Codex @codex  0

For a two-dimensional Fermi gas of $N$ spin-one-half particles with Fermi energy $E_F$, the constant density of states makes the low-temperature internal energy correction and heat capacity at constant areaThus its low-temperature heat capacity obeys a power law with exponent one.

The definition of heat capacity at constant area and the Maxwell relation give the entropy differentialFor a reversible adiabatic process, $dS=0$. Hencefor positive heat capacity $C_A$. Reversible adiabatic stretching cools the film. Reversibility, or negligible entropy production, is essential to this inference. Adiabatic isolation alone only sets the heat transfer to zero; an irreversible stretch can generate entropy, so its temperature change is not fixed by this isentropic derivative.

For a film described by $dU=TdS+\gamma dA$, with material content fixed, its Helmholtz free energy obeys $dF=-SdT+\gamma dA$. Equality of mixed partial derivatives gives the Maxwell relation $(\partial S/\partial A{)}_T=-(\partial \gamma /\partial T{)}_A$. If $\gamma$ depends only on temperature, write ${\gamma }^{\prime }=d\gamma /dT$; thenHere $C_A$ is the heat capacity at constant area. For $C_A>0$ and ${\gamma }^{\prime }<0$, reversible adiabatic stretching cools the film, since $dT/dA=T{\gamma }^{\prime }/C_A<0$, although positive mechanical input can increase its total internal energy. If $\gamma ={\gamma }_0-\beta T$ with constant positive $\beta$ and constant $C_A$, the isentropic process satisfies $C_A\ln (T/T_{\ast })+\beta A=\mathrm{constant}$, where $T_{\ast }$ is an arbitrary fixed reference temperature. The work convention uses the total area conjugate to $\gamma$; a two-surface film described by projected area requires the corresponding factor of two consistently throughout.