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Here $s(m)=\frac{1+m}{2}\ln \frac{1+m}{2}+\frac{1-m}{2}\ln \frac{1-m}{2}$. This free-energy density is the energy minus temperature times entropy of independent sublattice Ising spins. Since $s^{\prime }(m)=\mathrm{atanh}m$, stationarity yields the two-sublattice Ising mean-field approximation. Writing $M_{\pm }=(M_B\pm M_W)/2$, the quadratic coefficients are $(k_{B}T+qJ)/2$ for $M_{+}^2$ and $(k_{B}T-qJ)/2$ for $M_{-}^2$. The positive quartic coefficient $k_{B}T/12$ in the staggered direction stabilizes the continuous transition. Unlike a log-cosh auxiliary potential, this variational expression can be minimized away from self-consistency.

In a regular thermodynamic formulation, a state at absolute zero cannot be reached from positive temperature by finitely many cooling operations with finite control resources. The entropy-difference form of the third law of thermodynamics makes regular entropy curves approach a common limiting entropy. An isentropic process starting above that entropy cannot intersect the common zero-temperature endpoint.

For an illustrative pair of settings with $S-S_0=a_{1}T$ and $S-S_0=a_{2}T$, $0<a_1<a_2$, an isothermal entropy-reduction step followed by reversible adiabatic cooling gives $T_{n+1}=(a_1/a_2)T_n$. Every finite number of stages leaves $T_n>0$. The model illustrates the entropy geometry; a general finite-time impossibility statement additionally needs assumptions about available interactions and cooling rates.

The harmonic-oscillator entropy contribution for one vibrational coordinate, with $x={\theta }_{\mathrm{vib}}/T$. Contributions from independent normal modes add, with one term for each component of a degenerate mode. A temperature-independent zero-point energy does not change this entropy.

A vibrational partition function sums over molecular vibrational energy levels. A single quantum harmonic oscillator, with energies measured relative to its ground level and vibrational temperature ${\theta }_v=h\nu /k$, has $q_{\mathrm{vib}}=(1-e^{-{\theta }_v/T}{)}^{-1}$. Retaining its absolute zero-point energy multiplies this by $e^{-{\theta }_v/(2T)}$. That factor shifts the internal energy but contributes no entropy.

Two water molecules associated by a donor O–H···O hydrogen bond. Gas-phase pair energy does not include the complete solvent and entropy changes governing association in solution.

Zero-point motion is the nonzero quantum fluctuation of position and momentum in a system's ground state. For a quantum harmonic oscillator, the ground state has energy $\hslash \omega /2$ and nonzero position and momentum variances even at zero temperature. It does not represent a thermal mixture and need not carry entropy. The large zero-point motion of light helium atoms helps explain why helium can remain a liquid at low pressure even as temperature approaches absolute zero.