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Lecture: “Thermal Forces and Brownian Motion” by Ju Li. Given August 11, 2006 during the GEM4 session at MIT in Cambridge, MA. Please use the following citation format: Li, Ju. “Thermal Forces and Brownian Motion.” Lecture, GEM4 session at MIT, Cambridge, MA, August 11, 2006.
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Thermal Forces and Brownian Motion
Ju Li
GEM4 Summer School 2006
Cell and Molecular Mechanics in BioMedicine
August 7–18, 2006, MIT, Cambridge, MA, USA
Outline
• Meaning of the Central Limit Theorem
• Diffusion vs Langevin equation descriptions
(average vs individual)
• Diffusion coefficient and
fluctuation-dissipation theorem
2
Central Limit Theorem Y = X1 + X2 + … + XN
X1, X2, …, XN are random variables
E[Y] = E[X1] + E[X2] + … + E[XN]
If X1, X2, …, XN are independent random variables:
var[Y] = var[X1] + var[X2] + … + var[XN]
Note: var[X] = σ2X ≡ E[ (X-E[X])2 ]
3
If X1, X2, …, XN are independent random variables sampled from the same
distribution:
E[Y] = NE[X] var[Y] = N var[X1] = Nσ2X
Average of the sum: y ≡ Y/N
E[y] = E[X], var[y] = var[Y]/N2 = σ2X / N
Law of large numbers: as N gets large, the average of the sum becomes more and more deterministic, with variance σ2X / N.
4
X1, X2, …, XN may be sampled from
Probability
density
Probability
density
Probability
density
-1
2
X
X
X
5
We know the probability distribution of Y
is shifting (NE[X]), as well as getting fat (Nσ2X). But how about its shape ?
The central limit theorem says that irrespective of the shape of X,
Probability
density
Nσ2X
NE[X])
Y
6
Why Gaussian ?
large N
ρ (Y ) →
 (Y − NE[ X ]) 
exp 

2
2
2N σ X
2π N σ X


1
2
Gaussian is special (Maxwellian velocity distribution, etc). While proof is involved,
here we note that Gaussian is an invariant
shape (attractor in shape space) in the mathematical operation of convolution. 7
Diffusion Equation in 1D
J
2
∂ t ρ = − ∂ x ( − D∂ x ρ ) = D
∂ x ρ
Random walker view of diffusion: imagine
(a) We release the walker at x=0 at t=0,
(b) Walker makes a move of ±a, with equal
probability, every ∆t=1/ν from then on.
Mathematically, we say ρ(x,t=0)=δ(x).
N=
t
=ν t independent random steps
∆t
Then, x (t ) = ∆x1 + ∆x2 + ... + ∆xt
/ ∆t
8
When N=νt >>1,
the central limit theorem applies:
E[x(t)] = 0, var[x(t)] = νt var[∆x] = νta2
So we can directly write down ρ ( x (t )) as
2
 x 
1
ρ G ( x, t ) =
exp 
2 
2
2πν a t
 2ν a t 
It is the probability of finding the walker at x
at time t, knowing he was at 0 at time 0.
9
By plugging in, we can directly verify
ρ G ( x, t ) satisfies
2
∂ t ρ = D∂ x ρ , ρ ( x,0) = δ ( x).
2
va
with macroscopic D identified as
.
2
ρ G ( x, t ) =
 x

1
exp 

2π (2 Dt)
 2(2 Dt) 
2
is called Green's function solution to diffusion equation.
10
Brownian Motion
Courtesy of Microscopy-UK. Used with permission.
Fat droplets suspended in milk (from Dave Walker).
The droplets range in size from about 0.5 to 3 µm.
11
viscous oil
Stokes' law:
F=-6πrηv=-λv
v
mv = F = −λ v, v (t = 0) = v0
→ v (t ) = v0e
λ
− t
m
Einstein's Explanation of Brownian Motion
Also, equi-partition theorem:
mv
2
2
k BT
=
2
In addition to dissipative force, there must be another, stimulative force.
12
mv = Fdissipative + Fstimulative/fluctuation = −λ v + Ffluc (t
)
Ffluc (t ) = 0
Ffluc (t ) Ffluc (t ′) = b(t − t ′)
If b(t − t ′) = Bδ ( t − t ′) : white noise
Exact Green's function solution of v (t ):
λ
− (t −t ′ )
1 t
m
′
′
v (t ) =
dt
F
t
e
(
)
fluc
∫
−∞
m
13
v (t )v (t
) 1
= 2
m
1
= 2
m
∫
t
dt ′Ffluc (t ′)e
−∞
∫
t
−∞
1
= 2
m
dt ′e
∫
t
−∞
1
= 2
m
λ
− (t − t ′ )
m
∫
λ
− (t −t ′ )
m
t
∫
t
−∞
dt′e
λ
− (t − t′ )
m
−∞
dt ′e
∫
t
−∞
λ
− (t −t ′ )
m
∫
t
−∞
dt ′e
λ
− (t − t ′ )
m
dt′Ffluc (t′)e
dt′e
Ffluc (t ′) Ffluc (t′)
λ
− (t − t′
)
m
Bδ ( t ′ − t′)
H (t − t ′)e
B
e
=
2mλ
−
λ
m
λ
− (t − t′ )
m
λ
− (t −t ′ )
m
B
H ( x ) is Heaviside step function:
t −t
1
H ( x) = 
0
if x > 0
if x ≤ 0
14
In particular:
B
v (t )v (t ) =
2mλ
However, from equilibrium statistical mechanics:
equi-partition theorem:
m v (t )v (t ) = k BT B
→
= k B
T
2λ
The ratio between square of stimulative force and dissipative force is fixed, ∝ T
15
k
T
B
v (t )v (t ) =
e
m
−
λ
m
t −t
Previously, from the Gaussian solution to
2
∂ tρ = D∂ x ρ , ρ ( x,0) = δ ( x) :
2

 x
1
ρ G ( x, t ) =
exp 

2π (2 Dt)
 2(2 Dt) 
we know if the particle is released at x = 0 at t = 0 :
x (t ) x (t ) = 2Dt
t
x (t ) = 0 + ∫ dt ′v (t ′),
0
x (t ) = v (t ) 16
d
x (t ) x (t ) = 2 x (t ) x (t ) = 2 x (t )v (t )
dt
d
= (2Dt) = 2D
dt
D = x (t )v (t ) =
(∫ dt′v(t′)) v(t)
t
0
t
= ∫ dt ′ v (t ′
)v (t )
0
t
= ∫ dt ′ v (t ′
)v (0)
0
Velocity auto-correlation function: g (t ) ≡ v (t )v (0)
17
Actually, the onset of macroscopic diffusion (∂ t ρ = D∂ ρ ) is only valid only when 2
x
t intrinsic timescale of g (t ) ∝
m
λ
(Same as central limit theorem in random walk)
So the correct formula is
∞
D = ∫ dt ′ v (t ′)v (0)
0
The above is one of the fluctuation-dissipation theorems.
18
∞
1
Thermal conductivity: κ =
J q (t ) J q (0) dt
2 ∫0
Ωk BT
∞
1
Electrical conductivity: σ =
J (t ) J (0) dt
∫
Ωk BT 0
Ω ∞
Shear viscosity: η =
τ xy (t )τ xy (0) dt
∫
k BT 0
Fluctuation-dissipation theorem (Green-Kubo
formula) is one of the most elegant and
significant results of statistical mechanics. It
relates transport properties (system behavior if
linearly perturbed from equilibrium) to the
time-correlation of equilibrium fluctuations.
19
Coming back to diffusion (mass transport):
λ −
t −t
k
T
B
v (t )v (t ) =
e m
m
∞
k BT
So D = ∫ dt ′ v (t ′)v (0) =
.
0
1
λ
λ
is actually the mobility of the particle, when
driven by external (non-thermal) force.
D
= k BT is called the Einstein relation,
1/λ
first derived in 1905.
20
References
Kubo, Toda & Hashitume,
Statistical Physics II: Nonequilibrium Statistical Mechanics (Springer-Verlag, New York, 1992).
Zwanzig, Nonequilibrium Statistical Mechanics
(Oxford University Press, Oxford, 2001).
van Kampen, Stochastic processes in physics and chemistry, rev. and enl. ed. (North-Holland, Amsterdam, 1992).
Reichl, A modern course in statistical physics
(Wiley, New York, 1998).
21