# Spectral theory of metastability and extinction in birth-death systems.

@article{Assaf2006SpectralTO,
title={Spectral theory of metastability and extinction in birth-death systems.},
author={Michael Assaf and Baruch Meerson},
journal={Physical review letters},
year={2006},
volume={97 20},
pages={
200602
}
}
• Published 16 October 2006
• Mathematics
• Physical review letters
We suggest a general spectral method for calculating the statistics of multistep birth-death processes and chemical reactions of the type mA-->nA (m and n are positive integers) which possess an absorbing state. The method employs the generating function formalism in conjunction with the Sturm-Liouville theory of linear differential operators. It yields accurate results for the extinction statistics and for the quasistationary probability distribution, including large deviations, of the…
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## References

SHOWING 1-10 OF 60 REFERENCES

### Large fluctuations and optimal paths in chemical kinetics

• Mathematics
• 1994
The eikonal approximation (instanton technique) is applied to the problem of large fluctuations of the number of species in spatially homogeneous chemical reactions with the probability density

### Spectral formulation and WKB approximation for rare-event statistics in reaction systems.

• Physics
Physical review. E, Statistical, nonlinear, and soft matter physics
• 2006
We develop a spectral formulation and a stationary WKB approximation for calculating the probabilities of rare events (large deviations from the mean) in systems of reacting particles with

### Extinction Times for Birth-Death Processes: Exact Results, Continuum Asymptotics, and the Failure of the Fokker-Planck Approximation

• Mathematics
Multiscale Model. Simul.
• 2005
An exact expression is given for the mean time to extinction in the discrete case and its asymptotic expansion for large values of the population scale and it is observed that the Fokker--Planck approximation is valid only quite near the threshold.

### Stochastic models for chemical reactions: I. Theory of the unimolecular reaction process

A stochastic model for the basic unimolecular chemical reaction$$A\mathop \to \limits^\mu B$$ is derived. This model provides a mathematical basis, altogether missing in the current kinetic theory,

### Field Theory of Branching and Annihilating Random Walks

• Physics
• 1997
We develop a systematic analytic approach to the problem of branching and annihilating random walks, equivalent to the diffusion-limited reaction processes 2A → ∅ and A → (m + 1) A, where m ≥ 1.

### Rare event statistics in reaction-diffusion systems.

• Physics
Physical review. E, Statistical, nonlinear, and soft matter physics
• 2004
We present an efficient method to calculate probabilities of large deviations from the typical behavior (rare events) in reaction-diffusion systems. This method is based on a semiclassical treatment

### On the Stochastic Theory of Dissociation and Recombination of Diatomic Molecules in Inert Diluents

A formal stochastic theory is developed to describe the dissociation and recombination of diatomic molecules by inert third bodies, and the manner in which this reaction is coupled with internal

### Extinction and quasi-stationarity in the Verhulst logistic model.

• I. Nåsell
• Environmental Science
Journal of theoretical biology
• 2001
A stochastic version of the Verhulst deterministic model for density-dependent growth of a single population with three parameter regions with qualitatively different behaviours is formulated and analysed.

### Statistical Fluctuations in Autocatalytic Reactions

The differential equations describing the statistical fluctuations of a simple autocatalytic reaction mechanism are set up and solved completely. The fluctuations are found to approach a constant

### Decay of the metastable state in a chemical system: Different predictions between discrete and continuous models

• Mathematics
• 1996
We show, using a specific example of a chemical reaction, that the rate constants predicted from the discrete master equation and its continuum Fokker-Planck approximations differ in exponential