Nested superstatistics: When the fluctuation parameter fluctuates
Year: 2026
Type: article
Abstract: Many nonequilibrium systems exhibit a hierarchy of well-separated dynamical scales. Typically, microscopic variables such as energy or velocity evolve rapidly, whereas environmental parameters such as the inverse temperature β fluctuate more slowly. This separation naturally leads to superstatistics, in which local equilibrium factors are averaged over fluctuations of β , whose relative strength can be characterized by q = 〈 β 2 〉 / 〈 β 〉 2 . Here, we introduce an additional hierarchical level by allowing q itself to fluctuate on a still slower scale. Physically, such fluctuations may arise when different regions or time intervals are characterized by different fluctuation intensities, correlation structures, or effective numbers of microscopic degrees of freedom. The resulting nested construction produces effective distributions that depend on the full distribution g ( q ) and, in the weak-fluctuation regime, reduce to a universal correction governed only by the mean and variance of q . Using a Langevin model, we show explicitly that the nested construction emerges dynamically when the three relevant timescales are sufficiently well separated, and we provide a method for reconstructing g ( q ) from a time series of the fast variable. Finally, we apply the theory to solar-wind electron distributions and cosmic-ray electron spectra, and identify observational situations in which the systematic deviations from fixed- q superstatistics are consistent with a fluctuating superstatistical parameter.
Source: Physical Review Research
DOI: 10.1103/dtcy-db7k
Author: Kamel Ourabah
Language: English
Subfield: Statistical and Nonlinear Physics
Field: Physics and Astronomy
Domain: Physical Sciences
Open Access status: gold
Estimated APC paid: $2,625
OpenAlex ID: W7222985212