Introduction

Introduction

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Physical and chemical interactions between atoms and molecules manifest themselves in numerous phenomena, and are studied either using deterministic or stochastic models (an overview of deterministic and stochastic models is given in Vlachos, 2004). In the early years of chemical engineering, continuum deterministic equations pertaining to material, energy and momentum balances (Bird et al., 1960) were derived and formed the backbone of chemical engineering modeling. These models employ phenomenological constitutive equations and neglect the role of molecular scales. There are two other important assumptions in deterministic, continuum models, namely —

that discrete representation of matter and energy can be replaced by a continuum field, and

that the system evolves deterministically, i.e., the role of (thermal) fluctuations in the system is negligible.

These the assumptions are typically addressed by stochastic simulation. By comparison to deterministic, continuum models, stochastic models have received less attention. Continuum models are often inadequate to describe intermolecular forces and the role of fluctuations in nonlinear behavior such as pattern formation, phase transitions, oscillations, etc. (Raimondeau and Vlachos, 2002). Very often in biological cells, nanodevices, catalyst surfaces, (turbulent) eddies, etc. the number of atoms or molecules N of a particular species is small, rendering fluctuations significant. This is corroborated by statistical mechanics, which suggests that the fluctuations in a system, at least at equilibrium, are proportional to N^-1/2. For nonlinear processes and phenomena, temporal and spatial correlations stemming from intermolecular forces and/or small system sizes are important and fluctuations should be included (Van Kampen, 1992; Gardiner, 1985).
The statistical mechanics master equation approach is the starting point for a probabilistic representation of far from equilibrium Markovian processes (Van Kampen, 1992; Gardiner 1985). For example the chemical master equation (Gillespie, 1992)

describes the evolution of the probability distribution function for N species (S₁, S₂, …, S_N) in M reactions (R₁, R₂, …, R_M) in well mixed conditions. B_jdt and a_jPdt are the probabilities of moving away from and to a particular state (X₁, X₂, …, X_N), respectively in a time interval dt. The number of molecules of a species S_i is denoted by its population size X_i. Further, a_j is the transition probability per unit time for reaction R_j and n_ij is the stoichiometric coefficient, i.e., the number of molecules of species S_i in reaction R_j.

With the exception of simple reaction networks, such as the reversible isomerization reaction S₁=S₂, the chemical master equation cannot be solved analytically (Gillespie, 1977). Instead one solves the master equation numerically using the Monte Carlo method. The stochastic simulation algorithm (SSA) is one such Monte Carlo algorithm (Gillespie, 1976; 1977). SSA uses a discrete representation for particles (see figure). Continuum stochastic representations can also be found in the literature. For example, the chemical Langevin equation (see Gillespie, 2000, in the context of chemical reactions) employs continuum species populations with an added white noise term. However, this approach is valid for large population sizes only.

References

Bird, R. B., W. E. Stewart, et al. (1960). Transport phenomena. New York, Wiley.

Gardiner, C. W. (1985). Handbook of stochastic methods, Springer-Verlag.

Gillespie, D. T. (1976). "A general method for numerically simulating the stochastic evolution of coupled chemical reactions." Journal of Computational Physics 22: 403-434.

Gillespie, D. T. (1977). "Exact stochastic simulation of coupled chemical reactions." Journal of Physical Chemistry 81: 2340-2361.

Gillespie, D. T. (1992). "A rigorous derivation of the chemical master equation." Physica A 188: 404-425.

Gillespie, D. T. (2000). "The chemical Langevin equation." Journal of Chemical Physics 113(1): 297-306.

Kampen, N. G. V. (1992). Stochastic processes in physics and chemistry, North-Holland personal library.

Raimondeau, S. and D. G. Vlachos (2002). "Recent developments on multiscale, hierarchical modeling of chemical reactors." Chemical Engineering Journal 90(1-2): 3-23.

Vlachos, D. G. (2004). "A review of multiscale analysis: Examples from systems biology, materials engineering, and other fluid-surface interacting systems." Adv. Chem. Eng.: accepted.

Professor Dion G. Vlachos
Department of Chemical Engineering
University of Delaware
Newark, DE 19716 USA

Phone: 302-831-2830
Fax: 302-831-1048
Email: vlachos@che.udel.edu

Created and designed by Abhijit Chatterjee