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Last modified: January 26, 2018

Liouville's Theorem
-------------------

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

p
|       . .
|     .(p,q)
|   
|
|_________q


        ^
y       | point flow
|        --->
|      
|    -- c --
|   |       |             
|   |a     b|Δy           
|   |       |              
|    -- 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: vxaΔy - vxbΔy = (∂vx/∂x)ΔxΔy

y: vydΔx - vycΔx = (∂vy/∂y)ΔxΔy 

Total change: [∂vx/∂x + ∂vy/∂y]ΔaΔy

The flow in phase space is incompressible therefore the [] term 
must equal 0.  This is the divergence of the velocity.

In general Σi∂vi/∂xi = 0 or ∇.v = 0.  

Return to phase space.  Replace vx with vq and vy with vp.  Now
from Hamilton's equations we get:
.
p = -∂H/∂q = vp
.
q = ∂H/∂p = vq

Therefore,

∂vp/∂p = ∂/∂p(-∂H/∂q)   

and

∂vq/∂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)mx2  
                   .    .
Therefore px = ∂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)y2/α2 and py = y/α2 = x/α = px/α  Thus,

Py = px/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 = dxdydzdpxdpydpz then the smallest "cell" in phase 
space which we can consider, is constrained by the uncertainty 
principle to be duminimum = h3

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.

 

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