i. brownian motion - biubarkaie/chapter7.pdf · i. brownian motion the dynamics of small particles...
Post on 28-Oct-2020
6 Views
Preview:
TRANSCRIPT
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.
top related