论文标题
一类高斯工艺的熵生产率的大偏差
Large Deviations of the Entropy Production Rate for a Class of Gaussian Processes
论文作者
论文摘要
我们证明了较大的偏差原理（LDP）和以下$ d $ d $ dimensional随机微分方程\ begin \ begin \ begin {equation*x_ {t x_ {t} = ax_ = ax_ = ax_ {稳定的矩阵，$ Q $是正确定的，矩阵$ a $和$ q $通勤。 EPR的速率函数采用以下显式形式：\ begin {equination*} i（x）= \ left \ {\ begin {array} {ll} {ll} x \ frac {\ sqrt {\ sqrt {1+ \ ell_0（x）}+\ freac 12 \ sum} \左（\ sqrt {α_k^2-β_k^2 \ ell_0（x）}+α_k\ right），＆x \ ge 0，\\ \\ -x \ frac {\ sqrt {1+ \ ell_0（x） 12 \ sum \ limits_ {k = 1}^{d} \ left（\ sqrt {α_k^2-β_k^2 \ ell_0（x）}+α_k\ right），＆x <0，＆x <0，\ end end end {arnay}} \ right。 \ end {qore*}其中$α_{k} \ pm {\ rm i}β_{k} $是$ a $的特征值，而$ \ ell_0（x）$是方程的唯一解决方案： \ sum_ {k = 1}^{d} \ frac {β_k^2} {\ sqrt {α_k^2- \ellβ_k^2}}，\ qquad -1 \ le \ le \ ell \ ell \ min_ {k = 1，...，d} \ {\ frac {α_k^2} {β_k^2} \}。 \ end {Align*}简单的速率函数封闭式公式很少见，我们的工作确定了一类重要的大型偏差问题，可以使用此类公式。与LDP相关的对数力矩生成函数（波动函数）$λ$具有封闭形式（在纸张中查看）。功能$λ（λ）$和$ i（x）$满足Cohen-Gallavotti对称属性。 特别是，功能$ i $和$λ$不取决于扩散矩阵$ q $，并且完全取决于$ a $的特征值的真实和虚构部分。正式地，具有$ Q = 0 $的确定性系统的EPR为零，因此该模型表现出一个相变，因为EPR在$ q = 0 $时不连续变化。
We prove a large deviation principle (LDP) and a fluctuation theorem (FT) for the entropy production rate (EPR) of the following $d$ dimensional stochastic differential equation \begin{equation*} d X_{t}=AX_{t} d t+\sqrt{Q} d B_{t} \end{equation*} where $A$ is a real normal stable matrix, $Q$ is positive definite, and the matrices $A$ and $Q$ commute. The rate function for the EPR takes the following explicit form: \begin{equation*} I(x)=\left\{ \begin{array}{ll} x\frac{\sqrt{1+\ell_0(x)}-1}{2}+\frac 12\sum\limits_{k=1}^{d} \left(\sqrt{α_k^2- β_k^2\ell_0(x)}+α_k\right) , & x\ge 0 , \\ -x\frac{\sqrt{1+\ell_0(x)}+1}{2} +\frac 12\sum\limits_{k=1}^{d} \left(\sqrt{α_k^2- β_k^2\ell_0(x)}+α_k\right) , &x<0, \end{array} \right. \end{equation*} where $α_{k}\pm {\rm i} β_{k}$ are the eigenvalues of $A$, and $\ell_0(x)$ is the unique solution of the equation: \begin{align*} |x|={\sqrt{1+\ell}} \times \sum_{k=1}^{d} \frac{β_k^2}{\sqrt{α_k^2 -\ellβ_k^2} },\qquad -1 \le \ell< \min_{k=1,...,d}\{\frac{α_k^2}{β_k^2}\}. \end{align*} Simple closed form formulas for rate functions are rare and our work identifies an important class of large deviation problems where such formulas are available. The logarithmic moment generating function (the fluctuation function) $Λ$ associated with the LDP has a closed form (see it in the paper). The functions $Λ(λ)$ and $ I(x)$ satisfy the Cohen-Gallavotti symmetry properties. In particular, the functions $I$ and $Λ$ do not depend on the diffusion matrix $Q$, and are determined completely by the real and imaginary parts of the eigenvalues of $A$. Formally, the deterministic system with $Q=0$ has zero EPR and thus the model exhibits a phase transition in that the EPR changes discontinuously at $Q = 0$.
