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

Large deviations in statistics of the local time and occupation time for a run and tumble particle

We investigate the statistics of the local time T=∫_{0}^{T}δ(x(t))dt that a run and tumble particle (RTP) x(t) in one dimension spends at the origin, with or without an external drift. By relating the local time to the number of times the RTP crosses the origin, we find that the local time distribution P(T) satisfies the large deviation principle P(T)∼e^{-TI(T/T)} in the large observation time limit T→∞. Remarkably, we find that in the presence of drift the rate function I(ρ) is nonanalytic: we interpret its singularity as a dynamical phase transition of first order. We then extend these results by studying the statistics of the amount of time R that the RTP spends inside a finite interval (i.e., the occupation time), with qualitatively similar results. In particular, this yields the long-time decay rate of the probability P(R=T) that the particle does not exit the interval up to time T. We find that the conditional end-point distribution exhibits an interesting change of behavior from unimodal to bimodal as a function of the size of the interval. To study the occupation time statistics, we extend the Donsker-Varadhan large-deviation formalism to the case of RTPs, for general dynamical observables and possibly in the presence of an external potential.

Thermodynamics of chaotic relaxation processes

The established thermodynamic formalism of chaotic dynamics, valid at statistical equilibrium, is here generalized to systems out of equilibrium that have yet to relax to a steady state. A relation between information, escape rate, and the phase-space average of an integrated observable (e.g., Lyapunov exponent, diffusion coefficient) is obtained for finite time. Most notably, the thermodynamic treatment may predict the phase-space profile of any integrated observable for finite time, from the leading and subleading eigenfunctions of the Perron-Frobenius or Koopman transfer operator. Examples of that equivalence are shown, and the theory is tested analytically on the Bernoulli map while numerically on the perturbed cat map, the Hénon map, and the Ikeda map, all paradigms of chaos.

Confined run-and-tumble particles with non-Markovian tumbling statistics

Confined active particles constitute simple, yet realistic, examples of systems that converge into a nonequilibrium steady state. We investigate a run-and-tumble particle in one spatial dimension, trapped by an external potential, with a given distribution g(t) of waiting times between tumbling events whose mean value is equal to τ. Unless g(t) is an exponential distribution (corresponding to a constant tumbling rate), the process is non-Markovian, which makes the analysis of the model particularly challenging. We use an analytical framework involving effective position-dependent tumbling rates to develop a numerical method that yields the full steady-state distribution (SSD) of the particle's position. The method is very efficient and requires modest computing resources, including in the large-deviation and/or small-τ regime, where the SSD can be related to the the large-deviation function, s(x), via the scaling relation P_{st}(x)∼e^{-s(x)/τ}.

Stretched-exponential relaxation in weakly confined Brownian systems through large deviation theory

Stretched-exponential relaxation is a widely observed phenomenon found in ordered ferromagnets as well as glassy systems. One modeling approach connects this behavior to a droplet dynamics described by an effective Langevin equation for the droplet radius with an r^{2/3} potential. Here, we study a Brownian particle under the influence of a general confining, albeit weak, potential field that grows with distance as a sublinear power law. We find that for this memoryless model, observables display stretched-exponential relaxation. The probability density function of the system is studied using a rate-function ansatz. We obtain analytically the stretched-exponential exponent along with an anomalous power-law scaling of length with time. The rate function exhibits a point of nonanalyticity, indicating a dynamical phase transition. In particular, the rate function is double valued both to the left and right of this point, leading to four different rate functions, depending on the choice of initial conditions and symmetry.

Finding the Effective Dynamics to Make Rare Events Typical in Chaotic Maps

Dynamical fluctuations or rare events associated with atypical trajectories in chaotic maps due to specific initial conditions can crucially determine their fate, as the may lead to stability islands or regions in phase space otherwise displaying unusual behavior. Yet, finding such initial conditions is a daunting task precisely because of the chaotic nature of the system. In this Letter, we circumvent this problem by proposing a framework for finding an effective topologically conjugate map whose typical trajectories correspond to atypical ones of the original map. This is illustrated by means of examples which focus on counterbalancing the instability of fixed points and periodic orbits, as well as on the characterization of a dynamical phase transition involving the finite-time Lyapunov exponent. The procedure parallels that of the application of the generalized Doob transform in the stochastic dynamics of Markov chains, diffusive processes, and open quantum systems, which in each case results in a new process having the prescribed statistics in its stationary state. This Letter thus brings chaotic maps into the growing family of systems whose rare fluctuations-sustaining prescribed statistics of dynamical observables-can be characterized and controlled by means of a large-deviation formalism.

Dynamical phase transition in the occupation fraction statistics for noncrossing Brownian particles

We consider a system of N noncrossing Brownian particles in one dimension. We find the exact rate function that describes the long-time large deviation statistics of their occupation fraction in a finite interval in space. Remarkably, we find that, for any general N≥2, the system undergoes N-1 dynamical phase transitions of second order. The N-1 transitions are the boundaries of N phases that correspond to different numbers of particles which are in the vicinity of the interval throughout the dynamics. We achieve this by mapping the problem to that of finding the ground-state energy for N noninteracting spinless fermions in a square-well potential. The phases correspond to different numbers of single-body bound states for the quantum problem. We also study the process conditioned on a given occupation fraction and the large-N limiting behavior.