Fluctuations and Brownian Motion
Brownian motion in water Brownian motion of DNA
2  fluorescent spheres in water (left) and DNA solution (right)
(Movie Courtesy Professor Eric Weeks, Emory University: http://www.seas.harvard.edu/weitzlab/research/brownian.html)
Copyright (c) Stuart Lindsay 2008
Fluctuations
Simulated distribution of speeds in a population of 23 atoms of an
ideal gas: V=1.25·103 nm3 (50·50·50nm); T=300K.
One set of collisions
5·105 set of collisions
Calculating Fluctuations
E

 Er exp Er
r
Z
 ln Z
 ln Z Z
1



Z 
Z
 ln Z







E
exp


E
r
r 
 
r

Taking the second derivative of dlnZ/d:
 2 ln Z
  1 Z    Z  1  Z 1  2 Z

  



  .
2
2

  Z    Z   Z   Z 
  Er exp  Er
2
2
 r
1  Z  1  Z
  2   
 
2
Z
Z    Z 


  Er2 exp  Er

r
 E2  E
 
Z


2
2
This is just the mean square thermal average of E:
E
2
 E  E
E
2

2
 E
2
 E 2 E E  E
2
1
1 
2  E j
  E Pj   E j e

Q j
Q 
j
2
j
 Eje
j
 E
1 
 ln Q
Q E   

 E

Q 


 E j
2

 E
2
 kT
E
2
2
 kT
 E
T
2
 E
T
 E
2
 k BT 2CV
For an ideal gas:
3
E  Nk BT
2
E
E
2

3
CV  Nk
2
kT 2CV
1

E
N
The relative size of energy fluctuations scales as
1
N
  ln Z 

N  kT 
  V ,T
In an open system:
N
N
2
  N PN , j
2
N ,j
1

Z
2
N
N,j
 N
2  E N , j
e
 N
2
e
N
2
kT 
  E N , j  N

N
e
e


Z  N , j
 N
 N
kT 
 ln Z
Z N   kT

 N
 kT
 N
Z 



N
2
 N 

 kT 
  T ,V
2
  
 V   p 

    

 N T ,V
 N   V  N ,T
2
From thermodynamics:
N
2
kT 2

N 
V
1  V 
    
V  p  N ,T
For an ideal gas:
N 2  N
Isothermal compressibility

1
p
N
N
2

1
N
• The result just obtained for energy holds for all quantities
(extensive quantities) that, like energy, grow with N (T, E,
P, V, S).
• In general the root-mean-square value of the fluctuations
relative to the mean value of a quantity is given by
X RMS
1

X
N
N
10
Relative fluctuation
31.6%
1000
1026
3.2%
10-13
Copyright (c) Stuart Lindsay 2008
Brownian motion
Langevin equation:
dv
m
 v  F ( t )
dt
Stoke’s force = friction exerted on the particle by the fluid.
For small velocities, it is proportional to the velocity v.
α (friction coefficient )
For a sphere of radius a in a medium of viscosity η: =6πηa
F(t) is a random force representing the constant molecular
bombardment exerted by the surrounding fluid:
F (t )  0
Average is zero!
F( t )F( t  )  F ( t  t  )
Finite only over duration of single
“effective” collision
F(t) is independent of the velocity of the particle (v) and varies
extremely rapidly compared to the variations in v.
There is no correlation between F(t) and F(t+Δt) even though Δt
is expected to be very small.
Multiplying both sides of the Langevin eqn. by x and using
v  x
d
2
mxx  m  ( xx )  x   xx  xF (t )
 dt

Re-arranging and taking thermal averages:
d
m
xx  m x 2   xx  xF (t )
dt
1
1
2
m x  k BT
2
2
=0
so
k BT 
d
xx 

xx
dt
m
m
substituting
xx  A expBt   C
yields
C
k BT

B
To find A, note
1 d 2
xx 
x
2 dt

m
with <x2>=0 at t=0:
A
k BT

1 d 2
k BT 
 t  
x 
1  exp  

2 dt
 
 m 
substitute

t
m

dt  
m

d
Integrate with <x2>=0 at t=0:
x2 
2k B T  m 
t  
t

1

exp




  
m 
x2 
Long time solution:
x
2

2 k BT t

2k B T  m 
t  
t

1

exp




  
m 
t 
k BT t

3a
Mean square displacement
increases with t!
Copyright (c) Stuart Lindsay 2008
m

• Now we see why the sphere in a viscous DNA
solution moves more slowly!
(Movie Courtesy Professor Eric Weeks, Emory University:
http://www.seas.harvard.edu/weitzlab/research/brownian.html)
2  fluorescent spheres in water (left) and DNA solution (right)
x
Copyright (c) Stuart Lindsay 2008
2

2k BTt

k BTt

3a
The Diffusion Equation
Flow of solute or heat under action of random forces:
J is flux per second per unit area
 A.J ( x  x)t  AJ ( x)t 
C  

Ax

x , t  0
C ( x, t )
J ( x, t )

t
x
The flux is the change in concentration across a surface
multiplied by the speed with which particles arrive:
x
C x 2
J  C

t
x t
Using:
x 2
t
Diffusion
coefficient
D
C
J  D
x
m 2  s 1
Fick’s first law
From which
C ( x, t )
 C ( x, t )
D
t
x 2
2
In 3D:
Fick’s second law
Diffusion Equation
C( r ,t )
 2C( r ,t )
2
D

D

C( r ,t )
2
t
r
The solution in 1-D for a solute initially added as a point source is:
C ( x, t ) 
 x2 

exp  
Dt
 4 Dt 
A
A=1
As t→∞, the distribution
becomes uniform, the point of
‘half-maximum concentration’
x½ advancing
with time
according to:
x1 / 2  2Dt
Einstein-Smoluchowski Relation
Compare
x1 / 2  2Dt
to
x2 
2k BTt


k BTt
3a
k BT
D
6a
A surprising relation between thermal motion and driven motion:
the diffusion constant is the ratio of kT to the friction constant!
The mobility is defined as the
inverse of the friction coefficient:
Einstein-Smoluchowski relation
F
v
 F
6a
D  k BT
A fundamental relation between energy dissipation and diffusion
Viscosity is not an equilibrium property,
because viscous forces are generated only
by movement that transfers energy from
one part of the system to another.
Ex. Motion of a spherical large particle with respect to a large
number of small molecules.
F  6av
Stokes’ law
speed
Force
radius
viscosity
The introduction of bulk viscosity requires that the small
molecules rearrange themselves on very short times compared
with the time scale of the motion of the sphere.
Diffusion, fluctuations and chemical reactions
1.
If the reactants are not already mixed they need to come
together by diffusion.
2.
Once together, they need to be jiggled by thermal
fluctuations into a “transition state”
3.
If the free energy of the products is lower than that of the
reactants the products lose heat to the environment to form
stable end products.
Haber process for Ammonia
The transition state
 G 
k BT
k1 
exp  
h
 k BT



Eyring transition
state theory
Copyright (c) Stuart Lindsay 2008
The entropy adds a temperature-dependent component to the
energy differences that determine the final equilibrium state of
a system.
Kramers’ Theory of Chemical Reactions
• Noise driven escape:
Thermal fluctuations allow the particles in the well to rapidly
equilibrate with the surroundings.
The motion of the particles over the barrier is much slower.
The Kramers Model
Microscopic description
of the prefactor in terms
of potential curvature
 Eb 
ab

k 
exp 
2
 k BT 

Reaction coordinate
a2,b ,c
1  2U

m x 2
x  a ,b ,c

Unimolecular reactions
• Not very common, but can be described as a one step process
by the Kramers theory
k+
E2
E1
k-
d E1 
 k  E1   k  E2 
dt
• Example is isomerism of isonitrile
H3C
N
C
H3C
C
N
Copyright (c) Stuart Lindsay 2008
Thermodynamic Potentials for Nanosystems
• The Gibbs free energy in a multicomponent system is:
G  E  PV  TS  N
This equation contains no reference to system size (all
quantities are extensive: doubling the volume of a system,
doubles its free energy)
• Nanosystems at equilibrium derive their “stable” size from
surface and interface effects which are not extensive.
Ex. Self-assembly originates from a competition between bulk
and surface energies of the phases that self-assemble (stability
of colloidal systems).
Copyright (c) Stuart Lindsay 2008
• Hill has generalized thermodynamics to include a “subdivision
potential”
 dE 
d
dE  TdS  Vdp  dN  
 d 
Subdivision potential

number of independent parts of the system
The simplest approach add surface terms to the free energy.
Modeling nanosystems explicitly:
Molecular Dynamics
 2U (r )
Fij  
rir
2
rij
1
1
2
miVi , x , y , z  k BT
2
2
Limited to s timescales by required time step/processor speed.
Interatomic vibrations are rapid (≈1013 Hz) so that the time step in
the calculations must be much smaller (≈10-15 s).
Copyright (c) Stuart Lindsay 2008
Limitations of MD
• The number of calculations scales as N2 (it depends on the
problem).
If all possible pair interactions need to be considered , the number
of interactions would scale like 2N.
• Timescale is limited to ns or s at best. This is fine for harmonic
vibrations but nanosystems like enzymes work on ms timescales.
Ex. Viscously damped motion:
6a


m
1
Ex. Activated transitions over barriers:
Eb
 ab
k 
exp 
2
k BT


Example of MD
Movie link:
13mer.avi
Cyclic Sugar molecule being pulled over DNA
Qamar et al. Biophys J. (2008) 94, 1233
Thermal Fluctuations and Quantum
Mechanics
• The density matrix formulation of quantum mechanics allows
description of the time evolution of a system subject to time
dependent forces.

ˆ
i
 Ĥ , ˆ
t

 nm   n q   a*n aq
Copyright (c) Stuart Lindsay 2008
• Thermal fluctuations can be represented by the “random phase
approximation” – quantum interference effects are destroyed.
Uncertainty Principle and fluctuations
• Interaction energy, , will decay exponentially at large
distances
• When  kT

 4 10 14 eV  s
t 
 
 1 ps
E 
0.025eV
The time characteristic of a molecular vibration.
Copyright (c) Stuart Lindsay 2008