Relativity of Observation: Operational Intensive Variables in Nonequilibrium Thermodynamics

TL;DR

This work reframes nonequilibrium thermodynamics by treating intensive variables as operational parameters defined through an inference-based procedure relative to a reference distribution $Q(\omega)$, yielding $P_\mathbf{m}^*(\omega)=\frac{1}{Z(\boldsymbol{\lambda})}Q(\omega)\exp[-\sum_i \lambda_i M_i(\omega)]$ with conjugates $\lambda_i(\mathbf{m})=-\partial_{m_i} D(P_\mathbf{m}^*\|Q)$. It shows that a continuity equation $\partial_{m_i} P_\mathbf{m}^*(\omega)=-\partial_\omega J_i^\mathbf{m}(\omega)$ and a parallel definition of intensive observables $\Lambda_i^\mathbf{m}(\omega)$, satisfying $\langle \Lambda_i^\mathbf{m}\rangle_{P_\mathbf{m}^*}=\lambda_i$, enable direct measurement of conjugate variables via $\Lambda_i^\mathbf{m}(\omega)=-\frac{J_i^\mathbf{m}(\omega)}{P_\mathbf{m}^*(\omega)}\partial_\omega\ln\frac{P_\mathbf{m}^*(\omega)}{Q(\omega)}$. Onsager reciprocity is reinterpreted as a local flatness condition on the information manifold, expressed by $\partial_{m_i} J_j^\mathbf{m}-\partial_{m_j} J_i^\mathbf{m}=0$, valid in the tangent space and leading to a gauge-fixed, operational framework where $\boldsymbol{\lambda}$ are observable. An experimental protocol is proposed to realize these local inertial frames by calibrating at equilibrium, perturbing constraints, and tuning measurement devices to maintain reciprocity, thereby providing a practical route to access nonequilibrium intensive variables with potential extensions to fluctuation theory and quantum measurements.

Abstract

We formulate nonequilibrium thermodynamics in which intensive variables acquire operational meaning through measurement protocols consistent with local reciprocity. Using physical equilibrium as a reference, conjugate observables are constructed by continuously adjusting devices along the local tangent space of the statistical manifold. In this relativity of observation, Onsager reciprocity holds locally, allowing inference-based Lagrange multipliers to be directly measured. This provides a systematic method to extend operational definitions of intensive variables to nonequilibrium states, highlighting their context-dependent nature and offering a concrete experimental strategy.