A supersymmetric quantum perspective on the explicit large deviations for reversible Markov jump processes, with applications to pure and random spin chains

Abstract

The large deviations at various levels that are explicit for Markov jump processes satisfying detailed-balance are revisited in terms of the supersymmetric quantum Hamiltonian $H$ that can be obtained from the Markov generator via a similarity transformation. We first focus on the large deviations at level 2 for the empirical density $\hat{p}(C)$ of the configurations $C$ seen during a trajectory over the large time-window $[0,T]$ and rewrite the explicit Donsker-Varadhan rate function as the matrix element $I^{[2]}[\hat{p}(.)]=⟨\sqrt{\hat{p}}|H|\sqrt{\hat{p}}⟩$ involving the square-root ket $|\sqrt{\hat{p}}⟩$. [The analog formula is also discussed for reversible diffusion processes as a comparison.] We then consider the explicit rate functions at higher levels, in particular for the joint probability of the empirical density $\hat{p}(C)$ and the empirical local activities $\hat{a}(C,C^{\prime })$ characterizing the density of jumps between two configurations $(C,C^{\prime })$. Finally, the explicit rate function for the joint probability of the empirical density $\hat{p}(C)$ and of the empirical total activity $\hat{A}$ that represents the total density of jumps of a long trajectory is written in terms of the two matrix elements $⟨\sqrt{\hat{p}}|H|\sqrt{\hat{p}}⟩$ and $⟨\sqrt{\hat{p}}|H^{off}|\sqrt{\hat{p}}⟩$, where $H^{off}$ represents the off-diagonal part of the supersymmetric Hamiltonian $H$. This general formalism is then applied to pure or random spin chains with single-spin-flip or two-spin-flip transition rates, where the supersymmetric Hamiltonian $H$ correspond to quantum spin chains with local interactions involving Pauli matrices of two or three neighboring sites.

arXiv commentv2=revised version with new discussions (31 pages)

References in corpus (107)

• Supersymmetry and Quantum Mechanics
• The large deviation approach to statistical mechanics
• A Gallavotti-Cohen Type Symmetry in the Large Deviation Functional for Stochastic Dynamics
• Fluctuation theorem for stochastic dynamics
• Non equilibrium steady states: fluctuations and large deviations of the density and of the current
• Glassy dynamics of kinetically constrained models
• Macroscopic fluctuation theory
• The Fluctuation Theorem as a Gibbs Property
• 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
• Thermodynamic formalism for systems with Markov dynamics
• Direct evaluation of large-deviation functions
• Nonequilibrium Markov processes conditioned on large deviations
• Fluctuation Theorems
• Large deviations and ensembles of trajectories in stochastic models
• Nonequilibrium microcanonical and canonical ensembles and their equivalence
• Simulating rare events in dynamical processes
• Probability currents as principal characteristics in the statistical mechanics of non-equilibrium steady states
• Canonical structure of dynamical fluctuations in mesoscopic nonequilibrium steady states
• Stability of Frustration-Free Hamiltonians
• Introduction to dynamical large deviations of Markov processes
• Unravelling the large deviation statistics of Markovian open quantum systems
• Aspects of non-equilibrium in classical and quantum systems: slow relaxation and glasses, dynamical large deviations, quantum non-ergodicity, and open quantum dynamics
• Variational and optimal control representations of conditioned and driven processes
• Steady state statistics of driven diffusions
• Ergodicity and large deviations in physical systems with stochastic dynamics
• From quantum mechanics to classical statistical physics: generalized Rokhsar-Kivelson Hamiltonians and the "Stochastic Matrix Form" decomposition
• From classical to quantum dynamics at Rokhsar-Kivelson points
• Effective interactions and large deviations in stochastic processes
• Chaotic properties of systems with Markov dynamics
• A formal view on 2.5 large deviations and fluctuation relations
• Construction of a Coordinate Bethe Ansatz for the asymmetric simple exclusion process with open boundaries
• Trajectory phase transitions, Lee-Yang zeros, and high-order cumulants in full counting statistics
• The Physicist's Companion to Current Fluctuations: One-Dimensional Bulk-Driven Lattice Gases
• Note on Two Theorems in Nonequilibrium Statistical Mechanics
• Current large deviations for driven periodic diffusions
• Making rare events typical in Markovian open quantum systems
• A minimal model of dynamical phase transition
• Non-equilibrium work relations
• Quantum mechanical and information theoretic view on classical glass transitions
• Using matrix product states to study the dynamical large deviations of kinetically constrained models
• Transition probabilities and dynamic structure factor in the ASEP conditioned on strong flux
• Large deviations and dynamical phase transitions in stochastic chemical networks
• Dynamical phase transition in drifted Brownian motion
• Large deviations conditioned on large deviations I: Markov chain and Langevin equation
• Density profiles, dynamics, and condensation in the ZRP conditioned on an atypical current
• Microscopic structure of shocks and antishocks in the ASEP conditioned on low current
• Generic Dynamical Phase Transition in One-Dimensional Bulk-Driven Lattice Gases with Exclusion
• Second-order dynamic transition in a p=2 spin-glass model
• Level 2 large deviation functionals for systems with and without detailed balance
• Effective driven dynamics for one-dimensional conditioned Langevin processes in the weak-noise limit
• Diffusions conditioned on occupation measures
• Large-deviation theory for a Brownian particle on a ring: a WKB approach
• Large deviations conditioned on large deviations II: Fluctuating hydrodynamics
• Current fluctuations in periodically driven systems
• Spectral theory of fluctuations in time-average statistical mechanics of reversible and driven systems
• Large deviations at level 2.5 for Markovian open quantum systems: quantum jumps and quantum state diffusion
• Frustration free gapless Hamiltonians for Matrix Product States
• Activity phase transition for constrained dynamics
• Dynamical large deviations of reflected diffusions
• Bethe Ansatz for the Weakly Asymmetric Simple Exclusion Process and phase transition in the current distribution
• Localization of thermal packets and metastable states in Sinai model
• Large deviations of the dynamical activity in the East model: analysing structure in biased trajectories
• Thermodynamic formalism and large deviation functions in continuous time Markov dynamics
• Dynamics and large deviation transitions of the XOR-Fredrickson-Andersen kinetically constrained model
• Eigenvalue method to compute the largest relaxation time of disordered systems
• Exact Distributions of Currents and Frenesy for Markov Bridges
• A simple proof of the detectability lemma and spectral gap amplification
• Large deviations for Markov processes with stochastic resetting : analysis via the empirical density and flows or via excursions between resets
• Dynamical large deviations of linear diffusions
• Microscopic Fluctuation Theory (mFT) for interacting Poisson processes
• Large Deviations for the density and the current in Non-Equilibrium-Steady-States on disordered rings
• Large deviations of the Lyapunov exponent in 2D matrix Langevin dynamics with applications to one-dimensional Anderson Localization models
• Revisiting classical and quantum disordered systems from the unifying perspective of large deviations
• Large deviations of currents in diffusions with reflective boundaries
• Trajectory phase transitions in non-interacting spin systems
• Periodically driven jump processes conditioned on large deviations
• Optimal sampling of dynamical large deviations via matrix product states
• Nonequilibrium diffusion processes via non-Hermitian electromagnetic quantum mechanics with application to the statistics of entropy production in the Brownian gyrator
• Large Deviations for dynamical fluctuations of Open Markov processes, with application to random cascades on trees
• When a local Hamiltonian must be frustration-free
• Conditioned stochastic particle systems and integrable quantum spin systems
• Revisiting the Ruelle thermodynamic formalism for Markov trajectories with application to the glassy phase of random trap models
• Statistical physics of long dynamical trajectories for a system in contact with several thermal reservoirs
• Trajectory phase transitions and dynamical Lee-Yang zeros of the Glauber-Ising chain
• Inference of Markov models from trajectories via Large Deviations at Level 2.5 with applications to random walks in disordered media
• Large deviations at various levels for run-and-tumble processes with space-dependent velocities and space-dependent switching rates
• Effective Hamiltonians and Lagrangians for conditioned Markov processes at large volume
• Space-time phase transitions in the East model with a softened kinetic constraint
• One-dimensional classical diffusion in a random force field with weakly concentrated absorbers
• Large deviations for the Skew-Detailed-Balance Lifted-Markov processes to sample the equilibrium distribution of the Curie-Weiss model
• Jump-Drift and Jump-Diffusion Processes : Large Deviations for the density, the current and the jump-flow and for the excursions between jumps
• Anomalous dynamical large deviations of local empirical densities and activities in the pure and in the random kinetically-constrained East Model
• Large deviations for metastable states of Markov processes with absorbing states with applications to population models in stable or randomly switching environment
• Dynamical large deviations of diffusions
• The fluctuation theorem and Lyapunov weights
• BEST statistics of Markovian fluxes: a tale of Eulerian tours and Fermionic ghosts
• From dynamical systems to statistical mechanics: the case of the fluctuation theorem
• Large deviations for the Pearson family of ergodic diffusion processes involving a quadratic diffusion coefficient and a linear force
• Revisiting boundary-driven non-equilibrium Markov dynamics in arbitrary potentials via supersymmetric quantum mechanics and explicit large deviations at various levels
• Frustration-free Hamiltonians supporting Majorana zero edge modes
• Large deviations and conditioning for chaotic non-invertible deterministic maps: analysis via the forward deterministic dynamics and the backward stochastic dynamics
• Large deviations at level 2.5 and for trajectories observables of diffusion processes : the missing parts with respect to their random-walks counterparts
• Scaling approach to related disordered stochastic and free-fermion models
• Large deviations for trajectory observables of diffusion processes in dimension $d>1$ in the double limit of large time and small diffusion coefficient
• Inverse problem in the conditioning of Markov processes on trajectory observables : what canonical conditionings can connect two given Markov generators ?

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