### A stochastic diffusion process for Lochner's generalized Dirichlet distribution

The method of potential solutions of Fokker-Planck equations is used to develop a transport equation for the joint probability of N stochastic variables with Lochner’s generalized Dirichlet distribution as its asymptotic solution. Individual samples of a discrete ensemble, obtained from the system of stochastic differential equations, equivalent to the Fokker-Planck equation developed here, satisfy a unit-sum constraint at all times and ensure a bounded sample space, similarly to the process developed in for the Dirichlet distribution. Consequently, the generalized Dirichlet diffusion process may be used to represent realizations of a fluctuating ensemble of N variables subject to a conservation principle. Compared to the Dirichlet distribution and process, the additional parameters of the generalized Dirichlet distribution allow a more general class of physical processes to be modeled with a more general covariance matrix.

- Publication Date:

- Report Number(s):
- LA-UR-13-21573

Journal ID: ISSN 0022-2488; JMAPAQ

- Grant/Contract Number:
- AC52-06NA25396

- Type:
- Accepted Manuscript

- Journal Name:
- Journal of Mathematical Physics

- Additional Journal Information:
- Journal Volume: 54; Journal Issue: 10; Journal ID: ISSN 0022-2488

- Publisher:
- American Institute of Physics (AIP)

- Research Org:
- Los Alamos National Lab. (LANL), Los Alamos, NM (United States)

- Sponsoring Org:
- USDOE

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 97 MATHEMATICS AND COMPUTING; Fokker-Planck equation, Stochastic diffusion, Generalized Dirichlet distribution

- OSTI Identifier:
- 1233161

```
Bakosi, J., and Ristorcelli, J. R..
```*A stochastic diffusion process for Lochner's generalized Dirichlet distribution*. United States: N. p.,
Web. doi:10.1063/1.4822416.

```
Bakosi, J., & Ristorcelli, J. R..
```*A stochastic diffusion process for Lochner's generalized Dirichlet distribution*. United States. doi:10.1063/1.4822416.

```
Bakosi, J., and Ristorcelli, J. R.. 2013.
"A stochastic diffusion process for Lochner's generalized Dirichlet distribution". United States.
doi:10.1063/1.4822416. https://www.osti.gov/servlets/purl/1233161.
```

```
@article{osti_1233161,
```

title = {A stochastic diffusion process for Lochner's generalized Dirichlet distribution},

author = {Bakosi, J. and Ristorcelli, J. R.},

abstractNote = {The method of potential solutions of Fokker-Planck equations is used to develop a transport equation for the joint probability of N stochastic variables with Lochner’s generalized Dirichlet distribution as its asymptotic solution. Individual samples of a discrete ensemble, obtained from the system of stochastic differential equations, equivalent to the Fokker-Planck equation developed here, satisfy a unit-sum constraint at all times and ensure a bounded sample space, similarly to the process developed in for the Dirichlet distribution. Consequently, the generalized Dirichlet diffusion process may be used to represent realizations of a fluctuating ensemble of N variables subject to a conservation principle. Compared to the Dirichlet distribution and process, the additional parameters of the generalized Dirichlet distribution allow a more general class of physical processes to be modeled with a more general covariance matrix.},

doi = {10.1063/1.4822416},

journal = {Journal of Mathematical Physics},

number = 10,

volume = 54,

place = {United States},

year = {2013},

month = {10}

}