Disordered many-body systems are ubiquitous in condensed matter physics, materials science and biological systems. Examples include amorphous and glassy states of matter, granular materials, and tissues composed of packings of cells in the extra-cellular matrix (ECM). Understanding the collective emergent properties in these systems is crucial to improving the capability for controlling, engineering and optimizing their behaviors, yet it is extremely challenging due to their complexity and disordered nature. The main theme of the thesis is to address this challenge by characterizing and understanding a variety of disordered many-body systems via unique statistical geometrical and topological tools and the state-of-the-art simulation methods. Two major topics of the thesis are modeling ECM-mediated multicellular dynamics and understanding hyperuniformity in 2D material systems. Collective migration is an important mode of cell movement for several biological processes, and it has been the focus of a large number of studies over the past decades. Hyperuniform (HU) state is a critical state in a many-particle system, an exotic property of condensed matter discovered recently. The main focus of this thesis is to study the mechanisms underlying collective cell migration behaviors by developing theoretical/phenomenological models that capture the features of ECM-mediated mechanical communications in vitro and investigate general conditions that can be imposed on hyperuniformity-preserving and hyperuniformity-generating operations, as well as to understand how various novel transport physical properties arise from the unique hyperuniform long-range correlations.

Traditionally, allostery is perceived as the response of a catalytic pocket to perturbations induced by binding at another distal site through the interaction network in a protein, usually associated with a conformational change responsible for functional regulation. Here, I utilize dynamics-based metrics, Dynamic Flexibility Index and Dynamic Coupling Index to provide insight into how 3D network of interactions wire communications within a protein and give rise to the long-range dynamic coupling, thus regulating key allosteric interactions. Furthermore, I investigate its role in modulating protein function through mutations in evolution. I use Thioredoxin and β-lactamase enzymes as model systems, and show that nature exploits "hinge-shift'' mechanism, where the loss in rigidity of certain residue positions of a protein is compensated by reduced flexibility of other positions, for functional evolution. I also developed a novel approach based on this principle to computationally engineer new mutants of the promiscuous ancestral β-lactamase (i.e., degrading both penicillin and cephatoxime) to exhibit specificity only towards penicillin with a better catalytic efficiency through population shift in its native ensemble.I investigate how allosteric interactions in a protein can regulate protein interactions in a cell, particularly focusing on E. coli ribosome. I describe how mutations in a ribosome can allosterically change its associating with magnesium ions, which was further shown by my collaborators to distally impact the number of biologically active Adenosine Triphosphate molecules in a cell, thereby, impacting cell growth. This allosteric modulation via magnesium ion concentrations is coined, "ionic allostery''. I also describe, the role played by allosteric interactions to regulate information among proteins using a simplistic toy model of an allosteric enzyme. It shows how allostery can provide a mechanism to efficiently transmit information in a signaling pathway in a cell while up/down regulating an enzyme’s activity.
The results discussed here suggest a deeper embedding of the role of allosteric interactions in a protein’s function at cellular level. Therefore, bridging the molecular impact of allosteric regulation with its role in communication in cellular signaling can provide further mechanistic insights of cellular function and disease development, and allow design of novel drugs regulating cellular functions.

I describe the first continuous space nuclear path integral quantum Monte Carlo method, and calculate the ground state properties of light nuclei including Deuteron, Triton, Helium-3 and Helium-4, using both local chiral interaction up to next-to-next-to-leading-order and the Argonne $v_6'$ interaction. Compared with diffusion based quantum Monte Carlo methods such as Green's function Monte Carlo and auxiliary field diffusion Monte Carlo, path integral quantum Monte Carlo has the advantage that it can directly calculate the expectation value of operators without tradeoff, whether they commute with the Hamiltonian or not. For operators that commute with the Hamiltonian, e.g., the Hamiltonian itself, the path integral quantum Monte Carlo light-nuclei results agree with Green's function Monte Carlo and auxiliary field diffusion Monte Carlo results. For other operator expectations which are important to understand nuclear measurements but do not commute with the Hamiltonian and therefore cannot be accurately calculated by diffusion based quantum Monte Carlo methods without tradeoff, the path integral quantum Monte Carlo method gives reliable results. I show root-mean-square radii, one-particle number density distributions, and Euclidean response functions for single-nucleon couplings. I also systematically describe all the sampling algorithms used in this work, the strategies to make the computation efficient, the error estimations, and the details of the implementation of the code to perform calculations. This work can serve as a benchmark test for future calculations of larger nuclei or finite temperature nuclear matter using path integral quantum Monte Carlo.

This study aims to address the deficiencies of the Marcus model of electron transfer

(ET) and then provide modifications to the model. A confirmation of the inverted energy

gap law, which is the cleanest verification so far, is presented for donor-acceptor complexes.

In addition to the macroscopic properties of the solvent, the physical properties of the solvent

are incorporated in the model via the microscopic solvation model. For the molecules

studied in this dissertation, the rate constant first increases with cooling, in contrast to the

prediction of the Arrhenius law, and then decreases at lower temperatures. Additionally,

the polarizability of solute, which was not considered in the original Marcus theory, is included

by the Q-model of ET. Through accounting for the polarizability of the reactants, the

Q-model offers an important design principle for achieving high performance solar energy

conversion materials. By means of the analytical Q-model of ET, it is shown that including

molecular polarizability of C60 affects the reorganization energy and the activation barrier

of ET reaction.

The theory and Electrochemistry of Ferredoxin and Cytochrome c are also investigated.

By providing a new formulation for reaction reorganization energy, a long-standing disconnect

between the results of atomistic simulations and cyclic voltametery experiments is

resolved. The significant role of polarizability of enzymes in reducing the activation energy

of ET is discussed. The binding/unbinding of waters to the active site of Ferredoxin leads

to non-Gaussian statistics of energy gap and result in a smaller activation energy of ET.

Furthermore, the dielectric constant of water at the interface of neutral and charged

C60 is studied. The dielectric constant is found to be in the range of 10 to 22 which is

remarkably smaller compared to bulk water( 80). Moreover, the interfacial structural

crossover and hydration thermodynamic of charged C60 in water is studied. Increasing the

charge of the C60 molecule result in a dramatic structural transition in the hydration shell,

which lead to increase in the population of dangling O-H bonds at the interface.

Quantum Monte Carlo is one of the most accurate ab initio methods used to study nuclear physics. The accuracy and efficiency depend heavily on the trial wave function, especially in Auxiliary Field Diffusion Monte Carlo (AFDMC), where a simplified wave function is often used to allow calculations of larger systems. The simple wave functions used with AFDMC contain short range correlations that come from an expansion of the full correlations truncated to linear order. I have extended that expansion to quadratic order in the pair correlations. I have investigated this expansion by keeping the full set of quadratic correlations as well an expansion that keeps only independent pair quadratic correlations. To test these new wave functions I have calculated ground state energies of 4He, 16O, 40Ca and symmetric nuclear matter at saturation density ρ = 0.16 fm−3 with 28 particles in a periodic box. The ground state energies calculated with both wave functions decrease with respect to the simpler wave function with linear correlations only for all systems except 4He for both variational and AFDMC calculations. It was not expected that the ground state energy of 4He would decrease due to the simplicity of the alpha particle wave function. These correlations have also been applied to study alpha particle formation in neutron rich matter, with applications to neutron star crusts and neutron rich nuclei. I have been able to show that this method can be used to study small clusters as well as the effect of external nucleons on these clusters.

Fluctuations with a power spectral density depending on frequency as $1/f^\alpha$ ($0<\alpha<2$) are found in a wide class of systems. The number of systems exhibiting $1/f$ noise means it has far-reaching practical implications; it also suggests a possibly universal explanation, or at least a set of shared properties. Given this diversity, there are numerous models of $1/f$ noise. In this dissertation, I summarize my research into models based on linking the characteristic times of fluctuations of a quantity to its multiplicity of states. With this condition satisfied, I show that a quantity will undergo $1/f$ fluctuations and exhibit associated properties, such as slow dynamics, divergence of time scales, and ergodicity breaking. I propose that multiplicity-dependent characteristic times come about when a system shares a constant, maximized amount of entropy with a finite bath. This may be the case when systems are imperfectly coupled to their thermal environment and the exchange of conserved quantities is mediated through their local environment. To demonstrate the effects of multiplicity-dependent characteristic times, I present numerical simulations of two models. The first consists of non-interacting spins in $0$-field coupled to an explicit finite bath. This model has the advantage of being degenerate, so that its multiplicity alone determines the dynamics. Fluctuations of the alignment of this model will be compared to voltage fluctuations across a mesoscopic metal-insulator-metal junction. The second model consists of classical, interacting Heisenberg spins with a dynamic constraint that slows fluctuations according to the multiplicity of the system's alignment. Fluctuations in one component of the alignment will be compared to the flux noise in superconducting quantum interference devices (SQUIDs). Finally, I will compare both of these models to each other and some of the most popular models of $1/f$ noise, including those based on a superposition of exponential relaxation processes and those based on power law renewal processes.

Bio-molecules and proteins are building blocks of life as is known, and understanding

their dynamics and functions are necessary to better understand life and improve its

quality. While ergodicity and fluctuation dissipation theorem (FDT) are fundamental

and crucial concepts regarding study of dynamics of systems in equilibrium, biological

function is not possible in equilibrium.

In this work, dynamical and orientational structural crossovers in low-temperature

glycerol are investigated. A sudden and notable increase in the orientational Kirk-

wood factor and the dielectric constant is observed, which appears in the same range

of temperatures that dynamic crossover of translational and rotational dynamics oc-

cur.

Theory and electrochemistry of cytochrome c is also investigated. The seeming

discrepancy in reorganization energies of protein electron transfer produced by atom-

istic simulations and those reported by protein electrochemistry (which are smaller)

is resolved. It is proposed in this thesis that ergodicity breaking results in an effective

reorganization energy (0.57 eV) consistent with experiment.

Ergodicity breaking also affects the iron displacement in heme proteins. A model

for dynamical transition of atomic displacements in proteins is provided. Different

temperatures for rotational and translational crossovers of water molecules are re-

ported, which all are ergodicity breaking transitions depending on the corresponding

observation windows. The comparison with Mössbauer spectroscopy is presented.

Biological function at low temperatures and its termination is also investigated in

this research. Here, it is proposed that ergodicity breaking gives rise to the violation

of the FDT, and this violation is maintained in the entire range of physiological

temperatures for cytochrome c. Below the crossover temperature, the protein returns

to the FDT, which leads to a sudden jump in the activation barrier for electron

itransfer.

Finally the interaction of charges in dielectric materials is discussed. It is shown

that the potential of mean force between ions in polar liquids becomes oscillatory at

short distances.

The rigidity of a material is the property that enables it to preserve its structure when deformed. In a rigid body, no internal motion is possible since the degrees of freedom of the system are limited to translations and rotations only. In the macroscopic scale, the rigidity and response of a material to external load can be studied using continuum elasticity theory. But when it comes to the microscopic scale, a simple yet powerful approach is to model the structure of the material and its interparticle interactions as a ball$-$and$-$spring network. This model allows a full description of rigidity in terms of the vibrational modes and the balance between degrees of freedom and constraints in the system.

In the present work, we aim to establish a microscopic description of rigidity in \emph{disordered} networks. The studied networks can be designed to have a specific number of degrees of freedom and/or elastic properties. We first look into the rigidity transition in three types of networks including randomly diluted triangular networks, stress diluted triangular networks and jammed networks. It appears that the rigidity and linear response of these three types of systems are significantly different. In particular, jammed networks display higher levels of self-organization and a non-zero bulk modulus near the transition point. This is a unique set of properties that have not been observed in any other types of disordered networks. We incorporate these properties into a new definition of jamming that requires a network to hold one extra constraint in excess of isostaticity and have a finite non-zero bulk modulus. We then follow this definition by using a tuning by pruning algorithm to build spring networks that have both these properties and show that they behave exactly like jammed networks. We finally step into designing new disordered materials with desired elastic properties and show how disordered auxetic materials with a fully convex geometry can be produced.

The structure of glass has been the subject of many studies, however some

details remained to be resolved. With the advancement of microscopic

imaging techniques and the successful synthesis of two-dimensional materials,

images of two-dimensional glasses (bilayers of silica) are now available,

confirming that this glass structure closely follows the continuous random

network model. These images provide complete in-plane structural information

such as ring correlations, and intermediate range order and with computer

refinement contain indirect information such as angular distributions, and

tilting.

This dissertation reports the first work that integrates the actual atomic

coordinates obtained from such images with structural refinement to enhance

the extracted information from the experimental data.

The correlations in the ring structure of silica bilayers are studied

and it is shown that short-range and intermediate-range order exist in such networks.

Special boundary conditions for finite experimental samples are designed so atoms

in the bulk sense they are part of an infinite network.

It is shown that bilayers consist of two identical layers separated by a

symmetry plane and the tilted tetrahedra, two examples of

added value through the structural refinement.

Finally, the low-temperature properties of glasses in two dimensions

are studied. This dissertation presents a new approach to find possible

two-level systems in silica bilayers employing the tools of rigidity theory

in isostatic systems.

In this dissertation two kinds of strongly interacting fermionic systems were studied: cold atomic gases and nucleon systems. In the first part I report T=0 diffusion Monte Carlo results for the ground-state and vortex excitation of unpolarized spin-1/2 fermions in a two-dimensional disk. I investigate how vortex core structure properties behave over the BEC-BCS crossover. The vortex excitation energy, density profiles, and vortex core properties related to the current are calculated. A density suppression at the vortex core on the BCS side of the crossover and a depleted core on the BEC limit is found. Size-effect dependencies in the disk geometry were carefully studied. In the second part of this dissertation I turn my attention to a very interesting problem in nuclear physics. In most simulations of nonrelativistic nuclear systems, the wave functions are found by solving the many-body Schrödinger equations, and they describe the quantum-mechanical amplitudes of the nucleonic degrees of freedom. In those simulations the pionic contributions are encoded in nuclear potentials and electroweak currents, and they determine the low-momentum behavior. By contrast, in this work I present a novel quantum Monte Carlo formalism in which both relativistic pions and nonrelativistic nucleons are explicitly included in the quantum-mechanical states of the system. I report the renormalization of the nucleon mass as a function of the momentum cutoff, an Euclidean time density correlation function that deals with the short-time nucleon diffusion, and the pion cloud density and momentum distributions. In the two nucleon sector the interaction of two static nucleons at large distances reduces to the one-pion exchange potential, and I fit the low-energy constants of the contact interactions to reproduce the binding energy of the deuteron and two neutrons in finite volumes. I conclude by showing that the method can be readily applied to light-nuclei.