Master equations and the theory of stochastic path integrals

Abstract

This review provides a pedagogic and self-contained introduction to master equations and to their representation by path integrals. We discuss analytical and numerical methods for the solution of master equations, keeping our focus on methods that are applicable even when stochastic fluctuations are strong. The reviewed methods include the generating function technique and the Poisson representation, as well as novel ways of mapping the forward and backward master equations onto linear partial differential equations (PDEs). Spectral methods, WKB approximations, and a variational approach have been proposed for the analysis of the PDE obeyed by the generating function. After outlining these methods, we solve the derived PDEs in terms of two path integrals. The path integrals provide distinct exact representations of the conditional probability distribution solving the master equations. We exemplify both path integrals in analysing elementary chemical reactions. Furthermore, we review a method for the approximation of rare event probabilities and derive path integral representations of Fokker-Planck equations. To make our review accessible to a broad community, we have used the language of probability theory rather than quantum (field) theory. The probabilistic structures underpinning various technical concepts, such as coherent states, the Doi-shift, and normal-ordered observables, are thereby made explicit.

78 pages and 11 figures

References in corpus (58)

• Statistical physics of social dynamics
• The Magnus expansion and some of its applications
• Analytical distributions for stochastic gene expression
• Neutral Theory and Relative Species Abundance in Ecology
• Mobility promotes and jeopardizes biodiversity in rock-paper-scissors games
• Phase Coexistence in Driven One Dimensional Transport
• Approximation and inference methods for stochastic biochemical kinetics - a tutorial review
• Asymptotics in ASEP with Step Initial Condition
• Applications of Field-Theoretic Renormalization Group Methods to Reaction-Diffusion Problems
• Coexistence versus extinction in the stochastic cyclic Lotka-Volterra model
• The Totally Asymmetric Simple Exclusion Process with Langmuir Kinetics
• Self-Organization of Mobile Populations in Cyclic Competition
• Noise and Correlations in a Spatial Population Model with Cyclic Competition
• A Fredholm Determinant Representation in ASEP
• Incorporating postleap checks in tau-leaping
• Extinction Rates for Fluctuation-Induced Metastabilities : A Real-Space WKB Approach
• Mapping out of equilibrium into equilibrium in one-dimensional transport models
• Growth dynamics and the evolution of cooperation in microbial populations
• Evolutionary game theory in growing populations
• Comparison of different moment-closure approximations for stochastic chemical kinetics
• Chemical warfare and survival strategies in bacterial range expansions
• Spectral theory of metastability and extinction in birth-death systems
• Microtubule Length-Regulation by Molecular Motors
• Non-perturbative fixed point in a non-equilibrium phase transition
• Exclusion Processes with Internal States
• Spectral solutions to stochastic models of gene expression with bursts and regulation
• Dynamics of interacting particle systems: stochastic process and field theory
• Stochastic differential equations for evolutionary dynamics with demographic noise and mutations
• A stochastic spectral analysis of transcriptional regulatory cascades
• Towards Classification of Phase Transitions in Reaction--Diffusion Models
• Evolutionary games of condensates in coupled birth-death processes
• Validity conditions for moment closure approximations in stochastic chemical kinetics
• Population oscillations in spatial stochastic Lotka-Volterra models: A field-theoretic perturbational analysis
• Total Current Fluctuations in ASEP
• Quasispecies theory for finite populations
• The dynamics of simple gene-network motifs subject to extrinsic fluctuations
• Field theories and exact stochastic equations for interacting particle systems
• Coherent-state path integral versus coarse-grained effective stochastic equation of motion: From reaction diffusion to stochastic sandpiles
• Non-equilibrium steady states of ideal bosonic and fermionic quantum gases
• Spectral formulation and WKB approximation for rare-event statistics in reaction systems
• A variational approach to the stochastic aspects of cellular signal transduction
• Extinction of oscillating populations
• Peeling and Sliding in Nucleosome Repositioning
• Langevin equations for reaction-diffusion processes
• The Emergence of Cooperation from a Single Mutant during Microbial Life-Cycles
• System size expansion using Feynman rules and diagrams
• Long Range Hops and the Pair Annihilation Reaction A+A->0: Renormalization Group and Simulation
• Imaginary noise and parity conservation in the reaction A+A <--> 0
• Non-selective evolution of growing populations
• Predator-Prey Quasi-cycles from a Path Integral Formalism
• Complete Solution of the Kinetics in a Far-from-equilibrium Ising Chain
• Branching rate expansion around annihilating random walks
• One-parameter extension of the Doi-Peliti formalism and relation with orthogonal polynomials
• Non-equilibrium Lyapunov function and a fluctuation relation for stochastic systems: Poisson representation approach
• Two-loop calculation of anomalous kinetics of the reaction $A+A\to \varnothing$ in randomly stirred fluid
• Complex noise in diffusion-limited reactions of replicating and competing species
• Statistics of trajectories in two-state master equations
• Duality-based calculations for transition probabilities in stochastic chemical reactions

Cited by in corpus (61)

• WKB theory of large deviations in stochastic populations
• Effective thermodynamics for a marginal observer
• Phase-space geometry of mass-conserving reaction-diffusion dynamics
• Run-and-tumble motion: field theory and entropy production
• Large deviations and dynamical phase transitions in stochastic chemical networks
• Interlacing Relaxation and First-Passage Phenomena in Reversible Discrete and Continuous Space Markovian Dynamics
• Physically consistent numerical solver for time-dependent Fokker-Planck equations
• Effective fluctuation and response theory
• A stochastic and dynamical view of pluripotency in mouse embryonic stem cells
• Macroscopic Stochastic Thermodynamics
• Ecological feedback in quorum-sensing microbial populations can induce heterogeneous production of autoinducers
• Efficiency fluctuations of stochastic machines undergoing a phase transition
• Reliability and entropy production in non-equilibrium electronic memories
• Stochastic thermodynamics of all-to-all interacting many-body systems
• Crowding and pausing strongly affect dynamics of kinesin-1 motors along microtubules
• Transient chaotic dimensionality expansion by recurrent networks
• Large deviations theory for noisy non-linear electronics: CMOS inverter as a case study
• Effective Hamiltonians and Lagrangians for conditioned Markov processes at large volume
• The Generalized Stochastic Microdosimetric Model: the main formulation
• Efficient Low-Order Approximation of First-Passage Time Distributions
• Path Weight Sampling: Exact Monte Carlo Computation of the Mutual Information between Stochastic Trajectories
• Coherence of oscillations in the weak-noise limit
• Degree Dispersion Increases the Rate of Rare Events in Population Networks
• Self-consistent formulations for stochastic nonlinear neuronal dynamics
• Experimental measurement of relative path probabilities and stochastic actions
• Beat of a current
• Large deviation induced phase switch in an inertial majority-vote model
• Construction of stochastic hybrid path integrals using "quantum-mechanical'' operators
• Stochastic action for tubes: Connecting path probabilities to measurement
• Combinatorial Conversion and Moment Bisimulation for Stochastic Rewriting Systems
• Subdiffusive Activity Spreading in the Diffusive Epidemic Process
• Critical heat current fluctuations in Curie-Weiss model in and out of equilibrium
• A stochastic field theory for the evolution of quantitative traits in finite populations
• Applications of WKB and Fokker-Planck methods in analyzing population extinction driven by weak demographic fluctuations
• Field theories and quantum methods for stochastic reaction-diffusion systems
• Spin coherent states and stochastic hybrid path integrals
• Doi-Peliti Path Integral Methods for Stochastic Systems with Partial Exclusion
• Squeezing stationary distributions of stochastic chemical reaction systems
• Dynamical Phase Transitions in Open Quantum Walks
• Stochastic paths controlling speed and dissipation
• Diffusion equations from master equations -- A discrete geometric approach --
• Accelerated first-passage dynamics in a non-Markovian feedback Ornstein--Uhlenbeck process
• The chemical birth-death process with Gillespie noise
• Magnus expansions and pseudospectra of Master Equations
• Time-fractional kinetic equation for the non-Markovian kinetic processes
• Chemical master equation parameter exploration using DMRG
• Open reaction-diffusion systems: bridging probabilistic theory and simulations across scales
• Conformal Dimensions On Causal Random Geometry
• Noise equals control
• Stochastic Thermodynamics of Non-reciprocally Interacting Particles and Fields
• Effective one-dimension reduction of multi-compartment complex systems dynamics
• Cavity-induced bifurcation in classical rate theory
• Redundant basis interpretation of Doi-Peliti method and an application
• Mean-field equation for a stochastic many-particle model of quorum-sensing microbial populations
• Ubiquitous Dynamical Time Asymmetry in Measurements on Materials and Biological Systems
• Quantum state-preparation control in noisy environment via most-likely paths
• Sojourn probabilities in tubes and pathwise irreversibility for Itô processes
• Comprehensive review of models and methods for inferences in bio-chemical reaction networks
• Path integrals for stochastic hybrid reaction-diffusion processes
• From orders to prices: A stochastic description of the limit order book to forecast intraday returns
• Stochastic Control of Tolman-Oppenheimer-Snyder Collapse of Zero-Pressure Stars to Black Holes: Rigorous Criteria for Density Bounds and Singularity Smoothing