The fixed common temperature, rigid external container and fixed particle numbers on either side make the total Helmholtz free energy the appropriate thermodynamic potential. Let the barrier have cross-sectional area , and let increasing expand the left compartment. Then and . At fixed temperature and particle number, , soThus mechanical equilibrium, , makes stationary. The physical force on the barrier towards increasing is . If the barrier is displaced slightly from , strict mechanical stability requires a restoring force:ConsequentlyThe Taylor expansion proves that thermodynamic equilibrium is a strict local minimum of the Helmholtz free energy, rather than merely a stationary point.
For individually stable gases, the positive isothermal compressibilities make the sign especially transparent:If these inequalities hold throughout the accessible interval, is a strictly convex function and an interior stationary point is the unique global minimum of the Helmholtz free energy. Stability at one point alone proves only the local statement. A vanishing restoring derivative describes marginal stability and requires examining higher-order terms.
An additive-looking energy expression does not by itself establish two noninteracting subsystems. Their allowed microstates must also form a Cartesian product of independently available states. Here the second label's allowed values depend on the first. The four allowed pairs and their energies areThe canonical partition function, with , is thereforeThe conditional sum over is and depends on ; it cannot be pulled out as the partition function of an independent second subsystem. Summing independently over and all four values would invent four forbidden pairs and give eight states instead of four.
One can make the state labels independent by writing , but then contains a coupling between them. For genuinely additive independent energies, the difference between the and states would be independent of ; here it is . Hence the proposed and subsystems cannot be treated as noninteracting independent subsystems. The literal two-value reading of the PDF's endpoint notation has been used. If that notation were instead intended as shorthand for full angular-momentum multiplets, including every integer between the endpoints, the eight allowed states would giveThe conclusion about the failure of independent-subsystem factorization is unchanged. The two state-counting conventions should not be mixed.
A nonzero superconducting energy gap suppresses the population of thermally excited quasiparticles. When , their Boltzmann factor produces an approximately activated electronic heat capacity,where the prefactor varies algebraically rather than exponentially. For the rough estimate requested, neglect its change between the two quoted temperatures. Taking a ratio eliminates its unknown magnitude:ThereforeThe corresponding excitation energy is . The units of the two heat-capacity readings cancel in their ratio.
This is a heat-capacity activation estimate, not an exact determination of a gap from two data points. More generally, if , the same measurements giveFor example, the low-temperature BCS theory form for a fully gapped isotropic superconductor has and would give about . The stated approximate data and constant-gap assumption support the simpler estimate; they do not specify a detailed density of states or heat-capacity prefactor. Here means the minimum quasiparticle excitation energy, rather than automatically twice that value.
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