Large deviations and conditioning for chaotic non-invertible deterministic maps: analysis via the forward deterministic dynamics and the backward stochastic dynamics

Abstract

The large deviations properties of trajectory observables for chaotic non-invertible deterministic maps as studied recently by N. R. Smith, Phys. Rev. E 106, L042202 (2022) and by R. Gutierrez, A. Canella-Ortiz, C. Perez-Espigares, arXiv:2304.13754 are revisited in order to analyze in detail the similarities and the differences with the case of stochastic Markov chains. To be concrete, we focus on the simplest example displaying the two essential properties of local-stretching and global-folding, namely the doubling map $x_{t+1}=2x_t[\text{mod}1]$ on the real-space interval $x\in [0,1[$ that can be also analyzed via the decomposition $x={\sum }_{l=1}^{+\infty }\frac{{\sigma }_l}{2^l}$ into binary coefficients ${\sigma }_l=0,1$. The large deviations properties of trajectory observables can be studied either via deformations of the forward deterministic dynamics or via deformations of the backward stochastic dynamics. Our main conclusions concerning the construction of the corresponding Doob canonical conditioned processes are: (i) non-trivial conditioned dynamics can be constructed only in the backward stochastic perspective where the reweighting of existing transitions is possible, and not in the forward deterministic perspective ; (ii) the corresponding conditioned steady state is not smooth on the real-space interval $x\in [0,1[$ and can be better characterized in the binary space ${\sigma }_{l=1,2,..,+\infty }$. As a consequence, the backward stochastic dynamics in the binary space is also the most appropriate framework to write the explicit large deviations at level 2 for the probability of the empirical density of long backward trajectories.

v2=revised version with new discussions (34 pages)

References in corpus (10)

• The large deviation approach to statistical mechanics
• Non equilibrium steady states: fluctuations and large deviations of the density and of the current
• Dynamic first-order phase transition in kinetically constrained models of glasses
• Fluctuation theorems for stochastic dynamics
• First-order dynamical phase transition in models of glasses: an approach based on ensembles of histories
• Steady state statistics of driven diffusions
• Probing rare physical trajectories with Lyapunov weighted dynamics
• A minimal model of dynamical phase transition
• Dynamical large deviations of reflected diffusions
• Thermodynamic formalism and large deviation functions in continuous time Markov dynamics

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• A supersymmetric quantum perspective on the explicit large deviations for reversible Markov jump processes, with applications to pure and random spin chains
• 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
• Making rare events typical in $d$-dimensional chaotic maps