Classical Particles subject to a Gaussian-Distributed Random Force APS April Meeting Sunday, May 2 nd 2004 Aaron Plasek, Athanasios Petridis Drake University. Abstract.
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Classical Particles subject to a Gaussian-Distributed Random ForceAPS April Meeting Sunday, May 2nd 2004Aaron Plasek, Athanasios Petridis Drake University
The motion of a classical particle under the influence of a random gaussian-distributed force sampled at discrete time intervals is studied. In the case of unbound particles the expectation value and the standard deviation of the position and the kinetic energy are calculated analytically and numerically (by means of a Monte-Carlo simulation over an ensemble of identical particles and initial conditions). The system is shown to exhibit chaotic behavior. An ensemble of particles confined in a box is also studied and compared with the unbound case.
To model an ensemble of particles under these “random” forces, we make a few simplifying assumptions, namely:
1) the particles in the ensemble are point particles
2) the particles in the ensemble make completely elastic collisions with walls
The algorithm propagates the particles in the ensemble via the standard Newtonian equations of motion. In the case where the ensemble is contained by a one-dimensional box, the time it will take for the particle to reflect off a wall is calculated and the particle is accordingly propagated and reflected. It is a non-trivial problem to model this numerically (at last count, the code exceeded 600 lines).
The “random” force is generated via a very powerful random number generator known as “the Mersenne Twister” (has a period of 2^19937-1) and the Central Limit Theorem.
The code allows us to model an ensemble of N particles for M time steps, where the size of N and M are only limited by the memory of the computer and the maximum allowed array size in C++.
The code has error checking and will alert the user for any “missed” case or violation of the energy-work theorem.
For an ensemble of particles subject to “random” forces, the following equations can be written (see authors for derivations):
where n = t / Dt, T = Kinetic Energy, and s = standard deviation
Note: After 1 time unit, the distribution
becomes uniform and, thus, chaotic. For
this particular box, the distribution becomes
uniform at Sigma = .577, which is matched
nicely by the numerical results. The downward
fluctuations at small times are due to narrowing
of the distribution upon reflection.