# Classical non-Markovian Boltzmann equation

## Abstract

The modeling of particle transport involves anomalous diffusion, (x²(t) ) ∝ t{sup α} with α ≠ 1, with subdiffusive transport corresponding to 0 < α < 1 and superdiffusive transport to α > 1. These anomalies give rise to fractional advection-dispersion equations with memory in space and time. The usual Boltzmann equation, with only isolated binary collisions, is Markovian and, in particular, the contributions of the three-particle distribution function are neglected. We show that the inclusion of higher-order distribution functions give rise to an exact, non-Markovian Boltzmann equation with resulting transport equations for mass, momentum, and kinetic energy with memory in both time and space. The two- and the three-particle distribution functions are considered under the assumption that the two- and the three-particle correlation functions are translationally invariant that allows us to obtain advection-dispersion equations for modeling transport in terms of spatial and temporal fractional derivatives.

- Authors:

- Department of Physics and Physical Oceanography, University of North Carolina Wilmington, Wilmington, North Carolina 28403-5606 (United States)

- Publication Date:

- OSTI Identifier:
- 22306199

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Journal of Mathematical Physics; Journal Volume: 55; Journal Issue: 8; Other Information: (c) 2014 AIP Publishing LLC; Country of input: International Atomic Energy Agency (IAEA)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; ADVECTION; BOLTZMANN EQUATION; CORRELATION FUNCTIONS; DIFFUSION; DISTRIBUTION FUNCTIONS; MARKOV PROCESS; TRANSPORT THEORY

### Citation Formats

```
Alexanian, Moorad, E-mail: alexanian@uncw.edu.
```*Classical non-Markovian Boltzmann equation*. United States: N. p., 2014.
Web. doi:10.1063/1.4886475.

```
Alexanian, Moorad, E-mail: alexanian@uncw.edu.
```*Classical non-Markovian Boltzmann equation*. United States. doi:10.1063/1.4886475.

```
Alexanian, Moorad, E-mail: alexanian@uncw.edu. Fri .
"Classical non-Markovian Boltzmann equation". United States.
doi:10.1063/1.4886475.
```

```
@article{osti_22306199,
```

title = {Classical non-Markovian Boltzmann equation},

author = {Alexanian, Moorad, E-mail: alexanian@uncw.edu},

abstractNote = {The modeling of particle transport involves anomalous diffusion, (x²(t) ) ∝ t{sup α} with α ≠ 1, with subdiffusive transport corresponding to 0 < α < 1 and superdiffusive transport to α > 1. These anomalies give rise to fractional advection-dispersion equations with memory in space and time. The usual Boltzmann equation, with only isolated binary collisions, is Markovian and, in particular, the contributions of the three-particle distribution function are neglected. We show that the inclusion of higher-order distribution functions give rise to an exact, non-Markovian Boltzmann equation with resulting transport equations for mass, momentum, and kinetic energy with memory in both time and space. The two- and the three-particle distribution functions are considered under the assumption that the two- and the three-particle correlation functions are translationally invariant that allows us to obtain advection-dispersion equations for modeling transport in terms of spatial and temporal fractional derivatives.},

doi = {10.1063/1.4886475},

journal = {Journal of Mathematical Physics},

number = 8,

volume = 55,

place = {United States},

year = {Fri Aug 01 00:00:00 EDT 2014},

month = {Fri Aug 01 00:00:00 EDT 2014}

}