Information-fluctuation inequalities for collective response

Abstract

Hidden stochastic effects acting uniformly on a many-particle system can generate strong correlations and macroscopic relative fluctuations that persist at large system sizes, even when the particles themselves remain causally independent. Here we derive a universal upper bound on relative fluctuations for a large class of observables, formulated in terms of a generalized mutual information between observable states and the hidden variable. This information-fluctuation inequality provides general insights into the principles governing collective response induced by global disorder. We demonstrate the result with applications to non-interacting Brownian gases exposed to various types of dynamical disorder.

Figures (3)

• Figure 1: a) Sketch of the system, where the density of Brownian particles respond to a random external force. b) Variance of the number fluctuations near the origin as a function of particle number. In panel b), dimensionless variables are used where lengths are set by $\sqrt{k_BT/\kappa}$, times by the relaxation timescale $\gamma/\kappa$, and characteristic force scale is set to $\sqrt{k_BT \kappa}$. This way only a dimensionless noise amplitude $\tilde{B}$ and timescale $\tilde{\tau}$ for the external Ornstein-Uhlenbeck forcing remains as independent parameters.
• Figure 2: a) Independent Brownian particles evolve for a common time $\tau$, after which a potential is activated. Each particle gains an energy $w_1(x_j)$. We use exponentially distributed $\tau$ with rates $\lambda$ as indicated. b) Comparison of the general finite-$n$ bound (dashed line) and Eq. (\ref{['eq:trap_exact2']}) for a harmonic potential. Dots show numerical simulations. c) Disordered potential used in panel d), showing numerical data from simulations using a disordered cosine potential with $20$ random contributions. Temporal units are chosen such that the trap relaxation time $\gamma/\kappa = 1$, and we set $D= 1/10$.
• Figure 3: Under expectation values, a difference in probability densities as measured by generalized divergences $D(p\parallel q)$ on the space of densities $\mathcal{L}(\Omega)$ induces a change in an observable quantity $\Delta_{p,q}\mathcal{O}$.