Spectral Softening and the Structural Breakdown of Thermodynamic Equilibrium

TL;DR

This paper demonstrates that spectral softening in driven quadratic Hamiltonian systems causes a fundamental breakdown of thermodynamic reversibility by destroying the equilibrium state structure, even under slow driving.

Contribution

It reveals that spectral softening leads to a divergence of the partition function and the loss of adiabaticity, challenging the assumptions of quasistatic thermodynamics.

Findings

Spectral softening causes divergence of the partition function.

Adiabatic following fails once the soft-mode frequency drops below a threshold.

The breakdown of thermodynamic reversibility is linked to spectral softening, not critical slowing down.

Abstract

Under sufficiently slow driving, thermodynamics predicts reversible evolution through a sequence of equilibrium states. We show that this expectation fails near spectral degeneracy in driven quadratic Hamiltonian systems. As the soft-mode frequency collapses, the intrinsic dynamical timescale diverges and quadratic confinement is lost, leading to a breakdown of timescale separation and the failure of adiabatic following even under arbitrarily slow driving. More precisely, adiabaticity is lost once the soft-mode frequency falls below a finite, drive-dependent threshold, implying that the breakdown extends over a finite regime rather than being confined to a singular limit. Crucially, this dynamical instability is accompanied by a divergence of the canonical partition function, rendering equilibrium ensembles ill-defined and eliminating the foundation of quasistatic thermodynamic processes. This breakdown does not arise from unbounded Hamiltonians or critical slowing down, but emerges structurally from spectral softening within a bounded quadratic system. Analysis of the Wigner phase-space representation, together with its classical counterpart, reveals the same singular structure, demonstrating that this limitation is not uniquely quantum but originates from the underlying Hamiltonian phase-space geometry. These results show that thermodynamic reversibility is fundamentally constrained, as a direct consequence of the breakdown of equilibrium, whenever spectral softening removes an intrinsic frequency scale.