Dynamical phase transition for a logarithm observable in chaotic systems

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 $⟨A⟩$ and variance which are both of order $\sim N$. We find that the left tail of the distribution ($A<⟨A⟩$) scales as $P_N(A)\sim e^{-NI(A\mathrm{/}N)}$, and calculate the rate function $I(a)$ which depends on the map $f(x)$. Interestingly, we find that the right tail of the distribution ($A>⟨A⟩$) follows an anomalous scaling: $P_N(A)\sim e^{-N^{1\mathrm{/}3}F((A-⟨A⟩)\mathrm{/}{N}^{2\mathrm{/}3})}$. 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.