# Anomalous Diffusion Equations with Multiplicative Acceleration

## Abstract

A generalization of the model of Lévy walks with traps is considered. The main difference between the model under consideration and the already existing models is the introduction of multiplicative particle acceleration at collisions. The introduction of acceleration transfers the consideration of walks to coordinate–momentum phase space, which allows both the spatial distribution of particles and their spectrum to be obtained. The kinetic equations in coordinate–momentum phase space have been derived for the case of walks with two possible states. This system of equations in a special case is shown to be reduced to ordinary Lévy walks. This system of kinetic equations admits of integration over the spatial variable, which transfers the consideration only to momentum space and allows the spectrum to be calculated. An exact solution of the kinetic equations can be obtained in terms of the Laplace–Mellin transform. The inverse transform can be performed only for the asymptotic solutions. The calculated spectra are compared with the results of Monte Carlo simulations, which confirm the validity of the derived asymptotics.

- Authors:

- Ulyanovsk State University, Kapitza Research Technological Institute (Russian Federation)

- Publication Date:

- OSTI Identifier:
- 22749903

- Resource Type:
- Journal Article

- Journal Name:
- Journal of Experimental and Theoretical Physics

- Additional Journal Information:
- Journal Volume: 126; Journal Issue: 4; Other Information: Copyright (c) 2018 Pleiades Publishing, Inc.; Country of input: International Atomic Energy Agency (IAEA); Journal ID: ISSN 1063-7761

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; ASYMPTOTIC SOLUTIONS; COMPUTERIZED SIMULATION; DIFFUSION EQUATIONS; EXACT SOLUTIONS; KINETIC EQUATIONS; MONTE CARLO METHOD; PHASE SPACE; SPATIAL DISTRIBUTION

### Citation Formats

```
Saenko, V. V., E-mail: saenkoslava@mail.ru.
```*Anomalous Diffusion Equations with Multiplicative Acceleration*. United States: N. p., 2018.
Web. doi:10.1134/S1063776118030202.

```
Saenko, V. V., E-mail: saenkoslava@mail.ru.
```*Anomalous Diffusion Equations with Multiplicative Acceleration*. United States. doi:10.1134/S1063776118030202.

```
Saenko, V. V., E-mail: saenkoslava@mail.ru. Sun .
"Anomalous Diffusion Equations with Multiplicative Acceleration". United States. doi:10.1134/S1063776118030202.
```

```
@article{osti_22749903,
```

title = {Anomalous Diffusion Equations with Multiplicative Acceleration},

author = {Saenko, V. V., E-mail: saenkoslava@mail.ru},

abstractNote = {A generalization of the model of Lévy walks with traps is considered. The main difference between the model under consideration and the already existing models is the introduction of multiplicative particle acceleration at collisions. The introduction of acceleration transfers the consideration of walks to coordinate–momentum phase space, which allows both the spatial distribution of particles and their spectrum to be obtained. The kinetic equations in coordinate–momentum phase space have been derived for the case of walks with two possible states. This system of equations in a special case is shown to be reduced to ordinary Lévy walks. This system of kinetic equations admits of integration over the spatial variable, which transfers the consideration only to momentum space and allows the spectrum to be calculated. An exact solution of the kinetic equations can be obtained in terms of the Laplace–Mellin transform. The inverse transform can be performed only for the asymptotic solutions. The calculated spectra are compared with the results of Monte Carlo simulations, which confirm the validity of the derived asymptotics.},

doi = {10.1134/S1063776118030202},

journal = {Journal of Experimental and Theoretical Physics},

issn = {1063-7761},

number = 4,

volume = 126,

place = {United States},

year = {2018},

month = {4}

}