Dynamical phase transition for a logarithm observable in chaotic systems

Abstract: We consider the full distribution $P_N(A)$ of the dynamical observable $A = - \sum_{i=1}^N \ln x_i$ where $x_1, \dots, x_N$ is a sequence generated by a chaotic map $f(x)$ via $x_{n+1} = f(x_n)$. We study the particular cases where $f(x)$ is the doubling, tent or logistic map, and focus on the limit $N \gg 1$. Typical fluctuations of $A$ are Gaussian with mean $\left\langle A\right\rangle$ and variance which are both of order $\sim N$. We find that the left tail of the distribution ($A < \left\langle A\right\rangle$) scales as $P_{N}(A) \sim e^{-NI(A/N)}$, and calculate the rate function $I(a)$ which depends on the map $f(x)$. Interestingly, we find that the right tail of the distribution ($A > \left\langle A\right\rangle$) follows an anomalous scaling: $$P_{N}(A) \sim e^{-N^{1/3} F((A-\left\langle A\right\rangle) / N^{2/3})}$$. The large-deviation function $F(y)$ has a universal form for the three maps, differing only in the scaling constants. At a critical value $y=y_c$, it exhibits a corner singularity which has the character of a condensation transition. The transition separates between a subcritical phase $0< y<y_c$ in which the fluctuations are Gaussian and arise homogeneously from all of the $x_i$'s, and a supercritical phase $y>y_c$ in which a small fraction of the $x_i$'s give a macroscopic contribution to the fluctuation. Despite this anomalous behavior of the right tail, we find that the cumulants of the distribution $P_N(A)$ grow linearly with $N$ at large $N$, and calculate the associated coefficients.