Fermi–Ulam model - Citizendia

The Fermi–Ulam model (FUM) is a dynamical system that was introduced by Polish mathematician Stanislaw Ulam in 1961. The dynamical system concept is a mathematical Formalization for any fixed "rule" which describes the Time dependence of a point's position Poland (Polska officially the Republic of Poland A mathematician is a person whose primary area of study and research is the field of Mathematics. Stanisław Marcin Ulam ( April 13, 1909 &ndash May 13, 1984) was a Polish Mathematician who participated in the Manhattan Year 1961 ( MCMLXI) was a Common year starting on Sunday (link will display full calendar of the Gregorian calendar.

FUM is a variant of Enrico Fermi's primary work on acceleration of cosmic rays, namely Fermi acceleration. For the 1962 Bruce Conner film see Cosmic Ray (film Cosmic rays are energetic particles originating from space that impinge on Fermi acceleration, sometimes referred to diffusive shock acceleration (a subclass of Fermi acceleration is the Acceleration that charged particles 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 magnetic mirror is a Magnetic field configuration where the field strength changes when moving along a field line For the 1962 Bruce Conner film see Cosmic Ray (film Cosmic rays are energetic particles originating from space that impinge on

A. J. Lichtenberg and M. A. Lieberman provided a simplified version of FUM (SFUM) that derives from the Poincaré surface of section $x = c o n s t . The Fermi&ndashUlam model (FUM is a Dynamical system that was introduced by Polish Mathematician Stanislaw Ulam in 1961. In Mathematics, particularly in Dynamical systems, a first recurrence map or Poincaré map, named after Henri Poincaré, is the intersection$ 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 $u n$ 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, $U wall$ 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. The Kolmogorov–Arnold–Moser theorem is a result in Dynamical systems about the persistence of quasi-periodic motions under small perturbations 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. In Mathematics and related technical fields the term map or mapping is often a Synonym for function.

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).

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