# Coagulation kinetics beyond mean field theory using an optimised Poisson representation

## Abstract

Binary particle coagulation can be modelled as the repeated random process of the combination of two particles to form a third. The kinetics may be represented by population rate equations based on a mean field assumption, according to which the rate of aggregation is taken to be proportional to the product of the mean populations of the two participants, but this can be a poor approximation when the mean populations are small. However, using the Poisson representation, it is possible to derive a set of rate equations that go beyond mean field theory, describing pseudo-populations that are continuous, noisy, and complex, but where averaging over the noise and initial conditions gives the mean of the physical population. Such an approach is explored for the simple case of a size-independent rate of coagulation between particles. Analytical results are compared with numerical computations and with results derived by other means. In the numerical work, we encounter instabilities that can be eliminated using a suitable “gauge” transformation of the problem [P. D. Drummond, Eur. Phys. J. B 38, 617 (2004)] which we show to be equivalent to the application of the Cameron-Martin-Girsanov formula describing a shift in a probability measure. The cost ofmore »

- Authors:

- Department of Mathematics, UCL, Gower Street, London WC1E 6BT (United Kingdom)
- Department of Physics and Astronomy, UCL, Gower Street, London WC1E 6BT (United Kingdom)

- Publication Date:

- OSTI Identifier:
- 22415793

- Resource Type:
- Journal Article

- Journal Name:
- Journal of Chemical Physics

- Additional Journal Information:
- Journal Volume: 142; Journal Issue: 19; Other Information: (c) 2015 AIP Publishing LLC; Country of input: International Atomic Energy Agency (IAEA); Journal ID: ISSN 0021-9606

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; AGGLOMERATION; APPROXIMATIONS; COMPARATIVE EVALUATIONS; COMPUTERIZED SIMULATION; GAUGE INVARIANCE; MEAN-FIELD THEORY; MONTE CARLO METHOD; NOISE; PARTICLES; PROBABILITY; RANDOMNESS; REACTION KINETICS; TRANSFORMATIONS

### Citation Formats

```
Burnett, James, and Ford, Ian J.
```*Coagulation kinetics beyond mean field theory using an optimised Poisson representation*. United States: N. p., 2015.
Web. doi:10.1063/1.4921350.

```
Burnett, James, & Ford, Ian J.
```*Coagulation kinetics beyond mean field theory using an optimised Poisson representation*. United States. doi:10.1063/1.4921350.

```
Burnett, James, and Ford, Ian J. Thu .
"Coagulation kinetics beyond mean field theory using an optimised Poisson representation". United States. doi:10.1063/1.4921350.
```

```
@article{osti_22415793,
```

title = {Coagulation kinetics beyond mean field theory using an optimised Poisson representation},

author = {Burnett, James and Ford, Ian J.},

abstractNote = {Binary particle coagulation can be modelled as the repeated random process of the combination of two particles to form a third. The kinetics may be represented by population rate equations based on a mean field assumption, according to which the rate of aggregation is taken to be proportional to the product of the mean populations of the two participants, but this can be a poor approximation when the mean populations are small. However, using the Poisson representation, it is possible to derive a set of rate equations that go beyond mean field theory, describing pseudo-populations that are continuous, noisy, and complex, but where averaging over the noise and initial conditions gives the mean of the physical population. Such an approach is explored for the simple case of a size-independent rate of coagulation between particles. Analytical results are compared with numerical computations and with results derived by other means. In the numerical work, we encounter instabilities that can be eliminated using a suitable “gauge” transformation of the problem [P. D. Drummond, Eur. Phys. J. B 38, 617 (2004)] which we show to be equivalent to the application of the Cameron-Martin-Girsanov formula describing a shift in a probability measure. The cost of such a procedure is to introduce additional statistical noise into the numerical results, but we identify an optimised gauge transformation where this difficulty is minimal for the main properties of interest. For more complicated systems, such an approach is likely to be computationally cheaper than Monte Carlo simulation.},

doi = {10.1063/1.4921350},

journal = {Journal of Chemical Physics},

issn = {0021-9606},

number = 19,

volume = 142,

place = {United States},

year = {2015},

month = {5}

}