APM 530 - Lecture 10

29 Pages·2010·1.52 MB·English

by  Jay Taylor

LangevinandBrownianDynamics Overview
Langevin Dynamics of a Single Particle
We consider a spherical particle of radius r immersed in a viscous fluid and
suppose that the dynamics of the particle depend on two forces:
A drag force arising from friction between the particle and the
viscous fluid:
F = −γv
drag
where v is the velocity of the particle and γ is a friction constant.
For a spherical particle, Stokes’ law gives γ = 6πrη, where η is the
viscosity.
A random force R(t) arising from the collisions of the solvent
molecules with the particle. This is a phenomenological model in the
sense that the solvent molecules are not modeled explicitly.
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LangevinandBrownianDynamics Overview
To make further progress, we need to specify R(t). Our first assumption is
that on average the solvent fluctuations are non-directional:
E(cid:2)R(t)(cid:3) = 0.
Our second assumption concerns the covariance function of the solvent
fluctuations:
Cov(cid:0)R(t),R(t(cid:48))(cid:1) = κ2δ(t −t(cid:48))I,
where κ is a parameter and I is the 3×3 identity matrix. This is
motivated by two assumptions:
Solvent fluctuations are isotropic and therefore uncorrelated along
orthogonal directions.
The particle is much heavier than the individual solvent molecules
and so R(t) fluctuates much more rapidly than v.
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LangevinandBrownianDynamics Overview
Together, these assumptions imply that R(t) is a white noise process on
R3, i.e.,
R(t) = κW˙
t
where W is a three-dimensional Brownian motion. Substituting the drag
t
force and the white noise process into Newton’s laws of motions leads to
the following Langevin equation:
mv˙ = −γv+κW˙ ,
t
which can be rewritten as a SDE:
(cid:16)γ (cid:17) κ
dv = − v dt + dW
t t t
m m
≡ −ζv +σdW .
t t
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LangevinandBrownianDynamics Overview
Furthermore, if we write
(cid:0) (1) (2) (3)(cid:1)
v = v ,v ,v
t t t t
(cid:0) (1) (2) (3)(cid:1)
W = W ,W ,W ,
t t t
then each component of v satisfies a SDE of the form
(i) (i) (i)
dv = −ζv dt +σdW ,
t t t
which we recognize as the Ornstein-Uhlenbeck equation. From the
previous lecture, we know that the solution is a diffusion process given
by the stochastic integral
(cid:90) t
v = e−ζtv +σ e−ζ(t−s)dW .
t 0 t
0
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LangevinandBrownianDynamics Overview
Furthermore, if the initial velocity v is prescribed, then the distribution of
0
v at time t is a three-dimensional Gaussian distribution with the following
t
mean vector and covariance matrix:
m(t) = v e−ζt
0
σ2 (cid:16) (cid:17)
Σ(t) = 1−e−2ζt I.
2ζ
In particular, notice that as t → ∞, the limits
m(t) → 0 and Σ(t) → (σ2/2ζ)I,
are independent of the initial velocity. Since limits of sequences of
Gaussian variables are also Gaussian, it follows that the stationary
distribution of the velocity of the particle is just N(0,(σ2/2ζ)I).
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LangevinandBrownianDynamics Overview
This last result can be used to identify κ. If the solvent is held at constant
temperature T, then in the limit t → ∞, the distribution of v will tend to
t
the Maxwellian distribution N(0,(k T/m)I). Thus
B
σ2 k T
B
=
2ζ m
which along with ζ = γ/m implies that
2ζk T 2γk T
σ2 = B = B
m m2
κ2 = m2σ2 = 2γk T.
B
This is an example of the fluctuation-dissipation theorem: the energy
imparted to the particle by solvent fluctuations is on average balanced by
the energy lost to friction.
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LangevinandBrownianDynamics Overview
We can also solve for the trajectory of the particle by integrating the
velocity process:
(cid:90) t
X = X + v ds
t 0 s
0
(cid:90) t(cid:18) (cid:90) s (cid:19)
= X + e−ζsv +σ e−ζ(s−u)dW ds
0 0 u
0 0
1 (cid:16) (cid:17) (cid:90) t (cid:90) t
= X + 1−e−ζt v +σ eζudW e−ζsds
0 0 u
ζ
0 u
1 (cid:16) (cid:17) σ (cid:90) t(cid:16) (cid:17)
= X + 1−e−ζt v + 1−e−ζ(t−u) dW ,
0 0 u
ζ ζ
0
where we have interchanged the order of integration in passing to the
third line.
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LangevinandBrownianDynamics Overview
This shows that if the initial position X = x is fixed, then X is Gaussian
0 0 t
with mean vector and covariance matrix
1 (cid:16) (cid:17)
m(t) = x + 1−e−ζt v
0 0
ζ
(cid:40) (cid:41)
(cid:18)σ(cid:19)2(cid:90) t(cid:16) (cid:17)2
Σ = 1−e−ζ(t−s) ds I
t
ζ
0
σ2 (cid:104) (cid:105)
= 2ζt −3+4e−ζt −e−2ζt I
2ζ3
where I is the 3×3 identity matrix.
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LangevinandBrownianDynamics Overview
This last result can be used to calculate the mean squared displacement of
the particle in time t:
(cid:104) (cid:105) (cid:18)v (cid:19)2(cid:16) (cid:17)2
E (X −x )2 = 0 1−e−ζt +
t 0
ζ
3σ2 (cid:104) (cid:105)
2ζt −3+4e−ζt −e−2ζt .
2ζ3
When t is small, this is approximately
(cid:104) (cid:105)
E (X −x )2 = v2t2+O(t3)
t 0 0
which shows that the particle moves linearly over very short time intervals.
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LangevinandBrownianDynamics Overview
In contrast, for large t >> ζ−1, the mean squared displacement is
approximately
(cid:104) (cid:105) 3σ2 3k T
E (X −x )2 = t = B t
t 0 ζ2 γ
which shows that the particle moves diffusively over long time intervals,
with diffusion coefficient
k T k T
B B
D = = (Einstein-Stokes relation).
γ 6πrη
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