Ornstein-Uhlenbeck过程大偏差下的反常标度和一级动力学相变
Anomalous scaling and first-order dynamical phase transition in large deviations of the Ornstein-Uhlenbeck process
摘要：
研究了$A=\int_{0}^{T}x^{n}\left(t\right)dt$，$n=1,2,\dots$的全分布，其中$x\left(t\right)$是Ornstein-Uhlenbeck过程.我们发现，对于$n>2$的长时间（$T \to \infty$）标度形式的分布是反常形式的$P\left(A;T\right)\sim e^{-T^μf_{n}\left(ΔA/T^ν\right)}$，其中$ΔA$是$A$和它的平均值之间的差，和反常指数是$μ=2/\left(2n-2\right)$，和$ν=n/\left(2n-2\right)$。率函数$f_n\left(y\right)$，我们精确计算，表现出一阶动力学相变之间的均匀相，描述了高斯分布的典型波动，和一个“凝聚”的阶段，描述了分布的尾巴。我们还计算了最可能实现的$\mathcal{A}(t)=\int_{0}^{t}x^{n}\left(s\right)ds$和分布的$x(t)$在一个中间时间$t$条件下的一个给定值的$A$。扩展和其他连续时间系统的影响进行了讨论。
英文摘要：
We study the full distribution of $A=\int_{0}^{T}x^{n}\left(t\right)dt$, $n=1,2,\dots$, where $x\left(t\right)$ is an Ornstein-Uhlenbeck process. We find that for $n>2$ the long-time ($T \to \infty$) scaling form of the distribution is of the anomalous form $P\left(A;T\right)\sim e^{-T^μf_{n}\left(ΔA/T^ν\right)}$ where $ΔA$ is the difference between $A$ and its mean value, and the anomalous exponents are $μ=2/\left(2n-2\right)$, and $ν=n/\left(2n-2\right)$. The rate function $f_n\left(y\right)$, that we calculate exactly, exhibits a first-order dynamical phase transition which separates between a homogeneous phase that describes the Gaussian distribution of typical fluctuations, and a "condensed" phase that describes the tails of the distribution. We also calculate the most likely realizations of $\mathcal{A}(t)=\int_{0}^{t}x^{n}\left(s\right)ds$ and the distribution of $x(t)$ at an intermediate time $t$ conditioned on a given value of $A$. Extensions and implications to other continuous-time systems are discussed.
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