Sub-Gaussian and subexponential fluctuation-response inequalities

Abstract: Sub-Gaussian and subexponential distributions are introduced and applied to study the fluctuation-response relation out of equilibrium. A bound on the difference in expected values of an arbitrary sub-Gaussian or subexponential physical quantity is established in terms of its sub-Gaussian or subexponential norm. Based on that, we find that the entropy difference between two states is bounded by the energy fluctuation in these states. Moreover, we obtain generalized versions of the thermodynamic uncertainty relation in different regimes. Operational issues concerning the application of our results in an experimental setting are also addressed, and nonasymptotic bounds on the errors incurred by using the sample mean instead of the expected value in our fluctuation-response inequalities are derived.

• The paper extends fluctuation-response inequalities to account for sub-Gaussian and subexponential fluctuations, effectively bridging Gaussian and non-Gaussian regimes.
• It rigorously derives bounds linking observable differences to concentration norms and relative entropy, highlighting implications for thermodynamic uncertainty relations.
• The analysis offers practical finite-sample concentration inequalities and estimator frameworks, paving the way for robust experimental validation in nonequilibrium systems.

Sub-Gaussian and Subexponential Fluctuation-Response Inequalities: A Technical Summary

Introduction and Motivation

This work rigorously extends fundamental fluctuation-response inequalities in nonequilibrium statistical physics to regimes governed by sub-Gaussian and subexponential distributions. While linear response theory, underpinned by Gaussian fluctuations, forms the basis of equilibrium physics, real-world nonequilibrium processes often exhibit non-Gaussian, heavy-tailed, or skewed statistics. Sub-Gaussian and subexponential classes encompass these phenomena and thus enable a systematic treatment of nonlinear response.

The paper establishes sharp inequalities relating the expectation differences of physical observables (measured under two probability distributions, e.g., perturbed vs. reference state) to their concentration properties, measured through sub-Gaussian and subexponential norms. These results generalize classical concentration inequalities to operationally relevant settings in statistical physics, with explicit connections to entropy production, thermodynamical uncertainty relations (TURs), and pathwise fluctuation theorems.

Sub-Gaussian and Subexponential Distributions in Statistical Physics

A centered random variable $X$ is categorized as sub-Gaussian if its moment generating function is bounded (for all $s\in\mathbb{R}$) by that of a Gaussian with some proxy variance $\sigma^2$, formalized as $\mathbb{E}e^{sX} \leq e^{s^2\sigma^2/2}$. The infimum $\sigma$ for which this holds defines the sub-Gaussian norm. Subexponential random variables relax this requirement, controlling tails with an exponential envelope over a bounded interval of $s$, resulting in the subexponential norm. The hierarchy is strict: Gaussian $\subset$ sub-Gaussian $\subset$ subexponential.

Empirically, these distribution classes manifest in a variety of nonequilibrium processes where higher-order cumulants can't be neglected and path-level quantities like stochastic work and entropy production exhibit non-Gaussian statistics (e.g., the Crooks and Jarzynski relations for work in single-molecule experiments). Gamma and chi-squared distributed work or heat variables are prototypical examples where subexponential, not Gaussian, bounds control the error rates.

Central Fluctuation-Response Inequalities

The main technical results extend the fluctuation-response inequality of Dechant and Sasa (PNAS 2020) to the sub-Gaussian and subexponential regimes. For arbitrary random variables $X$ with sub-Gaussian norm $\|X\|_G$ (computed relative to a target distribution), for two distributions $P_0$ and $P_1$ with finite Kullback-Leibler divergence, the inequality

$$|\mathbb{E}_1 X - \mathbb{E}_0 X| \leq \|X-\mathbb{E}_1 X\|_G \, \sqrt{2 D_{KL}(P_0\|P_1)}$$

holds. For subexponential variables with norm $\|\cdot\|_E$, an analogous statement is valid under a regime constraint on $D_{KL}$.

The bound is notable in that it does not require explicit knowledge of higher cumulants or the process’ detailed dynamics—only a bound on concentration via the sub-Gaussian (or subexponential) norm and the relative entropy between the distributions.

For physical systems where $P_0$ and $P_1$ are Boltzmann distributions corresponding to different Hamiltonians, $D_{KL}$ reduces to combinations of entropy difference and energy fluctuations, tightening the connection between response, fluctuations, and entropy production. When considering forward and backward trajectories (as in fluctuation theorems), the KL divergence becomes the total entropy production, rendering the bound a universal nonlinear TUR.

Generalized Thermodynamic Uncertainty Relations (TURs)

A rigorous corollary is a nonlinear TUR: $$2 (\mathbb{E}X)^2 \leq \|X - \mathbb{E}X\|_G^2 \Delta S$$ where $\Delta S$ is total entropy production, holding for time-antisymmetric observables in sub-Gaussian regimes. This generalizes the standard TUR—previously proven only for specific Markovian dynamics or under normality assumptions—by permitting broader classes of random variables. The result is also contrasted with alternative TURs, e.g., those incorporating $e^{\Delta S}-1$ dependence on entropy.

For stochastic processes or path observables governed by discrete-time Markov chains (e.g., the finite-state ring model), this TUR remains valid as long as the process increments are sub-Gaussian, even when the variance-based TUR fails. For instance, explicit upper bounds on the sub-Gaussian norm can be derived, revealing transitions between tightness regimes compared to variance-based approaches.

Concentration Inequalities and Empirical Implications

The work derives non-asymptotic, finite-sample bounds for estimates of expectation differences using empirical averages, crucial for experimental protocols with limited data. The Hoeffding-type and Bernstein-type concentration inequalities characterize the probability that empirical deviations exceed theoretical bounds, directly in terms of sub-Gaussian or subexponential norms.

Estimators (plug-in and empirical concentration-based) for these norms are proposed, with numerical evidence for convergence and bias under realistic sampling scenarios. The practical utility and robustness of bounds in experimental settings are discussed, with caveats regarding estimator tightness, especially in high-variance or small-sample regimes.

Implications and Outlook

The presented framework enables a distribution-class-based analysis of nonequilibrium response and fluctuation limits, replacing restrictive Gaussian assumptions by verifiable (or, at minimum, conservative) concentration norm bounds. This permits the derivation of system-independent but distribution-class-specific constraints on measurable thermodynamic quantities, rendering the approach broadly applicable across wide classes of stochastic models, including under weakly specified or empirically determined systems.

On the theoretical front, these results motivate new investigations into the structure and tightness of nonequilibrium inequalities under various tail behaviors, as well as further exploration of norm estimation, minimax bounds, and the role of Wasserstein distances in statistical mechanics. Future directions include systematic analysis of the statistical properties of norm estimators and experimental verification in small systems, as well as theoretical work extending to the large deviations and moderate deviations regimes.

Conclusion

This paper rigorously extends fluctuation-response and uncertainty relations beyond the Gaussian paradigm, establishing operationally meaningful upper bounds for response and observable fluctuations in terms of sub-Gaussian and subexponential norms. This broadens the applicability of fluctuation theorems and thermodynamic uncertainty relations to a much wider range of nonequilibrium phenomena and experimental situations. The theoretical constructs provide a flexible yet rigorous basis for quantifying fluctuations, responses, and entropy production in systems far from equilibrium, under minimal model assumptions.