# Linear Boltzmann equation as the weak coupling limit of a random Schrödinger equation

Erdös L, Yau H. 2000. Linear Boltzmann equation as the weak coupling limit of a random Schrödinger equation. Communications on Pure and Applied Mathematics. 53(6), 667–735.

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Author

Erdös, László

^{ISTA}^{}; Yau, HorngAbstract

We study the time evolution of a quantum particle in a Gaussian random environment. We show that in the weak coupling limit the Wigner distribution of the wave function converges to a solution of a linear Boltzmann equation globally in time. The Boltzmann collision kernel is given by the Born approximation of the quantum differential scattering cross section.

Publishing Year

Date Published

2000-06-01

Journal Title

Communications on Pure and Applied Mathematics

Acknowledgement

Partially supported by U.S. National Science Foundation grants DMS-9403462, 9703752. We would like to thank H. Spohn for his several comments and discussions on this project. Part of this work was done during the time when L. E. visited the Erwin Schrödinger Institute in Vienna and when both authors visited the Center of Theoretical Sciences in Taiwan. We thank them for the hospitality and the support of this work.

Volume

53

Issue

6

Page

667 - 735

ISSN

IST-REx-ID

### Cite this

Erdös L, Yau H. Linear Boltzmann equation as the weak coupling limit of a random Schrödinger equation.

*Communications on Pure and Applied Mathematics*. 2000;53(6):667-735. doi:10.1002/(SICI)1097-0312(200006)53:6<667::AID-CPA1>3.0.CO;2-5Erdös, L., & Yau, H. (2000). Linear Boltzmann equation as the weak coupling limit of a random Schrödinger equation.

*Communications on Pure and Applied Mathematics*. Wiley-Blackwell. https://doi.org/10.1002/(SICI)1097-0312(200006)53:6<667::AID-CPA1>3.0.CO;2-5Erdös, László, and Horng Yau. “Linear Boltzmann Equation as the Weak Coupling Limit of a Random Schrödinger Equation.”

*Communications on Pure and Applied Mathematics*. Wiley-Blackwell, 2000. https://doi.org/10.1002/(SICI)1097-0312(200006)53:6<667::AID-CPA1>3.0.CO;2-5.L. Erdös and H. Yau, “Linear Boltzmann equation as the weak coupling limit of a random Schrödinger equation,”

*Communications on Pure and Applied Mathematics*, vol. 53, no. 6. Wiley-Blackwell, pp. 667–735, 2000.Erdös, László, and Horng Yau. “Linear Boltzmann Equation as the Weak Coupling Limit of a Random Schrödinger Equation.”

*Communications on Pure and Applied Mathematics*, vol. 53, no. 6, Wiley-Blackwell, 2000, pp. 667–735, doi:10.1002/(SICI)1097-0312(200006)53:6<667::AID-CPA1>3.0.CO;2-5.### Export

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### Sources

arXiv math-ph/9901020