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4. Brownian particles
LAST TIME
Started to discuss Fokker-Planck equations on the form
J=Probabiliy current
A(x)P(x,t)=drift term or transport term
diffusion term
Linear
Ornstein-Uhlenbeck process
obtained from jump moments
Today: Examples
MESOSCOPIC PARTICLE IN STATIONARY FLUID
"infinitesimal" time
mean time for collisions
velocity relaxation time
(see later)
time of interest
f=external force on particle
mobility
v= velocity
jump moments:
FOKKER-PLANCK EQUATION
BROWNIAN MOTION
This is just the diffusion equation, hence require
Fokker-Planck equation does not admit a stationary (steady) state
but can find
Gaussian processes like the boxed expression are also called
Wiener processes
PARTICLE IN FIELD OF GRAVITY
We now have
.
Presence of field changes a2 little
Why is this assumption reasonable?
The resulting Fokker-Planck equation is
No stationary solution for
.
We can solve time dependent equation:
Go to moving frame
Left with diffusion equation for !
REFLECTING BOUNDARY
Steady state exists if
!
Vessel with diffusing particle has a bottom.
Require that probability current vanishes in steady state
with solution
At equilibrium must have
Get Einstein relation
For time dependent problem apply zero current condition only at x=0!
RAYLEIGH PARTICLE
Consider fine time scale:
time between collisions
time for relaxation of velocity
velocity v independent variable
Macroscopic law
E.g. for spherical Stokes particle
r=radius,
viscosity
Jump moments
Assume
No external force!
Get Fokker Planck-equation
Stationary state (J(v)=0!)
with solution
Maxwell-Boltzmann distribution:
Comparing expressions we find
RAYLEIGH EQUATION
Equation is linear and describes an Ornstein-Uhlenbeck process. Solution:
velocity relaxation time
We have if
Can also calculate equilibrium velocity-velocity correlation
We find
GENERALIZE TO ARBITRARY FORCE
Overdamped motion in force field.
Macroscopic equation of motion Aristotelian:
not Newtonian.
Aristotle ok if
if
Substituting
we find
for the boxed expression to be valid
If condition
not satisfied both v and x
independent variables!
Jump moments
Fokker-Planck equation:
KRAMERS EQUATION
SUMMARY
Click here for 5. Several variables, SIR-model
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Birger Bergersen
1998-10-03