# Systematic method for solving transport equations derived from master equations

## Abstract

It is shown how a one-dimensional transport equation (derived from a master equation) for a time-dependent conditional average of an arbitrary function can be solved successively, if the moments of the transition probability can be expanded in a series of a special parameter. The transport equation for the time-dependent conditional average value can be transformed into a set of l linear partial differential equations of first order for the lth approximation of the conditional average value. The system of equations is closed in the sense that the lth partial differential equation is determined by the solutions of the l-1 partial differential equations. It is shown how the results obtained compare with the linear noise approximation. Two examples are chosen to illustrate the way to use the method. One is the classical example of the diffusion of a Brownian particle, the other is Alkemade's diode.

- Authors:

- Publication Date:

- Research Org.:
- Austrian Research Center, Seibersdorf, A-2444 Seibersdorf, Austria

- OSTI Identifier:
- 5733111

- Resource Type:
- Journal Article

- Journal Name:
- Phys. Rev. A; (United States)

- Additional Journal Information:
- Journal Volume: 28:2

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; TRANSPORT THEORY; ANALYTICAL SOLUTION; BROWNIAN MOVEMENT; CHAPMAN-KOLMOGOROV EQUATION; KUBO FORMULA; MARKOV PROCESS; ONE-DIMENSIONAL CALCULATIONS; PROBABILITY; SEMICONDUCTOR DIODES; TIME DEPENDENCE; DIFFERENTIAL EQUATIONS; EQUATIONS; SEMICONDUCTOR DEVICES; STOCHASTIC PROCESSES; 657006* - Theoretical Physics- Statistical Physics & Thermodynamics- (-1987)

### Citation Formats

```
Eder, O J, and Lackner, T.
```*Systematic method for solving transport equations derived from master equations*. United States: N. p., 1983.
Web. doi:10.1103/PhysRevA.28.952.

```
Eder, O J, & Lackner, T.
```*Systematic method for solving transport equations derived from master equations*. United States. doi:10.1103/PhysRevA.28.952.

```
Eder, O J, and Lackner, T. Mon .
"Systematic method for solving transport equations derived from master equations". United States. doi:10.1103/PhysRevA.28.952.
```

```
@article{osti_5733111,
```

title = {Systematic method for solving transport equations derived from master equations},

author = {Eder, O J and Lackner, T},

abstractNote = {It is shown how a one-dimensional transport equation (derived from a master equation) for a time-dependent conditional average of an arbitrary function can be solved successively, if the moments of the transition probability can be expanded in a series of a special parameter. The transport equation for the time-dependent conditional average value can be transformed into a set of l linear partial differential equations of first order for the lth approximation of the conditional average value. The system of equations is closed in the sense that the lth partial differential equation is determined by the solutions of the l-1 partial differential equations. It is shown how the results obtained compare with the linear noise approximation. Two examples are chosen to illustrate the way to use the method. One is the classical example of the diffusion of a Brownian particle, the other is Alkemade's diode.},

doi = {10.1103/PhysRevA.28.952},

journal = {Phys. Rev. A; (United States)},

number = ,

volume = 28:2,

place = {United States},

year = {1983},

month = {8}

}