Thermodynamic potential

Any one of the four functions defined on the set of macroscopic (thermodynamical) systems: the energy, the heat function (or enthalpy), the free Helmholtz energy, and the free Gibbs energy (sometimes called the thermodynamic potential in the restricted sense).

To formally construct a thermodynamical state of a (one-component) thermodynamical system, one describes any one of the pairs of parameters $ ( s, v) $, $ ( s, p) $, $ ( T, v) $, $ ( T, p) $, where $ s $ is the specific entropy of the system, $ T $ is its absolute temperature, $ p $ is the pressure, and $ v $ is the specific volume. To each of these pairs it is convenient to associate a thermodynamic potential: to $ ( s, v) $ the energy $ E = E ( s, v) $, to $ ( s, p) $ the heat function $ W = W ( s, p) $, to $ ( T, v) $ the free Helmholtz energy $ F = F ( T, v) $, and, finally, to $ ( T, p) $ the free Gibbs energy $ \Phi = \Phi ( T, p) $.

Here, if some pair of parameters is chosen to describe the system, then the other two parameters can be expressed as the partial derivatives of the corresponding thermodynamical potential (hence the name). The parameters $ s, T $ and $ p, v $ are conjugate in the sense that each can be expressed as a partial derivative with respect to the other; for example, choosing the pair $ ( s, v) $ with potential $ E ( s, v) $, the parameters $ T $ and $ p $ are:

$$ \tag{1 } T = \frac{\partial E }{\partial s } ,\ \ p = - \frac{\partial E }{\partial v } . $$

The transition from one pair of parameters with its potential to another pair of parameters with the corresponding potential is performed using the Legendre transform. Thus, going from the pair $ ( s, v) $ to the pair $ ( T, v) $, the potential $ F ( T, v) $ of this pair is

$$ F ( T, v) = E ( s ( T), v) - s ( T) T, $$

where $ s ( T) $ is obtained from equation (1), that is, $ F ( T, v) $ agrees up to sign with the Legendre transform of the function $ E ( s, v) $ regarded as a function of $ s $.

For a meaningful thermodynamic construction using equilibrium Gibbs ensembles, the thermodynamic potentials can be expressed in terms of the thermodynamic limit of the logarithm of the statistical sum (and its derivatives) of some or other of the Gibbs ensembles, divided by volume. For example, the free Helmholtz energy is given by

$$ F ( T, v) = \ \lim\limits _ {\begin{array}{c} N \rightarrow \infty \\ | \Lambda |/N \rightarrow \infty \end{array} } \ { \frac{1}{| \Lambda | } } \mathop{\rm ln} \ Z ( T, N, \Lambda ), $$

where $ Z ( T, N, \Lambda ) $ is the statistical sum of a small canonical ensemble for an $ N $- particle system, enclosed in a domain $ \Lambda $( of volume $ | \Lambda | $), at a fixed temperature $ T $( cf. [3]).

References

[1]	L.D. Landau, E.M. Lifshitz, "Statistical physics" , A course of theoretical physics , 5 , Pergamon (1969) (Translated from Russian)
[2]	I.M. Gel'fand, S.V. Fomin, "Calculus of variations" , Prentice-Hall (1963) (Translated from Russian)
[3]	D. Ruelle, "Statistical mechanics: rigorous results" , Benjamin (1974)