Network analysis for steady-state current fluctuations under finite affinity: Application to Brownian computation
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866909052978593792 |
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| author | Utsumi, Yasuhiro |
| author_facet | Utsumi, Yasuhiro |
| contents | 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 $\ell$, 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 $\ln \ell$, whereas reaching the transition point requires a thermodynamic force of order $\ln α$ per step to counteract backward branching. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_17838 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Network analysis for steady-state current fluctuations under finite affinity: Application to Brownian computation Utsumi, Yasuhiro Statistical Mechanics Mesoscale and Nanoscale Physics 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 $\ell$, 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 $\ln \ell$, whereas reaching the transition point requires a thermodynamic force of order $\ln α$ per step to counteract backward branching. |
| title | Network analysis for steady-state current fluctuations under finite affinity: Application to Brownian computation |
| topic | Statistical Mechanics Mesoscale and Nanoscale Physics |
| url | https://arxiv.org/abs/2605.17838 |