- #1

lamq_31

- 1

- 0

## Homework Statement

I am having problems with the following problem: "For a noninteracting gas of N particles in a cubic box of volume [itex]V = L^3 [/itex] where L is the length of the side of the box, find the solution, [itex]\rho (\mathbf{p}^{3N},\mathbf{q}^{3N},t) [/itex], of the Liouville equation at time t, where [itex] \mathbf{p}^{3N} = \left ( \mathbf{p_{1},...,\mathbf{p_N}} \right ) [/itex] and [itex] \mathbf{q}^{3N} = \left ( \mathbf{q_{1},...,\mathbf{q_N}} \right ) [/itex]. Assume periodic bondary conditions, and that the probability density at time t=0 is given by:

## Homework Equations

[tex] \rho \left ( \mathbf{p}^{3N},\mathbf{q}^{3N},t \right ) = \left ( \frac{\sqrt{\pi}}{2L} \right )^{3N}\prod_{i=1}^{3N}e^{-p_{i}^{2}/2m}sin\left ( \pi q_{i}/L \right ) [/tex]

with

[itex] 0\leq q_{i}\leq L [/itex]

The following equation may be helpful:

[itex] \int _{0}^{L}dx \sin \left ( \pi q/L \right )\ln \left [ \sin\left ( \pi q/L \right ) \right ] = \frac{L}{\pi}\left ( 2 - \ln 2\right ) [/itex]

## The Attempt at a Solution

So far, I started from the Liouville equation:

[itex] \frac{\partial \rho}{\partial t} = -\sum _{i}^{3N}\left [ \frac{\partial \rho}{\partial q}\frac{\partial H}{\partial p} + \frac{\partial \rho}{\partial p}\frac{\partial H}{\partial q}\right ] [/itex]

Since the gas is noninteracting, I can assume that the Hamiltonian is:

[itex]H = \sum _{i}^{3N}\frac{p_{i}^2}{2m} [/itex]

So, I arrive to the equation:

[itex]\frac{\partial \rho}{\partial t} = -\sum _{i}^{3N}\frac{p_{i}}{m}\frac{\partial \rho}{\partial q} [/itex]

I have read in a few sources that the formal solution to the Liouville equation is of the form:

[itex] \rho\left ( \mathbf{p},\mathbf{q},t \right ) = e^{-Lt}\rho\left ( \mathbf{p} ,\mathbf{q}\right,0 ) [/itex]

And, in this case:

[itex] L =\sum _{i}^{3N}\frac{p_{i}}{m}\frac{\partial }{\partial q} [/itex]

However , at this point I don't know how to proceed because I don't know how explicitly include boundary conditions. (Actually I am chemist and I am not used to solving partial differential equations).

Thank you very much for your help