# Deterministic and Stochastic Becker–Döring Equations: Past and Recent Mathematical Developments

@article{Hingant2016DeterministicAS, title={Deterministic and Stochastic Becker–D{\"o}ring Equations: Past and Recent Mathematical Developments}, author={Erwan Hingant and Romain Yvinec}, journal={arXiv: Mathematical Physics}, year={2016}, pages={175-204} }

Becker and Dorimy introduced their equations in 1935 to describes nucleation in supersaturated vapors. Since then, these equations have been popularized to a wide range of applications and Slemrod in 2000 said they “provide perhaps the simplest kinetic model to describe [...] phase transitions”. In this survey we attempt to give an overview of the results obtained on these equations in the parts decades until today. Particularly we gathered results on both deterministic and stochastic versions…

## 17 Citations

### Quasi-stationary distribution and metastability for the stochastic Becker-Döring model

- Mathematics
- 2020

We study a stochastic version of the classical Becker-Doring model, a well-known kinetic model for cluster formation that predicts the existence of a long-lived metastable state before a…

### Temporal oscillations in Becker–Döring equations with atomization

- MathematicsNonlinearity
- 2020

We prove that time-periodic solutions arise via Hopf bifurcation in a finite closed system of coagulation-fragmentation equations. The system we treat is a variant of the Becker–Döring equations, in…

### The Becker-D\"oring process: law of large numbers and non-equilibrium potential

- Mathematics
- 2018

In this note, we prove a law of large numbers for an infinite chemical reaction network for phase transition problems called the stochastic Becker-Doring process. Under a general condition on the…

### Oscillations in a Becker-Döring Model with Injection and Depletion

- PhysicsSIAM J. Appl. Math.
- 2022

We study the Becker–Döring bubblelator, a variant of the Becker–Döring coagulation-fragmentation system that models the growth of clusters by gain or loss of monomers. Motivated by models of gas…

### A Functional Central Limit Theorem for the Becker–Döring Model

- Mathematics
- 2018

We investigate the fluctuations of the stochastic Becker–Döring model of polymerization when the initial size of the system converges to infinity. A functional central limit problem is proved for the…

### A Functional Central Limit Theorem for the Becker–Döring Model

- MathematicsJournal of Statistical Physics
- 2018

We investigate the fluctuations of the stochastic Becker–Döring model of polymerization when the initial size of the system converges to infinity. A functional central limit problem is proved for the…

### On a modified Becker–Döring model for two-phase materials

- EngineeringContinuum Mechanics and Thermodynamics
- 2019

This work reconsiders the Becker–Döring model for nucleation under isothermal conditions in the presence of phase transitions. Based on thermodynamic principles, a modified model is derived where the…

### On a modified Becker–Döring model for two-phase materials

- EngineeringContinuum Mechanics and Thermodynamics
- 2019

This work reconsiders the Becker–Döring model for nucleation under isothermal conditions in the presence of phase transitions. Based on thermodynamic principles, a modified model is derived where the…

### Fokker-Planck Approach of Ostwald Ripening: Simulation of a Modified Lifshitz-Slyozov-Wagner System with a Diffusive Correction

- Materials ScienceSIAM J. Sci. Comput.
- 2020

A well-balanced scheme for the modified Lifshitz--Slyozov equation, which incorporates a size-diffusion term and uses the Fokker--Planck structure of the equation.

### The initial-boundary value problem for the Lifshitz–Slyozov equation with non-smooth rates at the boundary

- Mathematics
- 2020

We prove existence and uniqueness of solutions to the initial-boundary value problem for the Lifshitz–Slyozov equation (a nonlinear transport equation on the half-line), focusing on the case of…

## References

SHOWING 1-10 OF 100 REFERENCES

### The Becker-Döring Equations

- Mathematics
- 2000

This chapter deals with the modified Becker-Doring equations, which provide perhaps the simplest kinetic model to describe a number of issues in the dynamics of phase transitions, e.g. metastability,…

### Stochastic analysis of nucleation rates.

- MathematicsPhysical review. E
- 2016

Using a simple and general model for the attachment and detachment rates, it is found that the Ito choice approximates the nucleation rate best and also coincides with the Fokker-Planck equation resulting from the common way to Taylor expand the original set of rate equations.

### METASTABILITY IN THE CLASSICAL, TRUNCATED BECKER–DÖRING EQUATIONS

- MathematicsProceedings of the Edinburgh Mathematical Society
- 2002

Abstract We show that in the classical (fixed-monomer-concentration) Becker–Döring equations truncated at finite cluster size, the slow evolution (metastability) of solutions can be explained in…

### The Becker-Döring equations with exponentially size-dependent rate coefficients

- Mathematics
- 2004

This paper is concerned with an analysis of the Becker-Doring equations which lie at the heart of a number of descriptions of non-equilibrium phase transitions and related complex dynamical…

### From the Becker–Döring to the Lifshitz–Slyozov–Wagner Equations

- Mathematics
- 2002

Connections between two classical models of phase transitions, the Becker–Döring (BD) equations and the Lifshitz–Slyozov–Wagner (LSW) equations, are investigated. Homogeneous coefficients are…

### The Becker–Döring Equations and the Lifshitz–Slyozov Theory of Coarsening

- Materials Science
- 1998

In this paper the relation between the kinetic set of Becker–Döring (BD) equations and the classical Lifshitz–Slyozov (LS) theory of coarsening is studied. A model that resembles the LS theory but…

### On the Evolution of Large Clusters in the Becker-Döring Model

- MathematicsJ. Nonlinear Sci.
- 2003

This work rigorously derive for general coefficients that the evolution of these large clusters is described by a nonlocal transport equation, which is for specific coefficients the classical coarsening model by Lifshitz, Slyozov, and Wagner (LSW).

### A Scaling Limit of the Becker-Döring Equations in the Regime of Small Excess Density

- PhysicsJ. Nonlinear Sci.
- 2004

It is shown rigorously that the leading order dynamics are governed by another variant of the classical mean-field model in which total mass is preserved, not as large as in Penrose (1997) or Niethammer (2003).

### The Becker-Doring cluster equations

- Mathematics
- 1997

The Becker-Döring equations provide a model of the dynamics of a system consisting of a large number of identical particles. The particles can coagulate to form clusters, which in turn can fragment…

### Classical Becker-Döring cluster equations: rigorous results on metastability and long-time behaviour

- Mathematics
- 1993

We consider the classical Becker-Doring cluster equations with constant monomer concentration c1 = z > 0 and
as a model which describes the kinetics of a first-order phase transition.…