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Network analysis for steady-state current fluctuations under finite affinity: Application to Brownian computation

Yasuhiro Utsumi

  • Department of Electrical and Electronic Engineering, Faculty of Engineering, Mie University, Tsu, Mie 514-8507, Japan

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Phys. Rev. E 114, 044104 – Published 2 October, 2026

DOI: https://doi.org/10.1103/32qy-2v16

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 interferencelike 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 treelike 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 and 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 lnℓ, whereas reaching the transition point requires a thermodynamic force of order lnα per step to counteract backward branching.

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Outline

Information

Phys. Rev. E 114, 044104– Published 2 October, 2026
  • Received 17 May 2026
  • Accepted 18 September 2026
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