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Liouville's Theorem

Conider phase space of momentum, p, and position, q.
p
 . .
 .(p,q)


_________q
^
y  point flow
 >

  c 
  
 a bΔy
  
  d 
 Δx

 x
The number of points in phase space moving in/out of the box
is proportional to both the velocities at the boundaries and the
length of the boundaries.
x: v_{xa}Δy  v_{xb}Δy = (∂v_{x}/∂x)ΔxΔy
y: v_{yd}Δx  v_{yc}Δx = (∂v_{y}/∂y)ΔxΔy
Total change: [∂v_{x}/∂x + ∂v_{y}/∂y]ΔaΔy
The flow in phase space is incompressible therefore the [] term
must equal 0. This is the divergence of the velocity.
In general Σ_{i}∂v_{i}/∂x_{i} = 0 or ∇.v = 0.
Return to phase space. Replace v_{x} with v_{q} and v_{y} with v_{p}. Now
from Hamilton's equations we get:
.
p = ∂H/∂q = v_{p}
.
q = ∂H/∂p = v_{q}
Therefore,
∂v_{p}/∂p = ∂/∂p(∂H/∂q)
and
∂v_{q}/∂q = ∂/∂q(∂H/∂p)
If we add these 2 equations we get 0. This means that the flow
is incompressible. There is no convergence or divergence of
trajectories. This is LIOUVILLE's THEOREM.
.
Consider a Lagrangian L = (1/2)mx^{2}
. .
Therefore p_{x} = ∂L/∂x = mx
. .
Now perform a transformation y = αx so that y/α = x and y/α = x.
Now L becomes (assume m = 1):
. . .
L =(1/2)y^{2}/α^{2} and p_{y} = y/α^{2} = x/α = p_{x}/α Thus,
P_{y} = p_{x}/α
So, if the q coordinate is stretched by α then the p coordinate
must shrink by 1/α. In other words, the shape of an area can
change but the area must be conserved under a coordinate
transformation.
The counting of the number of states available to a particle
amounts to determining the available volume in phase space.
One might think that for a continuous phase space, any finite
volume would contain an infinite number of states. But the
uncertainty principle tells us that we cannot simultaneously
know both the position and momentum, so we cannot really
say that a particle is at a mathematical point in phase space.
So when we contemplate an element of "volume" in phase
space du = dxdydzdp_{x}dp_{y}dp_{z} then the smallest "cell" in phase
space which we can consider, is constrained by the uncertainty
principle to be du_{minimum} = h^{3}
In the case of a chaotic system the phase space becomes
fractal in nature. Unless you can measure this fractal with
infinite precision then its area will be larger than the original
one. This is referred to as COARSE GRAINING.