Session N55: Statistical and Nonlinear Physics II

Bipartiteness transition in a two-dimensional antiferromagnetic tetratic

We explore the effect of strong Ising antiferromagnetic correlations on dislocation-mediated melting in two dimensions. In the solid phase, fundamental dislocations disrupt the bipartiteness of the underlying square lattice; as a result, pairs of dislocations are linearly confined by string-like antiferromagnetic domain walls. It has previously been argued that an antiferromagnetic tetratic phase can arise when double dislocations proliferate while single dislocations remain bound. By identifying an emergent Ising gauge field and constructing an effective field theory, we demonstrate that the antiferromagnetic and paramagnetic tetratic phases are smoothly connected to each other. Despite the absence of a thermodynamic phase transition, we argue that a novel "bipartiteness" transition separates the antiferromagnetic and paramagnetic tetratic regimes: given an arrangement of particles and dislocations sampled from the Gibbs ensemble, we ask whether it is possible to unambiguously define antiferromagnetic correlations between the particles by constructing a nearly bipartite lattice. In the antiferromagnetic tetratic regime, dislocations can always be paired in a well-defined homology class, allowing for long-range antiferromagnetic correlations on the resulting lattice. In contrast, the paramagnetic tetratic does not admit a fixed bipartite structure, resulting in short-range antiferromagnetic correlations.   

Pattern Formation and Dendritic Crystal Growth of Ammonium Nitrate

Dendritic crystal growth is often observed when a non-faceted material crystalizes from a supercooled or supersaturated solution. Many metal alloys solidify in a dendritic pattern, as do some transparent inorganic and organic compounds, and the materials properties of those alloys are influenced by the underlying dendritic microsctructure.  In this work, we will present results for the dendritic solidification of ammonium nitrate from supersaturated aqueous solution.  This system has previously been studied by van Driel et al.[1], who identified several different phases of growth, depending on the saturation temperature.  In the highest-temperature phase (phase I), the dendrites have an approximately hemispherical cap followed almost immediately be a large set of sidebranches.  In the lower-temperature phase II growth, the main dendrite stem is approximately parabolic, but the sidebranches display a distinctive tip-splitting morphology.  We compare these results with those seen for other, similar systems.

[1] Van Driel, C.A., Van der Heijden, A.E.D.M, Van Rosmalen, G.M., "Growth of Ammonium-Nitrate Phase-I and Phase-II Dendrites." J. Cryst. Growth 128 (1993) 229–233.  https://doi.org/10.1016/0022-0248(93)90324-P   

Quantifying phase mixing and separation behaviors across length and time scales

Phase mixing and separation phenomena abound in a multiplicity of diverse material systems including composites, alloys, granular media, complex fluids, and biological tissues. While characterizing phase mixing is critical to understanding material microstructure formation, manufacturing methods, and physical properties, previous mixing metrics are only valid for special systems and fail to detect changes in mixing with respect to length scale, not to mention time scales. To improve upon the current state-of-the-art, we introduce and study a broadly applicable mixing metric that leverages the hyperuniformity concept, demonstrating that it provides the first unifying framework for systematically ranking and classifying mixing in materials across length and time scales. This task is accomplished by applying our metric to a diverse set of real and simulated material microstructures with varying degrees of order/disorder as well as distinct phase geometries and topologies. Our metric provides physically intuitive rankings of mixing in these example systems and is highly-sensitive to their salient dynamical features, including phase coarsening, separation, and transitions. We expect that our mixing metric can also be used to inform the design and discovery of materials with prescribed length- and time-dependent phase mixing or separation behaviors.   

Cusp singularities in the distribution of orientations of asymmetrically pivoted hard disks on a lattice

We study a system of equal-size circular disks, each with an asymmetrically placed pivot at a fixed distance from the center. The pivots are fixed at the vertices of a regular triangular lattice. The disks can rotate freely about the pivots, with the constraint that no disks can overlap with each other. Our Monte Carlo simulations show that the one-point probability distribution of orientations has multiple cusplike singularities. We determine the exact positions and qualitative behavior of these singularities. In addition to these geometrical singularities, we also find that the system shows order-disorder transitions, with a disordered phase at large lattice spacings, a phase with spontaneously broken orientational lattice symmetry at small lattice spacings, and an intervening Berezinskii-Kosterlitz-Thouless phase in between.   

General properties of classical tensor network methods in the thermodynamic limit

Studying emergent phenomena in classical statistical physics is known to be one of the most computationally difficult problems. One method to study these problems is with tensor networks, which renormalize the systems for the most relevant states ranked by entropy. In contrast with Monte Carlo sampling, tensor algorithms avoid any statistical errors from sampling and can compute billions of spins. An explosion of numerical algorithms which compute general properties of a statistical physics system such as specific heat, magnetization, and free energies are available. However, an overview of which tensor algorithms are best and where they must be improved would be highly advantageous for the scientific community and help with new modeling and discoveries. In this talk, we compare and contrast the algorithms found in literature, make recommendations of which algorithms to use, and speculate on improvements in future algorithms. We additionally present a unified coding framework within the DMRjulia library.   

Cluster tomography in percolation reveals universal information

Cluster formation is prevalent in complex systems, including magnetic domains, active matter, and cell migration. Characteristic properties of clusters often change when the system undergoes a phase transition. Consequently, phase transitions can, in principle, be detected through examining the cluster statistics. Here we demonstrate such an approach by investigating the statistics of cluster tomography, which measures the number of clusters N intersected by a line segment of length l through a finite sample. To leading order, N(l) scales as al, where a depends on microscopic details of the system. However, at criticality there is often an additional nonlinear term of the form bln(l), where b is universal. By investigating cluster tomography in 2d and 3d percolation using large-scale Monte Carlo simulations, we show that b depends only on the endpoint configurations of the line segment. In 2d the numerical results are further supported by analytic arguments from conformal field theory. More broadly, we demonstrate how cluster tomography can be an effective technique for identifying phase transitions in clustered systems and extracting universal information about the system at criticality.   

Inducing coexistence in vulnerable three-species ecosystems with cyclic competition

We employ agent-based Monte Carlo simulations on two-dimensional lattices to investigate spatially extended stochastic population dynamics for three cyclically competing species described by the May-Leonard model. Cyclic competition has been identified in diverse natural settings and appears as a characteristic motif in more complex food webs. In this work, we focus on finite model realizations with sufficiently asymmetric rates that would in isolation cause fixation for a single species, accompanied by extinction for the other two. By diffusively coupling a region prone to this finite-size fixation instability with another symmetric, hence stable May-Leonard system, we are able to induce qualitative changes in the behavior of the vulnerable subsystem, namely the stabilization of the three-species coexistence state through invasion fronts originating from the symmetric region. We aim to categorize this emergent behavior by analyzing the spatially inhomogeneous system's parameter space for which stabilization against finite-size fixation and extinction may be achieved, and to semi-quantitatively understand the necessary conditions for successfully implementing this novel control mechanism.   

Invasion fronts prevent extinction in a diffusively coupled inhomogeneous predator-prey ecosystem

Mathematical modeling of the effects of environmental variability on biodiverse ecosystems is of growing interest due to its potential applications in protecting endangered species or eradication of harmful organisms. The paradigmatic Lotka-Volterra model for predator-prey competition and coexistence is readily modified to include environmental effects by implementing a finite carrying capacity to represent limited resources for both competing species. This work studies a stochastic Lotka-Volterra model on a two-dimensional lattice (with periodic boundary conditions) subject to a spatially varying carrying capacity. Specifically, a region experiencing stable predator-prey coexistence is placed in diffusive contact with a similar subsystem that owing to high predation rates is prone to stochastic total extinction events. We investigate this coupled inhomogeneous system through agent-based Monte Carlo simulations. Due to traveling wave fronts emerging from the coexisting system into the region experiencing total extinction, the predator and prey population are sustained because of abundant prey. Our aim is to obtain (semi-) quantitative criteria to determine under what conditions a (finite) endangered ecosystem, which in isolation is likely to suffer stochastic extinction events, may be effectively stabilized through immigration waves emanating from a neighboring stable region.   

Spatial spread of epidemic with Allee effect

Public health measures can effectively fight the epidemic in its initial phase, when the fraction of infected is not large. A common method is tracing the chains of infections and possibly quarantining people who were in close contact with infected individuals. This method leads to a reduction in the effective transmission rate; thus, it may stabilize the state of no infection to small perturbations. In population dynamics, this phenomenon is called an Allee effect, and it was recently proposed to be an important factor in the spread of epidemic. In this talk, I will consider the spatial spread of epidemic in the case of bistable dynamics, where the effective transmission rate depends on the fraction of infected, and the state of no epidemic is linearly stable. The front propagation phenomenon is investigated both numerically and theoretically, and a good agreement between numerical and theoretical results is found both for the front profiles and for the speed of invasion. We discovered [1] a novel phenomenon of front stoppage: in some regime of parameters, the front solution ceases to exist, and the propagating pulse of infection decays despite the initial outbreak.

 

[1]. E. Khain, Phys. Rev. E 107, 064303 (2023).   

Fundamental Bound on Epidemic Overshoot in the SIR Model

We derive an exact upper bound on the epidemic overshoot for the Kermack-McKendrick SIR model. This maximal overshoot value of 0.2984... occurs at $R_0^*$ = 2.151... . In considering the utility of the notion of overshoot, a rudimentary analysis of data from the first wave of the COVID-19 pandemic in Manaus, Brazil highlights the public health hazard posed by overshoot for epidemics with $R_0$ near 2. Using the general analysis framework presented within, we then consider more complex SIR models that incorporate vaccination.