In S. Salinas, Introduction to Statistical Physics (Springer, 2001), the author states (p. 68):"...the thermodynamical limit is essential to allow the connection between the average values of statistical mechanics and the macroscopic values of the thermodynamic quantities". However, on p. 80 the author states: "As a matter of fact, it is not even necessary to invoke the thermodynamic limit to obtain the equations of state of the ideal gas" and goes on to show: $$S(E,V,N;\delta E)=k_B\ln\Omega(E,V,N;\delta E)=\frac{3}{2}k_B\ln E+k_B\ln V+f(N;\delta E),$$ where $f(N;\delta E)$ is a function of $N$ and $\delta E$. Therefore, $$\frac{1}{T}=\frac{\partial S}{\partial E}=\frac{3k_B}{2E}$$ and $$\frac{p}{T}=\frac{\partial S}{\partial V}=\frac{k_B}{V}.$$ Is there a contradiction here? How is the temperature (a macroscopic, averaged quantity) related to the derivative of the entropy without taking the thermodynamic limit?


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The thermodynamic limit is a formal procedure required to get the usual thermodynamic properties from statistical mechanics since, generally, not all of them are unconditionally true for finite-size systems. For example, equilibrium thermodynamics requires that the Helmholtz free energy is an extensive function of the volume. In general, for a finite system, this is not the case: there is a contribution to free energy coming from the boundary layers that does not scale as the volume. The thermodynamic limit takes care of this problem by making negligible the boundary contribution with respect to the bulk. Moreover, finite-size systems have completely analytic thermodynamic potentials, thus excluding the possibility of phase transitions. It is the thermodynamic limit that allows getting a non-analytic function as the limit of an analytic sequence. A third role of the thermodynamic limit is eliminating the dependence of thermodynamic results on the specific statistical ensemble.

However, there are cases where the thermodynamic limit is not required to restore the proper thermodynamic behavior and ensemble independence, at least for some properties. This is rooted in an interplay between the Hamiltonian of the system and the ensemble. For example, in the case of a one-component classical ideal gas (not interacting particles), the whole thermodynamics can be exactly reproduced even with a finite-size system in the canonical and grand canonical ensembles (some ensemble dependence may appear in the microcanonical ensemble depending on the specific flavor of it). The reason can be related to the absence of interaction, which has, among other consequences, the absence of different behavior of the boundary layers and the absence of phase transitions.

The existence of exceptional cases like the one you are reporting is not a contradiction. Actually, in your example, the critical observation is that the term containing the dependence on the ensemble is the function $f(N,\delta E)$. More precisely, the ensemble dependence is the dependence of the entropy on $\delta E$. It is generally possible to show that dependence gives a vanishing contribution to the entropy per particle (or the entropy per unit volume) at the thermodynamic limit if $\delta E$ does not scale with the size. However, for a perfect gas, the $\delta E$ dependence is confined to an additive term, which exactly disappears when a partial derivative of entropy is taken with respect to volume or energy.

The relations $\frac{1}{T} = \frac{\partial S}{\partial E}$ and $\frac{P}{T} = \frac{\partial S}{\partial V}$ are pure thermodynamic relations valid for every possible thermodynamic system.


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