Effective Thermodynamics of a Volume-Dependent Qubit

Abstract

We develop an effective thermodynamics for a qubit whose energy gap depends inversely on a volume-like control parameter, $Δ(V)=Δ_0/V$, where $V$ is a spectral-control coordinate rather than a literal mechanical volume. Starting from the canonical Gibbs state, we derive a complete thermodynamic structure from the Helmholtz potential and its Legendre and stability properties, including the fundamental relation $S(U,V)$, a generalized pressure conjugate to $V$, the equation of state $U=PV$, the first law, and the associated Maxwell relations. Reversible adiabats preserve the Gibbs populations and satisfy $TV=\mathrm{const.}$ and $PV^2=\mathrm{const.}$, corresponding to an effective exponent $γ=2$. The resulting effective Carnot cycle attains the Carnot efficiency, $η_{\mathrm{Carnot}}=1-T_C/T_H$, whereas the Otto cycle yields $η_{\mathrm{Otto}}=1-V_{\min}/V_{\max}$ and remains strictly sub-Carnot at finite work output. We identify the source of Otto irreversibility exactly: entropy is produced entirely during the isochoric thermalizations, and the cycle entropy production quantifies the informational mismatch between the adiabatically transported states and the reservoir-selected Gibbs states through the sum of the corresponding quantum relative entropies. Finally, we show that unitary-generated nonpassivity can act as a nonequilibrium battery, producing an apparent efficiency above Carnot when its preparation cost is omitted. At the same time, complete work accounting restores the thermodynamic bound.