Non-equilibrium dynamics in three-dimensional magnetic spin models and molecular motor-inspired one-dimensional exclusion processes

• Main Author: Nandi, Riya
• Other Authors: Physics
• Format: Others
• Published: Virginia Tech 2021
• Subjects: Non-equilibrium dynamics; critical dynamics; aging scaling; biological transport; molecular motors; etc.
• Online Access: http://hdl.handle.net/10919/102652

We investigate the relaxation dynamics of two distinct non-equilibrium processes: relaxation of three-dimensional antiferromagnetic lattice spin models with Heisenberg interaction following a critical quench, and a one-dimensional exclusion process inspired by the gear-like motion of molecular motors. In a system of three-dimensional Heisenberg antiferromagnets the non-conserved staggered magnetization components couple non-trivially to the conserved magnetization densities inducing fully reversible terms that enter the Langevin dynamic equation. We simulate the exact microscopic dynamics of such a system of antiferromagnets by employing a hybrid simulation algorithm that combines the reversible spin precession implemented by the fourth-order Runge-Kutta integration method with the standard relaxational dynamics at finite temperatures using Monte Carlo updates. We characterize the dynamic universality class of this system by probing the early temporal window where the system exhibits aging scaling properties. We also verify an earlier renormalization group prediction that the temporal decay exponent in the two-time spin autocorrelation function exhibits non-universality, specifically it depends on the width of the initial spin orientation distribution. We employ a similar numerical technique to study the critical dynamics of an anisotropic Heisenberg antiferromagnet in the presence of an external field. The phase diagram of this system exhibits two critical lines that meet at a bicritical point. We study the aging scaling dynamics for the model C critical line, probe the model F critical line by investigating the system size dependence of the characteristic spin-wave frequencies near criticality, and measure the dynamic critical exponents for the order parameter including its aging scaling at the bicritical point. We introduce a one-dimensional non-equilibrium lattice gas model representing the processive motion of dynein molecular motors over the microtubule. We study both dynamical and stationary state properties for the model consisting of hardcore particles hopping on the lattice with variable step sizes. We find that the stationary state gap-distribution exhibits striking peaks around gap sizes that are multiples of the maximum step size, for both open and periodic boundary conditions, and verify this using a mean-field calculation. For open boundary conditions, we observe intriguing damped oscillator-like distribution of particles over the lattice with a periodicity equal to the maximum step size. To characterize transient dynamics, we measure the mean square displacement that shows weak superdiffusive growth with exponent γ≈ 1.34 for periodic boundary and ballistic growth ( γ≈ 2) for open boundary conditions at early times. We also study the effect of Langmuir dynamics on the density profile. === Doctor of Philosophy === Most systems found in nature are out of equilibrium. In this dissertation we investigate the relaxation dynamics of two such non-equilibrium systems: 1. We investigate a three-dimensional antiferromagnetic system relaxing towards equilibrium from an initial state that is driven far away from equilibrium at the point in the parameter space where the system undergoes a second-order phase transition. We devise a novel simulation method that captures emerging dynamic universal features and scaling features at these points of continuous phase transition in the early times of relaxation when the system is still far away from equilibrium. 2. Cytoplasmic dyneins are one of three kinds of motor proteins that move on tubular structures called microtubules carrying and transporting cellular cargo inside the cells. Unlike the other molecular motors that move forward with fixed step sizes, the dyneins have been experimentally observed to vary their step size depending on the amount of cargo they are carrying. We model an exclusion process in a one-dimensional lattice inspired by the motion of the dynein molecular motors where the motors can hop from one to four steps depending on their internal states. We study the effect of this variable step size on the dynamics of a collection of dyneins. We observe intriguing oscillating density profiles and discrete peaks in the distribution of empty sites. Our results suggest self-organization among the motors and the empty sites.