arXiv subjects

Beate Schmittmann

Publications and source records attributed to Beate Schmittmann.

5 recordsLinked to original sources

arXiv2014

Modeling interacting dynamic networks: III. Extraordinary properties in a population of extreme introverts and extroverts

Recently, we introduced dynamic networks with preferred degrees, showing that interesting properties are present in a single, homogeneous system as well as one with two interacting networks. While simulations are readily performed, analytic studies are challenging, due mainly to the lack of detailed balance in the dynamics. Here, we consider the two-community case in a special limit: a system of extreme introverts and extroverts - the XIE model. Surprising phenomena appear, even in this minimal model, where the only control parameters are the numbers of each subgroup: $N_{I,E}$. Specifically, an extraordinary transition emerges when $N_I$ crosses $N_E$. For example, the fraction of total number of I-E links jumps from $\thicksim 0$ to $\thicksim 1$. In a $N_I=N_E$ system, this fraction performs a pure random walk so that its distribution displays a flat plateau across most of $[0,1]$, with the edges vanishing as $(N_{I,E})^{-0.38}$ for large systems. Thus, we believe the XIE model exhibits an extreme Thouless effect. For this limiting model, we show that detailed balance is restored and explicitly find the microscopic steady-state distribution. We then use a mean-field approach to find analytic expressions for the degree distributions that are in reasonably good agreement with simulations, provided $N_I$ is not too close to $N_{E}$.

physics.soc-ph↗

arXiv2013

Modeling interacting dynamic networks: I. Preferred degree networks and their characteristics

We study a simple model of dynamic networks, characterized by a set preferred degree, $κ$. Each node with degree $k$ attempts to maintain its $κ$ and will add (cut) a link with probability $w(k;κ)$ ($1-w(k;κ)$). As a starting point, we consider a homogeneous population, where each node has the same $κ$, and examine several forms of $w(k;κ)$, inspired by Fermi-Dirac functions. Using Monte Carlo simulations, we find the degree distribution in steady state. In contrast to the well-known Erdős-Rényi network, our degree distribution is not a Poisson distribution; yet its behavior can be understood by an approximate theory. Next, we introduce a second preferred degree network and couple it to the first by establishing a controllable fraction of inter-group links. For this model, we find both understandable and puzzling features. Generalizing the prediction for the homogeneous population, we are able to explain the total degree distributions well, but not the intra- or inter-group degree distributions. When monitoring the total number of inter-group links, $X$, we find very surprising behavior. $X$ explores almost the full range between its maximum and minimum allowed values, resulting in a flat steady-state distribution, reminiscent of a simple random walk confined between two walls. Both simulation results and analytic approaches will be discussed.

physics.soc-ph↗

arXiv2005

Anomalous nucleation far from equilibrium

We present precision Monte Carlo data and analytic arguments for an asymmetric exclusion process, involving two species of particles driven in opposite directions on a $2 \times L$ lattice. We propose a scenario which resolves a stark discrepancy between earlier simulation data, suggesting the existence of an ordered phase, and an analytic conjecture according to which the system should revert to a disordered state in the thermodynamic limit. By analyzing the finite size effects in detail, we argue that the presence of a single, seemingly macroscopic, cluster is an intermediate stage of a complex nucleation process: In smaller systems, this cluster is destabilized while larger systems allow the formation of multiple clusters. Both limits lead to exponential cluster size distributions which are, however, controlled by very different length scales.

cond-mat.stat-mech↗

arXiv2004

Bond percolation of polymers

We study bond percolation of $N$ non-interacting Gaussian polymers of $\ell$ segments on a 2D square lattice of size $L$ with reflecting boundaries. Through simulations, we find the fraction of configurations displaying {\em no} connected cluster which span from one edge to the opposite edge. From this fraction, we define a critical segment density $ρ_{c}^L(\ell)$ and the associated critical fraction of occupied bonds $p_{c}^L(\ell)$, so that they can be identified as the percolation threshold in the $L \to \infty$ limit. Whereas $p_{c}^L(\ell)$ is found to decrease monotonically with $\ell$ for a wide range of polymer lengths, $ρ_{c}^L(\ell)$ is non-monotonic. We give physical arguments for this intriguing behavior in terms of the competing effects of multiple bond occupancies and polymerization.

cond-mat.stat-mech↗

arXiv1999

Universal aspects of vacancy-mediated disordering dynamics: the effect of external fields

We investigate the disordering of an initially phase-segregated binary alloy, due to a highly mobile defect which couples to an electric or gravitational field. Using both mean-field and Monte Carlo methods, we show that the late stages of this process exhibit dynamic scaling, characterized by a set of exponents and scaling functions. A new scaling variable emerges, associated with the field. While the scaling functions carry information about the field and the boundary conditions, the exponents are universal. They can be computed analytically, in excellent agreement with simulation results.

cond-mat.stat-mech↗