Undergraduate Thesis Or Project

Using the Langevin Equation to Verify a Single-Particle Brownian Motion Simulation

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  • Brownian motion is the movement of a particle in a fluid resulting from collisions with surrounding molecules. It has modern applications in simulating fluid dynamics and the motion of proteins in biophysics. Brownian motion simulations generally evolve the initial state by applying Newton’s second law to every particle. These simulations can require large computation times depending on the chosen number of particles and time scale resolution. We tested an alternative simulation method that only keeps track of position and velocity information for a single particle. To create collisions without tracking surrounding particles, we used random number generators to generate velocities and times between collisions. We checked the behavior of the simulation by analyzing whether it produces a random walk and if it obeys the Langevin equation, which relates drag to thermal fluctuations according to the fluctuation-dissipation theorem. The drag coefficient for this particle was calculated from the velocity autocorrelation and a plot of the particle velocity change as a function of its velocity. These coefficients were verified by plugging them into the theoretical equation for the mean squared displacement (MSD) and comparing them to the MSD calculated from data. We demonstrated a relationship between the drag and thermal fluctuations using the particle velocity change vs. velocity plot and a modified form of the Langevin equation. An issue where the particle does not reach thermal equilibrium with its surroundings was identified, and we provided potential solutions to be addressed in future work.
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