Fluctuating Hydrodynamics and Wormholes

Abstract

We show that a recent reformulation of hydrodynamic equations for a large class of models consisting of q-dits on a graph with short range interactions is sufficient for understanding chaotic behavior. Any such system consists of large subsystems coupled together by interactions whose relative strength goes to zero with the subsystem size. In the absence of conservation laws other than energy, the Hamiltonians of the subsystems form a complete set of commuting operators. The hydrodynamic variables are the block diagonal matrix elements $ρ(e(X))$ of the density matrix in the joint eigenbasis of the subsystem Hamiltonians, averaged over energy bins. To leading order in the inverse subsystem size, $ρ(e(X); t)$ satisfies a classical stochastic equation, which for certain systems takes the form of a functional Fokker-Planck equation. In such systems the time averaged spectral form factors can be written as a two dimensional Euclidean functional integral, on a space with multiple disconnected boundaries. The failure of factorization in this representation is attributable to the time averaging necessary to apply the hydrodynamic approximation. The bulk Euclidean action is purely topological. We make tentative explorations of the special properties of the system that are required in order to have a representation as a functional integral over metrics.