KeywordsHydrodynamic fluctuations–Laminar fluid flow–Plane Couette flow–Orr-Sommerfeld equation We demonstrate the presence of aįlow-induced enhancement of the wall-normal velocity fluctuations and a resulting flow-induced energy amplification and provideĪ quantitative analysis how these quantities depend on wave number and Reynolds number. We show how an exact solution can be obtained fromĪn expansion in terms of the eigenfunctions of the Orr-Sommerfeld hydrodynamic operator. Specifically, fluctuating hydrodynamics yields a stochastic Orr-SommerfeldĮquation for the wall-normal velocity fluctuations, where spontaneous thermal noise acts as a random source.This stochasticĮquation needs to be solved subject to appropriate boundary conditions. Using plane Couette flow as a representative example. In this paper we consider the application of fluctuating hydrodynamics to laminar fluid flow, In recent years it has become evident that fluctuating hydrodynamics predicts that fluctuations in nonequilibrium states areĪlways spatially long ranged. Furthermore, we numerically show that the typical decay behaves asymptotically, for long times, as |A(t)|∼e−Dt in the same random circuit as well as in a prototypical nonintegrable model with diffusive energy transport but no disorder. We prove the following inequality for the disorder average of the amplitude, |A(t)|¯≥e−Dt, in a local spin-12 random circuit with a U(1) conservation law by mapping to the survival probability of a symmetric exclusion process. As long as |A(t)|≥e−Dt, which we argue holds true for generic diffusive nonintegrable systems, all nth Rényi entropies with n>1 (annealed averaged over initial product states) are bounded from above by t. To understand this phenomenon, we introduce an amplitude A(t), which is the overlap of the time evolution operator U(t) of the entire system with the tensor product of the two evolution operators of the subsystems of a spatial bipartition. Recent studies found that the diffusive transport of conserved quantities in nonintegrable many-body systems has an imprint on quantum entanglement: while the von Neumann entropy of a state grows linearly in time t under a global quench, all nth Rényi entropies with n>1 grow with a diffusive scaling t. Recent studies found that the diffusive transport of conserved quantities in non-integrable many-body systems has an imprint on quantum entanglement: while the von-Neumann entropy of a state grows linearly in time $t$ under a global quench, all $n$th R\'enyi entropies with $n > 1$ grow with a diffusive scaling $\sqrt $ in the same random circuit as well as in a prototypical non-integrable model with diffusive energy transport but no disorder.
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