Boltzmann transport equation and H-theorem
Aim of kinetic theory: find the distribution function f ( r , p, t )
for a given form of particle-particle interaction
Special case f ( r , p, t )
for t  
equilibrium distribution
equilibrium thermodynamics
Find the equation of motion for the distribution function
Consider a collisionless free flow (no force) in x direction for time t
x 1 x2
X’1 X’2
  x1,2  vx t
x1,2
x2  x1  x2  x1
x
# of particles in (x’,x’+dx’) with (px,px+dpx) # of particles in (x,x+dx) with (px,px+dpx)
flowing into dx’dpx
leaving phase space element dxdpx
px
f ( x   t , px , t   t ) dxdpx  f ( x, px , t ) dxdpx
m
dx’=dx
dx’dpx=dx dpx
vx
px
f ( x   t , p x , t   t )  f ( x, p x , t )
m
Through linear expansion
f px
f
f ( x, p x , t ) 
 t   t  f ( x, p x , t )
x m
t
f f px

0
t x m
Now let’s generalize into full r-dependence and allow for an external force F
f (r 
p
m
 t , p  F t , t ) d 3 r d 3 p  f ( r , p, t ) d 3 rd 3 p
𝜕𝑞′ 𝜕𝑞′
= Liouville’s theorem
𝜕𝑞
𝜕𝑝
𝑑𝑞 ′ 𝑑𝑝′ =
𝑑𝑞𝑑𝑝 = 𝐽 𝑑𝑞𝑑𝑝 Jacobian determinant J=1 for
𝜕𝑝′ 𝜕𝑝′
canonical transformations
(leaving Hamilton equations unchanged)
𝜕𝑞 𝜕𝑝
𝜕𝑞′ 𝜕𝑞′
𝑝
′
𝑞 = 𝑞 𝑡 + 𝛿𝑡 = 𝑞 + 𝛿𝑡
𝜕𝑞
𝜕𝑝
1 𝛿𝑡/𝑚
=
=1
𝑚
Here: ′
𝜕𝑝′ 𝜕𝑝′
0
1
𝑝 = 𝑝 𝑡 + 𝛿𝑡 = 𝑝 + 𝐹𝛿𝑡
f ( r , p, t )   r f
p
m
𝜕𝑞
𝜕𝑝
 t   p f F t 
f
 t  f ( r , p, t )
t
p
f
 r f   p f F  0
t
m
So far we ignored particle-particle collisions
change of f due to collisions
p
f
 f 
 r f   p f F   
t
m
 t coll
Celebrated Boltzmann
transport equation.
Very useful in CMP.
Detailed analysis of the transport equation is beyond the scope
of this course (see e.g., Kerson Huang, Statistical Mechanics, John Wiley&Sons,New York 1987, p. 60 )
We discuss briefly implications for statistical mechanics
Equilibrium:
-absence of an external force
-homogenous density
p
f
 f 
 r f   p f F   
t
m
 t coll
 f 
  0
 t coll
0
Solving
-only binary collisions (dilute gas)
 f 
  0
 t coll
- influence of container walls neglected
with assumptions
- velocities of colliding particles are
uncorrelated & independent of position
(molecular chaos or Stosszahlansatz)
Boltzmann distribution

f eq ( p) 
e
p2
2 mkBT
 2m kBT 
3/ 2
H-theorem

3
Boltzmann defined the functional H (t )  d p f ( p, t ) log f ( p, t )
H-theorem:
If at a given time t the state of a gas satisfies the assumption of
molecular chaos, then at t+ (->0)
dH
0
dt
dH
 0 if and only if f(p,t) is the Maxwell-Boltzmann distribution
dt
H
Note: it seems as if the H-theorem
explains the time asymmetry
from time inversion symmetric
H(t) calculated with f
microscopic descriptions.
solving Boltzmann
Points where molecular
chaos is fulfilled
Transport equation
t
This is not the case, assumption of
molecular chaos is the origin of time
asymmetry.