Theory and Applications of Stochastic Processes pp 257-301 | Cite as

# Diffusion Approximations to Langevin’s Equation

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## Abstract

In 1940, H. Kramers [140] introduced a diffusion model for chemical reactions, based on the Langevin equation for a Brownian particle in a potential
where γ is the dissipation constant (here normalized by the frequency of vibration at the bottom of the well),
is dimensionless temperature (normalized by the potential barrier height), and \(\dot{w}\)
is standard Gaussian white noise. Kramers sought to calculate the reaction rate \(\kappa\), which is the rate of escape of the Brownian particle from the potential well in which it is confined. In particular, he sought to determine the dependence of of
\(\kappa\) on temperature
\(T\) and on the viscosity (friction)
\(\gamma\), and to compare the values found with the results of transition state theory [87].

**U(x)**(per unit mass) that forms a well (see Figure 8.1)$$\ddot{x} + \gamma\dot{x} + U^\prime (x) = \sqrt{2\varepsilon \gamma} \dot{w},$$

(8.1)

$$ \varepsilon = \frac{kT}{\delta U}$$

## Keywords

Langevin Equation Planck Equation Diffusion Approximation Brownian Particle Eikonal Equation
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