Self-diffusion in simple models: Systems with long-range jumps

We review some exact results for the motion of a tagged particle in simple models. Then, we study the density dependence of the self-diffusion coefficientD _N() in lattice systems with simple symmetric exclusion in which the particles can jump, with equal rates, to a set ofN neighboring sites. We obtain positive upper and lower bounds onF _N()=N{(1–)–[D_N()/D_N(0)]}/[(1–)]x for [0, 1]. Computer simulations for the square, triangular, and one-dimensional lattices suggest thatF _N becomes effectively independent ofN forN20.     

Entropy,information theory,and the approach to equilibrium of coupled harmonic oscillator systems

Finite segments of infinite chains of classical coupled harmonic oscillators are treated as models of thermodynamic systems in contact with a heat bath, i.e., canonical ensembles. The Liouville function for the infinite chain is reduced by integrating over the outside variables to a function _N of the variables of theN-particle segment that is the thermodynamic system. The reduced Liouville function _N which is calculated from the dynamics of the infinite chain and the statistical knowledge of the coordinates and momenta att = 0, is a time-dependent probability density in the 2N-dimensional phase space of the system. A Gibbs entropy defined in terms of _N measures the evolution of knowledge of the system (more accurately, the growth of missing pertinent information) in the sense of information theory. As ¦t ¦ , energy is equipartitioned, the entropy evolves to the value expected from equilibrium statistical mechanics, and _N evolves to an equilibrium distribution function. The simple chain exhibits diffusion in coordinate space, i.e., Brownian motion, and the diffusivity is shown to depend only on the initial distribution of momenta (not of coordinates) in the heat bath. The harmonically bound chain, in the limit of weak coupling, serves as an excellent model for the approach to equilibrium of a canonical ensemble of weakly interacting particles.     

Analytic relativistic model for a superdense star

A new analytic relativistic model has been obtained for superdense stars by solving the Einstein field equations for the spherically symmetric and static case. The model stands all the tests of physical reality. The density,, remains positive under all conditions and decreases smoothly from the center to the surface of the structure. The pressure,P, the ratioP/ anddP/d decrease with decreasing density. For all the finite values of pressure, the configurations are stable under radial perturbation. FordP/d 1, the maximum mass of neutron star model is 4.17, and the surface and central red shifts are 0.63 and 1.60, respectively. For an infinite central pressure, the surface red shift is 1.61. The structures are bound and the binding coefficients increase with the increasing mass.     

Cosmic numbers and quantum theory

The cosmic numbers are considered, with emphasis on the relationN ². (HereN is the number of nucleons in the universe, and, its radius in atomic units.) This relation is interpreted in terms of a quantum mechanical model.     

What determinates the chemical changes of the internal conversion coefficients for inner atomic shells?

Calculations of internal conversion coefficients (ICC) of the E1–E4 and M1–M4 transitions for nuclei in ions show that the relative changes _i / _i of the ICC _i for deep inner subshells can differ significantly from the relative changes _i/_i of the electron densities _i at the nucleus. For the K conversion _i/ _i are many times greater than _i/_i. Especially large deviations of _i/ _i are characteristic of transitions of high multipolarity; however, for the M1 transitions they can also be significant. Illustrations of various dependencies of _i/ _iare presented for the conversion in the regionZ-50.     

Analysis of the order parameter for uniform nearest particle system

The uniform nearest particle system (UNPS) is studied, which is a continuoustime Markov process with state space . The rigorous upper bound ^(mf) = ( – 1)/ for the order parameter ₂, is given by the correlation identity and the FKG inequality. Then an improvement of this bound ^(mf) is shown in a similar fashion; C( – 1)/|log( – 1) for >1. Recently, Mountford proved that the critical value _c=1. Combining his result and our improved bound implies that if the critical exponent exists, it is strictly greater than the mean-field value 1 in the weak sense.     

Generalized Stefan-Boltzmann law

Thermodynamics arguments have been employed to derive how the energy density depends on the temperatureT for a fluid whose pressurep obeys the equation of statep = ( –1), where is a constant. Three different methods, among them the one considered by Boltzmann (Carnot cycle), lead to the expression = T/( –1), where is a constant. This result also appears naturally in the framework of general relativity for spacetimes with constant spatial curvature. Some particular cases are vacuum (p = –), cosmic strings (p= –1/3), radiation (p = 1/3), and stiff matter (p = ). It is also shown that such results can be adapted for blackbody radiation inN spatial dimensions.     

The GHS inequality for a large external field

We consider general even ferromagnetic systems with pair interactions in a nonnegative external magnetic fieldh. Classes of single-site measures are found such that the GHS inequality is valid for allh h, whereh 0 is a number depending on but independent of the size of the system. These measures include both absolutely continuous and discrete measures. For =a ₀+{(1–a)/2} · ( ₁ + _–1), somea [0, 1),h is determined exactly.Alfred P. Sloan Research Fellow. Research supported in part by National Science Foundation grant No. MCS 80-02149.Alfred P. Sloan Research Fellow. Research supported in part by National Science Foundation grant No. MCS 77-20683 and by the U.S.-Israel Binational Science Foundation.     

Radius,perimeter, and density profile for percolation clusters and lattice animals

Monte Carlo simulation and series expansion shows the radius of gyration of large clusters withs sites each to vary ass with0.56 in two and0.47 in three dimensions at the percolation threshold, and with(d=2)0.65 and(d=3)0.53 for random lattice animals (zero concentration). Clusters up tos=100 were used. The perimeter of random animals approaches 2.8s for larges on the simple cubic lattice. Monte Carlo simulation of the Eden process (growing animals) up tos=5,000 indicates a systematic variation of about ±0.05 for the effective exponent=(s) and thus suggests that the true asymptotic exponents may be compatible with the predictions of hyper-scaling.     

Metastable states in the van der Waals-Maxwell theory

A slight modification of the recent Penrose and Lebowitz treatment of thermodynamic metastable states is presented. For the case of periodic boundary conditions, this modification allows the condition of metastability to be extended to all the metastable states in the van der Waals-Maxwell theory of the liquid-vapor phase transition, that is, for all states satisfyingf ₀()+1/2 ²>f(, 0+) andf₀()+x>0 wheref(, 0+) is the (stable) Helmholtz free energy density of the generalized van der Waals-Maxwell theory andf ₀() is the Helmholtz free energy density of a reference system with no long-range interaction, is a mean field-type term arising from a long-range Kac interaction, is the overall mean particle density, andx is any positive number. For the case of rigid-wall boundary conditions, a more restrictive condition is placed onx.     

Electromagnetic effects in unified field theory

In unified field theory we derive expressions for the electric current densities j and. We show that j and depend on the intensitiesE andH;E andH possess a common limit 1/; and Coulomb's law is not compatible with the unified theory.     

Phase segregation dynamics in particle systems with long range interactions. I. Macroscopic limits

We present and discuss the derivation of a nonlinear nonlocal integrodifferential equation for the macroscopic time evolution of the conserved order parameter (r, t) of a binary alloy undergoing phase segregation. Our model is ad-dimensional lattice gas evolving via Kawasaki exchange with respect to the Gibbs measure for a Hamiltonian which includes both short-range (local) and long-range (nonlocal) interactions. The nonlocal part is given by a pair potential ^dJ(|x–y|), >0 x and y in ^d, in the limit 0. The macroscopic evolution is observed on the spatial scale ^–1 and time scale ^–2, i.e., the density (r, t) is the empirical average of the occupation numbers over a small macroscopic volume element centered atr=x. A rigorous derivation is presented in the case in which there is no local interaction. In a subsequent paper (Part II) we discuss the phase segregation phenomena in the model. In particular we argue that the phase boundary evolutions, arising as sharp interface limits of the family of equations derived in this paper, are the same as the ones obtained from the corresponding limits for the Cahn-Hilliard equation.     

A unified theory of matter. I. The fundamental idea

The Lorentz transformation is derived without assuming that the velocity of light is a constant. This suggests that the constantc which appears in the transformation has a deeper significance than heretofore commonly assumed. It is hypothesized that there exists, in all of physical reality, velocities of only one magnitude. The magnitude isc, the speed of light in vacuum. This hypothesis forces us to view a fundamental particle as an extended object and matter in general as a field (t, r, ), which we give the generic name stuff. An important feature of the field is that at each spacetime point(t, r) stuff travels in all directions with speedc. In order to elucidate the nature of (t, r, ), the equations determining for a one-dimensional world are derived and solved. Fundamental particles are shown to exist and their structure is obtained.A private communication; not an official publication of the National Bureau of Standards.     

Non-classical conductivity in a bulk semiconductor up to 35 K

We observed non-classical contributions to the conductivity in three-dimensional samples at temperatures up to 35 K. The experiments were carried out inn-type Hg_0.8Cd_0.2Te samples having dimensions as large as 8 ×28×25 m³. The two resistivity contributions, i.e. the normal metallic behaviour ₀ and a part provided by electron interference, are separated using a method disturbing the additional contribution by heating the electrons in electrical fields. The observed temperature and electric field dependence are compared with existing theories.     

Anisotropy of the low temperature resistivity of hexagonal single crystalline spin glass systems

The impurity contribution to the resistivity in zero field (T) of dilute hexagonal single crystals of ZnMn, CdMn and MgMn has been studied in the mK range on samples cut parallel () and perpendicular () to thec-axis, using a SQUID technique for the measurements. Typical spin glass behavior is found in (T) as well as (T) for all alloys, with Kondo like logarithmic increases at higher temperatures and maxima atT _m at lower temperatures, indicating the influence of impurity interactions. The differences in the corresponding isotropic resistivity _poly(T) between the three systems can qualitatively be understood within the framework of a theoretical model by Larsen, describing (T) as a function of universal quantitiesT/T _K and _RKKY/T _K , where _RKKY is the RKKY-interaction strength andT _K the Kondo temperature. With respect to the two lattice directions studied, the behavior of (T and (T is anisotropic in the Kondo regime as well as in the range where ordering becomes important. While the anisotropy in the Kondo slope can be understood by an anisotropic unitarity limit, the understanding of the anisotropy in region where impurity interactions are important remains problematic.Dedicated to Prof. Dr. S. Methfessel on the occasion of his 60th birthday     

Radius of clusters at the percolation threshold: A position space renormalization group study

Using a direct position-space renormalization-group approach we study percolation clusters in the limits , wheres is the number of occupied elements in a cluster. We do this by assigning a fugacityK per cluster element; asK approaches a critical valueK _c , the conjugate variables . All exponents along the path (K–K _c ) 0 are then related to a corresponding exponent along the paths . We calculate the exponent , which describes how the radius of ans-site cluster grows withs at the percolation threshold, in dimensionsd=2, 3. Ind=2 our numerical estimate of =0.52±0.02, obtained from extrapolation and from cell-to-cell transformation procedures, is in agreement with the best known estimates. We combine this result with previous PSRG calculations for the connectedness-length exponent , to make an indirect test of cluster-radius scaling by calculating the scaling function exponent using the relation =/. Our result for is in agreement with direct Monte-Carlo calculations of , and thus supports the cluster-radius scaling assumption. We also calculate ind=3 for both site and bond percolation, using a cell of linear sizeb=2 on the simple-cubic lattice. Although the result of such small-cell calculations are at best only approximate, they nevertheless are consistent with the most recent numerical estimates.Supported in part by grants from ARO and ONR     

Letter: Particle-like Behaviour of Some Bianchi Cosmologies: A Fundamental Unity in Nature?

We show that the equations of evolution of someBianchi cosmologies with equation of state p = (0 1) are formallysimilar to the equations of motion of a classical particle under the actionof suitably chosen potential functions. This appears topoint to a fundamental unity in Nature.     

The role of the distribution of nucleons in finite nuclei

The total energy of many-nucleon system is expressed as a functional E[ _p(r), _n(r)] of the proton and neutron densities _p(r) and _n(r), respectively. The distribution(r) of nucleons in the nucleus, which is essential to determine the energy functional, is chosen. The energy density formalism is applied to finite nuclei, and then the binding energies per nucleon together with the mean square radii, for some medium and heavy nuclei, are obtained. Finally the achieved results are compared with the corresponding experimental values.     

On Strong Superadditivity of the Entanglement of Formation

We employ a basic formalism from convex analysis to show a simple relation between the entanglement of formation E_F and the conjugate function E^* of the entanglement function E()=S(Tr_A). We then consider the conjectured strong superadditivity of the entanglement of formation E_F()E_F(_I)+E_F(_II), where _I and _II are the reductions of to the different Hilbert space copies, and prove that it is equivalent with subadditivity of E^*. Furthermore, we show that strong superadditivity would follow from multiplicativity of the maximal channel output purity for quantum filtering operations, when purity is measured by Schatten p-norms for p tending to 1.     

Spatial structure in diffusion-limited two-particle reactions

We analyze the limiting behavior of the densities _A(t) and _B(t), and the random spatial structure(r) = ( _A(t)., _B(t)), for the diffusion-controlled chemical reaction A+Binert. For equal initial densities _B(0) = _b(0) there is a change in behavior fromd 4, where _A(t) = _B(t) C/t^d/4, tod 4, where _A(t) = _b(t) C/t ast ; the termC depends on the initial densities and changes withd. There is a corresponding change in the spatial structure. Ind < 4, the particle types separate with only one type present locally, and , after suitable rescaling, tends to a random Gaussian process. Ind >4, both particle types are, after large times, present locally in concentrations not depending on type or location. Ind=4, both particle types are present locally, but with random concentrations, and the process tends to a limit.