Correlated randomness: Some examples of exotic statistical physics

One challenge of biology, medicine, and economics is that the systems treated by these sciences have no perfect metronome in time and no perfect spatial architecture-crystalline or otherwise. Nonetheless, as if by magic, out of nothing but randomness one finds remarkably fine-tuned processes in time and remarkably fine-tuned structures in space. To understand this ‘miracle’, one might consider placing aside the human tendency to see the universe as a machine. Instead, one might address the challenge of uncovering how, through randomness (albeit, as we shall see, strongly correlated randomness), one can arrive at many spatial and temporal patterns in biology, medicine, and economics. Inspired by principles developed by statistical physics over the past 50 years-scale invariance and universality-we review some recent applications of correlated randomness to fields that might startle Boltzmann if he were alive today.     

Economic fluctuations and statistical physics: Quantifying extremely rare and less rare events in finance

One challenge of economics is that the systems treated by these sciences have no perfect metronome in time and no perfect spatial architecture—crystalline or otherwise. Nonetheless, as if by magic, out of nothing but randomness one finds remarkably fine-tuned processes in time. We present an overview of recent research joining practitioners of economic theory and statistical physics to try to better understand puzzles regarding economic fluctuations. One of these puzzles is how to describe outliers, phenomena that lie outside of patterns of statistical regularity. We review evidence consistent with the possibility that such outliers may not exist. This possibility is supported by recent analysis of databases containing information about each trade of every stock.     

The puzzling unsolved mysteries of liquid water: Some recent progress

Water is perhaps the most ubiquitous, and the most essential, of any molecule on earth. Indeed, it defies the imagination of even the most creative science fiction writer to picture what life would be like without water. Despite decades of research, however, water's puzzling properties are not understood and 63 anomalies that distinguish water from other liquids remain unsolved. We introduce some of these unsolved mysteries, and demonstrate recent progress in solving them. We present evidence from experiments and computer simulations supporting the hypothesis that water displays a special transition point (which is not unlike the “tipping point” immortalized by Malcolm Gladwell). The general idea is that when the liquid is near this “tipping point,” it suddenly separates into two distinct liquid phases. This concept of a new critical point is finding application to other liquids as well as water, such as silicon and silica. We also discuss related puzzles, such as the mysterious behavior of water near a protein.     

How random is a random vector?

Over 80 years ago Samuel Wilks proposed that the “generalized variance” of a random vector is the determinant of its covariance matrix. To date, the notion and use of the generalized variance is confined only to very specific niches in statistics. In this paper we establish that the “Wilks standard deviation” –the square root of the generalized variance–is indeed the standard deviation of a random vector. We further establish that the “uncorrelation index” –a derivative of the Wilks standard deviation–is a measure of the overall correlation between the components of a random vector. Both the Wilks standard deviation and the uncorrelation index are, respectively, special cases of two general notions that we introduce: “randomness measures” and “independence indices” of random vectors. In turn, these general notions give rise to “randomness diagrams”—tangible planar visualizations that answer the question: How random is a random vector? The notion of “independence indices” yields a novel measure of correlation for Lévy laws. In general, the concepts and results presented in this paper are applicable to any field of science and engineering with random-vectors empirical data.     

Saturation field direction dependence of domain structures in NiFe elements

The domain structures in NiFe elements were studied by magnetic force microscopy measurement and micromagnetic modeling. The remanent states in the elements were dependent on the direction of the saturation field. The “S” and “U” states were observed at remanence by applying the saturation field at different directions. The “S” and “U” states are metastable: magnetic force microscopy tip field-induced switching from the “S” and “U” states to the flux closure configuration was observed.     

From shape to randomness: A classification of Langevin stochasticity

The Langevin equation–perhaps the most elemental stochastic differential equation in the physical sciences–describes the dynamics of a random motion driven simultaneously by a deterministic potential field and by a stochastic white noise. The Langevin equation is, in effect, a mechanism that maps the stochastic white-noise input to a stochastic output: a stationary steady state distribution in the case of potential wells, and a transient extremum distribution in the case of potential gradients. In this paper we explore the degree of randomness of the Langevin equation’s stochastic output, and classify it à la Mandelbrot into five states of randomness ranging from “infra-mild” to “ultra-wild”. We establish closed-form and highly implementable analytic results that determine the randomness of the Langevin equation’s stochastic output–based on the shape of the Langevin equation’s potential field.     

Spin Echo Attenuation of Restricted Diffusion as a Discord of Spin Phase Structure

By using the particle probability density we analyze the spin echo attenuation of particles, diffusing in a bounded region. It provides a means to expand a nonuniform spin phase distribution into a series of waves that characterize the geometry and boundary conditions of confinement. Random motion disrupts the initial phase structure created by applied gradients and consequently discords its structure waves. By assuming the spin phase fluctuation and/or the randomness of spin phase distribution in the subensemble as a Gaussian stochastic process, we derive a new analytical expression for the echo attenuation related to the particle velocity correlation. For a diffusion in porous structure we get the expression featuring the same “diffusive diffraction” patterns as those being found and explained by P. T. Callaghan and A. Coy (“Principles of Nuclear Magnetic Resonance Microscopy,” Oxford Univ. Press, Oxford (1991);J. Chem. Phys.101, 4599–4609 (1994)) with the use of propagator theory. With the new approach we cast a new light on the phenomena and derive analitically how the diffusive diffractions appear when the sequence of finite or even modulated gradients are applied. The method takes into account the non-Markovian character of restricted diffusion, and therefore the echo dependence on the diffusion lengths and on the strength of applied gradient differs from the results of authors assuming the Markovian diffusion either by dealing with the diffusion propagators or by the computer simulation of Fick's diffusion.     

Reproducible resistive switching effect for memory applications in heterocontacts based on strongly correlated electron systems

The transitional processes in heterocontacts based on strongly correlated electron systems (SCES) are studied for analyzing of the effect of resistive switching (ERS). It has been shown that the process is asymmetric with respect to switching into “on” and “off” states, the switching time is controlled by a voltage level, this time can be less than microseconds, on the other hand, relaxation processes can reach tens seconds. The switching is controlled by two processes: a change in the resistance state of the normal metal/SCES interface under effect of electric current field and by electrodiffusion of oxygen to vacancies, at that the doping level of the contact area and resistive properties of the heterocontact change. In particular, electrodiffusion of mobile oxygen induced by the electric field makes it possible to use a device with ERS as a memristor. On the other hand, a possibility to control the switching time and ON and OFF parameters show the possibilities to use these devices as memory elements “RAM”.     

Statistical physics of media processes: Mediaphysics

The processes of mass communications in complicated social or sociobiological systems such as marketing, economics, politics, animal populations, etc. as a subject for the special scientific subbranch—“mediaphysics”—are considered in its relation with sociophysics. A new statistical physics approach to analyze these phenomena is proposed. A keystone of the approach is an analysis of population distribution between two or many alternatives: brands, political affiliations, or opinions. Relative distances between a state of a “person's mind” and the alternatives are measures of propensity to buy (to affiliate, or to have a certain opinion). The distribution of population by those relative distances is time dependent and affected by external (economic, social, marketing, natural) and internal (influential propagation of opinions, “word of mouth”, etc.) factors, considered as fields. Specifically, the interaction and opinion-influence field can be generalized to incorporate important elements of Ising-spin-based sociophysical models and kinetic-equation ones. The distributions were described by a Schrödinger-type equation in terms of Green's functions. The developed approach has been applied to a real mass-media efficiency problem for a large company and generally demonstrated very good results despite low initial correlations of factors and the target variable.     

The geometry of bipartite qutrits including bound entanglement

We investigate the state space of bipartite qutrits, in particular we construct an analog to the “magic” tetrahedron for bipartite qubits—a magic simplex W. Due to the high symmetry it is enough to consider certain typical slices through W and optimal entanglement witnesses can be calculated. A whole region of bound entangled states is found.     

Minimal mechanisms for school formation in self-propelled particles

In the context of social organisms, a school refers to a cohesive group of organisms that share a common speed and direction of motion, as well as a common axis of body alignment or polarization. Schools are also noted for the relatively fixed nearest-neighbour distances between individuals. The rules of interaction that lead to the formation and maintenance of a school structure have been explored experimentally, analytically, and by simulation. Interest in biological examples, and non-biological “self-propelled particles” such as robots, vehicles, or autonomous agents leads to the question of what are the simplest possible sets of rules that can assure the formation and the stability of the “perfect school”: an aggregate in which the nearest-neighbour distances and speeds are identical.Here we explore mechanisms that lead to a perfect school structure in one and two dimensions. We consider distance-detection as well as velocity-detection between the interacting pairs of self-propelled particles. We construct interaction forces and formulate schooling equations. In the simplest cases, these equations have analytic solutions. In many cases, the stability of the perfect school can be explored. We then investigate how these structures form and evolve over time from various initial configurations using simulations. We study the relationship between the assumed interaction forces and the school patterns that emerge. While true biological schools are far from perfect, the insights gained from this investigation can help to understand some properties of real schools, and to suggest the appropriate properties of artificial schools where coordinated motion is desired.     

Quantum teleportation using entangled 3-qubit states and the ‘magic bases’

We study quantum teleportation of single qubit information state using 3-qubit general entangled states. We propose a set of 8 GHZ-like states which gives (i) standard quantum teleportation (SQT) involving two parties and 3-qubit Bell state measurement (BSM) and (ii) controlled quantum teleportation (CQT) involving three parties, 2-qubit BSM and an independent measurement on one qubit. Both are obtained with perfect success and fidelity and with no restriction on destinations (receiver) of any of the three entangled qubits. For SQT, for each designated one qubit which is one of a pair going to Alice, we obtain a magic basis containing eight basis states. The eight basis states can be put in two groups of four, such that states of one group are identical with the corresponding GHZ-like states and states of the other differ from the corresponding GHZ-like states by the same phase factor. These basis states can be put in two different groups of four-states each, such that if any entangled state is a superposition of these with coefficients of each group having the same phase, perfect SQT results. Also, for perfect CQT, with each set of given destinations of entangled qubits, we find a different magic basis. If no restriction on destinations of any entangled qubit exists, three magic semi-bases, each with four basis states, are obtained, which lead to perfect SQT. For perfect CQT, with no restriction on entangled qubits, we find four magic quarter-bases, each having two basis states. This gives perfect SQT also. We also obtain expressions for co-concurrences and conditional concurrences.     

A modified alignment method based on four-quadrant-grating moiré for proximity lithography

In this paper, we demonstrate a modified coarse-fine alignment scheme designed for proximity lithography. Both wafer alignment mark and mask alignment mark consists of linear grating arrays and “+” bar. Coarse alignment and fine alignment could work together to achieve the perfect alignment. Thereinto, coarse alignment, measured from two superposed “+” bars, guarantees the misalignment across wafer and mask within the measurement range of fine alignment, which is based on moiré fringes formed by the superposition of linear grating arrays. Then we conduct the experiments using a nanometer actuator to drive the wafer alignment mark meanwhile keeping the mask alignment mark motionless, which validates the feasibility and rationality of our designed scheme.     

Measuring statistical evenness: A panoramic overview

Motivated by the question “how equal is the distribution of wealth within a given human population?” economics devised an impressive toolbox of quantitative measures of societal egalitarianism including the Lorenz curve and the following indices: Gini, Pietra, Hoover, Amato, Hirschman, Theil and Atkinson. These quantitative measures-considered in the broader context of general data-sets with positive values-are, in effect, general gauges of statistical evenness. While the application of Gini’s index grew beyond economics and reached diverse fields of science, the aforementioned “evenness toolbox” has largely remained within the confines of the social sciences. The aim of this Paper is to expose this “evenness toolbox” to the physics community by presenting a comprehensive evenness-based approach to a fundamental problem in science—the measurement of statistical heterogeneity.     

Agent-based approach for generation of a money-centered star network

The history of trade is a progression from a pure barter system. A medium of exchange emerges autonomously in the market, a position currently occupied by money. We investigate an agent-based computational economics model consisting of interacting agents considering distinguishable properties of commodities which represent salability. We also analyze the properties of the commodity network using a spanning tree. We find that the “storage fee” is more crucial than “demand” in determining which commodity is used as a medium of exchange.     

Scattering in thick multifractal clouds, Part II: Multiple scattering

In Part I of this paper, we developed asymptotic approximations for single photon scattering in thick, highly heterogeneous, “Log-Lévy” multifractal clouds. In Part II, theoretical multiple scattering predictions are numerically tested using Monte Carlo techniques, which show that, due to long range correlations, the photon paths are “subdiffusive” with the corresponding fractal dimensions tending to increase slowly with mean optical thickness. We develop reasonably accurate statistical relations between N scatter statistics in thick clouds and single scatter statistics in thin clouds. This is explored further using discrete angle radiative transfer (DART) approach in which the radiances decouple into non-interacting families with only four (for 2-D clouds) radiance directions each. Sparse matrix techniques allow for rapid and extremely accurate solutions for the transfer; the accuracy is only limited by the spatial discretization.By “renormalizing” the cloud density, we relate the mean transmission statistics to those of an equivalent homogeneous cloud. This simple idea is remarkably effective because two complicating effects act in contrary directions: the “holes” which lead to increased single scatter transmission and the tendency for multiply scattered photons to become “trapped” in optically dense regions, thus decreasing the overall transmission.     

Rich-club and page-club coefficients for directed graphs

Rich-club and page-club coefficients and their null models are introduced for directed graphs. Null models allow for a quantitative discussion of the rich-club and page-club phenomena. These coefficients are computed for four directed real-world networks: Arxiv High Energy Physics paper citation network, Web network (released from Google), Citation network among US Patents, and email network from a EU research institution. The results show a high correlation between rich-club and page-club ordering. For journal paper citation network, we identify both rich-club and page-club ordering, showing that “elite” papers are cited by other “elite” papers. Google web network shows partial rich-club and page-club ordering up to some point and then a narrow declining of the corresponding normalized coefficients, indicating the lack of rich-club ordering and the lack of page-club ordering, i.e. high in-degree (PageRank) pages purposely avoid sharing links with other high in-degree (PageRank) pages. For UC patents citation network, we identify page-club and rich-club ordering providing a conclusion that “elite” patents are cited by other “elite” patents. Finally, for email communication network we show lack of both rich-club and page-club ordering. We construct an example of synthetic network showing page-club ordering and the lack of rich-club ordering.     

Eigenvalues of collective emission in multi-slice slab configurations

We compute the eigenmodes of collective emission from multi-slice slab configurations, using the transfer matrix formalism. We elucidate within this formalism the phenomena of “Invisible Gaps” in multiple-slice configuration and of “Precocious Superradiance” in periodic structures previously observed in numerical solutions of Maxwell-Bloch equations.     

Ballistic spin interferometer based on the Rashba and Dresselhaus spin-orbit interactions

By using the Al'tshuler-Aronov-Spivak (AAS) model, we give the amplitude changing with Rashba spin-orbit interaction (SOI) and Dresselhaus SOI strength. In the first idea 1D square loop (SL), Rashba SOI acts on two sides while Dresselhaus SOI acts on the other two sides. In the second SL, we consume Rashba SOI and Dresselhaus SOI act on four sides simultaneously. This model can be replaced by another one that Rashba SOI and Dresselhaus SOI act on every side independently, and each side is twice long. We theoretically illustrate the influence of the Dresselhaus SOI on node position and number. To explain the “half oscillation” phenomenon found in experiment, we apply Dresselhaus SOI to the ideal 1D SL. The conclusion is that the Dresselhaus SOI has a strong effect on the emergence of “half oscillation”.