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We investigate a dynamical system consisting of $N$ particles moving on a $d$-dimensional torus under the action of an electric field $E$ with a Gaussian thermostat to keep the total energy constant. The particles are also subject to stochastic collisions which randomize direction but do not change the speed. We prove that in the van Hove scaling limit, $Eto 0$ and $tto t/E^2$, the trajectory of the speeds $v_i$ is described by a stochastic differential equation corresponding to diffusion on a constant energy sphere. This verifies previously conjectured behavior. Our results are based on splitting the systems evolution into a slow process and an independent noise. We show that the noise, suitably rescaled, converges a Brownian motion, enhanced in the sense of rough paths. Then we employ the It^o-Lyons continuity theorem to identify the limit of the slow process.
Dynamical systems with $epsilon$ small random perturbations appear in both continuous mechanical motions and discrete stochastic chemical kinetics. The present work provides a detailed analysis of the central limit theorem (CLT), with a time-inhomoge
We present a variational formulation for the Navier-Stokes-Fourier system based on a free energy Lagrangian. This formulation is a systematic infinite dimensional extension of the variational approach to the thermodynamics of discrete systems using t
This paper compares the results of applying a recently developed method of stochastic uncertainty quantification designed for fluid dynamics to the Born-Infeld model of nonlinear electromagnetism. The similarities in the results are striking. Namely,
A potential representation for the subset of traveling solutions of nonlinear dispersive evolution equations is introduced. The procedure involves a reduction of a third order partial differential equation to a first order ordinary differential equat
We perform the study of the stability of the Lorenz system by using the Jacobi stability analysis, or the Kosambi-Cartan-Chern (KCC) theory. The Lorenz model plays an important role for understanding hydrodynamic instabilities and the nature of the t