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Cecile Monthus

Publications and source records attributed to Cecile Monthus.

At least 37 records · Page 2Linked to original sources

arXiv2023

On the Kemeny time for continuous-time reversible and irreversible Markov processes with applications to stochastic resetting and to conditioning towards forever-survival

For continuous-time ergodic Markov processes, the Kemeny time $τ_*$ is the characteristic time needed to converge towards the steady state $P_*(x)$ : in real-space, the Kemeny time $τ_*$ corresponds to the average of the Mean-First-Passage-Time $τ(x,x_0) $ over the final configuration $x$ drawn with the steady state $P_*(x)$, which turns out to be independent of the initial configuration $x_0$; in the spectral domain, the Kemeny time $τ_*$ corresponds to the sum of the inverses of all the non-vanishing eigenvalues $λ_n \ne 0 $ of the opposite generator. We describe many illustrative examples involving jumps and/or diffusion in one dimension, where the Kemeny time can be explicitly computed as a function of the system-size, via its real-space definition and/or via its spectral definition : we consider both reversible processes satisfying detailed-balance where the eigenvalues are real, and irreversible processes characterized by non-vanishing steady currents where the eigenvalues can be complex. In particular, we study the specific properties of the Kemeny times for Markov processes with stochastic resetting, and for absorbing Markov processes conditioned to survive forever.

cond-mat.stat-mech↗

arXiv2022

Markov trajectories : Microcanonical Ensembles based on empirical observables as compared to Canonical Ensembles based on Markov generators

The Ensemble of trajectories $x(0 \leq t \leq T)$ produced by the Markov generator $M$ can be considered as 'Canonical' for the following reasons : (C1) the probability of the trajectory $x(0 \leq t \leq T)$ can be rewritten as the exponential of a linear combination of its relevant empirical time-averaged observables $E_n$, where the coefficients involving the Markov generator are their fixed conjugate parameters; (C2) the large deviations properties of these empirical observables $E_n$ for large $T$ are governed by the explicit rate function $I^{[2.5]}_M (E_.) $ at Level 2.5, while in the thermodynamic limit $T=+\infty$, they concentrate on their typical values $E_n^{typ[M]}$ determined by the Markov generator $M$. This concentration property in the thermodynamic limit $T=+\infty$ suggests to introduce the notion of the 'Microcanonical Ensemble' at Level 2.5 for stochastic trajectories $x(0 \leq t \leq T)$, where all the relevant empirical variables $E_n$ are fixed to some values $E^*_n$ and cannot fluctuate anymore for finite $T$. The goal of the present paper is to discuss its main properties : (MC1) when the long trajectory $x(0 \leq t \leq T) $ belongs the Microcanonical Ensemble with the fixed empirical observables $E_n^*$, the statistics of its subtrajectory $x(0 \leq t \leq τ) $ for $1 \ll τ\ll T $ is governed by the Canonical Ensemble associated to the Markov generator $M^*$ that would make the empirical observables $E_n^*$ typical ; (MC2) in the Microcanonical Ensemble, the central role is played by the number $Ω^{[2.5]}_T(E^*_.) $ of stochastic trajectories of duration $T$ with the given empirical observables $E^*_n$, and by the corresponding explicit Boltzmann entropy $S^{[2.5]}( E^*_. ) = [\ln Ω^{[2.5]}_T(E^*_.)]/T $. This general framework is applied to continuous-time Markov Jump processes and to discrete-time Markov chains with illustrative examples.

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arXiv2022

Microcanonical conditioning of Markov processes on time-additive observables

The recent study by B. De Bruyne, S. N. Majumdar, H. Orland and G. Schehr [arXiv:2110.07573], concerning the conditioning of the Brownian motion and of random walks on global dynamical constraints over a finite time-window $T$, is reformulated as a general framework for the 'microcanonical conditioning' of Markov processes on time-additive observables. This formalism is applied to various types of Markov processes, namely discrete-time Markov chains, continuous-time Markov jump processes and diffusion processes in arbitrary dimension. In each setting, the time-additive observable is also fully general, i.e. it can involve both the time spent in each configuration and the elementary increments of the Markov process. The various cases are illustrated via simple explicit examples. Finally, we describe the link with the 'canonical conditioning' based on the generating function of the time-additive observable for finite time $T$, while the regime of large time $T$ allows to recover the standard large deviation analysis of time-additive observables via the deformed Markov operator approach.

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arXiv2022

Anomalous dynamical large deviations of local empirical densities and activities in the pure and in the random kinetically-constrained East Model

The East model is the simplest one-dimensional kinetically-constrained model of $N$ spins with a trivial equilibrium that displays anomalously large spatio-temporal fluctuations, with characteristic "space-time bubbles" in trajectory space, and with a discontinuity at the origin for the first derivative of the scaled cumulant generating function of the total activity. These striking dynamical properties are revisited via the large deviations at various levels for the relevant local empirical densities and activities that only involve two consecutive spins. This framework allows to characterize their anomalous rate functions and to analyze the consequences for all the time-additive observables that can be reconstructed from them, both for the pure and for the random East model. These singularities in dynamical large deviations properties disappear when the hard-constraint of the East model is replaced by the soft constraint.

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arXiv2021

Large deviations for metastable states of Markov processes with absorbing states with applications to population models in stable or randomly switching environment

The large deviations at Level 2.5 are applied to Markov processes with absorbing states in order to obtain the explicit extinction rate of metastable quasi-stationary states in terms of their empirical time-averaged density and of their time-averaged empirical flows over a large time-window $T$. The standard spectral problem for the slowest relaxation mode can be recovered from the full optimization of the extinction rate over all these empirical observables and the equivalence can be understood via the Doob generator of the process conditioned to survive up to time $T$. The large deviation properties of any time-additive observable of the Markov trajectory before extinction can be derived from the Level 2.5 via the decomposition of the time-additive observable in terms of the empirical density and the empirical flows. This general formalism is described for continuous-time Markov chains, with applications to population birth-death model in a stable or in a switching environment, and for diffusion processes in dimension $d$.

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arXiv2021

Inhomogeneous asymmetric exclusion processes between two reservoirs : large deviations for the local empirical observables in the Mean-Field approximation

For a given inhomogeneous exclusion processes on $N$ sites between two reservoirs, the trajectories probabilities allow to identify the relevant local empirical observables and to obtain the corresponding rate function at Level 2.5. In order to close the hierarchy of the empirical dynamics that appear in the stationarity constraints, we consider the simplest approximation, namely the Mean-Field approximation for the empirical density of two consecutive sites, in direct correspondence with the previously studied Mean-Field approximation for the steady state. For a given inhomogeneous Totally Asymmetric model (TASEP), this Mean-Field approximation yields the large deviations for the joint distribution of the empirical density profile and of the empirical current around the mean-field steady state; the further explicit contraction over the current allows to obtain the large deviations of the empirical density profile alone. For a given inhomogeneous Asymmetric model (ASEP), the local empirical observables also involve the empirical activities of the links and of the reservoirs; the further explicit contraction over these activities yields the large deviations for the joint distribution of the empirical density profile and of the empirical current. The consequences for the large deviations properties of time-additive space-local observables are also discussed in both cases.

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arXiv2021

Large deviations for the Skew-Detailed-Balance Lifted-Markov processes to sample the equilibrium distribution of the Curie-Weiss model

Among the Markov chains breaking detailed-balance that have been proposed in the field of Monte-Carlo sampling in order to accelerate the convergence towards the steady state with respect to the detailed-balance dynamics, the idea of 'Lifting' consists in duplicating the configuration space into two copies $σ=\pm$ and in imposing directed flows in each copy in order to explore the configuration space more efficiently. The skew-detailed-balance Lifted-Markov-chain introduced by K. S. Turitsyn, M. Chertkov and M. Vucelja [Physica D Nonlinear Phenomena 240 , 410 (2011)] is revisited for the Curie-Weiss mean-field ferromagnetic model, where the dynamics for the magnetization is closed. The large deviations at various levels for empirical time-averaged observables are analyzed and compared with their detailed-balance counterparts, both for the discrete extensive magnetization $M$ and for the continuous intensive magnetization $m=\frac{M}{N}$ for large system-size $N$.

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arXiv2021

Inference of Markov models from trajectories via Large Deviations at Level 2.5 with applications to random walks in disordered media

The inference of Markov models from data on stochastic dynamical trajectories over the large time-window $T$ is revisited via the Large Deviations at Level 2.5 for the time-empirical density and the time-empirical flows. The goal is to obtain the large deviations properties for the probability distribution of the inferred Markov parameters in order to characterize their possible fluctuations around the true Markov parameters for large $T$. The explicit rate functions are given for several settings, namely discrete-time Markov chains, continuous-time Markov jump processes, and diffusion processes in dimension $d$. Applications to various models of random walks in disordered media are described, where the goal is to infer the quenched disordered variables defining a given disordered sample.

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arXiv2021

Revisiting the Ruelle thermodynamic formalism for Markov trajectories with application to the glassy phase of random trap models

The Ruelle thermodynamic formalism for dynamical trajectories over the large time $T$ corresponds to the large deviation theory for the information per unit time of the trajectories probabilities. The microcanonical analysis consists in evaluating the exponential growth in $T$ of the number of trajectories with a given information per unit time, while the canonical analysis amounts to analyze the appropriate non-conserved $β$-deformed dynamics in order to obtain the scaled cumulant generating function of the information, the first cumulant being the famous Kolmogorov-Sinai entropy. This framework is described in detail for discrete-time Markov chains and for continuous-time Markov jump processes converging towards some steady-state, where one can also construct the Doob generator of the associated $β$-conditioned process. The application to the Directed Random Trap model on a ring of $L$ sites allows to illustrate this general framework via explicit results for all the introduced notions. In particular, the glassy phase is characterized by anomalous scaling laws with the size $L$ and by non-self-averaging properties of the Kolmogorov-Sinai entropy and of the higher cumulants of the trajectory information.

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arXiv2021

Large deviations at various levels for run-and-tumble processes with space-dependent velocities and space-dependent switching rates

One-dimensional run-and-tumble processes may converge towards some localized non-equilibrium steady state when the two velocities and/or the two switching rates are space-dependent. A long dynamical trajectory can be then analyzed via the large deviations at Level 2.5 for the joint probability of the empirical densities, of the empirical spatial currents and of the empirical switching flows. The Level 2 for the empirical densities alone can be then derived via the optimization of the Level 2.5 over the empirical flows. More generally, the large deviations of any time-additive observable can be also obtained via contraction from the Level 2.5, or equivalently via the deformed generator method and the corresponding Doob conditioned process. Finally, the large deviations for the empirical intervals between consecutive switching events can be obtained via the introduction of the alternate Markov chain that governs the series of all the switching events of a long trajectory.

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arXiv2021

Jump-Drift and Jump-Diffusion Processes : Large Deviations for the density, the current and the jump-flow and for the excursions between jumps

For one-dimensional Jump-Drift and Jump-Diffusion processes converging towards some steady state, the large deviations of a long dynamical trajectory are described from two perspectives. Firstly, the joint probability of the empirical time-averaged density, of the empirical time-averaged current and of the empirical time-averaged jump-flow are studied via the large deviations at Level 2.5. Secondly, the joint probability of the empirical jumps and of the empirical excursions between consecutive jumps are analyzed via the large deviations at Level 2.5 for the alternate Markov chain that governs the series of all the jump events of a long trajectory. These two general frameworks are then applied to three examples of positive jump-drift processes without diffusion, and to two examples of jump-diffusion processes, in order to illustrate various simplifications that may occur in rate functions and in contraction procedures.

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arXiv2021

Properties of the simplest inhomogeneous and homogeneous Tree-Tensor-States for Long-Ranged Quantum Spin Chains with or without disorder

The simplest Tree-Tensor-States (TTS) respecting the Parity and the Time-Reversal symmetries are studied in order to describe the ground states of Long-Ranged Quantum Spin Chains with or without disorder. Explicit formulas are given for the one-point and two-point reduced density matrices that allow to compute any one-spin and two-spin observable. For Hamiltonians containing only one-body and two-body contributions, the energy of the TTS can be then evaluated and minimized in order to obtain the optimal parameters of the TTS. This variational optimization of the TTS parameters is compared with the traditional block-spin renormalization procedure based on the diagonalization of some intra-block renormalized Hamiltonian.

cond-mat.dis-nn↗

arXiv2021

Large deviations for Markov processes with stochastic resetting : analysis via the empirical density and flows or via excursions between resets

Markov processes with stochastic resetting towards the origin generically converge towards non-equilibrium steady-states. Long dynamical trajectories can be thus analyzed via the large deviations at Level 2.5 for the joint probability of the empirical density and the empirical flows, or via the large deviations of semi-Markov processes for the empirical density of excursions between consecutive resets. The large deviations properties of general time-additive observables involving the position and the increments of the dynamical trajectory are then analyzed in terms of the appropriate Markov tilted processes and of the corresponding conditioned processes obtained via the generalization of Doob's h-transform. This general formalism is described in detail for the three possible frameworks, namely discrete-time/discrete-space Markov chains, continuous-time/discrete-space Markov jump processes and continuous-time/continuous-space diffusion processes, and is illustrated with explicit results for the Sisyphus Random Walk and its variants, when the reset probabilities or reset rates are space-dependent.

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arXiv2021

Large deviations of the Lyapunov exponent in 2D matrix Langevin dynamics with applications to one-dimensional Anderson Localization models

For the 2D matrix Langevin dynamics that corresponds to the continuous-time limit of the product of some $2 \times 2$ random matrices, the finite-time Lyapunov exponent can be written as an additive functional of the associated Riccati process submitted to some Langevin dynamics on the infinite periodic ring. Its large deviations properties can be thus analyzed from two points of view that are equivalent in the end by consistency but give different perspectives. In the first approach, one starts from the large deviations at level 2.5 for the joint probability of the empirical density and of the empirical current of the Riccati process and one performs the appropriate Euler-Lagrange optimization in order to compute the cumulant generating function of the Lyapunov exponent. In the second approach, this cumulant generating function is obtained from the spectral analysis of the appropriate tilted Fokker-Planck operator. The associated conditioned process obtained via the generalization of Doob's h-transform allows to clarify the equivalence with the first approach. Finally, applications to one-dimensional Anderson Localization models are described in order to obtain explicitly the first cumulants of the finite-size Lyapunov exponent.

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arXiv2020

Explicit properties of the simplest inhomogeneous Matrix-Product-State including the Riemann metric of the MPS manifold

We consider the simplest inhomogeneous Matrix-Product-State for an open chain of N quantum spins that involves only two angles per site and two angles per bond with the following direct physical meanings. The two angles associated to the site $k$ are the two Bloch angles that parametrize the two orthonormal eigenvectors of the reduced density matrix $ρ_k$ of the spin $k$ alone. The two angles associated to the bond $(k,k+1)$ parametrize the entanglement properties of the Schmidt decomposition across the bond $(k,k+1)$. Explicit results are given for the reduced density matrix $ρ_{k,k+1}$ of two consecutive sites that is needed to evaluate the energy of two-body Hamiltonians, and for the reduced density matrix $ρ_{k,k+r}$ of two sites at distance $r$ that is needed to evaluate the spin-spin correlations at distance $r$. The global structure of the MPS manifold as parametrized by these $(4N-2)$ angles is then characterized by its explicit Riemann metric. Finally, the generalizations to any tree-like structure without loops and to the chain with periodic boundary conditions are discussed.

cond-mat.str-el↗

arXiv2019

Real-Space Renormalization for disordered systems at the level of Large Deviations

The real-space renormalization procedures on hierarchical lattices have been much studied for many disordered systems in the past at the level of their typical fluctuations. In the present paper, the goal is to analyze instead the renormalization flows for the tails of probability distributions in order to extract the scalings of their large deviations and the tails behaviors of the corresponding rate functions. We focus on the renormalization rule for the ground-state energy of the Directed Polymer model in a random medium, and study the various renormalization flows that can emerge for the tails as a function of the tails of the initial condition.

cond-mat.dis-nn↗

arXiv2019

Construction of Many-Body-Localized Models where all the eigenstates are Matrix-Product-States

The inverse problem of 'eigenstates-to-Hamiltonian' is considered for an open chain of $N$ quantum spins in the context of Many-Body-Localization. We first construct the simplest basis of the Hilbert space made of $2^N$ orthonormal Matrix-Product-States (MPS), that will thus automatically satisfy the entanglement area-law. We then analyze the corresponding $N$ Local Integrals of Motions (LIOMs) that can be considered as the local building blocks of these $2^N$ MPS, in order to construct the parent Hamiltonians that have these $2^N$ MPS as eigenstates. Finally we study the Matrix-Product-Operator form of the Diagonal Ensemble Density Matrix that allows to compute long-time-averaged observables of the unitary dynamics. Explicit results are given for the memory of local observables and for the entanglement properties in operator-space, via the generalized notion of Schmidt decomposition for density matrices describing mixed states.

cond-mat.dis-nn↗

arXiv2019

Statistical physics of long dynamical trajectories for a system in contact with several thermal reservoirs

For a system in contact with several reservoirs $r$ at different inverse-temperatures $β_r$, we describe how the Markov jump dynamics with the generalized detailed balance condition can be analyzed via a statistical physics approach of dynamical trajectories $[{\cal C}(t)]_{0 \leq t \leq T} $ over a long time interval $T \to + \infty$. The relevant intensive variables are the time-empirical density $ρ(\cal C)$, that measures the fractions of time spent in the various configurations ${\cal C}$, and the time-empirical jump densities $k_r ({\cal C', \cal C}) $, that measure the frequencies of jumps from configuration ${\cal C} $ to configuration ${\cal C '} $ when it is the reservoir $r$ that furnishes or absorbs the corresponding energy difference ($E({\cal C '})- E({\cal C })$).

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