Entropy maximization implies equality of temperature

Statement (temperature equality from entropy maximization)

Let two weakly interacting subsystems 1 and 2 form an isolated composite system. Let $U_i$ be the internal energy of subsystem $i$, and let $S_i(U_i,V_i,N_i)$ be its thermodynamic entropy (with $V_i,N_i$ fixed).

Assume:

• the subsystems can exchange energy but not volume or particles,

• the total energy is fixed: $U_1+U_2=U_{\mathrm{tot}}$,

• the total entropy is additive:

  $$S_{\mathrm{tot}}(U_1)= S_1(U_1,V_1,N_1) + S_2(U_{\mathrm{tot}}-U_1,V_2,N_2),$$

• $S_1,S_2$ are differentiable in $U$ and the equilibrium corresponds to an interior maximizer of $S_{\mathrm{tot}}$.

Then at equilibrium,

Using the definition of temperature ,

it follows that the equilibrium condition is

Key hypotheses

• Two subsystems with negligible interaction energy so that entropy is additive.
• Exchange of energy only, with fixed $(V_i,N_i)$.
• Differentiability of $S_i$ in $U_i$.
• Equilibrium characterized by a (local) maximum of $S_{\mathrm{tot}}$ at fixed $U_{\mathrm{tot}}$.

Key conclusions

• Equality of temperatures is the equilibrium condition for thermal contact:

  $$T_1=T_2.$$

• If, additionally, the entropies are strictly concave in $U$ (a stability condition), the entropy maximizer (hence the equilibrium energy split) is unique and stable.

Cross-links to relevant definitions

• Entropy and its equilibrium role: thermodynamic entropy , thermodynamic equilibrium .
• Temperature as an intensive variable: temperature .
• Internal energy: internal energy .
• Stability/concavity viewpoint: thermodynamic stability .

Proof idea / significance (sketch)

Maximize $S_{\mathrm{tot}}(U_1)$ subject to $U_2=U_{\mathrm{tot}}-U_1$. At an interior maximizer,

Identifying $(\partial S/\partial U)_{V,N}$ with $1/T$ yields $T_1=T_2$. This is the thermodynamic origin of temperature as the intensive parameter that equalizes under thermal contact.