# Equality of expectation value integral over coordinate space and over energy

1. Aug 3, 2012

### Derivator

Dear all,

I'm wondering, how one could justify mathematically the equality
$\int O(E(\vec{x}_1,...\vec{x}_N)) exp(-\beta E(\vec{x}_1,...,\vec{x}_N)) d\vec{x}_1...d\vec{x}_N$ = $\int g(E) O(E) exp(-\beta E) dE$

where O(E(x)) is an observable and g(E) the density of states.

Is there a mathematical justification for the equality?

best,
derivator

2. Aug 3, 2012

### Jolb

This is a non-trivial question. There IS a mathematical justification (that you'd have to be really clever to figure out), but I think it's much easier to think about the PHYSICAL justification.

Regardless of whatever variable you choose to use in your integral, the probability of occupation of a state is dictated by Maxwell-Boltzmann statistics. Whether you express the Boltzmann factor as a function of E or a function of x and x's time derivative is irrelevant, since the Boltzmann factor only depends on the energy of that state (and the energy is typically a function of x and x's time derivatives). Here's a mathematical way of stating that:

$$P(E)=\frac{e^{\frac{-E}{kT}}}{\int _Ee^{\frac{-E}{kT}} dE}\Leftrightarrow P(x_1, x_2, ..., x_n, \dot{x}_1, \dot{x}_2, ..., \dot{x}_n)=\frac{e^{-\frac{E(x_1, x_2, ..., x_n, \dot{x}_1, \dot{x}_2, ..., \dot{x}_n)}{kT}}}{\int_{x_1}\int_{x_2}...\int_{x_n}\int_{\dot{x}_1}\int_{\dot{x}_2}...\int_{\dot{x}_n} e^{-\frac{E(x_1, x_2, ..., x_n, \dot{x}_1, \dot{x}_2, ..., \dot{x}_n)}{kT}}dx_1 dx_2 ... dx_n d\dot{x}_1d\dot{x}_2...d\dot{x}_n}$$
where I have assumed the energy is only a function of the generalized coordinates and their time derivatives. (Or more precisely, the x dots should be generalized momenta.)

The reason for this is physically obvious: whether you label a state based on its energy or its coordinates/momenta, you still have the same Boltzmann factor to dictate the occupation of that state.

Last edited: Aug 3, 2012
3. Aug 3, 2012

### Derivator

(By the way: the (unnormalized) probability density, when expressed as a function of the energy, is given by $g(E) e^{\frac{-E}{kT}}$, the density of states is somewhat like the Jacobian determinant that emerges when performing a variable-transformation on probability densities)