Editing H-theorem (section)

=== Classical mechanical ===

<!-- This section need to be very explicit about the underpinning assumptions of the Boltzmann equation, e.g. Stosszahlansatz etc.
 Technical math derivations are fine but they should go into a collapsible block, and technical terms like "μ-space" must be at least defined!

Other inadequately defined symbols: G, P, \ln \delta v_\gamma. The derivations (or isolated formulas, if you like) are incomprehensible 
by themselves without definitions.     -->

We let {{math|''q<sub>i</sub>''}} and {{math|''p<sub>i</sub>''}} be our [[Canonical coordinates|generalized canonical coordinates]] for a set of <math>r</math> particles. Then we consider a function <math>f</math> that returns the probability density of particles, over the states in [[phase space]]. Note how this can be multiplied by a small region in phase space, denoted by <math>\delta q_1 ... \delta p_r</math>, to yield the (average) expected number of particles in that region.

:<math>\delta n = f(q_1 ... p_r,t)\,\delta q_1\delta p_1 ... \delta q_r \delta p_r.\,</math>

Tolman offers the following equations for the definition of the quantity ''H'' in Boltzmann's original ''H'' theorem.

: <math>H= \sum_i f_i \ln f_i \,\delta q_1 \cdots \delta p_r</math><ref>Tolman 1938 pg. 135 formula 47.5</ref>

Here we sum over the regions into which phase space is divided, indexed by <math>i</math>. And in the limit for an infinitesimal phase space volume <math>\delta q_i \rightarrow 0, \delta p_i \rightarrow 0 \; \forall \, i</math>, we can write the sum as an integral.

: <math>H= \int \cdots \int f \ln f \,d q_1 \cdots dp_r</math><ref>Tolman 1938 pg. 135 formula 47.6</ref>

''H'' can also be written in terms of the number of molecules present in each of the cells.
: <math>
\begin{align}
H & = \sum( n_i \ln n_i -  n_i \ln \delta v_\gamma) \\
& = \sum n_i \ln n_i + \text{constant}
\end{align}
</math><ref name="Tolman">Tolman 1938 pg. 135 formula 47.7</ref>{{clarify|date=April 2014}}

An additional way to calculate the quantity ''H'' is:

: <math>H = -\ln P + \text{constant}\,</math><ref>Tolman 1938 pg. 135 formula 47.8</ref>

where ''P'' is the probability of finding a system chosen at random from the specified [[microcanonical ensemble]]. It can finally be written as:

: <math>H = -\ln G + \text{constant}\,</math><ref>Tolman 1939 pg. 136 formula 47.9</ref>

where ''G'' is the number of classical states.{{clarify|date=April 2014}}

The quantity ''H'' can also be defined as the integral over velocity space{{Citation needed|date=March 2009}} :

:{| style="width:100%" border="0"
|-
| style="width:95%" |
<math>  \displaystyle
H \ \stackrel{\mathrm{def}}{=}\  
\int { P ({\ln P}) \, d^3 v} 
= \left\langle \ln P \right\rangle
</math>
| style= | (1)
|}
where ''P''(''v'') is the probability distribution.

Using the Boltzmann equation one can prove that ''H'' can only decrease.

For a system of ''N'' statistically independent particles, ''H'' is related to the thermodynamic entropy ''S'' through:<ref> Huang 1987 pg 79 equation 4.33 </ref>

:<math>S \ \stackrel{\mathrm{def}}{=}\  - V k H + \text{constant}</math> 

So, according to the ''H''-theorem, ''S'' can only increase.