Anomalous scaling and first-order dynamical phase transition in large deviations of the Ornstein-Uhlenbeck process
Naftali R. Smith
Abstract
We study the full distribution of $A={\ensuremath{\int}}_{0}^{T}{x}^{n}(t)dt, n=1,2,\ensuremath{\cdots}$, where $x(t)$ is an Ornstein-Uhlenbeck process. We find that for $n>2$ the long-time ($T\ensuremath{\rightarrow}\ensuremath{\infty}$) scaling form of the distribution is of the anomalous form $P(A;T)\ensuremath{\sim}{e}^{\ensuremath{-}{T}^{\ensuremath{\mu}}{f}_{n}(\mathrm{\ensuremath{\Delta}}A/{T}^{\ensuremath{\nu}})}$ where $\mathrm{\ensuremath{\Delta}}A$ is the difference between $A$ and its mean value, and the anomalous exponents are $\ensuremath{\mu}=2/(2n\ensuremath{-}2)$ and $\ensuremath{\nu}=n/(2n\ensuremath{-}2)$. The rate function ${f}_{n}(y)$, which 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 $\mathcal{A}(t)={\ensuremath{\int}}_{0}^{t}{x}^{n}(s)ds$ 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.