Editing H-theorem (section)

== Tolman's ''H''-theorem ==

[[Richard C. Tolman]]'s 1938 book ''The Principles of Statistical Mechanics'' dedicates a whole chapter to the study of Boltzmann's ''H'' theorem, and its extension in the generalized classical statistical mechanics of [[Josiah Willard Gibbs|Gibbs]]. A further chapter is devoted to the quantum mechanical version of the ''H''-theorem.

=== 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.

=== Quantum mechanical ===

In quantum statistical mechanics (which is the quantum version of classical statistical mechanics), the H-function is the function:<ref>Tolman 1938 pg 460 formula 104.7</ref>
: <math>H= \sum_i p_i \ln p_i, \,</math>
where summation runs over all possible distinct states of the system, and ''p<sub>i</sub>'' is the probability that the system could be found in the ''i''-th state.

This is closely related to the [[Gibbs entropy|entropy formula of Gibbs]],
:<math>S = - k \sum_i p_i \ln p_i \;</math>
and we shall (following e.g., Waldram (1985), p.&nbsp;39) proceed using ''S'' rather than ''H''.

First, differentiating with respect to time gives

:<math>\begin{align}
\frac{dS}{dt} & =  - k \sum_i \left(\frac{dp_i}{dt} \ln p_i  + \frac{dp_i}{dt}\right) \\
& =  - k \sum_i \frac{dp_i}{dt} \ln p_i  \\
\end{align}</math>

(using the fact that Σ&nbsp;''dp''<sub>''i''</sub>/''dt'' = 0, since Σ&nbsp;''p''<sub>''i''</sub> = 1, so the second term vanishes. We will see later that it will be useful to break this into two sums.)

Now [[Fermi's golden rule]] gives a [[master equation]] for the average rate of quantum jumps from state α to β; and from state β to α. (Of course, Fermi's golden rule itself makes certain approximations, and the introduction of this rule is what introduces irreversibility. It is essentially the quantum version of Boltzmann's ''Stosszahlansatz''.) For an isolated system the jumps will make contributions
:<math>\begin{align}
\frac{dp_\alpha}{dt} & = \sum_\beta \nu_{\alpha\beta}(p_\beta - p_\alpha) \\
\frac{dp_\beta}{dt}  & = \sum_\alpha \nu_{\alpha\beta}(p_\alpha - p_\beta) \\
\end{align}</math>
where the reversibility of the dynamics ensures that the same transition constant ''ν''<sub>''αβ''</sub> appears in both expressions.

So

:<math>\frac{dS}{dt} =  \frac{1}{2} k \sum_{ \alpha,\beta} \nu_{\alpha\beta}(\ln p_{\beta}-\ln p_{\alpha})(p_{\beta}- p_{\alpha}).</math>

The two differences terms in the summation always have the same sign. For example:
:<math>\begin{align}
w_{\beta} < w_{\alpha} 
\end{align}</math>

then 

:<math>\begin{align}
\ln w_{\beta} < \ln w_{\alpha}
\end{align}</math>

so overall the two negative signs will cancel. 

Therefore,

:<math>\Delta S \geq 0 \, </math>

for an isolated system.

The same mathematics is sometimes used to show that relative entropy is a [[Lyapunov function]] of a [[Markov process]] in [[detailed balance]], and other chemistry contexts.