Abstract
The Langevin equation is reviewed as a fundamental equation for the Brownian motion. We first review the mathematical characters of the Langevin equation, and show its microscopic derivation by an asymptotic expansion of master equations (i.e., the system size expansion). By combining the kinetic theory, the Langevin equation is derived for various systems from microscopic dynamics. We finally discuss unsolved problems within the framework of the original system size expansion.
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Notes
- 1.
The variable transformation \(\tau \equiv \varepsilon t\) is introduced for renormalization of the divergence of the relaxation time, in this sense.
- 2.
This means that the system size expansion is not a uniform asymptotic expansion in terms of the velocity.
- 3.
To avoid these corrections, a more elegant choice of scaled variables is proposed in Ref. [9].
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Kanazawa, K. (2017). Langevin Equation and Its Microscopic Derivation. In: Statistical Mechanics for Athermal Fluctuation. Springer Theses. Springer, Singapore. https://doi.org/10.1007/978-981-10-6332-9_4
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DOI: https://doi.org/10.1007/978-981-10-6332-9_4
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