Two-scale expansions of quasinormal waves in nonregular,refractive waveguides and an estimate of their applicability at high frequencies

The problem of wave propagation in an inhomogeneous half space having an index of refraction which varies slowly in the horizontal coordinates and has a single maximum on the boundary of the half space (a surface waveguide) or in its interior (an interior waveguide) is solved by the method of two-scale expansions. It is shown that the method of two-scale expansions can also be applied at high frequencies if in the case of an interior waveguide p 1, where is a small parameter characterizing the slow variation of the properties of the medium in horizontal directions, and p is the dimensionless frequency. In the case of a presurface waveguide for a small number m of nodes of the eigenfunction the expansion can be used without assuming slow variation of the refractive index in the horizontal coordinates; if m p, then p 1.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 117, pp. 78–97, 1981.     

Statistical mechanics in biology: how ubiquitous are long-range correlations?

The purpose of this opening talk is to describe examples of recent progress in applying statistical mechanics to biological systems. We first briefly review several biological systems, and then focus on the fractal features characterized by the long-range correlations found recently in DNA sequences containing non-coding material. We discuss the evidence supporting the finding that for sequences containing only coding regions, there are no long-range correlations. We also discuss the recent finding that the exponent alpha characterizing the long-range correlations increases with evolution, and we discuss two related models, the insertion model and the insertion-deletion model, that may account for the presence of long-range correlations. Finally, we summarize the analysis of long-term data on human heartbeats (up to 10(4) heart beats) that supports the possibility that the successive increments in the cardiac beat-to-beat intervals of healthy subjects display scale-invariant, long-range "anti-correlations" (a tendency to beat faster is balanced by a tendency to beat slower later on). In contrast, for a group of subjects with severe heart disease, long-range correlations vanish. This finding suggests that the classical theory of homeostasis, according to which stable physiological processes seek to maintain "constancy," should be extended to account for this type of dynamical, far from equilibrium, behavior.     

Applications of the Stell–Hemmer Potential to Understanding Second Critical Points in Real Systems

We consider the novel properties of the Stell–Hemmer core-softened potentials. First we explore how the theoretically predicted second critical point for these potentials is related to the occurrence of the experimentally observed solid–solid isostructural critical point. We then discuss how this class of potentials can generate anomalies analogous to those found experimentally in liquid water.     

Application of Statistical Physics to Understand Static and Dynamic Anomalies in Liquid Water

We present an overview of recent research applying ideas of statistical mechanics to try to better understand the statics and especially the dynamic puzzles regarding liquid water. We discuss recent molecular dynamics simulations using the Mahoney–Jorgensen transferable intermolecular potential with five points (TIP5P), which is closer to real water than previously-proposed classical pairwise additive potentials. Simulations of the TIP5P model for a wide range of deeply supercooled states, including both positive and negative pressures, reveal (i) the existence of a non-monotonic temperature of maximum density line and a non-reentrant spinodal, (ii) the presence of a low-temperature phase transition. The take-home message for the static aspects is that what seems to matter more than previously appreciated is local tetrahedral order, so that liquid water has features in common with SiO₂ and P, as well as perhaps Si and C. To better understand dynamic aspects of water, we focus on the role of the number of diffusive directions in the potential energy landscape. What seems to matter most is not values of thermodynamic parameters such as temperature T and pressure P, but only the value of a parameter characterizing the potential energy landscape—just as near a critical point what matters is not the values of T and P but rather the values of the correlation length.     

Avalanche dynamics of crackle sound in the lung

We analyze a sequence of short transient sound waves, called "crackles," which are associated with explosive openings of airways during lung inflation. The distribution of time intervals between consecutive crackles Delta(t) shows two regimes of power law behavior. We develop an avalanche model which fits the data over five decades of Delta(t). We find that the regime for large Delta(t) is related to the dynamics of distinct avalanches in a Cayley tree, and the regime for small Delta(t) is determined by the dynamics of crackle propagation within a single avalanche. We also obtain a mean-field solution of the model which provides information about lung inflation.     

Traveling time and traveling length in critical percolation clusters

We study traveling time and traveling length for tracer dispersion in two-dimensional bond percolation, modeling flow by tracer particles driven by a pressure difference between two points separated by Euclidean distance r. We find that the minimal traveling time t(min) scales as t(min) approximately r(1.33), which is different from the scaling of the most probable traveling time, t* approximately r(1.64). We also calculate the length of the path corresponding to the minimal traveling time and find l(min) approximately r(1.13) and that the most probable traveling length scales as l* approximately r(1.21). We present the relevant distribution functions and scaling relations.     

Robustness of a network of networks

Network research has been focused on studying the properties of a single isolated network, which rarely exists. We develop a general analytical framework for studying percolation of n interdependent networks. We illustrate our analytical solutions for three examples: (i) For any tree of n fully dependent Erd?s-Rényi (ER) networks, each of average degree k, we find that the giant component is P∞ =p[1-exp(-kP∞)](n) where 1-p is the initial fraction of removed nodes. This general result coincides for n = 1 with the known second-order phase transition for a single network. For any n>1 cascading failures occur and the percolation becomes an abrupt first-order transition. (ii) For a starlike network of n partially interdependent ER networks, P∞ depends also on the topology-in contrast to case (i). (iii) For a looplike network formed by n partially dependent ER networks, P∞ is independent of n.     

The acoustic field of a high-frequency source moving in a waveguide

The acoustic field of a source moving at a subsonic velocity in a regular waveguide with perfectly reflecting boundaries is considered. The acceleration of the source is assumed to be small. In a moving coordinate system, the asymptotics of the wave field is obtained. This asymptotics is inapplicable near the critical cross sections, for which the Doppler frequency of the source coincides with the frequency of the waveguide mode under consideration. It is demonstrated that, in this case, the wave field can be represented locally by a special type of integral, which is analyzed by the saddle-point method.Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 239, 1997, pp. 71–78.This work was supported in part by the Russian Foundation for Basic Research under grant 95-01-01285a.