Fermi–Ulam model

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The Fermi–Ulam model (FUM) is a dynamical system that was introduced by Polish mathematician Stanislaw Ulam in 1961.

FUM is a variant of Enrico Fermi's primary work on acceleration of cosmic rays, namely Fermi acceleration. The system consists of a particle that collides elastically between a fixed wall and a moving one, of infinite mass each. The walls represent the magnetic mirrors with whom the cosmic particles collide.

A. J. Lichtenberg and M. A. Lieberman provided a simplified version of FUM (SFUM) that derives from the Poincaré surface of section x = const. and writes

u_{n+1}=|u_n+U_\mathrm{wall}(\varphi_n)| \,


\varphi_{n+1}=\varphi_n+\frac{kM}{u_{n+1}} \pmod k,

where un is the velocity of the particle after the n-th collision with the fixed wall, \varphi_n is the corresponding phase of the moving wall, Uwall is the velocity law of the moving wall and M is the stochasticity parameter of the system.

If the velocity law of the moving wall is differentiable enough, according to KAM theorem invariant curves in the phase space (\varphi,u) exist. These invariant curves act as barriers that do not allow for a particle to further accelerate and the average velocity of a population of particles saturates after finite iterations of the map. For instance, for sinusoidal velocity law of the moving wall such curves exist, while they do not for sawtooth velocity law that is discontinuous. Consequently, at the first case particles cannot accelerate infinitely, reversely to what happens at the last one.

FUM became over the years a prototype model for studying non-linear dynamics and coupled mappings.

[edit] External links

  • [1] Regular and Chaotic Dynamics : A widely acknowledged scientific book that treats FUM, written by A. J. Lichtenberg and M. A. Lieberman (Appl. Math. Sci. vol 38) (New York: Springer).