Collision Integrals for Attractive Potentials

Motivated by previous discussions of particle interactions under the Manev potential U(r)=–/r–/r ², we construct the collision integrals for attractive potentials U(r) satisfying the condition U(r) r ²– as r0 with 0. For =0, we obtain a Boltzmann-type integral with a collision law allowing spiral interactions and nonunique correspondence between impact parameter and scattering angle. For >0, an additional Smoluchowski-type coagulation integral arises. All these integrals are derived and possible applications are discussed.     

Source-free electromagnetic fields in the Einstein-Gödel universe

A metric containing a parameter (² = 1) has been derived which represents axially symmetric source-free electromagnetic fields in a static Einstein universe when is put equal to 1. The same metric represents the source-free electromagnetic fields in a Gödel rotating universe when is put equal to-1. Many known solutions are obtained as particular cases.     

Power law scaling of the top Lyapunov exponent of a Product of Random Matrices

A sequence of i.i.d. matrix-valued random variables with probabilityp and with probability 1–p is considered. Leta() = a ₀ + O(), c() = c ₀ + O() lim ₀ b() = Oa ₀,c ₀, >0, andb()>0 for all >0. It is shown show that the top Lyapunov exponent of the matrix productX _n X _n-1...X ₁, = lim_n (1/n) n X _n X _n-1...X ₁ satisfies a power law with an exponent 1/2. That is, lim ₀(ln /ln ) = 1/2.     

Dissipation Statistics of a Passive Scalar in a Multidimensional Smooth Flow

We compute analytically the probability distribution function () of the dissipation field =()² of a passive scalar advected by a d-dimensional random flow, in the limit of large Peclet and Prandtl numbers (Batchelor–Kraichnan regime). The tail of the distribution is a stretched exponential: for , ln ()–(d ² )^1/3.     

s-Polarized nonlinear surface polaritons

An exact solution of Maxwell's equations is found, corresponding to ans-polarized nonlinear surface polarition, at the planar interface between two dielectric media, one of which is optically unaxial and is characterized by a diagonal dielectric tensor whose elements depend on the amplitude of the electric field according to ₁₁=₂₂=₀()+a() (|E ₁|²+|E ₂|²), ₃₃=(). Such modes have no counterpart in the corresponding linear system.     

Colored noise in activated rate processes

We consider bistable systems driven by stationary wideband Gaussian colored noise. We construct uniform asymptotic expansions of the stationary probability density function and of the activation rate, for small intensity and short correlation time of the noise. We find that for different values of the total power output / of the noise, different terms in the asymptotic expansions become dominant. For we recover previously derived results, while for =O() and new results are obtained.     

Dielectric constant characterization of large-area substrates in millimeter wave band

A possibility of the dielectric constant measurement for substrates with permittivity=+i without an essential restriction on their area has been shown experimentally. The method uses frequency measurement of quasioptical dielectric resonator (QDR) with two slots oriented along the QDR radius with a dielectric substrate in one of them. Taking QDR of teflon in 8mm waveband as an example it is found that measurable values of can ran up 15 _q , where _q is the QDR material permittivity. Absolute error of the measurements is determined by an accuracy with which the permittivity of calibrated (standard) samples is known. The relative measuring error is determined by the accuracy of the QDR frequency measurement and can be quite a small. As an example the method is demonstrated forLaAlO ₃ single crystals.     

Reflection and Absorption Spectra of Fluorite in the UV Spectral Region

We have calculated the R(E) and ₁(E) spectra from the theoretical ₂(E) spectra of five models in the region 8–27 eV and the ₂(E) and ₁(E) spectra from the experimental R(E) spectrum in the region 6–35 eV. The results are compared with the known theoretical ₂(E) spectra of five models. The basic features of all of the R(E), ₂(E), and ₁(E) spectra have been revealed. It is established that the experimental R(E) spectrum and the ₂(E) and ₁(E) spectra calculated with the use of experimental data are in good agreement with the results of theoretical calculations for the models of ₂(E). On the basis of the known theoretical calculations of the fluorite zones, a scheme of the nature of the principal maxima of R(E) and ₂(E) is suggested.     

Finite size scaling at first order phase transitions?

It is demonstrated that finite size scaling at first order phase transitions is something basically very simple: As the number of particlesN in the system goes to infinity,s _N (), the entropy per particle, rapidly approaches its limiting behaviours (). Onces () has been determined, the thermal behaviour of the infinite system is completely known and in case of a first order phase transition the specific heat exhibitis a -function singularity. If, however, the specific heatc _N (T) per particle is calculated from the canonical partition functionZ _N ()=d exp {N[s _N ()-]}, then even ifs _N () is replaced by its limiting forms (),c _N (T) only exhibits a peak with a finite maximum value proportional toN which is due to the explicit factorN in front of the angular bracket in the exponent. This is theN-dependence which has recently been called finite size scaling at first order phase transitions. The entropys _N () can very efficiently be determined in the dynamical ensemble.     

Unmixing of binary alloys by a vacancy mechanism of diffusion: a computer simulation

The initial stages of phase separation are studied for a model binary alloy (AB) with pairwise interactions _AA , _AB , _BB between nearest neighbors, assuming that there is no direct interchange of neighboring atoms possible, but only an indirect one mediated by vacancies (V) occurring in the system at a concentrationc _v and which are strictly conserved, as are the concentrationsc _A andc _B of the two species.A-atoms may jump to vacant sites with jump rate _A , B-atoms with jump rate _B (in the absence of interactions). Particular attention is paid to the question to what extent nonuniform distribution of vacancies affects the unmixing kinetics. Our study focuses on the special case _A = _B on a square lattice, considering three different choices of interactions with the same = _AB – ( _AA + _BB )/2: (i) _AB =, _AA = _BB = 0; (ii) _AA = 0, _AA = _BB ; = ; (iii) _AB = _BB = 0, _AA = –2. We obtain both the time evolution of the structure factorS(k,t) following a quench from infinite temperature to the considered temperature, and the timedependence of the mean cluster size and the various neighborhood probabilities of a vacancy. While in case (i) forc _V 0.16 the distribution of vacancies in the system stays nearly random, in case (ii) the vacancies cluster in theA-B interfacial region, and in case (iii) they get nearly completely expelled from theA-rich regions. While phase separation proceeds in case (i) only slightly faster than in case (ii), a significant slowing down of the relaxation is observed for case (iii), which shows up in a strong reduction of the effective exponents describing the growth.     

Particle-like configurations of the electromagnetic field: An extension of de Broglie's ideas

Localised configurations of the free electromagnetic field are constructed, possessing properties of massive, spinning, relativistic particles. In an inertial frame, each configuration travels in a straight line at constant speed, less than the speed of lightc, while slowly spreading. It eventually decays into pulses of radiation travelling at speedc. Each configuration has a definite rest mass and internal angular momentum, or spin. Each can be of electric or magnetic type, according as the radial component of the magnetic or electric field vanishes in the rest frame, and each has an antiparticle. Any such configuration, of electric or magnetic type, is characterized in part by a set of labels (, ₀, ,l, m), where ₀ is the mean of the angular frequencies of the plane waves making up the configuration, is the variance of those frequencies, is a positive constant with dimensions of action, andl, m are angular momentum quantum numbers withl a positive integer andm an integer such that ml. The rest energy of the particle is ₀, its spin is m, and its lifetime is of the order of 1/. Its antiparticle has ₀ replaced by –₀.     

Betatron-synchrotron resonance in free electron laser

In terms of Bogoliubov's nonlinear theory, the first five betatron-synchrotron resonances of _s=(p q)2K, p/q=1/2, 1, 3/2, 1/3, 2/3 which may occur during laser satulation establishing process have been generally examined. Contrast to previous studies we found that the most dangerous resonance does not take place in the vicinity of _s=2k, but in that of _s=k. The width of region of _s=k is proportional to with (k _w ² /k) remittance, the passing time over the region due to the laser field being amplified is only one pass over the undulator at most and depends on the gain, and the increment of synchrotron amplitude in passing time is proportional to .In summary we conclude that the betatron-synchrotron resonances do not have any significant effect for detrapping off-axis electrons in a practical case.     

Electron Transitions of Cadmium Oxide

The bands, densities of states, and spectrum of the permittivity ₂ of a CdO crystal have been calculated by the FP–LMTO method. The ₂, ₁, –Im ^–1, and Re ^–1 spectra for the CdO crystal in the range 1–30 eV have been obtained on the basis of the experimental spectrum of reflection. The spectra were decomposed into elementary components and their main parameters were calculated. The obtained spectrum of the transverse components was compared with the theoretical spectrum of ₂.     

The correlation function in Wilson's theory of phase transitions: Above and below the transition

The correlation function and the correlation length are discussed in the theoretical framework of the Wilson-Feynman diagram expansion for small =4–d. It is shown explicitly that to order ² the scaling relation = (2–) is satisfied and that the correlation function is a homogeneous function ofk and . The explicit form of the scaled correlation function is exhibited.     

The Migdal-Kadanoff renormalization group scheme for the Ising model with a free surface

The Migdal-Kadanoff scheme is applied to the Ising model with a free surface. The resulting renormalization group transformation and the duality transformation commute in any dimension. Two simple recursion relations are obtained which reproduce the global phase diagram for the semi-infinite Ising model. The surface critical exponents space methods. In dimensiond=2+, we find the exponentsy t ₁ ^(SB)= andy h ₁ ^(SB)=1+ for the multicritical surface-bulk transition. We also derive and discuss approximate differential recursion relations for the bulk and the surface free energies.     

Hyperbolic initial-boundary-value problem for magnetic flux compression by plane liners

Based on the (relativistic) Maxwell equations with displacement current E/t, the initial-boundary-value problem for the compression of an initially homogeneous magnetic fieldB={0,B(x,t),0} between a fixed liner atx=0 and a detonation-driven liner atx=s(t) is solved analytically. By homogenizing the boundary conditions at the moving boundary, the transient electromagnetic fields are shown to be a superposition of quasistatic elliptic (E/t=0) and hyperbolic (E/t0) wave solutions. The wave equation is solved by a Fourier expansion in time-dependent eigenfunctionsf _n =f _n [nx/s(t)] for the variable region 0xs(t), where the Fourier amplitudes _n (t) are determined by coupled differential equations of second order. It is concluded that the conventional elliptic flux compression theories (E/t=0) hold approximately for nonrelativistic liner speeds , whereas the hyperbolic theory (E/t0) is valid for arbitrary liner speeds .     

Nonstationary Markov chains and convergence of the annealing algorithm

We study the asymptotic behavior as timet + of certain nonstationary Markov chains, and prove the convergence of the annealing algorithm in Monte Carlo simulations. We find that in the limitt + , a nonstationary Markov chain may exhibit phase transitions. Nonstationary Markov chains in general, and the annealing algorithm in particular, lead to biased estimators for the expectation values of the process. We compute the leading terms in the bias and the variance of the sample-means estimator. We find that the annealing algorithm converges if the temperatureT(t) goes to zero no faster thanC/log(t/t ₀) ast+, with a computable constantC andt ₀ the initial time. The bias and the variance of the sample-means estimator in the annealing algorithm go to zero likeO(t^–1+) for some 0<1, with =0 only in very special circumstances. Our results concerning the convergence of the annealing algorithm, and the rate of convergence to zero of the bias and the variance of the sample-means estimator, provide a rigorous procedure for choosing the optimal annealing schedule. This optimal choice reflects the competition between two physical effects: (a) The adiabatic effect, whereby if the temperature is loweredtoo abruptly the system may end up not in a ground state but in a nearby metastable state, and (b) the super-cooling effect, whereby if the temperature is loweredtoo slowly the system will indeed approach the ground state(s) but may do so extremely slowly.     

Optimal multigrid algorithms for the massive Gaussian model and path integrals

Multigrid algorithms are presented which, in addition to eliminating the critical slowing down, can also eliminate the volume factor. The elimination of the volume factor removes the need to produce many independent fine-grid configurations for averaging out their statistical deviations, by averaging over the many samples produced on coarse grids during the multigrid cycle. Thermodynamic limits of observables can be calculated to relative accuracy _r in justO( _r ^-2 ) computer operations, where _r is the error relative to the standard deviation of the observable. In this paper, we describe in detail the calculation of the susceptibility in the one-dimensional massive Gaussian model, which is also a simple example of path integrals. Numerical experiments show that the susceptibility can be calculated to relative accuracy _r in about 8 _r ^-2 random number generations, independent of the mass size.     

Singular perturbations of piecewise monotonic maps of the interval

Let t: [0, 1] [0, 1] be a piecewise monotonic, C₂, and expanding map. In computing an orbit { ⁱ (x ₀)} _i=0 , we model the roundoff error at each iteration by a singular perturbation; i.e.,X _n+1=(X _n )+W , whereW is a random variable taking on discrete values in an interval (-&#x03B5;, ). The main result proves that this process admits an absolutely continuous invariant measure which approaches the absolutely continuous measure invariant under the deterministic map t as the precision of computation 0.     

Band-gap structure of the spectrum of periodic Maxwell operators

We investigate the band-gap structure of some second-order differential operators associated with the propagation of waves in periodic two-component media. Particularly, the operator associated with the Maxwell equations with position-dependent dielectric constant (x),xR ³, is considered. The medium is assumed to consist of two components: the background, where (x) = _b , and the embedded component composed of periodically positioned disjoint cubes, where (x) = _a . We show that the spectrum of the relevant operator has gaps provided some reasonable conditions are imposed on the parameters of the medium. Particularly, we show that one can open up at least one gap in the spectrum at any preassigned point provided that the size of cubesL, the distancel=L betwen them, and the contrast = _b / _a are chosen in such a way thatL ^–2, and quantities ^-1-3/2 and ² are small enough. If these conditions are satisfied, the spectrum is located in a vicinity of widthw(^3/2)^-1 of the set {² L ^-2 k ²:kZ³}. This means, in particular, that any finite number of gaps between the elements of this discrete set can be opened simultaneously, and the corresponding bands of the spectrum can be made arbitrarily narrow. The method developed shows that if the embedded component consists of periodically positioned balls or other domains which cannot pack the space without overlapping, one should expect pseudogaps rather than real gaps in the spectrum.