Interacting Stochastic Processes: from Viciousness to Caging to Force Chains

This thesis documents a quest to develop and study several novel interacting stochastic processes. As for the first example, we generalize a system of vicious random walkers in which the only interaction between any two random walkers is that when they intersect, both walkers are annihilated. We define a system of N vicious accelerating walkers with each walker undergoing random acceleration and compute the survival probability distribution for this system. We also define and study a system of N vicious Levy flights in which any two Levy flights crossing one another annihilate each other. The average mean-squared displacement of a Levy flight is not proportional to time, but scales with what is known as the Levy index divided by two. In both cases, vicious accelerating walkers and vicious Levy flights, we are motivated by ultimately generalizing our understanding of Gaussian random matrices via non-Markovian and non-Gaussian extensions respectively. Moreover, inspired by recent experiments on periodically sheared colloids at low densities, we define and investigate several new contact processes, or interacting stochastic processes, with conserved particle number and threeor-more-body interactions. We do so to characterize the periodically sheared colloidal system at higher densities. We find two new dynamical phase transitions between an active phase, where some fraction of the colloids are always being displaced from their position at the beginning and end of each shear cycle, and an inactive phase in which all colloids return to their initial positions at the end of each shear cycle. One of the transitions is discontinuous, while the second, which is due to a caging, or crowding, effect at high densities, appears to be continuous and in a new universality from what is known as conserved directed percolation. The latter transition may have implications for the onset of glassiness in dense, particulate systems. In addition, this thesis also includes analysis of the heterogeneous force network present in amorphous solids near the onset of rigidity, or jamming. The onset of rigidity can, in some sense, be viewed an interacting stochastic process with added constraints to enforce force-balance on each particle, for example. Our analysis yields string-like correlations in the locally-large forces in the system. Such correlations are reminiscent of force chains. While force chains have been readily observed in experiments, it is the first time these correlations have been observed in conjugate gradient simulations of repulsive soft spheres. We also study the contact geometry of the force network and explore a link with spin systems, namely spin glasses, to search for signatures of chaos due to marginal stability, for instance. Connections between jamming systems and spin glass systems will hopefully open up new avenues of theoretical investigation for both systems. Finally, we explore the quantum version of an individual stochastic process, namely the fractional Schrodinger equation. We prove that previously claimed exact solutions for certain potentials are incorrect and determine a new exact solution for a Levy index of unity and the harmonic oscillator potential. While our results contribute to the realm of mathematical physics, a physical realization of the fractional Schrodinger equation will indeed launch a new subfield of quantum mechanics. Interacting Stochastic Processes: From Viciousness to Caging to Force Chains