UC Riverside

The work presented in this dissertation is centered around the phenomena of correlations between members of a system, and how these correlations lead to unique effects which the system is in a non-equilibrium state. The systems described in this work are divided into two sections: dynamical systems and networks. The work on dynamical systems is focused on how correlated systems can gradually evolve when strongly driven. The work on networks is focused on how percolation is affected by correlations.

Dynamical systems focuses on the motion of particles in the presence of an external driving force, specifically focusing on nonlinear effects observable in three different examples of correlated systems which are strongly driven by an external field which is rapidly oscillating in time. The first system is a classical system of coupled oscillators, the second is a quantum system with avoided level crossings and the third is the conduction level of a semiconductor. All three systems are strongly driven and specifically driven far from an initial equilibrium. It is posited that these strongly driven systems will reach quasi-stationary states, where the effects of the external driving can be classified on two time scales, slow evolution and rapid oscillation. The classical system is studied as a classical field theory, the effects of the strong driving force is observed in the qualitative properties of the system's effective potential. In the presence of a strong driving force this effective potential will exhibit points whose stability will change depending on the parameters of the driving force. In the quantum systems the effects of the external driving are seen in the Floquet quasienergy spectrum. In the third system, the strong driving force results in a renormalization of the bandwidth. If the driving force is strong enough the band can be completely inverted. Again working in the Floquet representation, the inverted band will exhibit multiplication of the driving frequency, specifically the third harmonic will become the dominant frequency and the fundamental frequency will be suppressed.

Correlations in networks have many different aspects, correlations can affect how a network is formed (via attachment models) or how a random process is able to percolate across a random graph. The effects of correlated percolation is discussed in the second part of this dissertation, where a process can affect single sites on a graph or a cluster of sites simultaneously. It is shown that a lower bound on the percolation threshold can be established, and this bound is dependent on the size of clusters present.