### Technical Report 2005-023

# Dimensional Reduction of the Fokker-Planck Equation for Stochastic Chemical Reactions

### Per Lötstedt and Lars Ferm

August 2005

**Abstract:**The Fokker-Planck equation models chemical reactions on a mesoscale. The solution is a probability density function for the copy number of the different molecules. The number of dimensions of the problem can be large making numerical simulation of the reactions computationally intractable. The number of dimensions is reduced here by deriving partial differential equations for the first moments of some of the species and coupling them to a Fokker-Planck equation for the remaining species. With more simplifying assumptions, another system of equations is derived consisting of integro-differential equations and a Fokker-Planck equation. In this way, the simulation of the chemical networks is possible without the exponential growth in computatational work and memory of the original equation and with better modelling accuracy than the macroscopic reaction rate equations. Some terms in the equations are small and are ignored. Conditions are given for the influence of these terms to be small on the equations and the solutions. The difference between different models is illustrated in a numerical example.

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