Editing Second law of thermodynamics (section)

=== Derivation of the entropy change for reversible processes ===
The second part of the second law states that the entropy change of a system undergoing a reversible process is given by:
: <math>dS =\frac{\delta Q}{T}</math>
where the temperature is defined as:
: <math>\frac{1}{k_{\mathrm B} T}\equiv\beta\equiv\frac{d\ln\left[\Omega\left(E\right)\right]}{dE}</math>

See ''[[Microcanonical ensemble]]'' for the justification for this definition. Suppose that the system has some external parameter, ''x'', that can be changed. In general, the energy eigenstates of the system will depend on ''x''. According to the [[adiabatic theorem]] of quantum mechanics, in the limit of an infinitely slow change of the system's Hamiltonian, the system will stay in the same energy eigenstate and thus change its energy according to the change in energy of the energy eigenstate it is in.

The generalized force, ''X'', corresponding to the external variable ''x'' is defined such that <math>X dx</math> is the work performed by the system if ''x'' is increased by an amount ''dx''. For example, if ''x'' is the volume, then ''X'' is the pressure. The generalized force for a system known to be in energy eigenstate <math>E_{r}</math> is given by:
: <math>X = -\frac{dE_{r}}{dx}</math>

Since the system can be in any energy eigenstate within an interval of <math>\delta E</math>, we define the generalized force for the system as the expectation value of the above expression:
: <math>X = -\left\langle\frac{dE_{r}}{dx}\right\rangle\,</math>

To evaluate the average, we partition the <math>\Omega\left(E\right)</math> energy eigenstates by counting how many of them have a value for <math>\frac{dE_{r}}{dx}</math> within a range between <math>Y</math> and <math>Y + \delta Y</math>. Calling this number <math>\Omega_{Y}\left(E\right)</math>, we have:
: <math>\Omega\left(E\right)=\sum_{Y}\Omega_{Y}\left(E\right)\,</math>

The average defining the generalized force can now be written:
: <math>X = -\frac{1}{\Omega\left(E\right)}\sum_{Y} Y\Omega_{Y}\left(E\right)\,</math>

We can relate this to the derivative of the entropy with respect to ''x'' at constant energy ''E'' as follows. Suppose we change ''x'' to ''x'' + ''dx''. Then <math>\Omega\left(E\right)</math> will change because the energy eigenstates depend on ''x'', causing energy eigenstates to move into or out of the range between <math>E</math> and <math>E+\delta E</math>. Let's focus again on the energy eigenstates for which <math display="inline">\frac{dE_{r}}{dx}</math> lies within the range between <math>Y</math> and <math>Y + \delta Y</math>. Since these energy eigenstates increase in energy by ''Y dx'', all such energy eigenstates that are in the interval ranging from ''E'' – ''Y'' ''dx'' to ''E'' move from below ''E'' to above ''E''. There are
: <math>N_{Y}\left(E\right)=\frac{\Omega_{Y}\left(E\right)}{\delta E} Y dx\,</math>
such energy eigenstates. If <math>Y dx\leq\delta E</math>, all these energy eigenstates will move into the range between <math>E</math> and <math>E+\delta E</math> and contribute to an increase in <math>\Omega</math>. The number of energy eigenstates that move from below <math>E+\delta E</math> to above <math>E+\delta E</math> is given by <math>N_{Y}\left(E+\delta E\right)</math>. The difference
: <math>N_{Y}\left(E\right) - N_{Y}\left(E+\delta E\right)\,</math>
is thus the net contribution to the increase in <math>\Omega</math>. If ''Y dx'' is larger than <math>\delta E</math> there will be the energy eigenstates that move from below ''E'' to above <math>E+\delta E</math>. They are counted in both <math>N_{Y}\left(E\right)</math> and <math>N_{Y}\left(E+\delta E\right)</math>, therefore the above expression is also valid in that case.

Expressing the above expression as a derivative with respect to ''E'' and summing over ''Y'' yields the expression:
: <math>\left(\frac{\partial\Omega}{\partial x}\right)_{E} = -\sum_{Y}Y\left(\frac{\partial\Omega_{Y}}{\partial E}\right)_{x}= \left(\frac{\partial\left(\Omega X\right)}{\partial E}\right)_{x}\,</math>

The logarithmic derivative of <math>\Omega</math> with respect to ''x'' is thus given by:
: <math>\left(\frac{\partial\ln\left(\Omega\right)}{\partial x}\right)_{E} = \beta X +\left(\frac{\partial X}{\partial E}\right)_{x}\,</math>

The first term is intensive, i.e. it does not scale with system size. In contrast, the last term scales as the inverse system size and will thus vanish in the thermodynamic limit. We have thus found that:
: <math>\left(\frac{\partial S}{\partial x}\right)_{E} = \frac{X}{T}\,</math>

Combining this with
: <math>\left(\frac{\partial S}{\partial E}\right)_{x} = \frac{1}{T}\,</math>
gives:
: <math>dS = \left(\frac{\partial S}{\partial E}\right)_{x}dE+\left(\frac{\partial S}{\partial x}\right)_{E}dx = \frac{dE}{T} + \frac{X}{T} dx=\frac{\delta Q}{T}\,</math>