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Posts Tagged ‘phase space density’

An updated compilation of notes, for ‘PHY452H1S Basic Statistical Mechanics’, Taught by Prof. Arun Paramekanti

Posted by peeterjoot on March 3, 2013

In A compilation of notes, so far, for ‘PHY452H1S Basic Statistical Mechanics’ I posted a link this compilation of statistical mechanics course notes.

That compilation now all of the following too (no further updates will be made to any of these) :

February 28, 2013 Rotation of diatomic molecules

February 28, 2013 Helmholtz free energy

February 26, 2013 Statistical and thermodynamic connection

February 24, 2013 Ideal gas

February 16, 2013 One dimensional well problem from Pathria chapter II

February 15, 2013 1D pendulum problem in phase space

February 14, 2013 Continuing review of thermodynamics

February 13, 2013 Lightning review of thermodynamics

February 11, 2013 Cartesian to spherical change of variables in 3d phase space

February 10, 2013 n SHO particle phase space volume

February 10, 2013 Change of variables in 2d phase space

February 10, 2013 Some problems from Kittel chapter 3

February 07, 2013 Midterm review, thermodynamics

February 06, 2013 Limit of unfair coin distribution, the hard way

February 05, 2013 Ideal gas and SHO phase space volume calculations

February 03, 2013 One dimensional random walk

February 02, 2013 1D SHO phase space

February 02, 2013 Application of the central limit theorem to a product of random vars

January 31, 2013 Liouville’s theorem questions on density and current

January 30, 2013 State counting

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Liouville’s theorem questions on density and current

Posted by peeterjoot on February 5, 2013

[Click here for a PDF of this post with nicer formatting and figures if the post had any]

Liouville’s theorem questions on density and current

In the midterm we were asked to state and prove Liouville’s theorem. I couldn’t remember the proof, having only a recollection that it had something to do with the continuity equation

\begin{aligned}0 = \frac{\partial {\rho}}{\partial {t}} + \frac{\partial {j}}{\partial {x}},\end{aligned} \hspace{\stretch{1}}(1.1.1)

but unfortunately couldn’t remember what the j was. Looking up the proof, it’s actually really simple, just the application of chain rule for a function \rho that’s presumed to be a function of time, position and momentum variables. It didn’t appear to me that this proof has anything to do with any sort of notion of density, so I posed the following questions.

Context

The core of the proof can be distilled to one dimension, removing all the indexes that obfuscate what’s being one. For that case, application of the chain rule to a function \rho(t, x, p), we have

\begin{aligned}\frac{d{{\rho}}}{dt} &= \frac{\partial {\rho}}{\partial {t}} + \frac{\partial {x}}{\partial {t}} \frac{\partial {\rho}}{\partial {x}} + \frac{\partial {p}}{\partial {t}} \frac{\partial {\rho}}{\partial {p}} \\ &= \frac{\partial {\rho}}{\partial {t}} + \dot{x} \frac{\partial {\rho}}{\partial {x}} + \dot{p} \frac{\partial {\rho}}{\partial {p}} \\ &= \frac{\partial {\rho}}{\partial {t}} + \frac{\partial {\left( \dot{x} \rho \right)}}{\partial {x}} + \frac{\partial {\left( \dot{x} \rho \right)}}{\partial {p}} - \rho \left(  \frac{\partial {\dot{x}}}{\partial {x}} + \frac{\partial {\dot{p}}}{\partial {p}}  \right) \\ &= \frac{\partial {\rho}}{\partial {t}} + \frac{\partial {\left( \dot{x} \rho \right)}}{\partial {x}} + \frac{\partial {\left( \dot{x} \rho \right)}}{\partial {p}} - \rho \underbrace{\left(\frac{\partial {}}{\partial {x}} \left(  \frac{\partial {H}}{\partial {p}}  \right)+\frac{\partial {}}{\partial {p}} \left(  -\frac{\partial {H}}{\partial {x}}  \right)\right)}_{= 0}\end{aligned} \hspace{\stretch{1}}(1.1.2)

Wrong interpretation

From this I’d thought that the theorem was about steady states. If we do have a steady state, where d\rho/dt = 0 we have

\begin{aligned}0 = \frac{\partial {\rho}}{\partial {t}} + \frac{\partial {\left( \dot{x} \rho \right)}}{\partial {x}} + \frac{\partial {\left( \dot{p} \rho \right)}}{\partial {p}}.\end{aligned} \hspace{\stretch{1}}(1.1.3)

That would answer the question of what the current is, it’s this tuple

\begin{aligned}\mathbf{j} = \rho (\dot{x}, \dot{p}),\end{aligned} \hspace{\stretch{1}}(1.1.4)

so if we introduce a “phase space” gradient

\begin{aligned}\boldsymbol{\nabla} = \left(  \frac{\partial {}}{\partial {x}}, \frac{\partial {}}{\partial {p}}  \right)\end{aligned} \hspace{\stretch{1}}(1.1.5)

we’ve got something that looks like a continuity equation

\begin{aligned}0 = \frac{\partial {\rho}}{\partial {t}} + \boldsymbol{\nabla} \cdot \mathbf{j}.\end{aligned} \hspace{\stretch{1}}(1.1.6)

Given this misinterpretation of the theorem, I had the following two questions

  • This function \rho appears to be pretty much arbitrary. I don’t see how this connects to any notion of density?
  • If we pick a specific Hamiltonian, say the 1D SHO, what physical interpretation do we have for this “current” \mathbf{j}?

The clarification

Asking about this, the response was “Actually, equation 1.1.3 has to be assumed for the proof. This equation holds if \rho is the phase space density and since the pair in 1.1.4 is the current density in phase space. The theorem then states that d\rho/dt = 0 whether or not one is in the steady state. This means even as the system is evolving in time, if we sit on a particular phase space point and follow it around as it evolves, the density in our neighborhood will be a constant.”

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