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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
Abstract 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.
the Algorithm 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.
No Wall Case: Analytic Results 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
Comparison between Analytic and Numerical Results for No Walls Case • According to equation 1, the average position of the ensemble will increase linearly with time. As seen in AVERAGE X, the program produces numerical results that are (accounting for statistical scatter) linear. The average position of the ensemble behaves like one particle under constant velocity subject to Newton’s second law. According to equation 3, the average kinetic energy increases linearly with time and, as seen in AVERAGE T, the program does give this result. • According to equation 3, the standard deviation of the average position of the ensemble should vary as time to the 3/2. A quick glance at the STANDARD DEVIATION OF POSITION confirms this result numerically. • Equation 4 predicts that the standard deviation of the average kinetic energy will go as the square root of time. The numerical results, as seen in STANDARD DEVIATION OF T, agree nicely.
Walls Case: Numerical Results • Satisfied that the code predicts numerically what we have derived analytically, we now run the program for a case which is very difficult to derive analytically: that of an ensemble of particles subject to “random” forces in a one-dimensional box. • As can be seen in AVERAGE T, the average value of kinetic energy with respect to time is (statistically) linear and thus in agreement with equation 3. • As shown in the STANDARD DEVIATION OF T, the standard deviation of the kinetic energy goes as the square root of time and in agreement with equation 4. • Although the standard deviation of position, shown in STANDARD DEVIATION OF X, initially goes as time to the 3/2, it becomes narrower as the ensemble “packet” is reflected off walls (See POSITION HISTOGRAM). As the packet distribution becomes “flat” in the box, it has the same standard deviation as the no walls case (which makes sense since the standard deviation of a uniform distribution is a constant). • Unlike the average position of the ensemble for the No Wall case, the average position value of the ensemble for the Walls case (see AVERAGE X) does NOT behave like a particle in a one-dimensional box under Newton’s 2nd Law.
Closing Discussion • In both the No Wall case and in the Walls case, particles in the ensemble occupy the same point in phase space. As the position distribution of ensemble flattens (see POSITION HISTOGRAM), every point in phase space will be accessed, so both the No Walls and Walls cases are chaotic systems. • Although the average position of the ensemble for the no walls case grows linearly (i.e., like a particle with constant velocity), its average kinetic energy is not a constant. How is this possible? The additional kinetic energy added to the ensemble increases the mean speed of the particles rather than increasing the mean velocity of the ensemble. See POSITION HISTOGRAM. • Different widths of the gaussian that the “random” force is picked from will affect how the ensemble of particles propagates through the box. The wider the gaussian, the more quickly the ensemble becomes chaotic. • Future research will be conducted to study how different force distributions affect the ensemble.
Walls Case: STANDARD DEVIATION OF X 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.
