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E W Montroll

Publications and source records attributed to E W Montroll.

16 recordsLinked to original sources
PubMed1984

On the Williams-Watts function of dielectric relaxation.

It has been observed over the past 15 years that experimental frequency-dependent dielectric constants of broad classes of materials including polymeric systems and glasses may be interpreted in terms of the Williams-Watts polarization decay function [Formula: see text] The exponent alpha and the time constant T depend on the material and fixed external conditions such as temperature and pressure. We derive this form of varphi(alpha)(t) from the following random-walk model. Suppose that an electric field has been applied for some time to a medium containing many polar molecules (or polar groups in complex molecules) and the direction of their dipole moments remains frozen as the field is removed. Furthermore, suppose that the medium contains mobile defects that on reaching the site of a frozen dipole relax the medium to the degree that the dipole may reorient itself. If the diffusion of defects toward dipoles is executed as a continuous-time random walk composed of an alternation of steps and pauses and the pausing-time distribution function has a long tail of the form psi(t) infinity t(-1-alpha), then the relaxation function has the above fractional exponential form.

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PubMed1982

On 1/f noise and other distributions with long tails.

It is shown, following Shockley [Shockley, W. (1957) Proc. IRE 45, 279-290], that, when a population is engaged in tasks whose completion requires the successful conclusion of many independent subtasks, the distribution function for successes in the primary task is log normal. It is also shown that, when the dispersion of the log-normal distribution is large, the distribution is mimicked by a 1/x distribution over a wide range of x. This argument provides a generic set of processes that yields the much observed 1/x distribution, and will also lead to a 1/f noise spectrum. It is commonly found that distributions that seem to be log normal over a broad range (say to the 95th percentile of a population) change to an inverse fractional power (Pareto) distribution for the last few percentile. Annual income distributions are examples with this structure. The very wealthy generally achieve their superwealth through amplification processes that are not available to most. We have introduced a simple amplification model to characterize the transition from a log-normal distribution to an inverse-power Pareto tail.

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PubMed1981

On the dynamics of the Ising model of cooperative phenomena.

A two-dimensional (and to some degree three-dimensional) version of Glauber's one-dimensional spin relaxation model is described. The model is constructed to yield the Ising model of cooperative phenomena at equilibrium. A complete hierarchy of differential equations for multispin correlation functions is constructed. Some remarks are made concerning the solution of them for the initial value problem of determining the relaxation of an initial set of spin distributions.

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PubMed1981

On the entropy function in sociotechnical systems.

The entropy function H = -Sigmap(j) log p(j) (p(j) being the probability of a system being in state j) and its continuum analogue H = integralp(x) log p(x) dx are fundamental in Shannon's theory of information transfer in communication systems. It is here shown that the discrete form of H also appears naturally in single-lane traffic flow theory. In merchandising, goods flow from a whole-saler through a retailer to a customer. Certain features of the process may be deduced from price distribution functions derived from Sears Roebuck and Company catalogues. It is found that the dispersion in logarithm of catalogue prices of a given year has remained about constant, independently of the year, for over 75 years. From this it may be inferred that the continuum entropy function for the variable logarithm of price had inadvertently, through Sears Roebuck policies, been maximized for that firm subject to the observed dispersion.

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PubMed1981

Phase transition versus disorder: A criterion derived from a two-dimensional dynamic ferromagnetic model.

The dynamical equations for spin correlation functions for the two-dimensional Ising model are discussed for rapid-quench conditions. The temperature of the bath in which the Ising spins are immersed is postulated to be driven by the hyperbolic cooling program T(-1) = T(0) (-1) + bt (T(0) being the initial temperature, t the time, and b a constant). It is shown that for sufficiently large b the term in the dynamical equations that would normally drive the system into an ordered state would not have time to become effective so that the system freezes into a disordered state. The correlation length for the disordered state is characterized by the distance a random walker on a two-dimensional lattice would have been able to diffuse in the time required for the quench to reduce the temperature to some preassigned value.

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PubMed1981

Random walks with self-similar clusters.

We construct a random walk on a lattice having a hierarchy of self-similar clusters built into the distribution function of allowed jumps. The random walk is a discrete analog of a Lévy flight and coincides with the Lévy flight in the continum limit. The Fourier transform of the jump distribution function is the continuous nondifferentiable function of Weierstrass. We show that, for cluster formation, it is necessary that the mean-squared displacement per jump be infinite and that the random walk be transient. We interpret our random walk as having an effective dimension higher than the spatial dimension available to the walker. The difference in dimensions is related to the fractal (Hausdorff-Besicovitch) dimension of the self-similar clusters.

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PubMed1980

Field-induced trapping as a probe of dimensionality in molecular crystals.

The combined effect of an electric field and reduced dimensionality on the hopping motion of a charge carrier in an anisotropic molecular crystal is determined. New, exact expressions for S(t), the number of distinct sites occupied by a carrier in time t, are presented for two-dimensional and three-dimensional anisotropic lattices. The general results are used to model the unusual phenomenon of field-induced charge carrier trapping reported to occur in a quasi-one-dimensional molecular crystal. An analytic expression for the carrier lifetime tau, as a function of the anisotropy in the molecular hopping rates and electric field, is developed and is shown to be in excellent agreement with experimental results. A major conclusion of the theory is that E(c), the electric field characterizing the E dependence of tau, can be a more direct probe of the intrinsic anisotropy than is the observed mobility ratio.

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PubMed1979

Dynamics of technological evolution: Random walk model for the research enterprise.

Technological evolution is a consequence of a sequence of replacements. The development of a new technology generally follows from model testing of the basic ideas on a small scale. Traditional technologies such as aerodynamics and naval architecture involved feasibility experiments on systems characterized by only one or two dimensionless constants. Technologies of the "future" such as magnetically confined fusion depend upon many coupled dimensionless constants. Research and development is modeled and analyzed in terms of random walks in appropriate dimensionless constant space.

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PubMed1978

Social dynamics and the quantifying of social forces.

SOCIAL AND INDUSTRIAL EVOLUTIONARY PROCESSES ARE CONSIDERED TO BE A SEQUENCE OF REPLACEMENTS OR SUBSTITUTIONS: new ideas for old, new labor patterns for old, new technologies for old. The logistic equation has often been used to describe population growth processes and replacement processes. It sometimes suffers from contradicting observational data. It is shown here that the deviations are often associated with unusual intermittent events-wars, strikes, economic panics, etc.-and that in many cases a few years after the event it can be abstracted as an instantaneous delta function impulse. After the event, the evolutionary process continues along its normal course. A formula is derived to use the observational data to determine the strength of the impulse modeling an event.

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PubMed1977

On the mapping of chains of first order chemical reactions on random walks.

We present the transient solution to a random walk problem which characterizes chemical kinetic processes in which a particular state may have a finite number of internal states. While there are many chemical and physical examples, we introduce the solution through models of biophysical interest.

Biological Transport↗
PubMed1977

Random walks and generalized master equations with internal degrees of freedom.

We present an extension of the continuous-time random walk formalism to include internal states and to establish the connection to generalized master equations with internal states. The theory allows us to calculate physical observables from which we can extract the characteristic parameters of the internal states of the system under study.

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PubMed1972

A manner of characterizing the development of countries.

An attempt is made to characterize the socio-economic state and the temporal evolution of countries by the use of labor force distribution data on a multidimensional phase plot so that the development of a country is represented by an evolutionary track of a phase point. The evolutionary tracks of several countries are examined, and the phase points of many countries are compared. Snowflake diagrams, which make easy comparison of the socio-economic character of various countries, have also been developed.

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PubMed1972

On coupled rate equations with quadratic nonlinearities.

Rate equations with quadratic nonlinearities appear in many fields, such as chemical kinetics, population dynamics, transport theory, hydrodynamics, etc. Such equations, which may arise from basic principles or which may be phenomenological, are generally solved by linearization and application of perturbation theory. Here, a somewhat different strategy is emphasized. Alternative nonlinear models that can be solved exactly and whose solutions have the qualitative character expected from the original equations are first searched for. Then, the original equations are treated as perturbations of those of the solvable model. Hence, the function of the perturbation theory is to improve numerical accuracy of solutions, rather than to furnish the basic qualitative behavior of the solutions of the equations.

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PubMed1971

A note on the ground state energy of an assembly of interacting electrons.

The ground state energy of an assembly of charged particles of density rho imbedded in a continuum of charge of the other sign in an electrically neutral system is considered. Asymptotic formulae for the ground state energy of such systems are known in the high- and low-density regimes. An interpolation formula covering the entire density range is derived using the method of two-point Padé approximants. A phase transition from an electron lettice to an electron gas seems to occur at r(3) congruent with 14, r(3) being the radius of a sphere which, on the average, contains a single charge, in units of the Bohr radius of the electron in a hydrogen atom.

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