In a brief review of thermodynamics, our lecture notes read

Thermodynamic potentials are concave in their extensive variables and convex in their intensive variables.

Alright, we start with $U(S,V)$, with both its proper variables being extensive. Now all other thermodynamic potentials that we can derive by means of the Legendre transform are intensive in the new, intensive variables. By the properties of the transform, the transformed function is concave in the new variable, if it was convex in the old variable, and vice-versa.

But can I show that $S(U,V)$ is concave in its proper variables, and $U(S,V)$ is convex in its proper variables and that $S$ is maximal for the system in equilibrium with only the formulation of the 2$^{\text{nd}}$ Law of Thermodynamics

In an isolated system ($\delta Q=0$), the entropy can only increase.

or do I need an additional "though experiment" or postulate for that?


An equilibrium system at constant $U$ and $V$ has a well defined and constant value of entropy. The only possibility of increasing that value is as a consequence of modifying some internal constraint (removing walls, making them permeable to heat, etc.). However, the fully unconstrained system at fixed $U$ and $V$ (if it is a thermodynamic system) should have a finite value of the entropy. Therefore the equilibrium condition after constraint removal is the maximum value compatible with the new conditions.

The concavity of $S(U,V)$ can be obtained immediately if extensiveness is taken for granted.

Let's consider a system of energy $U$ and volume $V$, internally constrained by fixed adiabatic walls, to be made by two subsystems of the same substance, with energy $\tilde U_1$ and volume $\tilde V_1$, and $\tilde U_2$ and $\tilde V_2$ respectively. Additivity of entropy implies that the entropy of the constrained system is $$ S(\tilde U_1,\tilde V_1)+S(\tilde U_2,\tilde V_2) $$ If we remove the walls, we expect to get an equilibrium system of the same substance at the energy $U=\tilde U_1+\tilde U_2$ and at volume $V=\tilde V_1+\tilde V_2$. The entropy of such a system is $$ S(\tilde U_1+\tilde U_2,\tilde V_1+\tilde V_2)>S(\tilde U_1,\tilde V_1)+S(\tilde U_2,\tilde V_2) $$ where the $>$ relation follows from the maximum of entropy; in the the case of phase coexistence, one should keep a $\geq$ relation). If now we chose to write $\tilde U_1 = \lambda U_1$, $\tilde V_1 = \lambda V_1$, $\tilde U_2 = (1-\lambda) U_2$ and $\tilde V_2 = (1-\lambda) V_2$, and using the extensiveness ($S(\lambda U, \lambda V) = \lambda S(U,V)$) in the previous inequality, we get the condition of strict concavity of entropy. In a similar way for convexity of $U(S,V)$


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