Network analysis for steady-state current fluctuations under finite affinity: Application to Brownian computation
Abstract
A graph-theoretic analysis of the steady-state current noise in master equations under a finite thermodynamic force (affinity) is presented. The incidence matrix twisted by a finite affinity is not orthogonal to the standard cycle space, motivating the introduction of twisted circuit matrices to restore the orthogonality. The resulting twisted-cycle matrix yields an interference-like effect, enabling us to express the signal-to-noise ratio as a quadratic optimization problem in terms of twisted-cycle currents. We apply this framework to a Brownian computation model on a tree-like state-transition diagram with exponential backward branching, finite affinity at each step, and a single reset cycle. In the limit of an infinitely long intended computation path ℓ, the Fano factor of the reset current undergoes a transition from noiseless to Poissonian behavior at an affinity equal to the logarithm of the number of immediate predecessors α. This corresponds to an easy-hard transition in the computational time complexity [K. Okajima, K. Hukushima, arXiv:2512.24728 ], which is not captured by the thermodynamic uncertainty relation. This transition point precisely characterizes the thermodynamic costs of logically irreversible computation: in the absence of affinity, the reset cost scales as ln ℓ, whereas reaching the transition point requires a thermodynamic force of order ln α per step to counteract backward branching.
Versions
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BibTeX
@misc{utsumi2026network,
title = {{Network analysis for steady-state current fluctuations under finite affinity: Application to Brownian computation}},
author = {Yasuhiro Utsumi},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2605.17838},
eprint = {2605.17838},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2605.17838},
}
RIS
TY - UNPB
TI - Network analysis for steady-state current fluctuations under finite affinity: Application to Brownian computation
AU - Yasuhiro Utsumi
PY - 2026
DA - 2026/10/04
PB - arXiv
DO - 10.48550/arxiv.2605.17838
UR - https://arxiv.org/abs/2605.17838
AB - A graph-theoretic analysis of the steady-state current noise in master equations under a finite thermodynamic force (affinity) is presented. The incidence matrix twisted by a finite affinity is not orthogonal to the standard cycle space, motivating the introduction of twisted circuit matrices to restore the orthogonality. The resulting twisted-cycle matrix yields an interference-like effect, enabling us to express the signal-to-noise ratio as a quadratic optimization problem in terms of twisted-cycle currents. We apply this framework to a Brownian computation model on a tree-like state-transition diagram with exponential backward branching, finite affinity at each step, and a single reset cycle. In the limit of an infinitely long intended computation path ℓ, the Fano factor of the reset current undergoes a transition from noiseless to Poissonian behavior at an affinity equal to the logarithm of the number of immediate predecessors α. This corresponds to an easy-hard transition in the computational time complexity [K. Okajima, K. Hukushima, arXiv:2512.24728 ], which is not captured by the thermodynamic uncertainty relation. This transition point precisely characterizes the thermodynamic costs of logically irreversible computation: in the absence of affinity, the reset cost scales as ln ℓ, whereas reaching the transition point requires a thermodynamic force of order ln α per step to counteract backward branching.
ER -