A mean-field equation of motion for the dynamic Ising model

A mean-field type of approximation is used to derive two differential equations, one approximately representing the average behavior of the Ising model with Glauber (spin-flip) stochastic dynamics, and the other doing the same for Kawasaki (spin-exchange) dynamics. The proposed new equations are compared with the Cahn-Allen and Cahn-Hilliard equations representing the same systems and with information about the exact behavior of the microscopic models.     

The dynamics of random interfaces in phase transitions

A review is given of recent developments involving the dynamics of random interfaces formed in the evolution of metastable and unstable systems. Topics which are discussed include interface growth and nonequilibrium dynamical scaling. The possibility of there being dynamical universality classes in first-order phase transitions is also discussed. There are a large number of systems of experimental interest which include binary alloys, binary fluids, and polymer mixtures. Other systems studied by computer simulation include the kinetic Ising, Potts, andZ _N models.Work supported by NSF grant No. DMR-8013700.     

On the mechanical stability of bcc transition elements and alloys

A volume independent and a volume dependent lattice energy function involving short-range interatomic potentials were able to be fitted to the elastic constants, cohesive energy, lattice parameter and for the latter function to the vacancy formation energy and bcc-fcc lattice stability energy, as well, for fcc metals and bcc alkali metals, but not to the cohesive energy and C' elastic constant of bcc transition metals. The assumption that directional, but partial, covalent bonds exist between nearest-neighbors in the bcc transition elements provides an explanation for the latter results and in addition explains the identical dependence of C' and the bcc-fcc lattice stability upon N_d, where N_d is the average number of d electrons, for the bcc transition metals and alloys. Both the mechanical and thermodynamic stability of the bcc structure for transition metals and all transition metal alloys disappears for 5 ? N_d > 2 and <?1.     

A study of self-organizing processes of nonlinear stochastic variables

A general theory is given for the time evolution of nonlinear stochastic variables a(t) = {a_i(t)} whose statistical distribution is changing due to the self-organization of “macroscopic” order. The dynamics of a(t) is conveniently expressed by self-consistent equations for the ensemble average x(t) = 〈a(t)〉, the supersystem, and for the deviations ξ(t) = a(t)?x(t), the subsystem; the systems are connected to each other by feedback loops in their dynamics. The time dependence of the variance and the correlation function ofξ(t) are studied in terms of relaxation toward local equilibrium underx(t) and dynamical coupling withx(t). A special example shows that the stochastic motions of subsystems are pulled together by the motion of the supersystem through feedback loops, and that this pull-together phenomenon occurs when symmetry-breaking instability exists in nonlinear systems.     

Canonical structure of soliton equations. I

The canonical structure of the nonlinear evolution equations in 1 + 1 dimensions solvable in terms of an N × N inverse scattering problem is discussed. The simplest form of the scattering problems, that is those containing the spectral parameter linearly, is considered. It applies to most of the known soliton equations, like the Korteweg-de Vries eq., the sine-Gordon eq. and the Boussinesq eq. Discussion of various possible reductions of the number of dependent variables by imposing constraints consistent with the Hamiltonian flows is given together with the canonical structure of the reduced systems. A direct proof of the involutive character of the infinite number of conserved quantities is given for the general case as well as the reduced case. The relation between the conserved quantities and symmetry transformations (Lie-Bäcklund transformations) becomes very simple in this framework.     

Elliptic PDE formulation and boundary conditions of the spherical harmonics method of arbitrary order for general three-dimensional geometries

The inherent complexity of the radiative transfer equation makes the exact treatment of radiative heat transfer impossible even for idealized situations and simple boundary conditions. Therefore, a wide variety of efficient solution methods have been developed for the RTE. Among these solution methods the spherical harmonics method, the moment method, and the discrete ordinates method provide means to obtain higher-order approximate solutions to the equation of radiative transfer. Although the assembly of the governing equations for the spherical harmonics method requires tedious algebra, their final form promises great accuracy for any given order, since it is a spectral method (rather than finite difference/finite volume in the case of discrete ordinates). In this study, a new methodology outlined in a previous paper on the spherical harmonics method (P_N) is further developed. The new methodology employs successive elimination of spherical harmonic tensors, thus reducing the number of first-order partial differential equations needed to be solved simultaneously by previous P_N approximations (=(N+1)²). The result is a relatively small set (=N(N+1)/2) of second-order, elliptic partial differential equations, which can be solved with standard PDE solution packages. General boundary conditions and supplementary conditions using rotation of spherical harmonics in terms of local coordinates are formulated for the general P_N approximation for arbitrary three-dimensional geometries. Accuracy of the P_N approximation can be further improved by applying the “modified differential approximation” approach first developed for the P₁-approximation. Numerical computations are carried out with the P₃ approximation for several new two-dimensional problems with emitting, absorbing, and scattering media. Results are compared to Monte Carlo solutions and discrete ordinates simulations and a discussion of ray effects and false scattering is provided.     

Remarks on the Gibbs adsorption equation and the van der Waals,Cahn-Hilliard theory of interfaces

The common statements of the Gibbs adsorption equation, and a phenomenological theory of interfaces (van der Waals, Cahn-Hilliard theory), are summarized, and then two remarks are made. The first is that only certain identifications of the local energy and entropy densities are admissible in the phenomenological theory if it is to be consistent with the adsorption equation. The second is a generalization to any of the relative adsorptions Γ_i(j) in a c-component system, of a microscopic interpetation that Leng, Rowlinson, and Thompson recently gave to Γ₂₍₁₎ in a one-component system, where 2 and 1 refer to the densities of energy and matter, respectively.     

Hamiltonian and Brownian systems with long-range interactions: III. The BBGKY hierarchy for spatially inhomogeneous systems

We study the growth of correlations in systems with weak long-range interactions. Starting from the BBGKY hierarchy, we determine the evolution of the two-body correlation function by using an expansion of the solutions of the hierarchy in powers of 1/N in a proper thermodynamic limit N→+∞, where N is the number of particles. These correlations are responsible for the “collisional” evolution of the system beyond the Vlasov regime due to finite N effects. We obtain a general kinetic equation that can be applied to spatially inhomogeneous systems and that takes into account memory effects. These peculiarities are specific to systems with unshielded long-range interactions. For spatially homogeneous systems with short memory time like plasmas, we recover the classical Landau (or Lenard-Balescu) equations. An interest of our approach is to develop a formalism that remains in physical space (instead of Fourier space) and that can deal with spatially inhomogeneous systems. This enlightens the basic physics and provides novel kinetic equations with a clear physical interpretation. However, unless we restrict ourselves to spatially homogeneous systems, closed kinetic equations can be obtained only if we ignore some collective effects between particles. General exact coupled equations taking into account collective effects are also given. We use this kinetic theory to discuss the processes of violent collisionless relaxation and slow collisional relaxation in systems with weak long-range interactions. In particular, we investigate the dependence of the relaxation time with the system size N and try to provide a coherent discussion of all the numerical results obtained for these systems.     

Static properties of two-component kink in a non-linear discrete polarizability model

We study analytically, in the low-velocity regime, the static properties of two-component kink in a discrete polarizability model with a quartic electron–ion interaction in one ionic species. Using a discretized Hamiltonian formalism, we derive the exact equations of motion for the two-component kink centre of the mass X(t) and coupled field variables. Numerical analysis is performed to estimate the amplitude of the dressing and its effects on the static properties of the discrete lattice for the potassium selenate (K₂SeO₄) at T=40 K. We find that dressing has important effects on dynamical quantities such as the pinning frequency and the depth of the Peierls–Nabarro potential. Two-component kink contributions to the specific heat have been also analysed both in the continuum and the discrete limits.     

On exact N-loop soliton solution to nonlinear coupled dispersionless evolution equations

We investigate some nonlinear coupled dispersionless evolution equations (NLCDEE) modelling the dynamics of a current-fed string within an external magnetic field in 2D-space. Using a blend of transformations of independent variables, we derive from the previous equations a Schäfer-Wayne short pulse equation (SWSPE). By means of a transformation back to the original independent variables, we find the N-loop soliton solution to the coupled equations. We give some detail on the scattering behavior of two-loop solitons.     

Orientational dynamics of superfluid 3He: A “two-fluid” model. II. Orbital dynamics

We present a phenomenological theory of the homogeneous orbital dynamics of the class of “separable” anisotropic superfluid phases which includes the ABM state generally identified with ³He-A. The theory is developed by analogy with the spin dynamics described in the first paper of this series; the basic variables are the orientation of the Cooper-pair wavefunction (in the ABM phase, the l-vector) and a quantity K which we visualize as the “pseudo-angular momentum” of the Cooper pairs but which must be distinguished, in general, from the total orbital angular momentum of the system. In the ABM case l is the analog of d in the spin dynamics and K of the “superfluid spin” S_p. Important points of difference from the spin case which are taken into account include the fact that a rotation of l without a simultaneous rotation of the normal-component distribution strongly increases the energy of the system (“normal locking”), and that the equilibrium value of K is zero even for finite total angular momentum. The theory does not claim to handle correctly effects associated with any intrinsic angular momentum arising from particle-hole asymmetry, but it is shown that the magnitude of this quantity can be estimated directly from experimental data and is extremely small; also, the Landau damping does not emerge automatically from the theory, but can be put in in an ad hoc way. With these provisos the theory should be valid for all frequencies $\text{ω ?}\text{Δ(T)}\text{h}\text{?}$ irrespective of the value of ωτ. (Δ = gap parameter, τ = quasi-particle relaxation time.) It disagrees with all existing phenomenological theories of comparable generality, although the disagreement with that of Volovik and Mineev is confined to the “gapless” region very close to T_c.The phenomenological equations of motion, which are similar in general form to those of the spin dynamics with damping, involve an “orbital susceptibility of the Cooper pairs” χ_orb(T). We give a possible microscopic definition of the variable K and use it to calculate χ_orb(T) for a general phase of the “separable” type. The theory is checked by inserting the resulting formula in the phenomenological equations for ωτ ? 1 and comparing with the results of a fully microscopic calculation based on the collisionless kinetic equation; precise agreement is obtained for both the ABM and the (real) polar phase, showing that the complex nature of the ABM phase and the associated “pair angular momentum” is largely irrelevant to its orbital dynamics. We note also that the phenomenological theory gives a good qualitative picture even when ω ～ Δ(T), e.g., for the flapping mode near T_c. Our theory permits a simple and unified calculation of (1) the Cross-Anderson viscous torque in the overdamped regime, (2) the flapping-mode frequency near zero temperature, (3) orbital effects on the NMR, both at low temperatures and near T_c, (4) the orbit wave spectrum at zero temperature (this requires a generalization to inhomogeneous situations which is possible at T = 0 but probably not elsewhere). We also discuss the possibility of experiments of the Einstein-de Haas type. Generally speaking, our results for any one particular application can be also obtained from some alternative theory, but in the case of orbital and spin relaxation very close to T_c (within the “gapless” region) our predictions, while somewhat tentative and qualitative, appear to disagree with those of all existing theories. We discuss briefly how our approach could be extended to apply to more general phases.     

Grüneisen parameter of cerium compounds with unstable 4f shells

We give experimental results of magnetic susceptibility, thermal expansion and elastic constant measurements for CeSn₃ and CePd₃. We interpret these results with a phenomenological scaling function for the free energy and we include a brief discussion of CeAl₃. The coupling between the electronic system and the lattice can be described by an electronic Grüneisen parameter Ω_g associated with the unstable 4f system which turns out to become a system specific constant far below the fluctuation temperatureT ₀. The coupling parameter T₀Ω_g is an order of magnitude larger than the coupling constant for stable valentRE systems extracted from crystal field effects. A generalization to include magnetic field dependent effects is also given.     

Electronic phase separation and the inhomogeneities in Bi2212

Recent works on scanning tunneling microscopy (STM) have shown how the measured local gaps vary in a Bi2212 family of samples with different hole doping levels and also, how they vary with the temperature. Here we use the Cahn-Hilliard (CH) equation of phase segregation in alloys to describe the phase separations of holes in HTS. This method allows us to determine how the local Landau free energy changes in a given sample as a function of temperature, assuming that the line of anomalies or signals related with the upper pseudogap is the phase separation temperature T_ps. The free energy and the hole density form regions of low and high values separated by a potential barrier, which we propose to be the origin of the superconducting attraction.     

Comparing O(N) and SU(N) × SU(N) spin systems in 1 + 1 dimensions to SU(N) gauge theories in 3 + 1 dimensions

We have computed the scale breaking Λ parameters of the euclidean and hamiltonian formulations of the lattice regulated O(N) and SU(N) × SU(N) spin systems in 1 + 1 dimensions in terms of the Λ_PV parameters of the Pauli-Villars regulated continuum models. Using lattice perturbation theory, the renormalized mass gap has been determined in terms of Λ_PV for each model. These results are compared to analogous calculations in SU(N) gauge theories.     

Analysis of long crack lines in paper webs

A differential cluster variation method (DCVM) is proposed for analysis of spinoidal decomposition in alloys. In this method, lattice symmetry operations in the presence of an infinitesimal composition gradient are utilized to deduce the connection equations for the correlation functions and to reduce the number of independent variables in the cluster variation analysis. Application of the method is made to calculate the gradient energy coefficient in the Cahn-Hilliard free energy function and the fastest growing wavelength for spinodal decomposition in Al-Li alloys. It is shown that the gradient coefficient of congruently ordered Al-Li alloys is much larger than that of the disordered system. In such an alloy system, the calculated fastest growing wavelength is approximately 10 nm, which is an order of magnitude larger than the experimentally observed domain size. This may provide a theoretical explanation why spinodal decomposition after a congruent ordering is dominated by the antiphase boundaries.Received: 17 November 2003, Published online: 2 April 2004PACS: 64.75. + g Solubility, segregation, and mixing; phase separation - 81.30.-t Phase diagrams and microstructures developed by solidification and solid-solid phase transformations - 05.70.Ln Nonequilibrium and irreversible thermodynamics     

Robust and accurate filtered spherical harmonics expansions for radiative transfer

We present a novel application of filters to the spherical harmonics (P_N) expansion for radiative transfer problems in the high-energy-density regime. The filter we use is based on non-oscillatory spherical splines and a filter strength chosen to (i) preserve the equilibrium diffusion limit and (ii) vanish as the expansion order tends to infinity. Our implementation is based on modified equations that are derived by applying the filter after every time step in a simple first-order time integration scheme. The method is readily applied to existing codes that solve the P_N equations. Numerical results demonstrate that the solution to the filtered P_N equations are (i) more robust and less oscillatory than standard P_N solutions and (ii) more accurate than discrete ordinates solutions of comparable order. In particular, the filtered P₇ solution demonstrates comparable accuracy to an implicit Monte Carlo solution for a benchmark hohlraum problem in 2D Cartesian geometry.     

Effect of isothermal treatments on the magnetic behavior of nanocrystalline Cu–Ni–Fe alloy prepared by mechanical alloying

The present study concerns magnetic behavior of nanocrystalline Cu–Ni, Cu–Fe and Cu–Ni–Fe alloys prepared by mechanical alloying. It has been found that the magnetic properties e.g. H_c, M_r and M_s of the nanocrystalline alloys were significantly influenced by the changes in microstructural constituents, grain size and evolution of phases. Microstructural changes in the alloys have been effected by carrying out isothermal treatments on the mechanically alloyed products in the temperature range of 450–650 °C. Phase evolution in the samples after the isothermal treatments were identified and characterized by X-ray diffraction (XRD) and differential scanning calorimetric (DSC) techniques and the results were correlated with the magnetic properties of the alloys.