Thermodynamic incompleteness of state dynamics in Markovian transport

TL;DR

This paper reveals that Markovian state dynamics alone are insufficient for complete thermodynamic predictions in transport systems, highlighting the need for reservoir-resolved transport records.

Contribution

It introduces a thermodynamic completeness criterion showing when state dynamics can fully determine thermodynamic observables in Markovian transport.

Findings

Different reservoir assignments can produce identical state dynamics but different heat currents.

State tomography alone cannot predict heat noise and cross-correlations in Markovian transport.

A practical diagnostic for the completeness of Markovian transport models is proposed.

Abstract

Markovian transport is often described by a master equation for the system state. The thermodynamic information measured in transport experiments, however, is carried by reservoir-resolved transfer records, such as particle currents, heat currents, entropy production, and current noise. We identify a thermodynamic incompleteness of state dynamics: a Markovian state generator can fix the occupation probabilities, stationary response, and relaxation without specifying how the underlying transitions are assigned to reservoirs and energy filters. We study a multi-terminal Coulomb-blockaded quantum dot coupled to energy-filtered reservoirs, for which different assignments of reservoir channels can generate the same state master equation. These assignments give identical occupation dynamics, stationary state, and linear response of the dot, but different heat currents, entropy production, and current noise. We formulate a thermodynamic completeness criterion: a transport observable can be reconstructed from state dynamics only when it is invariant under all changes of reservoir-channel assignments that leave the state generator unchanged. The criterion gives a practical diagnostic for Markovian transport models and a measurable prediction: state tomography can be insufficient to predict heat-noise and cross-correlation measurements, even when the full Markovian state dynamics is known. The analysis identifies a concrete limitation of state-only Markovian thermodynamics and shows which additional transport records must be specified to make thermodynamic predictions experimentally complete.