Editing Second law of thermodynamics (section)

=== Derivation for systems described by the canonical ensemble ===
If a system is in thermal contact with a heat bath at some temperature ''T'' then, in equilibrium, the probability distribution over the energy eigenvalues are given by the [[canonical ensemble]]:
: <math>P_{j}=\frac{\exp\left(-\frac{E_{j}}{k_{\mathrm B} T}\right)}{Z}</math>

Here ''Z'' is a factor that normalizes the sum of all the probabilities to 1, this function is known as the [[Partition function (statistical mechanics)|partition function]]. We now consider an infinitesimal reversible change in the temperature and in the external parameters on which the energy levels depend. It follows from the general formula for the entropy:
: <math>S = -k_{\mathrm B}\sum_{j}P_{j}\ln\left(P_{j}\right)</math>
that
: <math>dS = -k_{\mathrm B}\sum_{j}\ln\left(P_{j}\right)dP_{j}</math>

Inserting the formula for <math>P_{j}</math> for the canonical ensemble in here gives:
: <math>dS = \frac{1}{T}\sum_{j}E_{j}dP_{j}=\frac{1}{T}\sum_{j}d\left(E_{j}P_{j}\right) - \frac{1}{T}\sum_{j}P_{j}dE_{j}= \frac{dE + \delta W}{T}=\frac{\delta Q}{T}</math>