Elliott W. Montroll

Random Walks on Lattices. II

Formulas are obtained for the mean first passage times (as well as their dispersion) in random walks from the origin to an arbitrary lattice point on a periodic space lattice with periodic boundary conditions. Generally this time is proportional to the number of lattice points.The number of distinct points visited after n steps on a k‐dimensional lattice (with k ≥ 3) when n is large is a1n + a2n½ + a3 + a4n−½ + …. The constants a1 − a4 have been obtained for walks on a simple cubic lattice when k = 3 and a1 and a2 are given for simple and face‐centered cubic lattices. Formulas have also been obtained for the number of points visited r times in n steps as well as the average number of times a given point has been visited.The probability F(c) that a walker on a one‐dimensional lattice returns to his starting point before being trapped on a lattice of trap concentration c is F(c) = 1 + [c/(1 − c)] log c.Most of the results in this paper have been derived by the method of Greens functions.

Random Walks on Lattices. III. Calculation of First‐Passage Times with Application to Exciton Trapping on Photosynthetic Units

The following statistical problem arises in the theory of exciton trapping in photosynthetic units: Given an infinite periodic lattice of unit cells, each containing N points of which (N − 1) are chlorophyll molecules and one is a trap; if an exciton is produced with equal probability at any nontrapping point, how many steps on the average are required before the exciton reaches a trapping center for the first time? It is shown that, when steps can be taken to near‐neighbor lattice points only, as N → ∞, our required number of steps is 〈n〉∼{N2/6,       linear chain,π−1NlogN,  square lattice,1.5164N,     single cubic lattice. The correction terms for medium and relatively small N are obtained for a number of lattices.

Studies in Nonequilibrium Rate Processes. I. The Relaxation of a System of Harmonic Oscillators

As a part of an investigation of nonequilibrium phenomena in chemical kinetics a theoretical study has been made of the collisional and radiative relaxation of a system of harmonic oscillators contained in a constant temperature heat bath and prepared initially in a vibrational nonequilibrium distribution. An exact solution has been obtained for the general relaxation equation applicable to this system and expressions have been derived for the relaxation of initial Boltzmann distributions, Poisson distributions, and δ‐function distributions as well as for the relaxation of the moments of the distributions. Using the latter result, explicit expressions are given for the relaxation of the internal energy of the system of oscillators and for the time dependence of the dispersion of the distributions.

Maximum entropy formalism, fractals, scaling phenomena, and 1/f noise : a tale of tails

In this report on examples of distribution functions with long tails we (a) show that the derivation of distributions with inverse power tails from a maximum entropy formalism would be a consequence only of an unconventional auxilliary condition that involves the specification of the average value of a complicated logarithmic function, (b) review several models that yield log-normal distributions, (c) show that log normal distributions may mimic 1/f noise over a certain range, and (d) present an amplification model to show how log-normal personal income distributions are transformed into inverse power (Pareto) distributions in the high income range.

Generalized master equations for continuous-time random walks

An equivalence is established between generalized master equations and continuous-time random walks by means of an explicit relationship betweenψ(t), which is the pausing time distribution in the theory of continuous-time random walks, andφ(t), which represents the memory in the kernel of a generalized master equation. The result of Bedeaux, Lakatos-Lindenburg, and Shuler concerning the equivalence of the Markovian master equation and a continuous-time random walk with an exponential distribution forψ(t) is recovered immediately. Some explicit examples ofφ(t) andψ(t) are also presented, including one which leads to the equation of telegraphy.

Random walks on lattices. IV. Continuous-time walks and influence of absorbing boundaries

The general study of random walks on a lattice is developed further with emphasis on continuous-time walks with an asymmetric bias. Continuous time walks are characterized by random pauses between jumps, with a common pausing time distributionψ(t). An analytic solution in the form of an inverse Laplace transform for P(l, t), the probability of a walker being atl at timet if it started atlo att=0, is obtained in the presence of completely absorbing boundaries. Numerical results for P(l, t) are presented for characteristically different ψ(t), including one which leads to a non-Gaussian behavior for P(l, t) even for larget. Asymptotic results are obtained for the number of surviving walkers and the mean 〈l〉 showing the effect of the absorption at the boundary.

Theory of Depolymerization of Long Chain Molecules

A theory of depolymerization of long chain molecules is developed on a statistical basis. It is assumed that all bonds connecting monomeric elements in the system have the same probability of being broken regardless of their position in a given polymer and regardless of the size of the polymer in which they are found. Expressions are derived for the distribution of molecular sizes in the depolymerized system as a function of the initial chain length and the average number of bonds split per molecule. Also, relationships are established between the average molecular weight of the degraded product and the average number of bonds split per molecule. Experiments on the acetolytic degradation of cellulose acetate are briefly discussed.

Quantum theory of anharmonic oscillators. I. Energy levels of oscillators with positive quartic anharmonicity

This is an investigation of the energy levels of an anharmonic oscillator characterized by the potential (1/2) x2+λx4. Two regions of λ and n are distinguishable (n being the quantum number of the energy level) one in which the harmonic oscillator levels En=n+1/2 are only slightly distorted and the other in which the purely quartic oscillator form En?cλ1/3(n+1/2)4/3 (c being a constant) is only slightly distorted. Rapidly converging algorithms have been developed, using the Bargmann representation, from which energy levels in any (λ,n) (with λ≳0) regime can easily be computed. Simple formulas are also derived which give excellent approximations to the energy levels in various (λ,n) regimes.

Statistical Mechanics of Nearest Neighbor Systems

In many solids the intermolecular forces are sufficiently short ranged so that practically the entire potential energy of the system results from interactions between nearest neighbors. The thermodynamic properties of a solid with respect to a given coordinate, α, (for example, in a ferromagnetic system α might represent the excess unpaired electron spin at a given lattice point; or in a substitutional binary alloy α might denote which of the two possible kinds of atoms are at a given lattice point) can be found from the factor of the partition function which averages over all possible configurations of α at all lattice points.Here the evaluation of such a factor of the partition function is reduced to the calculation of the largest characteristic value of a linear homogeneous operator equation involving the potential energy between two adjacent layers of lattice points in the solid. By assuming that all possible configurations of a layer are equally probable, a lower bound for the partition function for ...

Poincaré Cycles, Ergodicity, and Irreversibility in Assemblies of Coupled Harmonic Oscillators

The transport coefficients (diffusion constant, electrical conductivity, etc.) associated with irreversible processes in an assembly of particles can be expressed as integrals over certain time relaxed correlation functions between small numbers of variables of the assembly. The scattering of slow neutrons is also a measure of time relaxed correlation functions.Irreversibility is a consequence of the vanishing of the correlation coefficients as the relaxation time becomes infinite. On the other hand these coefficients have Poincare cycles so that any value which they take on is repeated an infinite number of times. It is shown that, in the case of fluctuations of 0(N−½) from zero (N being the number of degrees of freedom), the period of Poincare cycles is of the order of the mean period of normal mode vibrations while for fluctuations of a magnitude independent of N the period is of the order of CN where C is a constant which is greater than 1.The time relaxed correlation coefficients of a pair of particl...