非平衡Stationary States中的拟势或Hamilton-Jacobi方程显式解的一种方法
Quasi-potentials in the Nonequilibrium Stationary States or a method to get explicit solutions of Hamilton-Jacobi equations
摘要：
我们假设一个系统在介观尺度描述的场$ϕ(x,t)$的朗之万方程的白噪声，其强度由一个参数$1/\sqrtΩ$控制的演变。在小噪声极限（$Ω\rightarrow\infty$）下的系统定态分布具有$P_{st}[ϕ]\simeq\exp(-ΩV_0[ϕ])$的形式，其中$V_0[ϕ]$称为{ \it准势}。$V_0$是Hamilton-Jacobi方程的未知数。因此，$V_0$可以被写为沿着路径计算的动作，该路径是通常不能显式求解的汉密尔顿方程的解。本文提出了一种理论方案，建立一个合适的规范变换，使我们能够这样做的整合变形成一条直线的原始路径。我们表明，这是可以做到的，当一组条件的规范转换和模型的动态满足。在这种情况下，我们可以用代数方法得到拟势。我们将该格式应用于扩散和反应扩散系统等几个一维非平衡模型。
英文摘要：
We assume that a system at a mesoscopic scale is described by a field $ϕ(x,t)$ that evolves by a Langevin equation with a white noise whose intensity is controlled by a parameter $1/\sqrtΩ$. The system stationary state distribution in the small noise limit ($Ω\rightarrow\infty$) is of the form $P_{st}[ϕ]\simeq\exp(-ΩV_0[ϕ])$ where $V_0[ϕ]$ is called the {\it quasipotential}. $V_0$ is the unknown of a Hamilton-Jacobi equation. Therefore, $V_0$ can be written as an action computed along a path that is the solution from Hamilton's equation that typically cannot be solved explicitly. This paper presents a theoretical scheme that builds a suitable canonical transformation that permits us to do such integration by deforming the original path into a straight line. We show that this can be done when a set of conditions on the canonical transformation and the model's dynamics are fulfilled. In such cases, we can get the quasipotential algebraically. We apply the scheme to several one-dimensional nonequilibrium models as the diffusive and reaction-diffusion systems.
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