Large deviations in chaotic systems: exact results and dynamical phase transition

Abstract

Large deviations in chaotic dynamics have potentially significant and dramatic consequences. We study large deviations of series of finite lengths $N$ generated by chaotic maps. The distributions generally display an exponential decay with $N$, associated with large-deviation (rate) functions. We obtain the exact rate functions analytically for the doubling, tent, and logistic maps. For the latter two, the solution is given as a power series whose coefficients can be systematically calculated to any order. We also obtain the rate function for the cat map numerically, uncovering strong evidence for the existence of a remarkable singularity of it that we interpret as a second order dynamical phase transition. Furthermore, we develop a numerical tool for efficiently simulating atypical realizations of sequences if the chaotic map is not invertible, and we apply it to the tent and logistic maps.

8 + 7 pages, 2 figures

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• Dynamical phase transition in the occupation fraction statistics for non-crossing Brownian particles
• Confined run and tumble particles with non-Markovian tumbling statistics
• Anomalous scalings of fluctuations of the area swept by a Brownian particle trapped in a $|x|$ potential
• Large deviations in statistics of the local time and occupation time for a run and tumble particle
• Finding the effective dynamics to make rare events typical in chaotic maps
• Large deviations and conditioning for chaotic non-invertible deterministic maps: analysis via the forward deterministic dynamics and the backward stochastic dynamics
• Thermodynamics of chaotic relaxation processes
• Nonequilibrium steady state of Brownian motion in an intermittent potential
• Explicit dynamical properties of the Pelikan random map in the chaotic region and at the intermittent critical point towards the non-chaotic region
• Lecture notes on large deviations in non-equilibrium diffusive systems
• Optimal finite-differences discretization for the diffusion equation from the perspective of large-deviation theory
• Large deviations for trajectory observables of diffusion processes in dimension $d>1$ in the double limit of large time and small diffusion coefficient
• Subleading-order theory for condensation transitions in large deviations of sums of independent and identically distributed random variables
• Remarkable similarities in distributions of dynamical observables in chaotic systems
• Full distribution of the ground-state energy of potentials with weak disorder
• Making rare events typical in $d$-dimensional chaotic maps