Editing H-theorem (section)

=== 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.