated squared displacements of free Brownian particles (semi-transparent wiggly lines) as a function of time, for three selected choices of initial squared velocity which are 0, 3kT/m, and 6kT/m respectively, with 3kT/m being the equipartition value in thermal equilibrium. The colored solid curves denote the mean squared displacements for the corresponding parameter choices.
 

where we have used the property that the variables   and   become uncorrelated for time separations  . Besides, the value of   is set to be equal to   such that it obeys the equipartition theorem. Note that if the system is initially at thermal equilibrium already with  , then   for all  , meaning that the system remains at equilibrium at all times.

The velocity   of the Brownian particle can be integrated to yield its trajectory (assuming it is initially at the origin)

 

Hence, the resultant average displacement   asymptotes to   as the system relaxes and randomness takes over. In addition, the mean squared displacement can be determined similarly to the preceding calculation to be

 

It can be seen that  , indicating that the motion of Brownian particles at timescales much shorter than the relaxation time   of the system is (approximately) time-reversal invariant. On the other hand,  , which suggests that the long-term random motion of Brownian particles is an irreversible dissipative process. Here we have made use of the Einstein–Smoluchowski relation  , where   is the diffusion coefficient of the fluid.

  1. ^ Langevin, P. (1908). "Sur la théorie du mouvement brownien [On the Theory of Brownian Motion]". C. R. Acad. Sci. Paris. 146: 530–533.; reviewed by D. S. Lemons & A. Gythiel: Paul Langevin’s 1908 paper "On the Theory of Brownian Motion" [...], Am. J. Phys. 65, 1079 (1997), doi:10.1119/1.18725
  2. ^ Pathria RK (1972). Statistical Mechanics. Oxford: Pergamon Press. pp. 443, 474–477. ISBN 0-08-018994-6.