1

I. BROWNIAN MOTION

The dynamics of small particles whose size is roughly 1 µmt or

smaller, in a fluid at room temperature, is extremely erratic, and is

called Brownian motion. The velocity of such particles is governed

by thermal fluctuations. The particle will interact with the bath, i.e.

the fluid it is embedded in, gaining and losing kinetic energy from the

bath, and thus the velocity of the particle and its position are stochas-

tic variables. Usually there is no hope of solving the problem from a

microscopical point of view, since the interactions of the Brownian par-

ticle with the surrounding particles are extremely complicated, hence

stochastic theories are very useful. Brownian type of dynamics de-

scribes dynamics of large molecules in solution, and hence is important

in many applications of Chemistry, Biology, and Physics. One orig-

inal motivation of investigation of Brownian motion by Einstein was

to prove the existence of atoms. Today theory of Brownian motion

is used as an example of non-equilibrium dynamics which is still close

to thermal equilibrium. Many aspects of theory of Brownian motion

can be generalized to other types of stochastic dynamics, for example

the so called Einstein relations, linear response theory or fluctuation

dissipation relation, we study here in the context of Brownian motion

have general importance in non-equilibrium systems.

We now construct a phenomenological theory for the velocity of a

Brownian motion. Our aim is to write an equation of motion for P (V, t)

2

the velocity PDF, given that the velocity of the particle at time t = 0

is V0. We use four main assumptions:

The velocity distribution of a Brownian particle with mass M , in equi-

librium is the Maxwell distribution

Peq(V ) =M1/2

√2πkbT

exp

(

− v2

2MkbT

)

, (1)

where T is the temperature. This equation does not give us any in-

formation of the dynamics, though it does state that the variance of

velocity fluctuations is given by the thermal velocity vth =√

kbT/M .

The fact that the velocity distribution is Gaussian is strongly related

to the central limit theorem as we show later in the context of kinetic

theory of Brownian motion.

The second assumption is that the average velocity satisfies the relax-

ation law

〈v〉 = −γ〈v〉 (2)

where γ is the relaxation or damping coefficient which has units of

1/Sec. For example for a spherical particle whose radius is a we know

from hydrodynamics the γ = 6πaη/M where η is the viscosity of the

fluid. The hydrodynamic Eq. (2) neglects fluctuations and it gives us

simple relaxation 〈v(t)〉 = v0e−γt. From hydrodynamics we know that

for Eq. (2) to hold the velocity must not be too high, e.g. if compared

to sound velocity.

The third assumption is that the velocity of the Brownian particle

is continuous. Mathematically this means the a partial differential

3

equation describes the dynamics.

The final assumption is that the dynamics is Markovian, namely to

determine the velocity distribution in time t + δt, we need to know

only the velocity distribution at time t. Physically this means that we

neglect the influence of the Brownian particle on the bath.

To construct an equation of motion we consider

∂P (v, t)

∂t=

B

2

∂2P (v, t)

v2+ A

∂

∂vvP (v, t). (3)

The coefficients A and B, which seem independent of each other, must

be determined from our assumptions. The first term described a dif-

fusion process in velocity space, it is called a fluctuation term. It is

clear that B > 0. The role of the second term is to restore the velocity

to equilibrium. To see this note that if A = 0 then clearly the vari-

ance of velocity is going to increase with time, which is in conflict with

Boltzmann’s equilibrium statistical mechanics and nature.

Multiplying Eq. (3) from the left with v and integrating with respect

to v, using the boundary conditions that P (v, t) = 0 when |v| → ∞ we

have 〈v(t)〉 = −A〈v〉 hence we find A = γ.

7.1 Assume the more general equation

∂P (v, t)

∂t=

Bm

2

∂2vmP (v, t)

∂v2+ An

∂

∂vvnP (v, t), (4)

show that the law 〈v〉 = −γ〈v〉 implies n = 1 but does not teach

us anything on m. Then show that for the equilibrium velocity

distribution to be Gaussian we must demand m = 0.

4

We now impose on the dynamics the request that in equilibrium,

namely when t → ∞ the velocity distribution is Maxwellian which is

independent of time. Inserting Eq. (1) in Eq. (2)

∂

∂v

[

B

2

∂

∂vPeq(V ) + γvPeq(v)

]

= 0 (5)

we find that the condition B/2 = γkbT/M must hold. To conclude we

find the equation of motion for the velocity of a Brownian particle

∂P (v, t)

∂t= γ

[

kbT

M

∂2P (v, t)

∂v2+

∂

∂vvP (v, t)

]

. (6)

This equation was derived by Rayleigh using a kinetic approach. The

approach used in the section to derive Rayleigh’s equation, based on

a macroscopical (or hydro-dynamical) relaxation law to which fluctua-

tions are added is due to Einstein and also Langevin.

7.2 Find solution to Eq. (6) assuming initial condition is V0. Check

that the solution is normalized, non-negative, and approaches

thermal equilibrium in the long time limit.

7.3 Show that

〈v2〉 = v20e

−2γt +kbT

M

(

1 − e−2γt)

.

7.4 Consider an ensemble of one dimensional Brownian particle,

which at time t = 0 are on the origin. Initially the particles

are in thermal equilibrium with temperature T . Consider the

random position of the Brownian particle x(t) =∫ t0 v(t′)dt where

5

v(t′) satisfies the dynamics in Eq. (6) show that for 0 ≤ t1 ≤ t2

〈x(t1)x(t2)〉 =kbT

M

[

2

γt1 −

1

γ2+

1

γ2

(

e−γt1 + e−γt2 − e−γ(t2−t1))

]

.

(7)

Since this equation implies that when t → ∞ 〈x2(t)〉 ∼ 2kbTMγ

t you

have proven the Einstein relation between diffusion constant

D = limt→∞

〈x2(t)〉t

= 2D

and damping γ, namely D = kbT/(Mγ).

A. The Fokker–Planck Equation

The Fokker Planck equation describes dynamics of a continuous

stochastic process y(t) whose dynamics is Markovian. It is used to

model many processes where a coordinate y(t) has small jumps, the

Rayleigh equation for the velocity of a Brownian particle being a spe-

cial case. The Fokker-Planck equation is

∂P (y, t)

∂t= − ∂

∂yA(y)P (y, t) +

1

2

∂2

∂y2B(y)P (y, t). (8)

The only conditions are that A(y), B(y) are real functions and B(y) >

0. Roughly speaking, the first term on the left hand side describes a

drift term, the second describes the fluctuations. The drift term de-

scribes the dynamics of the average of y namely it is easy to show

〈y〉 = 〈A(y)〉. For the Brownian motion we used the fact that the

average velocity obeys a linear law 〈y〉 = −γ〈y〉 to identify the phys-

ical meaning of A(y). However when the hydrodynamic law is not

6

linear we encounter a difficulty. Assume that for some fluid the aver-

age velocity of a particle obeys the non linear equation 〈v〉 = −β〈v〉2.

Then naively we might identify A = βv2. However this is wrong since

then it is easy to see that according to the Fokker-Planck equation

〈v〉 = −β〈v2〉 6= −β〈v〉2. This implies that if the dynamics of the

average 〈y〉 is described by a non-linear equation we cannot use the

simple approach we used in the previous section. Instead one must

derive the Fokker–Planck equation from some underlying dynamics,

wick should then be consistent with the non-linear phenomenological

relaxation laws.

7.5 van Kampen shows how to find A(y) and B(y) at least in prin-

ciple. Let P (y, ∆t|y0, t0) be the solution of the Fokker Planck

equation for particle starting on y0 at time t0. Take t = t0 + ∆t

and y − y0 = ∆y with ∆t and ∆y being small. Show that for

∆t → 0

〈∆y〉∆t

= A(y0),〈(∆y)2〉

∆t= B(y0),

〈(∆y)n〉∆t

= 0n ≥ 3. (9)

7.6 Find the stationary solution of the Fokker Planck equation (8).

7.7 Show that solution of Eq. (8) is normalizable. It is more difficult

to show that the solution is non-negative.

Consider the binomial random walk on a lattice, with jumps to

nearest neighbors only, and with probability of 1/2 to jump left or

right. We approximate the dynamics using a Fokker Planck equation

7

approach. First we forget about the underlying lattice and treat the

problem as a continuum. This means that while the jumps in the

random walk model have finite size (the size of the lattice spacing) and

hence the position of the particle is not a continuous function of time,

we still assume that on a coarse grained level a Fokker- Planck equation

might work well. The Fokker-Planck approach gives 〈∆x/∆t〉 = 0,

since we do not have bias. Also

lim∆x→0,∆t→0

∆x2/2∆t ≡ D

where 〈∆x2〉 is the lattice spacing square, or more generally the vari-

ance of the microscopical jump length, ∆t is the time between jumps,

and D by definition is the diffusion constant. We immediately find the

Fokker-Planck equation,

∂P (x, t)

∂t= D

∂2P (x, t)

∂x2, (10)

which is the famous diffusion equation. From central limit theorem

we know that this equation works well only in the long time limit.

Still in many Physical situations we are interested in intermediate time

scales where t � ∆t where ∆t is the time between jumps, however the

probability packet is still not in equilibrium (e.g., particles did not reach

yet the walls of the container) and then diffusion and Fokker-Planck

equations are very useful as excellent approximations.

8

B. Einstein Relations

We consider a Brownian motion under the influence of a driving

force field F . F does not depend on time or space, it is a constant

driving force, for example if the Brownian particle is charged then F is

the charge of the particle times a uniform electric field which is applied

on the system, or F could be due to the gravitational field of earth

F = mg. The particle will experience a net drift and we assume it will

reach a finite velocity (the effects of the size of the container are not

important now).

The linear equation of motion for the average velocity of a Brownian

particle is

〈v〉 = −γ〈v〉 +F

M(11)

and hence in the long time limit 〈v〉 = F/(Mγ). Now starting with

this law for the average velocity we wish to model the behavior of the

coordinate x of the particle. We clearly have 〈∆x/∆t〉 = F/(Mγ).

We also assume that the force is weak, in such a way that the diffusion

process is not altered by the external force field, besides the global drift

of-course. This means that 〈∆x2/δt〉 = D is the diffusion constant of

the Brownian particle in the absence of the external force. And hence

according to the Fokker-Planck equation approach we have

∂P (x, t)

∂t= D

∂2P (x, t)

∂x2− F

Mγ

∂P

∂x. (12)

Now Einstein adds the effect of the container to the process, and con-

siders the equilibrium of such process (in the absence of a container

9

the process never reaches equilibrium, it just exhibits a drift and dif-

fusion). Assume the force is positive, hence the drift is from left to

right. The particle are driven towards a wall which we put on x = 0.

In equilibrium we have according to Boltzmann

Peq(x) = Const exp(

Fx

kbT

)

forx < 0, (13)

where we used the fact that the potential energy of the particle is

U(x) = −Fx and the canonical ensemble. Inserting Eq. (13) in Eq.

(12) we find a relation between the damping or the dissipation and the

diffusion constant D which is the Einstein relation

D =kbT

Mγ. (14)

We also define the mobility µ which is a measure of the response of the

particle to a weak external field, by definition 〈v〉 = µF . Then clearly

µ = 1/(Mγ) and hence we find the second Einstein relation

D = kbTµ. (15)

We see that the response of the particles to external driving field, which

yields a net current, is related to the fluctuations in the absence of

the force fields. Thus µ (transport) D (diffusion) and γ (relaxation,

or dissipation) are all related. Theories of transport are many times

based on these ideas, though usually a more general framework called

linear response theory is used. The idea in many theoretical works is to

calculate D from some microscopical theory, and then give a prediction

on the response of the system to an external weak perturbation. The

10

advantage is that we do not have to consider the external force field in

the first place. Though to apply this scheme we obviously must assume

that the response to the external field is linear.

II. BROWNIAN MOTION: A SIMPLE KINETIC

APPROACH

We will consider a simple kinetic approach to obtain Maxwell’s ve-

locity distribution. Briefly we consider a one dimensional tracer parti-

cle of mass M randomly colliding with gas particles of mass m << M .

Four main assumptions are used: (i) molecular chaos holds, imply-

ing lack of correlations in the collision process (Stoszzahlansantz), (ii)

collision are elastic and impulsive, (iii) gas particles maintain their

equilibrium during the collision process, and (iv) rate of collisions is

independent of the energy of the colliding particles. Let the probabil-

ity density function (PDF) of velocity of the gas particles be f (vm).

Our goal is to obtain the equilibrium velocity PDF of the tracer par-

ticle Weq(VM). Questions: (i) Does Weq(VM) depend on m? (ii) Does

Weq(VM) depend on rate of the collisions R? (iii) Does Weq(VM) depend

on the precise shape of the velocity PDF of the gas particles f (vm)?

Answers: (i) no, (ii) no, and (iii) no. We will show that the equilibrium

PDF of the tracer particle Weq(VM) is the Maxwell velocity PDF.

11

A. Model and Time Dependent Solution

We consider a one dimensional tracer particle with the mass M

coupled with bath particles of mass m (these are treated as ideal gas

particles). The tracer particle velocity is VM . At random times the

tracer particle collides with bath particles whose velocity is denoted

with vm. Collisions are elastic hence from conservation of momentum

and energy

V +M = ξ1V

−

M + ξ2vm, (16)

where

ξ1 =1 − ε

1 + εξ2 =

2ε

1 + ε(17)

and ε ≡ m/M is the mass ratio. In Eq. (16) V +M (V −

M ) is the velocity

of the tracer particle after (before) a collision event. The duration of

the collision events is much shorter than any other time scale in the

problem. The collisions occur at a uniform rate R independent of the

velocities of colliding particles. The probability density function (PDF)

of the bath particle velocity is f(vm). This PDF does not change during

the collision process, indicating that re-collisions of the bath particles

and the tracer particle are neglected.

We now consider the equation of motion for the tracer particle ve-

locity PDF W (VM , t) with initial conditions concentrated on VM(0).

Kinetic considerations yield

∂W (VM , t)

∂t=

12

−RW (VM , t)+R∫

∞

−∞

dV −

M

∫

∞

−∞

dvmW(

V −

M , t)

f (vm)×δ(

VM − ξ1V−

M − ξ2vm

)

,

(18)

where the delta function gives the constrain on energy and momentum

conservation in collision events. The first (second) term, on the right

hand side of Eq. (18), describes a tracer particle leaving (entering)

the velocity point VM at time t. Eq. (18) yields the linear Boltzmann

equation

∂W (VM , t)

∂t= −RW (V, t) +

R

ξ1

∫

∞

−∞

dvmW

(

VM − ξ2vm

ξ1

)

f (vm) .

(19)

In Eq. (19) the second term on the right hand side is a convolution

in the velocity variables, hence we will consider the problem in Fourier

space. Let W (k, t) be the Fourier transform of the velocity PDF

W (k, t) =∫

∞

−∞

W (VM , t) exp (ikVM) dVM , (20)

we call W (k, t) the tracer particle characteristic function. Using Eq.

(19), the equation of motion for W (k, t) is a finite difference equation

∂W (k, t)

∂t= −RW (k, t) + RW (kξ1, t) f (kξ2) , (21)

where f (k) is the Fourier transform of f(vm). In Appendix A the

solution of the equation of motion Eq. (21) is obtained by iterations

W (k, t) =∞∑

n=0

(Rt)n exp (−Rt)

n!eikVM (0)ξn

1 Πni=1f

(

kξn−i1 ξ2

)

, (22)

with the initial condition W (k, 0) = exp[ikVM (0)].

The solution Eq. (22) has a simple interpretation. The probability

that the tracer particle has collided n times with the bath particles is

13

given according to the Poisson law

Pn(t) =(Rt)n

n!exp (−Rt) , (23)

reflecting the assumption of uniform collision rate. Let Wn(VM) be the

PDF of the tracer particle conditioned that the particle experiences n

collision events. It can be shown that the Fourier transform of Wn(VM)

is

Wn(k) = eikVM (0)ξn

1 Πni=1f

(

kξn−i1 ξ2

)

. (24)

Thus Eq. (22) is a sum over the probability of having n collision events

in time interval (0, t) times the Fourier transform of the velocity PDF

after exactly n collision event

W (k, t) =∞∑

n=0

Pn(t)Wn(k). (25)

It follows immediately that the solution of the problem is

W (VM , t) =∞∑

n=0

Pn(t)Wn(VM), (26)

where Wn(VM) is the inverse Fourier transform of Wn(k) Eq. (24).

B. Equilibrium

In the long time limit, t → ∞ the tracer particle characteristic

function reaches an equilibrium

Weq(k) ≡ limt→∞

W (k, t). (27)

This equilibrium is obtained from Eq. (22). We notice that when Rt →

∞, Pn(t) = (Rt)n exp(−Rt)/n! is peaked in the vicinity of 〈n〉 = Rt

14

hence it is easy to see that

Weq (k) = limn→∞

Πni=1f

(

kξn−i1 ξ2

)

. (28)

In what follows we investigate properties of this equilibrium.

We will consider the weak collision limit ε → 0. This limit is impor-

tant since number of collisions needed for the tracer particle to reach

an equilibrium is very large. Hence in this case we may expect the

emergence of a general equilibrium concept which is not sensitive to

the precise details of the velocity PDF f(vm) of the bath particles.

Remark 1 According to Eq. (24), after a single collision event

the PDF of the tracer particle in Fourier space is W1(k) = f (kξ2)

provided that VM(0) = 0. After the second collision event W2(k) =

f (kξ1ξ2) f (kξ2) and after n collision events

Wn(k) = Πni=1f

(

kξn−i1 ξ2

)

. (29)

This process is described in Fig. 1, where we show Wn(k) for n =

1, 3, 10, 100, 1000. In this example we use a uniform distribution for

the bath particles velocity Eq. (44), with ε = 0.01, and T = 1. After

roughly 100 collision events the characteristic function Wn(k) reaches

a stationary state, which as we will show is well approximated by a

Gaussian (i.e., the Maxwell velocity PDF is obtained).

15

C. Maxwell Velocity Distribution

We consider the case where all moments of f (vm) are finite and that

the following behavior holds:

f (vm) =1

√

T/mq(vm/

√

T/m). (30)

q(x) is a non-negative normalized function.

The second moment of the bath particle velocity is

〈v2m〉 =

T

m

∫

∞

−∞

x2q(x)dx. (31)

Without loss of generality we set∫

∞

−∞x2q(x)dx = 1. The scaling be-

havior Eq. (30) and the assumption of finiteness of moments of the

PDF yields

〈v2nm 〉 =

(

T

m

)n

q2n, (32)

where the moments of q(x) are defined according to

q2n =∫

∞

−∞

x2nq(x)dx, (33)

and we assume that odd moments of q(x) are zero. Thus the small k

expansion of the characteristic function is

f (k) = 1 − Tk2

2m+ q4

(

T

m

)2 k4

4!+ O(k6). (34)

For simplicity we consider only the first three terms in the expansion

in Eq. (34).

We now obtain the velocity distribution of the tracer particle using

Eq. (28)

ln[

Weq (k)]

= limn→∞

n∑

i=1

ln[

f(

kξn−i1 ξ2

)]

. (35)

16

Inserting Eq. (34) in Eq. (35) we obtain

ln[

Weq (k)]

= − T

2mεk2 +

q4 − 3

4!

(

T

m

)2 2ε3

1 + ε2k4 + O(k6). (36)

When ε is small we find (using ε = m/M)

ln[

Weq (k)]

= −Tk2

2M+(

T

M

)2 q4 − 3

4!2εk4 + O

(

k6)

. (37)

It is important to see that the k4 term approaches zero when ε → 0.

Hence we find

limε→0

ln[

Weq(k)]

= −Tk2

2M, (38)

inverting to velocity space we obtain the Maxwell velocity PDF

limε→0

Weq (VM) =

√M√

2πTexp

(

−MV 2M

2T

)

. (39)

We see that the parameters q2n with n > 1 are the irrelevant parameters

of the problem, and hence the Maxwell distribution is stable in the sense

that it does not depend on the detailed shape of f(vm).

Remark 1 To complete the proof we will show that the k6, k8 and

higher order terms in Eq. (37) also approach zero when ε → 0. Let

κm,2n (κM,2n) be the 2n th cumulant of bath particle (tracer particle)

velocity. The cumulants describing the bath particle are related to the

moments q2n in the usual way κm,2 = T/m, κm,4 = (q4 − 1)(T/m)2 etc.

Then using Eq. (28) one can show that

κM,2n = g∞

2n (ε) κm,2n. (40)

From the scaling function Eq. (30) we have κm,2n = c2nT n/mn, where

c2n are dimensionless parameters which depend on f(vm), n = 1, 2, · · ·,

17

e.g c2 = 1, c4 = q4 − 1 etc. The parameters c2n for n > 1 are the

irrelevant parameters of the model in the limit of weak collisions. To

see this note that when ε → 0 we have

κM,2n = (T2/M)δn1. (41)

Thus, besides the second cumulant, all cumulants of the tracer particle

velocity distribution function are zero. As well known the cumulants

of the Gaussian PDF with zero mean are all zero besides second. Eq.

(41) shows that the tracer particle reached the Maxwell equilibrium.

1. Numerical Examples

We now investigate numerically exact solutions of the problem, and

compare these solutions to the stable equilibrium which becomes exact

when ε → 0. We investigate three types of bath particle velocity PDFs:

(i) The exponential

f (vm) =

√2m

2√

T2

exp

(

−√

2m|vm|√T2

)

, (42)

which yields

f(k) =1

1 + T2k2

2m

. (43)

(ii) The uniform PDF

f(vm) =

√

m12T2

if |vm| <√

3T2

m

0 otherwise

(44)

18

which yields

f(k) =sin

(√

3T2

mk)

√

3T2

mk

. (45)

(iii) The Gaussian PDF

f(k) = exp

(

−k2T2

2m

)

. (46)

The small k expansion of Eqs. (43,45,46) is f(k) ∼ 1−k2T2/(2m)+ · · ·,

indicating that the second moment of velocity of bath particles 〈v2m〉 is

identical for the three PDFs.

To obtain numerically exact solution of the problem we use Eq.

(28) with large though finite n. In all our numerical examples we used

M = 1 hence m = ε. Thus for example for the uniform velocity PDF

Eq. (45) we have

Weq (k) ' exp

n∑

i=1

ln

√

ε

3T2

sin(

k√

3T2

m

(

1−ε1+ε

)n−i2ε

1+ε

)

k(

1−ε1+ε

)n−i2ε

1+ε

. (47)

To obtain equilibrium we increase n for a fixed ε and temperature until

a stationary solution is obtained.

According to our analytical results the bath particle velocity PDFs

Eqs. (42,44,46), belong to the domain of attraction of the Maxwellian

equilibrium. In Fig. 2 we show Weq(k) obtained from numerical solu-

tion of the problem. The numerical solution exhibits an excellent agree-

ment with Maxwell’s equilibrium. Thus details of the precise shape of

velocity PDF of bath particles are unimportant, and as expected the

Maxwell distribution is stable. We note that the convergence rate to

equilibrium depends on the value of k. To obtain the results in Fig. 2

I used ε = 0.01, T2 = 2, M = 1 and n = 2000.

19

III. APPENDIX A

In this Appendix the solution of the equation of motion for W (k, t)

Eq. (21) is obtained, the initial condition is W (k, 0) = exp[ikVM(0)].

The inverse Fourier transform of this solution yields W (VM , t) with

initial condition W (VM , 0) = δ[VM − VM(0)].

Introduce the Laplace transform

W (k, s) =∫

∞

0W (k, t) exp (−st) dt. (48)

Using Eq. (21) we have

sW (k, s) − eikVM (0) = −RW (k, s) + RW (kξ1, s) f (kξ2) , (49)

this equation can be rearranged to give

W (k, s) =eikVM (0)

R + s+

R

R + sW (kξ1, s) f (kξ2) . (50)

This equation is solved using the following procedure. Replace k with

kξ1 in Eq. (50)

W (kξ1, s) =eikξ1VM (0)

R + s+

R

R + sW(

kξ21 , s

)

f (kξ2ξ1) . (51)

Eq. (51) may be used to eliminate W (kξ1, s) from Eq. (50), yielding

W (k, s) =eikVM (0)

R + s+

Reikξ1VM (0)

(R + s)2f (kξ2) +

R2

(R + s)2 W(

k2ξ1, s)

f (kξ2ξ1) f (kξ2) . (52)

Replacing k with kξ21 in Eq. (50)

W (kξ21, s) =

eikξ2

1VM (0)

R + s+

R

R + sW(

kξ31 , s

)

f(

kξ2ξ21

)

. (53)

20

Inserting Eq. (53) in Eq. (52) and rearranging

W (k, s) =eikVM (0)

R + s+

Reikξ1VM (0)

(R + s)2 f (kξ2) +

R2eikξ2

1VM (0)

(R + s)3f (kξ2ξ1) f (kξ2)+

(

R

R + s

)3

W(

kξ31, s

)

f(

kξ2ξ21

)

f (kξ2ξ1) f (kξ2) .

(54)

Continuing this procedure yields

W (k, s) =eikVM (0)

R + s+

∞∑

n=1

Rn

(R + s)n+1 eikξn

1VM (0)Πn

i=1f(

kξn−i1 ξ2

)

. (55)

Inverting to the time domain, using the inverse Laplace s → t transform

yields Eq. (22). The solution Eq. (22) may be verified by substitution

in Eq. (21).

21

-30 -20 -10 0 10 20 30

0

0.2

0.4

0.6

0.8

1

Wn(k

)

-20 -10 0 10 20

0

-20 -10 0 10 20

k

0

0.5

1

Wn(k

)

-4 -2 0 2 4

k

0

0.5

1

FIG. 1: We show the dynamics of the collision process: the tracer parti-

cle characteristic function, conditioned that exactly n collision events have

occurred, Wn(k) versus k. The velocity PDF of the bath particle is uni-

form and ε = 0.01. We show n = 1 (top left), n = 3 (top right), n = 10

(bottom left) and n = 100 n = 1000 (bottom right). For the latter case

we have W100(k) ' W1000(k), hence the process has roughly converged after

100 collision events. The equilibrium is well approximated with a Gaussian

characteristic function indicating that a Maxwell–Boltzmann equilibrium is

obtained.

22

-2 0 2k

0

0.5

1

Weq

(k)

FIG. 2: The equilibrium characteristic function of the tracer particle, Weq(k)

versus k. We consider three types of bath particles velocity PDFs (i) ex-

ponential (squares), (ii) uniform (circles), and (iii) Gaussian (diamonds).

The velocity distribution of the tracer particle M is well approximated by

Maxwell’s distribution plotted as the solid curve Weq(k) = exp(

−|k|2)

. For

the numerical results I used: M = 1, T2 = 2, n = 2000, and ε = 0.01.