Publication Date

Spring 2026

Degree Type

Master's Project

Degree Name

Master of Arts (MA)

Department

Mathematics and Statistics

First Advisor

Liam G. Stanton

Second Advisor

Daniel Brinkman

Third Advisor

Patrick Murphy

Keywords

Mori-Zwanzig formalism, Generalized Langezin Equation, Fokker-Planck equation, Reduction of many body systems, Mathematical modeling

Abstract

This thesis investigates memory effects in many-body systems through the MoriZwanzig Formalism for projected dynamics of a Hamiltonian System which yields the Generalized Langevin Equation (GLE). The GLE is a stochastic differential equation (SDE) that studies the dynamics of observables under the effects of many other observables in the system. Although satisfying, the GLE has a term called the Memory Kernel that encodes the past of the system and introduces a computational challenge by introducing a non-Markovian property to the equation. The kernel is often approximated by introducing a delta function, which simplifies the computation, but at the loss of the memory. This paper will investigate a method to retain memory, while still producing an equation that describes the time evolution the system’s dynamics via a Fokker-Planck probability density function.

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