# A More Systematic View¶

## Ensemble¶

A standard procedure of solving mechanics problems is

Initial condition / Description of states -> Time evolution -> Extraction of observables

### States¶

Density of states in phase space

Continuity equation

$\partial _ t \rho + \nabla \cdot (\rho \vec u) =0$

This conservation law can be more simpler if dropped the term $$\nabla\cdot \vec u = 0$$ for incompressibility.

Or more generally,

$\partial _ t \rho + \nabla \cdot \vec j = 0$

and here $$\vec j$$ can take other definitions like $$\vec j = - D \partial_x \rho$$.

This second continuity equation can represent any conservation law provided the proper $$\vec j$$.

From continuity equation to Liouville theorem

From continuity equation to Liouville theorem:

We start from

$\frac{\partial}{\partial t} \rho + \vec \nabla \cdot (\rho \vec v)$

Divergence means

$\vec \nabla \cdot = \sum_i \left( \frac{\partial}{\partial q_i} + \frac{\partial}{\partial p_i} \right) .$

Then we will have the initial expression written as

$\frac{\partial}{\partial t} \rho + \sum_i \left( \frac{\partial}{\partial q_i} (\rho \dot q_i) + \frac{\partial}{\partial \dot p_i} \right) .$

Expand the derivatives,

$\frac{\partial}{\partial t} \rho + \sum_i \left[ \left( \frac{\partial}{\partial q_i} \dot q_i + \frac{\partial}{\partial p_i} \dot p_i\right) \rho + \dot q_i \frac{\partial}{\partial q_i} \rho + \dot p_i \frac{\partial}{\partial p_i} \rho \right] .$

Recall that Hamiltonian equations

\begin{align}\begin{aligned}\dot q_i = \frac{\partial H}{\partial p_i}\\\dot p_i = - \frac{\partial H}{\partial q_i}\end{aligned}\end{align}

Then

$\left( \frac{\partial}{\partial q_i} \dot q_i + \frac{\partial}{\partial p_i} \dot p_i\right) \rho .$

Finally convective time derivative becomes zero because $$\rho$$ is not changing with time in a comoving frame like perfect fluid.

$\frac{d}{d t} \rho \equiv \frac{\partial}{\partial t}\rho + \sum_i \left[ \dot q_i \frac{\partial}{\partial q_i} \rho + \dot p_i \frac{\partial}{\partial p_i} \rho \right] =0$

### Time evolution¶

Apply Hamiltonian dynamics to this continuity equation, we can get

$\partial_t \rho = \{H, \rho\}$

which is very similar to quantum density matrix operator

$\mathrm i \hbar \partial_t \hat \rho = [ \hat H, \hat \rho ]$

That is to say, the time evolution is solved if we can find out the Poisson bracket of Hamiltonian and probability density.

### Requirements for Liouville Density¶

1. Liouville theorem;

2. Normalizable;

Hint

What about a system with constant probability for each state all over the phase space? This is not normalizable. Such a system can not really pick out a value. It seems that the probability to be on states with a constant energy is zero. So no such system really exist. I guess?

Like this?

Someone have 50% probability each to stop on one of the two Sandia Peaks for a picnic. Can we do an average for such a system? Example by Professor Kenkre.

And one more for equilibrium systems, $$\partial_t \rho =0$$.

### Extraction of observables¶

It’s simply done by calculating the ensemble average

$\langle O \rangle = \int O(p_i; q_i;t) \rho(p_i;q_i;t) \sum_i dp_i dq_i dt$

where $$i=1,2,..., 3N$$.

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