# Preservation of phase space volume and Liouville's theorem

Consider a phase space volume element $$dx_0$$ at t=0, containing a small collection of initial conditions on a set of trajectories. The trajectories evolve in time according to Hamilton's equations of motion, and at a time t later will be located in a new volume element $$dx_t$$ as shown in the figure below:

Figure 1:

How is $$dx_0$$ related to $$dx_t$$﻿dxdd ? To answer this, consider a trajectory starting from a phase space vector $$x_0$$ in $$dx_0$$ and having a phase space vector $$x_t$$ at time t in $$dx_t$$. Since the solution of Hamilton's equations depends on the choice of initial conditions, $$x_t$$ depends on $$x_0$$ :

$x_0 = \left ( p_1 (0), \cdots , p_N(0), r_1(0), \cdots , r_N (0) \right )$

$x_0 = \left ( p_1 (t), \cdots , p_N(t), r_1(t), \cdots , r_N (t) \right )$

$x^i_t = x^i_t \left ( x^1_0 , \cdots , x^{6N}_0 \right )$

Thus, the phase space vector components can be viewed as a coordinate transformation on the phase space from t=0 to time t. The phase space volume element then transforms according to

$dx_t = J (x_t ; x_0 ) dx_0$

where $$J (x_t ; x_0 )$$ is the Jacobian of the transformation:

$J (x_t ; x_0 ) = \frac {\partial (x^1_t \cdots x^n_t )}{\partial (x^1_0 \cdots x^n_0 )}$

where n=6N. The precise form of the Jacobian can be determined as will be demonstrated below.

The Jacobian is the determinant of a matrix M,

$J (x_t ; x_0 ) = \text {det} (M) = e^{TrlnM}$

whose matrix elements are

$M_{ij} = \frac {\partial x^i_t}{\partial x^j_0}$

Taking the time derivative of the Jacobian, we therefore have

$\frac {dJ}{dt} = Tr \left ( M^{-1} \frac {dM}{dt} \right ) e^{TrlnM}$

$= J \sum _{i=1}^n \sum _{j=1}^n M^{-1}_{ij} \frac {dM_{ij}}{dt}$

The matrices M $$_{-1}$$ and $$\frac {dM}{dt}$$ can be seen to be given by

$M^{-1}_{ij} = \frac {\partial x^i_0}{\partial x^j_t}$

$\frac {dM_{ji}}{dt} = \frac {\partial \dot {x}^i_t}{\partial x^i_0}$﻿}}

Substituting into the expression for dJ/dt gives

$\frac {dJ}{dt} = J \sum _{i,j=1}^n \frac {\partial x^i_0}{\partial x^j_t} \frac {\partial \dot {x}^i_t}{\partial x^i_0}$

$= J \sum _{i,j,k=1}^n \frac {\partial x^i_0}{\partial x^j_t} \frac {\partial \dot {x}^i_t}{\partial x^k_t} \frac {\partial x^k_t}{\partial x^i_0}$

where the chain rule has been introduced for the derivative $$\frac {\partial x^j_t}{\partial x^i_0}$$. The sum over i can now be performed:

$\sum _{i=1}^n \frac {\partial x^i_0}{\partial x^j_t} \frac {\partial x^k_t}{\partial x^i_0} = \sum ^n_{i=1} M^{-1}_{ij} M_{ki} = \sum ^n_{i=1} M_{ki}M^{-1}_{ij} = \delta _{kj}$

Thus,

$\frac {dJ}{dt} = J \sum ^n_{j,k=1} \delta _{jk} \frac {\partial \dot {x}^j_t}{\partial x^k_0}$

$J \sum ^n_{j=1} \frac {\partial \dot {x}^j_t}{\partial x^j_t} = J \nabla _x \cdot \dot {x}$

or

$\frac {dJ}{dt} = J \nabla _x \cdot \dot {x}$

The initial condition on this differential equation is $$J (0) \equiv J (x_0; x_0) = 1$$. Moreover, for a Hamiltonian system $$\nabla _x \cdot \dot {x} = 0$$. This says that dJ/dt=0 and J(0)=1. Thus, $$J (x_t ; x_0 ) = 1$$. If this is true, then the phase space volume element transforms according to

$dx_o = dx_t$

which is another conservation law. This conservation law states that the phase space volume occupied by a collection of systems evolving according to Hamilton's equations of motion will be preserved in time. This is one statement of Liouville's theorem.

Combining this with the fact that df/dt=0, we have a conservation law for the phase space probability:

$f(x_o, o) dx_o = f(x_t,t)dx_t$

which is an equivalent statement of Liouville's theorem.