Friend asked me this question and I didn't manage to solve it with basic thermodynamic reasoning. I believe this problem is easly solvable via numeric methods by choosing specific systems, though I prefer an analytic, more general and more intuitive solution.
Two different and isolated systems (which specified by S1(E1,V1,N1) and S2(E2,V2,N2)) are seperatly prepeard to satisfy particular (P,T) requirements, so that P1=P2=P but T1≠T2. Afterwards the two systems are brough one near the other, with a single piston (unmovable at first) seperating them. The piston doesn't allow transfer of heat or particles at any stage. After the two systems were properly juxtaposed the restriction on the movement of the piston is removed. Will the piston move from its original position?
One way of treatment suggests that since P1=P2 and and since only mechanichal work (exchange) is allowed - the piston will not move.
Other way sugest that by forcing maximum entropy (thermodynamic equilibrium) for the combined system, we will get dStot=dS1+dS2=0, and in particular (since there is only one degree of freedom here) ∂S1∂V1=∂S2∂V1 so at equilibrium P1T1=P2T2, hence the piston will move.
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