A Heuristic Study of Temperature: Quantum Circuitry in Thermal Systems

TL;DR

This work introduces Complexity Window Thermodynamics (CWT), a resource-aware generalization of thermodynamics that replaces the classical equiprobability assumption with a finite complexity budget $\Xi$ controlling quantum state accessibility. Central constructs include the complexity-windowed entropy $S_\Xi(E,\Xi)$, the windowed temperature $T_\Xi$, and the complexity generation potential $\Pi_\Xi$, which together yield an extended first law $\mathrm{d}E = T_\Xi \mathrm{d}S_\Xi + \Pi_\Xi \mathrm{d}\Xi$ with an information-processing work term. The theory shows that a finite $\Xi$ regularizes classical singularities in phase transitions and negative temperatures, derives physical resource constraint principles linking complexity to action and time, and provides a dynamical bound on the rate of complexity growth tied to $T_\Xi$ (and $\Pi_\Xi$). It further demonstrates cross-disciplinary applications in condensed matter, gravity, and computation, and even extends these ideas to cosmology, where the universe’s maximal generatable complexity is argued to be of the same order as holographic horizon entropy, suggesting deep links between computation, information, and fundamental physics.

Abstract

The singularities prevalent in classical thermodynamics largely stem from the "postulate of equal a priori probabilities" neglecting the physical constraints imposed by computational complexity. This paper introduces Complexity Window Thermodynamics (CWT), a framework that characterizes the observer's "ignorance" via a finite complexity budget, thereby naturally smoothing out singular behaviors associated with phase transitions and negative temperatures within this window. We derive a generalized First Law of Thermodynamics driven by a complexity generation potential, which incorporates "information processing work," and demonstrate a universal action-time bound constraining the growth of complexity. CWT not only offers a unified perspective on critical phenomena in condensed matter and the black hole information problem but also suggests that the total generatable complexity of the universe is comparable in order of magnitude to its holographic entropy. Thus, it paves a new pathway for a resource-theoretic unification of thermodynamics, quantum computation, and gravity.