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Math, physics, perl, and programming obscurity.

PHY452H1S Basic Statistical Mechanics. Lecture 5: Motion in phase space. Liouville and Poincar’e theorems. Taught by Prof. Arun Paramekanti

Posted by peeterjoot on January 22, 2013

[Click here for a PDF of this post with nicer formatting and figures if the post had any (especially if my latex to wordpress script has left FORMULA DOES NOT PARSE errors.)]

Disclaimer

Peeter’s lecture notes from class. May not be entirely coherent.

Motion in phase space

Classical system: \mathbf{x}_i, \mathbf{p}_i with dimensionality

\begin{aligned}\underbrace{2}_{x, p}\underbrace{d}_{\text{space dimension}}\underbrace{N}_{\text{Number of particles}}\end{aligned} \hspace{\stretch{1}}(1.2.1)

Hamiltonian H is the “energy function”

\begin{aligned}H = \underbrace{\sum_{i = 1}^N \frac{\mathbf{p}_i^2}{2m} }_{\text{Kinetic energy}}+ \underbrace{\sum_{i = 1}^N V(\mathbf{x}_i) }_{\text{Potential energy}}+ \underbrace{\sum_{i < j}^N \Phi(\mathbf{x}_i - \mathbf{x}_j)}_{\text{Internal energy}}\end{aligned} \hspace{\stretch{1}}(1.2.2)

\begin{aligned}\mathbf{\dot{p}}_i = \mathbf{F} = \text{force}\end{aligned} \hspace{\stretch{1}}(1.0.3a)

\begin{aligned}\mathbf{\dot{x}}_i = \frac{\mathbf{p}_i}{m}\end{aligned} \hspace{\stretch{1}}(1.0.3b)

Expressed in terms of the Hamiltonian this is

\begin{aligned}\dot{p}_{i_\alpha} = - \frac{\partial {H}}{\partial {x_{i_\alpha}}}\end{aligned} \hspace{\stretch{1}}(1.0.4a)

\begin{aligned}\dot{x}_{i_\alpha} = \frac{\partial {H}}{\partial {p_{i_\alpha}}}\end{aligned} \hspace{\stretch{1}}(1.0.4b)

In phase space we can have any number of possible trajectories as illustrated in (Fig1).

Fig1: Disallowed and allowed phase space trajectories

Liouville’s theorem

We are interested in asking the question of how the density of a region in phase space evolves as illustrated in (Fig2)

Fig2: Evolution of a phase space volume

We define a phase space density

\begin{aligned}\rho(p_{i_\alpha}, x_{i_\alpha}, t),\end{aligned} \hspace{\stretch{1}}(1.0.5)

and seek to demonstrate Liouville’s theorem, that the phase space density does not change. To do so, consider the total time derivative of the phase space density

\begin{aligned}\frac{d{{\rho}}}{dt} &= \frac{\partial {\rho}}{\partial {t}} + \sum_{i_\alpha} \frac{\partial {p_{i_\alpha}}}{\partial {t}} \frac{\partial {\rho}}{\partial {p_{i_\alpha}}} + \frac{\partial {x_{i_\alpha}}}{\partial {t}} \frac{\partial {\rho}}{\partial {x_{i_\alpha}}} \\ &= \frac{\partial {\rho}}{\partial {t}} + \sum_{i_\alpha} \frac{\partial {p_{i_\alpha}}}{\partial {t}} \frac{\partial {\rho \dot{p}_{i_\alpha}}}{\partial {p_{i_\alpha}}} + \frac{\partial {x_{i_\alpha}}}{\partial {t}} \frac{\partial {\rho \dot{x}_{i_\alpha}}}{\partial {x_{i_\alpha}}} - \rho \left(\frac{\partial {\dot{p}_{i_\alpha}}}{\partial {p_{i_\alpha}}} +\frac{\partial {\dot{x}_{i_\alpha}}}{\partial {x_{i_\alpha}}} \right) \\ &= \frac{\partial {\rho}}{\partial {t}} + \underbrace{\sum_{i_\alpha} \left(\frac{\partial {p_{i_\alpha}}}{\partial {t}} \frac{\partial {(\rho \dot{p}_{i_\alpha})}}{\partial {p_{i_\alpha}}} + \frac{\partial {x_{i_\alpha}}}{\partial {t}} \frac{\partial {(\rho \dot{x}_{i_\alpha})}}{\partial {x_{i_\alpha}}} \right)}_{\equiv \boldsymbol{\nabla} \cdot \mathbf{j}}- \rho \sum_{i_\alpha} \underbrace{\left(-\frac{\partial^2 H}{\partial p_{i_\alpha} \partial x_{i_\alpha}}+\frac{\partial^2 H}{\partial x_{i_\alpha} \partial p_{i_\alpha}}\right)}_{= 0}\end{aligned} \hspace{\stretch{1}}(1.0.6)

We’ve implicitly defined a current density \mathbf{j} above by comparing to the continuity equation

\begin{aligned}\frac{\partial {\rho}}{\partial {t}} + \boldsymbol{\nabla} \cdot \mathbf{j} = 0\end{aligned} \hspace{\stretch{1}}(1.0.7)

Here we have

\begin{aligned}\rho(\dot{x}_{i_\alpha}, \dot{p}_{i_\alpha}) \rightarrow \mbox{current in phase space}\end{aligned} \hspace{\stretch{1}}(1.0.8)

Usually we have

\begin{aligned}\mathbf{j}_{\mathrm{usual}} \sim \rho \mathbf{v}\end{aligned} \hspace{\stretch{1}}(1.0.9a)

\begin{aligned}\mathbf{j} = -D \boldsymbol{\nabla} \rho\end{aligned} \hspace{\stretch{1}}(1.0.9b)

but we don’t care about this diffusion relation, just the continuity equation equivalent

\begin{aligned}v \frac{\partial {\rho}}{\partial {t}} + \boldsymbol{\nabla} \cdot \mathbf{j} = 0\end{aligned} \hspace{\stretch{1}}(1.0.10)

The implication is that

\begin{aligned}\frac{d{{\rho}}}{dt} = 0.\end{aligned} \hspace{\stretch{1}}(1.0.11)

Flow in phase space is very similar to an “incompressible fluid”.

Time averages, and Poincar\’e recurrence theorem

We want to look at how various observables behave over time

\begin{aligned}\overline{A} = \frac{1}{{T}} \int_0^T dt \rho(x, p, t) A(p, x)\end{aligned} \hspace{\stretch{1}}(1.0.12)

We’d like to understand how such averages behave over long time intervals, looking at \rho(x, p, t \rightarrow \infty).

This long term behaviour is described by the Poincar\’e recurrence theorem. If we wait long enough a point in phase space will come arbitrarily close to its starting point, recurring or “closing the trajectory loops”.

A simple example of a recurrence is an undamped SHO, such as a pendulum. That pendulum bob when it hits the bottom of the cycle will have the same velocity (and position) each time it goes through a complete cycle. If we imagine a much more complicated system, such as N harmonic oscillators, each with different periods, we can imagine that it will take infinitely long for this cycle or recurrence to occur, and the trajectory will end up sweeping through all of phase space.

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