Diffusion Approximations to Langevin’s Equation

  • Zeev Schuss
Part of the Applied Mathematical Sciences book series (AMS, volume 170)


In 1940, H. Kramers [140] introduced a diffusion model for chemical reactions, based on the Langevin equation for a Brownian particle in a potential U(x) (per unit mass) that forms a well (see Figure 8.1)
$$\ddot{x} + \gamma\dot{x} + U^\prime (x) = \sqrt{2\varepsilon \gamma} \dot{w},$$
where γ is the dissipation constant (here normalized by the frequency of vibration at the bottom of the well),
$$ \varepsilon = \frac{kT}{\delta U}$$
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].


Langevin Equation Planck Equation Diffusion Approximation Brownian Particle Eikonal Equation 
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© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  • Zeev Schuss
    • 1
  1. 1.Department of Applied MathematicsSchool of Mathematical Science Tel Aviv UniversityTel AvivIsrael

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