Critical Dynamics : The Expansion of the Master Equation Including a Critical Point
doctoral thesis
In this thesis it is shown how to solve the master equation for a Markov process including a critical point by means of successive approximations in terms of a small parameter. A critical point occurs if, by adjusting an externally controlled quantity, the system shows a transition from normal monostable to bistable behaviour. Examples of the external quantity (the pump parameter) are temperature, electric discharge current, chemical concentrations and mechanical force. The appropriate small parameter may be either the diffusion coefficient or the inverse size of the system. The latter is usually given by the volume or by the total number of constituents such as spins, photons or molecules.
The fundamental idea of the theory is to separate the master equation into its proper irreducible part and a corrective remainder. The irreducible or zeroth order stochastic approximation will be a relatively simple Fokker-Planck equation that contains the essential features of the process. Once the solution of this irreducible equation is known, the higher order corrections in the original master equation can be incorporated in a systematic manner.
In part I of this thesis we consider the problem of diffusion in an externally applied potential showing a monostable to bistable transition. The appendix of part I presents a discussion of the irreducible solutions. In part II we examine the general Markov process. The appendix of part II is devoted to an example, namely the magnetic mean field Ising model.
The fundamental idea of the theory is to separate the master equation into its proper irreducible part and a corrective remainder. The irreducible or zeroth order stochastic approximation will be a relatively simple Fokker-Planck equation that contains the essential features of the process. Once the solution of this irreducible equation is known, the higher order corrections in the original master equation can be incorporated in a systematic manner.
In part I of this thesis we consider the problem of diffusion in an externally applied potential showing a monostable to bistable transition. The appendix of part I presents a discussion of the irreducible solutions. In part II we examine the general Markov process. The appendix of part II is devoted to an example, namely the magnetic mean field Ising model.
TNO Identifier
352145
Publisher
TNO
Collation
50 p.
Place of publication
Den Haag