Robert J. Rubin

Bio: Robert J. Rubin is an academic researcher. The author has contributed to research in topics: Thermal conduction & Thermal conductivity. The author has an hindex of 6, co-authored 8 publications receiving 260 citations.

Abnormal Lattice Thermal Conductivity of a One‐Dimensional, Harmonic, Isotopically Disordered Crystal

Abstract: Energy transport is investigated in a model system for which exact analytic results can be obtained. The system is an infinite, one‐dimensional harmonic crystal which is perfect everywhere except in a finite segment which contains N isotopic defects. Initially, the momenta and displacements of all atoms to the left of the defect region are canonically distributed at a temperature T, and the right half of the crystal is at a lower temperature. This initial nonequilibrium state evolves according to the equations of motion, and ultimately a steady state is established in the vicinity of the region containing the defects. The thermal conductivity is calculated from exact expressions for the steady state energy flux and thermal gradient. For a crystal in which the N isotopic defects are distributed at random but in which the overall defect concentration is fixed, we demonstrate that the thermal conductivity approaches infinity as least as fast as N1/2. A Monte Carlo evaluation of the thermal conductivity for a given defect‐to‐host mass ratio and concentration is carried out for a series of random configurations of N defects for N in the range, 25 ≤ N ≤ 600. The thermal conductivity is proportional to N1/2 within the statistical uncertainty except for slight deviations at the smallest values of N.

Statistical Dynamics of Simple Cubic Lattices. Model for the Study of Brownian Motion. II

Abstract: The system considered is an n‐dimensional cubic crystal with nearest‐neighbor central and noncentral harmonic forces in which the mass M of one of the lattice particles is relatively large. It is assumed that the velocities and positions of the light particles in the system (mass m) are normally distributed, at time t=0, as in thermal equilibrium. The conditional velocity distribution for the heavy particle at time t is then a normal distribution with a time‐dependent mean value. This mean value is the velocity autocorrelation function. The dispersion of the distribution is shown to be a simple function of the autocorrelation. In the limit M/m≫1 in the one‐ and two‐dimensional lattices, the autocorrelation function is, respectively, a damped exponential and a damped oscillating exponential. These different types of statistical behavior are related to the different dynamic properties of the medium with which the heavy particle interacts.

Transmission Properties of an Isotopically Disordered One‐Dimensional Harmonic Crystal

Abstract: The amplitude SN(ω) of a wave of frequency ω which is transmitted by a disordered array of N isotopic defects in a one‐dimensional crystal has been investigated in the limit in which N → ∞ while the over‐all concentration of the defects in the array remains fixed The transmitted amplitude SN(ω) is proportional to the reciprocal of the magnitude of an Nth‐order determinant whose elements depend explicitly upon the spacings between defects, the incident frequency ω, and the relative mass difference Q = (M − m)/m between the defect particles and the particles of the host crystal SN(ω) is represented as exp [−NαN(ω, Q, C)], where C is the over‐all fractional concentration of defects; two types of estimates of αN(ω, Q, C) are obtained First, assuming that the spacings between nearest‐neighbor pairs of defects are independent random variables, upper and lower bounds are obtained on αN(ω, Q, C) which are independent of N Provided that C is sufficiently small, the lower bound is positive Second, Monte Carlo

Random Walk with an Excluded Origin

Abstract: The mean square end‐to‐end distance RN2 is calculated for the subset of all random walk configurations on a D‐dimensional simple cubic lattice which do not return to the starting point. Explicit results are obtained in the limit N ≫ 1 for the one‐, two‐, and three‐dimensional lattices. The values of the first two terms in the asymptotic series for RN2 are, respectively, N + N, N + N/log N, and N + 0.435/N−½. An unexpected relation is obtained between RN2 and SN, the average number of different lattice sites visited in an N‐step random walk on a perfect lattice. It is RN2=SN(SN+1−SN)−1.

Transmission Properties of an Isotopically Disordered One-Dimensional Harmonic Crystal. II. Solution of a Functional Equation

Abstract: The amplitude of a wave of frequency ω which is transmitted by a disordered array of N isotopic defects in a 1‐dimensional harmonic crystal is investigated in the limit N → ∞. In particular, the ratio TN(ω) of the amplitude of the Nth defect to the amplitude of the first defect is represented as exp [−NαN(ω,Q,{an})], where {an}, n = 2, ⋯, N, is the sequence of nearest‐neighbor spacings and Q = (M − m)/m. It is known from earlier work that αN(ω,Q,{an}) is the logarithm of the Nth root of the magnitude of a continuant determinant of order N. The value of the continuant is expressed formally as a product of N factors ĝn which are recursively related. In the present case, the ĝn happen to lie on a circle K0 in the complex ĝ plane. Assuming that the spacings between defects are independent identically distributed random variables with the mean value c−1 and going to the limit N → ∞, a functional equation for the limiting distribution function of the ĝn on K0 is derived. The limiting value α(ω,Q,c)=limαN(ω,...

Kinetic equations from Hamiltonian dynamics: Markovian limits

Abstract: Dynamical processes in macroscopic systems are often approximately described by kinetic and hydrodynamic equations. One of the central problems in nonequilibrium statistical mechanics is to understand the approximate validity of these equations starting from a microscopic model. We discuss a variety of classical as well as quantum-mechanical models for which kinetic equations can be derived rigorously. The probabilistic nature of the problem is emphasized: The approximation of the microscopic dynamics by either a kinetic or a hydrodynamic equation can be understood as the approximation of a non-Markovian stochastic process by a Markovian process.

Heat transport in low-dimensional systems

Abstract: Recent results on theoretical studies of heat conduction in low-dimensional systems are presented. These studies are on simple, yet non-trivial, models. Most of these are classical systems, but some quantum-mechanical work is also reported. Much of the work has been on lattice models corresponding to phononic systems, and some on hard-particle and hard-disc systems. A recently developed approach, using generalized Langevin equations and phonon Green's functions, is explained and several applications to harmonic systems are given. For interacting systems, various analytic approaches based on the Green–Kubo formula are described, and their predictions are compared with the latest results from simulation. These results indicate that for momentum-conserving systems, transport is anomalous in one and two dimensions, and the thermal conductivity κ diverges with system size L as κ ∼ L α. For one-dimensional interacting systems there is strong numerical evidence for a universal exponent α = 1/3, but there is no e...