Heuristic Arguments Research Articles

Interface Dynamics in Discrete Forward-Backward Diffusion Equations

We study the motion of phase interfaces in a diffusive lattice equation with bistable nonlinearity and derive a free boundary problem with hysteresis to describe the macroscopic evolution in the parabolic scaling limit. The first part of the paper deals with general bistable nonlinearities and is restricted to numerical experiments and heuristic arguments. We discuss the formation of macroscopic data and present numerical evidence for pinning, depinning, and annihilation of interfaces. Afterwards we identify a generalized Stefan condition along with a hysteretic flow rule that characterize the dynamics of both standing and moving interfaces. In the second part, we rigorously justify the limit dynamics for single-interface data and a special piecewise affine nonlinearity. We prove persistence of such data, derive upper bounds for the macroscopic interface speed, and show that the macroscopic limit can indeed be described by the free boundary problem. The fundamental ingredient to our proofs is a representation formula that links the solutions of the nonlinear lattice to the discrete heat kernel and enables us to derive macroscopic compactness results in the space of continuous functions.

Correlated percolation and tricriticality

The recent proliferation of correlated percolation models--models where the addition of edges and/or vertices is no longer independent of other edges and/or vertices--has been motivated by the quest to find discontinuous percolation transitions. The leader in this proliferation is what is known as explosive percolation. A recent proof demonstrates that a large class of explosive percolation-type models does not, in fact, exhibit a discontinuous transition [Riordan and Warnke, Science, 333, 322 (2011)]. Here, we discuss two lesser known correlated percolation models--the k ≥ 3-core model on random graphs and the counter-balance model in two-dimensions--both exhibiting discontinuous transitions. To search for tricriticality, we construct mixtures of these models with other percolation models exhibiting the more typical continuous transition. Using a powerful rate equation approach, we demonstrate that a mixture of k = 2-core and k = 3-core vertices on the random graph exhibits a tricritical point. However, for a mixture of k-core and counter-balance vertices in two dimensions, as the fraction of counter-balance vertices is increased, numerics and heuristic arguments suggest that there is a line of continuous transitions with the line ending at a discontinuous transition, i.e., when all vertices are counter-balanced. Interestingly, these heuristic arguments may help identify the ingredients needed for a discontinuous transition in low dimensions. In addition, our results may have potential implications for glassy and jamming systems.

On Design of Metal-Matrix Composites Lighter than Air

This paper discusses some theoretical aspects of design of ultralight metallic materials using analytical and heuristic arguments. Potential application of syntactic foams to obtain metal-matrix composites lighter than air is also analyzed. Carbon allotropes (fullerenes, colossal carbon tubes) and some non-carbon materials are considered as components of ultralight metal-matrix composites. Calculations for the size of fullerenes, number of atoms in their structure, and coating thickness required to produce ultralight composites are presented. It is concluded that 3D carbon molecules (fullerenes) and colossal carbon tubes are the most promising components to design ultralight metallic materials which can be lighter than air.

Discrete analogue of the Burgers equation

We propose the set of coupled ordinary differential equations dnj/dt = n2j − 1 − n2j as a discrete analogue of the classic Burgers equation. We focus on traveling waves and triangular waves, and find that these special solutions of the discrete system capture major features of their continuous counterpart. In particular, the propagation velocity of a traveling wave and the shape of a triangular wave match the continuous behavior. However, there are some subtle differences. For traveling waves, the propagating front can be extremely sharp as it exhibits double exponential decay. For triangular waves, there is an unexpected logarithmic shift in the location of the front. We establish these results using asymptotic analysis, heuristic arguments, and direct numerical integration.

Quantum Phenomenology of Conjunction Fallacy

A quantum-like description of human decision process is developed, and a heuristic argument supporting the theory as sound phenomenology is given. It is shown to be capable of quantitatively explaining the conjunction fallacy in the same footing as the violation of sure-thing principle.

Comments on correlation functions of large spin operators and null polygonal Wilson loops

We discuss the relation between correlation functions of twist-two large spin operators and expectation values of Wilson loops along light-like trajectories. After presenting some heuristic field theoretical arguments suggesting this relation, we compute the divergent part of the correlator in the limit of large ʼt Hooft coupling and large spins, using a semi-classical world-sheet which asymptotically looks like a GKP rotating string. We show this diverges as expected from the expectation value of a null Wilson loop, namely, as (lnμ−2)2, μ being a cut-off of the theory.

GRAVITY'S WEIGHT ON WORLDLINE FUZZINESS

We investigate a connection between recent results in three-dimensional (3D) quantum gravity, providing an effective noncommutative-spacetime description, and some earlier heuristic descriptions of a quantum-gravity contribution to the fuzziness of the worldlines of particles. We show that 3D-gravity-inspired spacetime noncommutativity reflects some of the features suggested by previous heuristic arguments. Most notably, gravity-induced worldline fuzziness, while irrelevantly small on terrestrial scales, could be observably large for propagation of particles over cosmological distances.

PROBING COMPLEX NETWORKS FROM MEASURED TIME SERIES

We propose a method to detect nodes of relative importance, e.g. hubs, in an unknown network based on a set of measured time series. The idea is to construct a matrix characterizing the synchronization probabilities between various pairs of time series and examine the components of the principal eigenvector. We provide a heuristic argument indicating the existence of an approximate one-to-one correspondence between the components and the degrees of the nodes from which measurements are obtained. The striking finding is that such a correspondence appears to be quite robust, which holds regardless of the detailed node dynamics and of the network topology. Our computationally efficient method thus provides a general means to address the important problem of network detection, with potential applications in a number of fields.

The Riemann Zeta Function on Arithmetic Progressions

We prove asymptotic formulas for the first discrete moment of the Riemann zeta function on certain vertical arithmetic progressions inside the critical strip. The results give some heuristic arguments for a stochastic periodicity that we observed in the phase portrait of the zeta function.

Theory and algorithms for hop-count-based localization with random geometric graph models of dense sensor networks

Wireless sensor networks can often be viewed in terms of a uniform deployment of a large number of nodes in a region of Euclidean space. Following deployment, the nodes self-organize into a mesh topology with a key aspect being self-localization . Having obtained a mesh topology in a dense, homogeneous deployment, a frequently used approximation is to take the hop distance between nodes to be proportional to the Euclidean distance between them. In this work, we analyze this approximation through two complementary analyses. We assume that the mesh topology is a random geometric graph on the nodes; and that some nodes are designated as anchors with known locations. First, we obtain high probability bounds on the Euclidean distances of all nodes that are h hops away from a fixed anchor node. In the second analysis, we provide a heuristic argument that leads to a direct approximation for the density function of the Euclidean distance between two nodes that are separated by a hop distance h . This approximation is shown, through simulation, to very closely match the true density function. Localization algorithms that draw upon the preceding analyses are then proposed and shown to perform better than some of the well-known algorithms present in the literature. Belief-propagation-based message-passing is then used to further enhance the performance of the proposed localization algorithms. To our knowledge, this is the first usage of message-passing for hop-count-based self-localization.

Stochastic Dynamics on Hypergraphs and the Spatial Majority Rule Model

This article starts by introducing a new theoretical framework to model spatial systems which is obtained from the framework of interacting particle systems by replacing the traditional graphical structure that defines the network of interactions with a structure of hypergraph. This new perspective is more appropriate to define stochastic spatial processes in which large blocks of vertices may flip simultaneously, which is then applied to define a spatial version of the Galam’s majority rule model. In our spatial model, each vertex of the lattice has one of two possible competing opinions, say opinion 0 and opinion 1, as in the popular voter model. Hyperedges are updated at rate one, which results in all the vertices in the hyperedge changing simultaneously their opinion to the majority opinion of the hyperedge. In the case of a tie in hyperedges with even size, a bias is introduced in favor of type 1, which is motivated by the principle of social inertia. Our analytical results along with simulations and heuristic arguments suggest that, in any spatial dimensions and when the set of hyperedges consists of the collection of all n×⋯×n blocks of the lattice, opinion 1 wins when n is even while the system clusters when n is odd, which contrasts with results about the voter model in high dimensions for which opinions coexist. This is fully proved in one dimension while the rest of our analysis focuses on the cases when n=2 and n=3 in two dimensions.

Corrections to scaling for watersheds, optimal path cracks, and bridge lines

We study the corrections to scaling for the mass of the watershed, the bridge line, and the optimal path crack in two and three dimensions (2D and 3D). We disclose that these models have numerically equivalent fractal dimensions and leading correction-to-scaling exponents. We conjecture all three models to possess the same fractal dimension, namely, d(f) =1.2168 ± 0.0005 in 2D and d(f) = 2.487 ± 0.003 in 3D, and the same exponent of the leading correction, Ω = 0.9 ± 0.1 and Ω=1.0 ± 0.1, respectively. The close relations between watersheds, optimal path cracks in the strong disorder limit, and bridge lines are further supported by either heuristic or exact arguments.

On the number of integers in a generalized multiplication table

Abstract. Motivated by the Erdős multiplication table problem we study the following question: Given numbers N 1 , ... , N k + 1 $N_1,\ldots ,N_{k+1}$ , how many distinct products of the form n 1 ⋯ n k + 1 $n_1\cdots n_{k+1}$ with 1 ≤ n i ≤ N i $1\le n_i\le N_i$ for i ∈ { 1 , ... , k + 1 } $i\in \lbrace 1,\ldots ,k+1\rbrace $ are there? Call A k + 1 ( N 1 , ... , N k + 1 ) $A_{k+1}(N_1,\ldots ,N_{k+1})$ the quantity in question. Ford established the order of magnitude of A 2 ( N 1 , N 2 ) $A_2(N_1,N_2)$ and the author the one of A k + 1 ( N , ... , N ) $A_{k+1}(N,\ldots ,N)$ for all k ≥ 2 $k\ge 2$ . In the present paper we generalize these results by establishing the order of magnitude of A k + 1 ( N 1 , ... , N k + 1 ) $A_{k+1}(N_1,\ldots ,N_{k+1})$ for arbitrary choices of N 1 , ... , N k + 1 $N_1,\ldots ,N_{k+1}$ when 2 ≤ k ≤ 5 $2\le k\le 5$ . Moreover, we obtain a partial answer to our question when k ≥ 6 $k\ge 6$ . Lastly, we develop a heuristic argument which explains why the limitation of our method is k = 5 $k=5$ in general and we suggest ways of improving the results of this paper.

Outbreak size distributions in epidemics with multiple stages

Multiple-type branching processes that model the spread of infectious diseases are investigated. In these stochastic processes, the disease goes through multiple stages before it eventually disappears. We mostly focus on the critical multistage susceptible–infected–recovered (SIR) infection process. In the infinite-population limit, we compute the outbreak size distributions and show that asymptotic results apply to more general multiple-type critical branching processes. Finally, using heuristic arguments and simulations, we establish scaling laws for a multistage SIR model in a finite population.

Strichartz Estimates on Asymptotically de Sitter Spaces

In this article we prove a family of local (in time) weighted Strichartz estimates with derivative losses for the Klein–Gordon equation on asymptotically de Sitter spaces and provide a heuristic argument for the non-existence of a global dispersive estimate on these spaces. The weights in the estimates depend on the mass parameter and disappear in the “large mass” regime. We also provide an application of these estimates to establish small-data global existence for a class of semilinear equations on these spaces.

Self-tuning and the derivation of a class of scalar-tensor theories

We have recently proposed a special class of scalar-tensor theories known as the Fab Four. These arose from attempts to analyze the cosmological constant problem within the context of Horndeski's most general scalar-tensor theory. The Fab Four together give rise to a model of self-tuning, with the relevant solutions evading Weinberg's no-go theorem by relaxing the condition of Poincar\'e invariance in the scalar sector. The Fab Four are made up of four geometric terms in the action with each term containing a free potential function of the scalar field. In this paper we rigorously derive this model from the general model of Horndeski, proving that the Fab Four represents the only classical scalar-tensor theory of this type that has any hope of tackling the cosmological constant problem. We present the full equations of motion for this theory, and give an heuristic argument to suggest that one might be able to keep radiative corrections under control. We also give the Fab Four in terms of the potentials presented in Deffayet et al's version of Horndeski.

Scaling behavior of threshold epidemics

We study the classic Susceptible-Infected-Recovered (SIR) model for the spread of an infectious disease. In this stochastic process, there are two competing mechanism: infection and recovery. Susceptible individuals may contract the disease from infected individuals, while infected ones recover from the disease at a constant rate and are never infected again. Our focus is the behavior at the epidemic threshold where the rates of the infection and recovery processes balance. In the infinite population limit, we establish analytically scaling rules for the time-dependent distribution functions that characterize the sizes of the infected and the recovered sub-populations. Using heuristic arguments, we also obtain scaling laws for the size and duration of the epidemic outbreaks as a function of the total population. We perform numerical simulations to verify the scaling predictions and discuss the consequences of these scaling laws for near-threshold epidemic outbreaks.

The unified method: II. NLS on the half-line with t-periodic boundary conditions

Boundary value problems for integrable nonlinear evolution PDEs formulated on the half-line can be analyzed by the unified method introduced by one of the authors and used extensively in the literature. The implementation of this general method to this particular class of problems yields the solution in terms of the unique solution of a matrix Riemann–Hilbert problem formulated in the complex k-plane (the Fourier plane), which has a jump matrix with explicit (x, t)-dependence involving four scalar functions of k, called spectral functions. Two of these functions depend on the initial data, whereas the other two depend on all boundary values. The most difficult step of the new method is the characterization of the latter two spectral functions in terms of the given initial and boundary data, i.e. the elimination of the unknown boundary values. For certain boundary conditions, called linearizable, this can be achieved by simply using algebraic manipulations. Here, we first present an effective characterization of the spectral functions in terms of the given initial and boundary data for the general case of non-linearizable boundary conditions. This characterization is based on the analysis of the so-called global relation and on the introduction of the so-called Gelfand–Levitan–Marchenko representations of the eigenfunctions defining the spectral functions. We then concentrate on the physically significant case of t-periodic Dirichlet boundary data. After presenting certain heuristic arguments which suggest that the Neumann boundary values become periodic as t → ∞, we show that for the case of the NLS with a sine-wave as Dirichlet data, the asymptotics of the Neumann boundary values can be computed explicitly at least up to third order in a perturbative expansion and indeed at least up to this order are asymptotically periodic.

On the energy conversion efficiency in magnetic hyperthermia applications: A new perspective to analyze the departure from the linear regime

The success of magnetic hyperthermia cancer treatments rely strongly on the magnetic properties of the nanoparticles and their intricate dependence on the externally applied field. This is particularly more so as the response departs from the low field linear regime. In this paper we introduce a new parameter, referred to as the efficiency in converting electromagnetic energy into thermal energy, which is shown to be remarkably useful in the analysis of the system response, especially when the power loss is investigated as a function of the applied field amplitude. Using numerical simulations of dynamic hysteresis, through the stochastic Landau-Lifshitz model, we map in detail the efficiency as a function of all relevant parameters of the system and compare the results with simple—yet powerful—predictions based on heuristic arguments about the relaxation time.

Marine ice sheet stability

AbstractWe examine the stability of two-dimensional marine ice sheets in steady state. The dynamics of marine ice sheets is described by a viscous thin-film model with two Stefan-type boundary conditions at the moving boundary or ‘grounding line’ that marks the transition from grounded to floating ice. One of these boundary conditions constrains ice thickness to be at a local critical value for flotation, which depends on depth to bedrock at the grounding line. The other condition sets ice flux as a function of ice thickness at the grounding line. Depending on the shape of the bedrock, multiple equilibria may be possible. Using a linear stability analysis, we confirm a long-standing heuristic argument that asserts that the stability of these equilibria is determined by a simple mass balance consideration. If an advance in the grounding line away from its steady-state position leads to a net mass gain, the steady state is unstable, and stable otherwise. This also confirms that grounding lines can only be stable in positions where bedrock slopes downwards sufficiently steeply.