Langevin

N.BORGHINI Topics in nonequilibrium physics

VI. Brownian motion

In this Chapter, we study the very general paradigm provided by Brownian motion. Originally,this motion is that a heavy particle, called Brownian particle, immersed in a fluid of much lighterparticlesin Browns first observations, this was some pollen particle in water. The Brownianparticle is constantly moving, in always changing random directions, due to the successive collisionswith the fluid constituents: the position x(t) and the velocity v(t) of the Brownian particles arethen stochastic processes, driven by a fluctuating force.

More generally, the Brownian particle might actually represent some stochastic collective prop-erty of a macroscopic system, and the techniques used for solving the initial question apply to amuch wider class of problems in nonequilibrium statistical physics and even beyond. Nevertheless,we shall throughout this Chapter retain the terminology of the original physical problem.

In Sec. VI.1, we introduce the approach pioneered by Langevin, which describes the evolutionof the Brownian particle velocity on time scales larger than the typical correlation time of the fluc-tuating force acting on the particle. We then investigate a generalization of the Langevin model, inwhich the friction force exerted by the fluid on the Brownian motion is non-local in time (Sec. VI.2).In Sec. VI.3, we adopt an alternative description, in which instead of solving the equation of motionfor given initial conditions, we rather focus on the time evolution of the velocity distribution.

Throughout this Chapter, we shall for the sake of simplicity consider a one-dimensional Brownianmotion. The generalization to motion in two or more dimensions is straightforward.

VI.1 Langevin dynamics

The dynamics of a Brownian particle much heavier than the constituents of the medium in whichit evolves can be described as resulting from the influence of two forces, namely an instantaneousfriction force and a fluctuating force. After establishing the corresponding equation for the velocityof the Brownian particle (Sec. VI.1.1), we study its solution for given initial condition (Sec. VI.1.2),as well as the resulting evolution of the position (Sec. VI.1.3). We then turn in Sec. VI.1.4 to thedynamics of the fluctuations of the velocity for a Brownian particle at equilibrium with its environ-ment. Eventually, anticipating on applications of the Brownian-motion paradigm to other problems,we introduce the spectral function associated to the Langevin model at equilibrium (Sec. VI.1.5),as well as the linear response of the model to an external force (Sec. VI.1.6).

VI.1.1 Langevin equation

::::::::VI.1.1 a

::::::::::Langevin

::::::model

The classical model introduced by Langevin consists in splitting the force exerted by the fluidparticles on the Brownian particles in two contributions:

First, a Brownian particle in motion with a velocity v with respect to the fluid sees more fluidparticles coming from the direction in which it is moving as from the other direction. Thelarger v is, the more pronounced the imbalance is.

To account for this effect, one introduces a force opposed to the instantaneous direction ofmotioni.e. to the velocity at the same instantand increasing with velocity. The simplestpossibility is that of a force proportional to v(t), which will be denoted as mv(t) with > 0.This actually corresponds to the viscous force exerted by a Newtonian fluid on an immersedbody, in which case the friction coefficient m is proportional to the shear viscosity of thefluid.

The fluid particles also exert a fluctuating force F (t), due to their random individual collisionswith the Brownian particle. This Langevin force, also referred to as noise term, will be

VI. Brownian motion 63


assumed to be independent of the kinematical variables (position and velocity) of the Brownianparticle.

Note that since both the friction and the noise terms introduced by this decomposition are ac-tually two aspects of a single underlying phenomenonthe microscopic scatterings of fluid particleson the Brownian particle, one can reasonably anticipate the existence of a relationship betweenthem, i.e. between and a characteristic of F (t), as we shall find in Sec. VI.1.2 b.

Assuming in addition that there is no external potential, i.e. that the Brownian particle is free,the equation of motion reads

mdv(t)

dt= mv(t) + F (t) with v(t) = dx(t)

dt. (VI.1)

This Langevin equation is an instance of a linear stochastic differential equation, i.e. an equationincluding a randomly varying termhere F (t)with given statistical propertieswhich we shallspecify in the next paragraph in the case of the Langevin force. The solution v(t) to such an equationfor a given initial condition is itself a stochastic process.

:::::::::VI.1.1 b

::::::::::Properties

:::of

:::the

::::::noise

:::::term

The fluid in which the Brownian particle is immersed is assumed to be in a stationary state, forinstance in thermodynamic equilibrium. Accordingly, the force acting upon the particle is describedby a stationary process, that is, the single-time average F (t) of the noise term is time-independent,while the two-point average F (t)F (t) only depends on the difference t t.

In order for the particle to remain (in average) motionless when it is at rest, the single-timeaverage should actually vanish:

F (t) = 0. (VI.2a)As a consequence, the autocorrelation function (C.3) simplifies to

() = F (t)F (t+ ) . (VI.2b)As always for stationary processes, () only depends on | |. () is assumed to be integrable, with

() d 2Dvm2, (VI.2c)

which defines Dv.Let c be the autocorrelation time over which () decreases. c is typically of the order of the

time interval between two collisions of the fluid particles on the Brownian particle. If c happensto be much smaller than all other time scales in the system, then the autocorrelation function canmeaningfully be approximated by a Dirac distribution

() = 2Dvm2(). (VI.2d)

Remark: In the case of multidimensional Brownian motion, one usually assumes that the correla-tion matrix ij() for the Cartesian components of fluctuating force is diagonal.

VI.1.2 Relaxation of the velocity

We now wish to solve the Langevin equation (VI.1), assuming that at the initial time t = 0, thevelocity of the Brownian particle is fixed, v(0) = v0.

With that choice of initial condition, one easily finds that the solution to the Langevin equationfor t > 0 reads

v(t) = v0 et +

1

m

t0F (t) e(tt

) dt for t > 0. (VI.3)

Since F (t) is a stochastic process, so is v(t). We can easily compute the first moments of itsdistribution.



::::::::VI.1.2 a

:::::::::Average

::::::::velocity

Averaging Eq. (VI.3) over an ensemble of realizations, one finds thanks to property (VI.2a)

v(t) = v0 et for t > 0. (VI.4)

That is, the average velocity relaxes exponentially to 0 with a characteristic relaxation time r 1.Note that this average is not time-independent, which shows at once that v(t) is not a stationary

process.

:::::::::VI.1.2 b

::::::::Variance

:::of::::the

::::::::velocity.

::::::::::::::::::::::::Fluctuationdissipation

::::::::theorem

Recognizing the average velocity (VI.4) in the first term in the right-hand side of Eq. (VI.3),one also obtains at once the variance

v(t)2

[v(t)v(t)]2 = 1

m2

t0

t0

F (t)F (t)

e(tt

) e(tt) dt dt for t > 0. (VI.5)

If the simplified form (VI.2d) for the autocorrelation function of the Langevin force holdswhichfor the sake of consistency necessitates at least c r, this variance becomes

v(t)2 = 2Dv

t0

e2(tt) dt =

Dv

(1 e2t) for t > 0. (VI.6)

2v thus vanishes at t = 0the initial condition is the same for all realizations, then grows, atfirst almost linearly

v(t)2 ' 2Dvt for 0 t r, (VI.7)

before saturating at large times

v(t)2 ' Dv

for t r. (VI.8)

Equation (VI.7) shows that Dv is a diffusion coefficient in velocity space.

From Eq. (VI.4), the average velocity vanishes at large times, so that the variance (VI.8) equalsthe mean square velocity. That is, the average kinetic energy of the Brownian particle tends atlarge times towards a fixed value

E(t) ' mDv2

for t r. (VI.9)

In that limit, the Brownian particle is in equilibrium with the surrounding fluid. If the latteris in thermodynamic equilibrium at temperature Tone then refers to it as a thermal bath orthermal reservoirthen the average energy of the Brownian particle is according to the equipartitiontheorem equal to 12kBT , which yields

Dv =kBT

m. (VI.10)

This identity relates a quantity associated with fluctuationsthe diffusion coefficient Dv, seeEq. (VI.7)with a coefficient modeling dissipation, namely . This is thus an example of the moregeneral fluctuationdissipation theorem discussed in Sec. ?? in the following chapter.

Since Dv characterizes the statistical properties of the stochastic Langevin force [Eq. (VI.2c)],Eq. (VI.10) actually relates the latter to the friction force

VI.1.3 Evolution of the position of the Brownian particle. Diffusion

Until now, we have only investigated the velocity of the Brownian particle. Instead, one couldstudy its position x(t), or equivalently its displacement from an initial position x(0) = x0 at timet = 0.



Integrating the velocity (VI.3) from the initial instant until time t, one finds

x(t) = x0 +v0

(1 et)+ 1

m

t0F (t)

1 e(tt)

dt for t > 0. (VI.11)

x(t), and in turn the displacement x(t) x0, is also a stochastic process, whose first two momentswe shall now compute.

::::::::VI.1.3 a

:::::::::Average

:::::::::::::displacement

First, the average position is simply

x(t) = x0 + v0

(1 et) for t > 0, (VI.12)

i.e. the mean displacement x(t) x0 tends exponentially towards v0

for t r.

:::::::::VI.1.3 b

::::::::Variance

:::of::::the

:::::::::::::displacement

The first two terms in the right member of Eq. (VI.11) are exactly the average position, that is,the last term is precisely x(t) x(t), or equivalently [x(t) x0] x(t) x0. The variance of theposition is thus equal to the variance of the displacement, and is given by

x(t)2 =

1

m22

t0

t0

F (t)F (t)

[1 e(tt)][1 e(tt)]dt dt for t > 0. (VI.13)

When the autocorrelation function of the Langevin force can be approximated by the simplifiedform (VI.2d), this yields

v(t)2 =

2Dv2

t0

[1 e(tt)]2 dt = 2Dv

2

(t 2 2e

t

+

1 e2t2

)for t > 0. (VI.14)

This variance vanishes at t = 0the initial condition is known with certainty, grows as t3 fortimes 0 t r, then linearly at large times

x(t)2 ' 2Dv

2t for t r. (VI.15)

Since Eq. (VI.14) is also the variance of the displacement, one finds under consideration ofEq. (VI.11)[

x(t) x0]2

= x(t)2 + x(t) x02 = x(t)2 + v

20

2(1 et)2 ' 2Dv

2t for t r. (VI.16)

The last two equations show that the position of the Brownian particle behaves as the solution ofa diffusion equation at large times, with a diffusion coefficient (in position space)

D =Dv2, (VI.17)

(cf. Sec. I.2.3 b).

::::::::VI.1.3 c

::::::::Viscous

::::::limit.

:::::::::Einstein

::::::::relation

In the limitm 0, at constant product v m, which physically amounts to neglectingthe effect of inertia (mdv/dt) compared to that of friction (vv)hence the denomination viscouslimit, the Langevin equation (VI.1) becomes

vdx(t)

dt= F (t). (VI.18a)

In that limit, the autocorrelation function of the fluctuating force in the limit of negligibly smallautocorrelation time is denoted as

() = 2D2v (), (VI.18b)

which defines the coefficient D.



In this approach, the displacement can directly be computed by integrating Eq. (VI.18a) withthe initial condition x(0) = x0, yielding

x(t) x0 = 1v

t0F (t) dt for t > 0. (VI.19)

With the autocorrelation function (VI.18b), this gives at once[x(t) x0

]2= 2Dt for t 0, (VI.20)

which now holds at any time t 0, not only in the large time limit as in Eq. (VI.16). That is, themotion of the Brownian particle is now a diffusive motion at all times.

Combining now Eq, (VI.17) with the fluctuationdissipation relation (VI.10) and the relationv = m, one obtains

D =kBT

v. (VI.21)

If the Brownian particles are charged, with electric charge q, then the friction coefficient v is relatedto the electrical mobility el. by el.= q/v (see Sec. VI.1.6 below), so that Eq. (VI.21) becomes theEinstein relation

D =kBT

qel.. (VI.22)


VI Brownian motionLangevin dynamicsLangevin equationRelaxation of the velocityEvolution of the position of the Brownian particle. Diffusion

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