LANGEVIN EQUATION
Let us consider a particle moving in a medium which continually bombards it. The net
effect may be modeled by giving the particle additional vibratory degrees of freedom over
and above its classical motion. Let the motion of the particle be described by (p(t), x(t))
and the additional degrees of freedom by (pi (t), qi (t)). The Hamiltonian is given by :
„
«2
X p2
X
1
γ
p2
i
i
+
+
ωi2 qi − 2 x
H=
2
2
2
ωi
i
The Hamiltonian equations of motion are
ṗ
=
q˙i
=
p˙i
=
X
∂H
−
=
γi
∂x
„
«
γi
qi − 2 x
ωi
∂H
= pi
∂pi
«
„
γ
∂H
i
= ωi2 qi − 2 x
∂qi
ωi
. – p.1/10
LANGEVIN EQUATION
From the last two equations :
q̈i + ωi2 qi = γi x
From which noting that ẋ = p (m = 1) we have a solution, using
qi (t) − (γi /ωi2 )x(t) = q̃i (t) :
q̃i (t) = q̃i (0) cos(ωi t) +
−
Z
0
t
pi (0)
sin(ωi t) + . . .
ωi
γi
cos[ωi (t − s)]p(s)ds
ωi2
Substituting :
ṗ(t) =
X
γi

ff
pi (0)
q̃i (0) cos(ωi t) +
sin(ωi t) + . . .
ωi
ff
Z t X 2
γi
−
cos[ωi (t − s)] p(s)ds
2
ω
0
i
. – p.2/10
LANGEVIN EQUATION
Thus,
ṗ = −
(1)
Z
t
0
K(t − s)ds + f (t)
Let us now assume that the extra degrees of freedom are random and that both pi (0) and
q̃i (0) have Maxwellian distribution which are independent of one another.
hpi (0)i
=
hpi (0)pj (0)i
=
0 = hq̃i (0)i
kT δij = ωi2 hq̃i (0)q̃j (0)i
On substitution in the expression for f (t) we get :
hf (t)i
=
X
hpi (0)i
sin(ωi t)
γi hq̃i i cos(ωi t) +
ωi
ff
=0
. – p.3/10
LANGEVIN EQUATION
hf (t)f (s)i
=
X
γi γj

hq̃i (0)q̃j (0)i cos(ωi t) cos(ωj s)+ . . .
1
hpi (0)pj (0)i sin(ωi t) sin(ωj s)
+
ωi ωj
=
=
ff
+ ...
+cross terms which vanish
X γ2
i
kT
{cos(ωi t) cos(ωi s) + sin(ωi t) sin(ωi s)}
ωi2
X γ2
i
cos[ωi (t − s)] = kT K(t − s)
kT
ωi2
The above equation (??) is called the Langevin equation. f (t) defines a ‘noise bath’ which
has the kind of correlation funstions shown above. K(t − s) is called the ‘memory function’
and f (t) called ‘colored noise’.
. – p.4/10
DIGRESSION
Let a function f (t) have a Gaussian distribution. That is,
» Z
P [f (t)] = A exp −
∞
dtf 2 (t)/σ 2
−∞
–
where,
A−1 =
Z
» Z
D[f (t)] exp −
∞
dtf 2 (t)/σ 2
−∞
–
what is the meaning of these functional integrals. To understand this, define
f (t) =
X
eiωt f (ω)
ω
then
"
P [f (t)] = A exp −
X
ω
|f (ω)|2 /σ 2
#
and
A−1
YZ n
˜o
ˆ
2
2
=
df (ω) exp −|f (ω)| /σ
ω
. – p.5/10
DIGRESSION
Consequence of above becomes :
hf (ω1 )f (ω2 ) . . . f (ω2n+1 )i
=
0
hf (ω1 )f (ω2 )i
=
hf (ω1 )f (ω2 ) . . . f (ω2n )i
=
σ 2 δ(ω1 + ω2 )
X
2n
δ(ω1 + ω2 )δ(ω3 + ω4 ) . . .
σ
pairwise
Thus
hf (t1 )f (t2 ) . . . f (t2n+1 )i
=
0
hf (t1 )f (t2 )i
=
hf (t1 )f (t2 ) . . . f (t2n )i
=
σ 2 δ(t1 − t2 )
X
2n
δ(t1 − t2 )δ(t3 − t4 ) . . .
σ
pairwise
There are (2n!)/(n!2n ) pairwise arrangements in the sum.
. – p.6/10
LANGEVIN EQUATION
For our case, let the number of extra degrees of freedom become a continuous variable
(infinitely many),
Z ∞ 2
X γ2
γ
i
cos[ω(t − s)]p(ω)dω
cos[ω
(t
−
s)]
=
K(t − s) =
i
2
ωi2
ω
0
Assume p(ω) = Γ(ω 2 /γ 2 ), then :
K(t − s) = Γ
Z
0
∞
dω cos[ω(t − s)] = lim Γ
ω→∞
sin[ω(t − s)]
= Γδ(t − s)
t−s
Substituting this in the general Langevin equation, we get :
ṗ(t) + Γp(t) = f (t)
hf (t)i = 0; hf (t1 )f (t2 )i = Dδ(t1 − t2 ); D = kT Γ
This is linear Langevin equation with ‘white noise’.
. – p.7/10
EQUILIBRIUM
The solution of he linear Langevin equation follows easily :
p(t) = e
−Γt
Z
t
f (s)eΓs ds = p(0)e−Γt
0
It follows immediately that :
hp(t)i = p(0)e−Γt → 0(t → ∞)
and, if t1 > t2
hp(t1 )p(t2 )i
=
=
hp2 (t)i
→
2
p (0)e
−Γ(t1 +t2 )
+ De
»
−(t1 +t2 )Γ
–
Z
t1
0
Z
t2
0
′
dsds′ eΓ(s+s ) δ(s − s′ )
D
D −Γ|t1 −t2 |
e
+ p2 (0) −
eΓ(t1 +t2 )
2Γ
2Γ
„ «2n
D
2n!
D
; hp2n (t)i →
as t → ∞
2Γ
2Γ
2n n!
. – p.8/10
EQUILIBRIUM
This is exactly the property of a Gaussian or Maxwellian distribution of the momentum of
the particle. We have again shown that in dynamical equilibrium the probability density of
a particle in contact with a heat bath is Maxwellian.
lim P (p) = √
t→∞
1
2πkT
exp{−p2 /2kT }
. – p.9/10
DIFFUSION
Again starting from the linear Langevin equation with white noise : since
R
∆x = x(t) − x(0) = 0t p(s)ds :
lim h∆xi
t→∞
h(∆x)2 i
=
=
=
lim h(∆x)2 i
t→∞
=
0
fiZ
„
t
p(s)ds
0
Z
t
0
−Γt «2
1−e
D
D
t
Γ
p(s′ )ds′
fl
[hp2 (0)i − D] + (D/Γ)t −
D
(1 − e−Γt )
2
Γ
This is characteristic of diffusion where the venture distance defined by :
ˆ
˜1/2
d = h(∆x)2 i
∼ t1/2
This is different from a ballistic particle forwhich d = ut + at2 /2 ∼ t2 for large t
. – p.10/10