Showing papers on "Randomness published in 1983"

Combinatorial foundations of information theory and the calculus of probabilities

Abstract: CONTENTS ??1. The growing role of finite mathematics ??2. Information theory ??3. The definition of complexity ??4. Regularity and randomness ??5. The stability of frequencies ??6. Infinite random sequences ??7. Relative complexity and quantity of information ??8. Barzdin's theorem ??9. ConclusionReferences

Microstructure of two-phase random media.III: The n-point matrix probability functions for fully penetrable spheres

Abstract: We examine the two‐ and three‐point matrix probability functions for a two‐phase random and homogeneous system of impenetrable spheres. For such a system, we give an exact analytical expression for the two‐point matrix function S2 through second order in the number density of particles. Moreover, the two‐point matrix function is evaluated, for the first time, for a very wide range of densities. We also discuss the evaluation of the three‐point matrix function S3 for an impenetrable‐sphere system and provide new expressions that may be used to estimate it.

Deterministic model of spin and statistics

Abstract: A deterministic model that accounts for the statistical behavior of random samples of identical particles is presented. The model is based on some nonmeasurable distribution of spin values in all directions. The mathematical existence of such distributions is proved by set-theoretical techniques, and the relation between these distributions and observed frequencies is explored within an appropriate extension of probability theory. The relation between quantum mechanics and the model is specified. The model is shown to be consistent with known polarization phenomena and the existence of macroscopic magnetism. Finally the possibility of a thought experiment which indicates a deviation from the predictions of quantum mechanics is described.

Schrödinger operators with an electric field and random or deterministic potentials

Abstract: We prove that the Schrodinger operatorH=−d2/dx2+V(x)+F·x has purely absolutely continuous spectrum for arbitrary constant external fieldF, for a large class of potentials; this result applies to many periodic, almost periodic and random potentials and in particular to random wells of independent depth for which we prove that whenF=0, the spectrum is almost surely pure point with exponentially decaying eigenfunctions.

One-dimensional Ising model in a random field

Abstract: The one-dimensional Ising model in a random field is studied with use of a functional recursion relation. For temperatures exceeding a given value, the fixed function of the relation is found and shown to be a devil's staircase. From this result it is possible to evaluate the free energy to arbitrary precision. In the field-strength--temperature plane, a crossover line corresponding to the onset of frustration is found.

On a Classical Spin Glass Model

Abstract: A simple, exactly soluble, model of a spin-glass with weakly correlated disorder is presented. It includes both randomness and frustration, but its solution can be obtained without replicas. As the temperatureT is lowered, the spin-glass phase is reached via an equilibrium phase transition atT=T f . The spin-glass magnetization exhibits a distinctS-shape character, which is indicative of a field-induced transition to a state of higher magnetization above a certain threshold field. For suitable probability distributions of the exchange interactions. The physical origin of the dependence upon the probability distributions is explained, and a careful analysis of the ground state structure is given.

Space‐time correlation properties of the magnetization process and eddy current losses: Theory

Abstract: The relationship between eddy current losses and magnetization dynamics is investigated on general bases, starting directly from Maxwell equations. The intrinsically stochastic character of the magnetization process is conveniently dealt with by describing the magnetization rate I(r,t) as a random sequence of elementary magnetization jumps, each corresponding to a sudden and localized displacement of a domain wall segment in the material. A general equation is obtained, in which the loss is expressed in terms of the energy spectrum ‖I(k,ω)‖2 of I(r,t). The loss calculation is thus reduced to the statistical problem of determining the energy spectrum of the random quantity I(r,t) as a function of the shape and correlation properties of the elementary magnetization events. It turns out that the hysteresis loss is related to the internal structure of the elementary jumps, while the dynamic and anomalous losses are instead determined by the space‐time correlation properties of the jump sequence. An explicit expression for the dynamic loss is obtained in the important case where the magnetization process is described as a Markov process, leading to a clear‐cut, direct link between the loss and the fundamental quantities which characterize the magnetization dynamics at a microscopic level. The model predicts the exact value of the Pry and Bean loss, as a particular case corresponding to very specialized correlation properties of the jump sequence. However, the model permits one to deal with actually more general and realistic descriptions of the domain structure dynamics. We believe that future applications to practical cases will provide a deeper understanding of the physical origin of loss anomalies.

Randomness and the density of hard problems

Abstract: A language L is random with respect to a given complexity class C if for all ′ ∈ C L and ′ disagree on half of all strings. It is known that for any complexity class there are recursive languages that are random with respect to that class. Here it is shown that there are tight space and time hierarchies of random languages, and that EXPTIME contains P-isomorphism classes containing only languages that are random with respect to polynomial-time computations. The technique used is extended to show that for any constructible bound on time or space it is possible to deterministically generate binary sequences that appear random to all prediction algorithms subject to the given resource bound. Furthermore, the generation of such a sequence requires only slightly more resources than the given bound.

Renormalisation-group study of self-avoiding walks on the random lattice

Abstract: Using a field theoretic method based on the replica trick, it is proved that the three-parameter renormalisation group for an n-vector model with quenched randomness reduces to a two-parameter one in the limit n to 0 which corresponds to self-avoiding walks (SAWS). This is also shown by the explicit calculation of the renormalisation-group recursion relations to second order in epsilon . From this reduction the author finds that SAWS on the random lattice are in the same universality class as SAWS on the regular lattice. By analogy with the case of the n-vector model with cubic anisotropy in the limit n to 1, the fixed-point structure of the n-vector model with randomness is analysed in the SAW limit, so that a physical interpretation of the unphysical fixed point is given. Corrections of the values of critical exponents of the unphysical fixed point published previously are also given.

Exact solution of a one-dimensional Ising model in a random magnetic field

Abstract: Analytic results for the free energy, magnetic structure factor, and Edwards-Anderson order parameter of a one-dimensional ferromagnetic Ising model in a random magnetic field are obtained. The structure factor consists of both Lorentzian and Lorentzian-squared terms at all temperatures ($T$) greater than zero; the Lorentzian-squared terms vanish at $T=0$. The calculated correlation length agrees with predictions of the Imry-Ma domain argument.

Intrinsic Randomness and Intrinsic Irreversibility in Classical Dynamical Systems

Abstract: We continue our previous work on dynamic “intrinsically random” systems for which we can derive dissipative Markov processes through a one-to-one change of representation. For these systems, the unitary group of evolution can be transformed in this way into two distinct Markov processes leading to equilibrium for either t→ + ∞ or t→ - ∞. To lift the degeneracy, we first formulate the second principle as a selection rule that is meaningful in intrinsically random systems. For these systems, this excludes a set of unrealizable states. As a result of this exclusion, permitted initial conditions correspond to a set of states that is not invariant through velocity inversion. In this way, the time-reversal symmetry of dynamics is broken and these systems acquire a new feature we may call “intrinsic irreversibility.” The set of admitted initial conditions can be characterized by an entropy displaying the amount of information necessary for their preparation. The initial conditions selected by the second law correspond to a finite amount of information, while the initial conditions that are rejected correspond to an infinite amount of information and are therefore “impossible.” We believe that our formulation permits a microscopic formulation of the second law of thermodynamics for well-defined classes of dynamical systems.

Phase diagrams of random Ising models : simple and systematic successive approximations

Abstract: 2014 We present a simple and systematic method to calculate phase diagrams of random Ising models. It applies to a wide class of systems in which, however, randomness has to be described by discrete random varia-bles. We calculated, at different orders of approximation, various quantities like critical temperatures and perco- lation thresholds in good agreement with exact known results. J. Physique 44 (1983) 1143-1147 OCTOBRE 1983, : Classification Physics Abstracts 64.60 Most papers dealing with random Ising systems make use either of real space renormalization group methods [1] or the replica trick combined with themean-field approximation [2]. This paper presents a much less sophisticated method due to one of us [3] which is, conceptually, as simple as an ordinary mean- field approximation and has the advantage of leading to systematic successive approximations which, even at the lowest order, give, from a numerical point of view, better results than the mean-field

Exact electronic spectra and inverse localization lengths in one-dimensional random systems: I. Random alloy, liquid metal and liquid alloy

Abstract: Analytic continuations into the complex energy plane of Dyson-Schmidt type of equations for the calculation of the density of states are constructed for a random alloy model, a liquid metal and for a liquid alloy In all these models the characteristic function follows from the solution of this equation Its imaginary part yields the accumulated density of states and its real part is a measure for the inverse of the localization length of the eigenfunctions The equations have been solved exactly for some distributions of the random variables In the random alloy case the strengths of the delta-potentials have an exponential distribution They may also have finite, exponentially distributed values with probability 0 ⪕ p ⪕ 1 and be infinite with probability q = 1 −p In the liquid metal the liquid particles are assumed to behave like hard rods This implies an exponential distribution of the distances between the particles The common electronic potential may be arbitrary, but is assumed to vanish outside the rods In the one-dimensional liquid alloy there is, apart from positional randomness of the liquid particles, a distribution of the strengths of the electronic delta-potentials For Cauchy distributions an argument of Lloyd is extended to obtain the characteristic function from the one in the model with equal strengths For the case of a liquid of point particles a three parameter class of distributions of the strengths is shown to yield a solution in the form of known functions of the equation mentioned above For several cases numerical calculations of the density of states and the inverse localization length of the eigenfunctions are presented and discussed New results are found: exponential decay of the density of states near special energies in the random alloy and liquid metal; divergence of the density of states at certain energies with non-classical exponent 13 in the random alloy if the average of the potential strengths vanishes; exponentially small broadening of the bound-state levels for low concentrations of the liquid particles; peak in the localization length at the bound-state energies, which becomes exponentially narrow for low concentrations; different exponent in the decay of the inverse localization length at large energies for delta potentials and square-well potentials Further an expression for the grand potential is given, involving a sum over the characteristic function at certain points and divergence of the zero-point energy is found for Cauchy distributions of the delta potential strengths

Stochastic processes originating in deterministic microscopic dynamics

Abstract: We investigate the probability distribution of the scaled trajectory of a test particle moving in an equilibrium fluid according to the laws of classical mechanics, i.e., ifQ(t) is the displacement of the test particle we letQA(t) =Q(At)/√A and consider the distribution of the trajectory QA(t) in the limit A→∞. The randomness of the motion is due entirely to the randomness of the initial state of the fluid, test particle, or both, and the process is generally non-Markovian. Nevertheless, it can be proven in some cases and we expect it to be true in many more that QA (t) looks like Brownian motion in the limit A→∞. Some results for simple model systems are presented.

Random Numbers Generated by a Physical Device

Abstract: A new method of generating random digits using radiations from radioactive nuclide is described. The digits are stored on magnetic tape which allows repeated use if required. One hundred million such digits have been subjected to stringent tests for randomness. These tests, some of which are described, show that the random digits avoid the well‐documented defects of digits generated by arithmetic means. They perform well for applications where sequences of independent random digits are required.

Majorisation, randomness and some discrete distributions

Abstract: Majorisation is used to compare discrete distributions in terms of randomness or, equivalently, informativeness. Some properties of measures of randomness are discussed. The effect on degree of randomness of a change in parameter value is investigated for some distributions including the binomial and Poisson.

Tests of randomness for pseudorandom number generators

Abstract: This study examined the “randomness” of the numbers generated by the VIC-20 computer. Using eight standard tests, it was determined that the numbers provided by the VIC-20 are adequately random for practical purposes. The tests are applicable to other computer systems and provide a general means for evaluating random-number generators.

Equilibrium statistics of two‐dimensional viscous flows with arbitrary random forcing

Abstract: A method to solve the Liouville equation for an ensemble of two‐dimensional viscous flows that are driven by random forcing with arbitrary statistics is outlined. By appropriate transformations of both the dependent variable (probability distribution) and independent variables, and by expansion of the solution in eigenfunctions of the separable part of the Liouville operator, it is found possible to reduce the problem to that of solving a simultaneous system of nonhomogeneous linear algebraic equations. The equilibrium kinetic energy and energy‐transfer spectra can be calculated directly from the equilibrium probability distribution.

Chaos and randomness in dynamical systems

Abstract: In this paper, we discuss the irregularity found in the trajectories of "chaotic" differential equations from a new point of view--in terms of predictability of future values by optimal filters. This study, together with numerical results, indicates that as far as models for engineering phenomena are concerned, rather simple deterministic equations can be a completely satisfactory alternative to stochastic models in treating some aspects of estimation and stochastic control.

A spatial smoothing formulation for location systems

Abstract: Two extensions of the classic passive location problem are considered. The first examines the ranging ability of passive receivers. The second presents alternative ways of describing the geometric content in positioning problems. The implications of the different approaches on the structure and performance of the location receiver are discussed. Because range determination by passive means is missing, the classical formulation of passive location may be viewed as a local geometry demodulation problem. With the explicit consideration of range, passive location becomes a global problem. At stake is what may be gained by processing the small but valuable amount of information carried by the wavefront curvature of the signals. Relevant questions relate to the design of passive receivers that aptly demodulate the range and the remaining quantities defining the geometry. There are passive applications where models exhibiting a high degree of (geometric) regularity are viable from a practical point of view. These occur, for example, when one can assume that the array sensors are collinear and that the moving target follows a deterministic linear path. In these models, the geometry is completely determined by a finite set of (unknown) parameters (e.g., range, bearing, speed, etc.). Accordingly, it is said that the regular models use an integral or ensemble approach for the description of the geometry. In many other problems, the geometry is more adequately described by statistical processes. Examples arise when the source follows a disturbed path, or when, due to towing, the array shape deforms, acquiring a not-completely-known shape. The paper models these constraints via a set of stochastic differential equations. The resulting representation is termed a differential description. It is emphasized that the differential approach is not only applied to the time content (relative dynamics), but also to the spatial dimension (array shape). The technique dualizes the space and time aspects of the problem. It provides a more flexible framework than the previous one. More general motions and array shapes than the traditional collinear ones can be considered by the analysis, e.g., irregular line arrays or arrays where the sensors are located at positions with a certain degree of randomness. Each approach fits a different design framework. The ensemble description is associated with the maximum likelihood technique. The differential representation uses recursive estimation methods (as provided by the Kalman-Bucy filtering theory). The paper discusses the main aspects of the structure of the resulting receivers and the associated measures of error performance. A second advantage of the differential model is immediately apparent The recursiveness of the differential receiver reduces its computational load. The speed-up obtained is fully appreciated in tracking applications, where the observations are sequentially updated. Finally, it is interesting to note that the time/space duality provided by the differential approach exhibits a remarkable distinction: the location recursive receiver behaves in time as a filter, while it behaves in space as a smoother.

Classical versus bayesian models—on the dangers of a little bit of knowledge

Abstract: In control applications, economists typically assume the model that constrains the objective function has fixed hut unknown parameters. The- unknown parameters are replaced with their random estimators—the classical method. Engineers typically assume the model has random parameters drawn from a known distribution—the bayesian method. We contrast control rules for random parameter models with two alternatives and show that if the true model has fixed but unknown parameters, ignoring the randomness in the estimator is better than adopting a random parameter type control rule. We also present a control rule that is superior to either a random parameter type control rule or a control rule that ignores the variance of the estimated coefficients

- Randomness and Stochastic Independence - on the Relationship Between Intuitive Motion Mathematical Definition

Abstract: Publisher Summary In stochastic education, one of the most important questions is ways in which students handle the concepts of chance and of randomness. This chapter examines the relationship between two aspects: (1) the preconcepts and intuitions of probability and (2) the mathematical calculi and techniques used to adapt and to develop these intuitions further. The concept of chance or of randomness is central for the stochastic education for the very reason that this concept, as opposed to other mathematical theories and concepts, is the specific feature of probability theory. Chance and randomness are common fundamental assumptions for the stochastic education and a great part of psychological research in decision making under uncertainty. In the attempt to precisely define the concept of randomness, the theory of computational complexity adopts a quite similar view of chance. A finite binary sequence is called “random” if the smallest algorithm capable of specifying it to a computer has about the same number of information bits as the sequence itself.

Measurements and Improvements of Randomness Deteriorations in the Random Pulser

Abstract: The random pulser utilizing the avalanche noise has been studied on its randomness deteriorations.

Relaxation time and randomness in phase space

Abstract: The stochastic region in the phase space of a classical nonlinear system, a Lennard-Jones chain, is proven to be nonuniform with respect to a random access, through the sensitivity of the relaxation time to initial conditions. The dependence on various parameters is analysed and the results are interpreted geometrically as the effect of residual invariant tori in the stochastic domain.