We consider a lattice gas on the discrete d-dimensional torus$(\mathbb{Z}/N\mathbb{Z})^d$ with a generic translation invariant, finite rangeinteraction satisfying a uniform strong mixing condition. The lattice gasperforms a Kawasaki dynamics in the presence of a weak external field E/N. Weshow that, under diffusive rescaling, the hydrodynamic behavior of the latticegas is described by a nonlinear driven diffusion equation. We then prove theassociated dynamical large deviation principle. Under suitable assumptions onthe external field (e.g., E constant), we finally analyze the variationalproblem defining the quasi-potential and characterize the optimal exittrajectory. From these results we deduce the asymptotic behavior of thestationary measures of the stochastic lattice gas, which are not explicitlyknown. In particular, when the external field E is constant, we prove astationary large deviation principle for the empirical density and show thatthe rate function does not depend on E.
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机译：我们考虑离散d维圆环$（\ mathbb {Z} / N \ mathbb {Z}）^ d $上的晶格气，它具有通用的平移不变，有限范围的相互作用，满足均匀的强混合条件。在弱外部电场E / N的存在下，晶格气体会执行川崎动力学。我们表明，在扩散重标度下，晶格气体的流体动力学行为由非线性驱动扩散方程描述。然后证明了相关的动力学大偏差原理。在外部场的适当假设下（例如E常数），我们最终分析定义准电势的变分问题并表征最佳出口轨迹。从这些结果中，我们推导出了随机晶格气的平稳度量的渐近行为，这是未知的。特别地，当外场E恒定时，我们证明了经验密度的平稳大偏差原理，并证明了速率函数不依赖于E。
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