Stochastic Diversity Stabilization in Nonlinear Competitive Dynamics: A Statistical Mechanics Approach to Fluctuation Suppression and Finite-Size Scaling
Abstract
Stochastic forcing can prevent diversity collapse in competitive dynamical systems, yet the quantitative conditions and the dynamical mechanism responsible remain poorly characterized. We address both questions in a class of nonlinear stochastic replicator dynamics on the N-dimensional probability simplex, parameterized by a nonlinearity exponent gamma and additive Gaussian noise of intensity sigma. Under the baseline interaction, the minimum noise required to sustain diversity decays as a power law, sigma_c ~ N^(-alpha); replacing the baseline interaction with a superlinear variant shifts alpha from about 1.1 to about 2.0, in agreement with a linear-stability prediction alpha = (gamma+2)/2, identifying alpha as a model-specific quantity rather than a universal index. Scanning the (gamma, sigma) plane reveals a smooth dynamical crossover: the fluctuation susceptibility chi_H = N Var(H) peaks at gamma ~ 0.7 (chi_H ~ 0.6) and is suppressed at high gamma, with no divergence near gamma ~ 1.4; matched finite-size analysis finds no systematic divergence of the peak with N, ruling out a genuine second-order phase transition. The entropy relaxation time tau_relax(gamma) remains approximately constant (about 2.9 time units) across the full range of gamma, even as chi_H decreases sixfold; this decoupling between relaxation timescale and fluctuation amplitude implies that suppression arises from convergence of diverse pathways to similar endpoints rather than from accelerated relaxation along a single channel. These findings identify nonlinear fluctuation suppression via pathway multiplicity -- distinct from both noise-induced transitions and equilibrium criticality -- as the operative stabilization pattern in this class of nonequilibrium stochastic systems.
Versions
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BibTeX
@misc{ou2026stochastic,
title = {{Stochastic Diversity Stabilization in Nonlinear Competitive Dynamics: A Statistical Mechanics Approach to Fluctuation Suppression and Finite-Size Scaling}},
author = {Ruocun Ou},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.30169},
eprint = {2610.30169},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.30169},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Stochastic Diversity Stabilization in Nonlinear Competitive Dynamics: A Statistical Mechanics Approach to Fluctuation Suppression and Finite-Size Scaling
AU - Ruocun Ou
PY - 2026
DA - 2026/10/16
PB - arXiv
DO - 10.48550/arxiv.2610.30169
UR - https://arxiv.org/abs/2610.30169
AB - Stochastic forcing can prevent diversity collapse in competitive dynamical systems, yet the quantitative conditions and the dynamical mechanism responsible remain poorly characterized. We address both questions in a class of nonlinear stochastic replicator dynamics on the N-dimensional probability simplex, parameterized by a nonlinearity exponent gamma and additive Gaussian noise of intensity sigma. Under the baseline interaction, the minimum noise required to sustain diversity decays as a power law, sigma_c ~ N^(-alpha); replacing the baseline interaction with a superlinear variant shifts alpha from about 1.1 to about 2.0, in agreement with a linear-stability prediction alpha = (gamma+2)/2, identifying alpha as a model-specific quantity rather than a universal index. Scanning the (gamma, sigma) plane reveals a smooth dynamical crossover: the fluctuation susceptibility chi_H = N Var(H) peaks at gamma ~ 0.7 (chi_H ~ 0.6) and is suppressed at high gamma, with no divergence near gamma ~ 1.4; matched finite-size analysis finds no systematic divergence of the peak with N, ruling out a genuine second-order phase transition. The entropy relaxation time tau_relax(gamma) remains approximately constant (about 2.9 time units) across the full range of gamma, even as chi_H decreases sixfold; this decoupling between relaxation timescale and fluctuation amplitude implies that suppression arises from convergence of diverse pathways to similar endpoints rather than from accelerated relaxation along a single channel. These findings identify nonlinear fluctuation suppression via pathway multiplicity -- distinct from both noise-induced transitions and equilibrium criticality -- as the operative stabilization pattern in this class of nonequilibrium stochastic systems.
ER -