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Variational formula for the relaxation time in the Boltzmann equation
L Giuggioli1, P E Parris, V M Kenkre
1Consortium of the Americas for Interdisciplinary Science, University of New Mexico, Albuquerque, New Mexico 87131, USA.
The Journal of Physical Chemistry. B
|September 22, 2006
Summary
The study introduces a new variational principle to improve the relaxation time approximation (RTA) for the Boltzmann equation. This method uses initial-condition-dependent relaxation times, offering a more accurate approach for nonequilibrium statistical mechanics.
Area of Science:
- Statistical Mechanics
- Computational Physics
- Quantum Transport
Background:
- The Boltzmann equation is fundamental in describing systems far from equilibrium.
- The standard relaxation time approximation (RTA) simplifies Boltzmann equation solutions but has limitations.
- RTA's fixed relaxation times neglect crucial initial condition dependencies.
Purpose of the Study:
- To develop an improved relaxation time approximation for the Boltzmann equation.
- To incorporate initial condition dependencies into the RTA.
- To enhance the accuracy of nonequilibrium statistical mechanics calculations.
Main Methods:
- Derivation of a variational principle for Boltzmann equation solutions.
- Extension of the standard RTA to include initial-distribution-dependent relaxation times.
- Application to calculate mobility in a 1D tight-binding model.
Main Results:
- The developed variational principle yields initial-condition-dependent relaxation times.
- The extended RTA provides a more accurate approximation compared to the standard RTA.
- Improved mobility calculations for a 1D tight-binding band were achieved.
Conclusions:
- The new variational approach enhances the accuracy of the RTA for nonequilibrium systems.
- Initial condition dependence is crucial for accurate relaxation time calculations.
- This method offers a more refined tool for statistical mechanics and transport phenomena.
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