Renormalization of branching models of triggered seismicity from total to observable seismicity

Several recent works point out that the crowd of small unobservable earthquakes (with magnitudes below the detection threshold m_d) may play a significant and perhaps dominant role in triggering future seismicity. Using the ETAS branching model of triggered seismicity, we apply the formalism of generating probability functions to investigate how the statistical properties of observable earthquakes differ from the statistics of all events. The ETAS (epidemic-type aftershock sequence) model assumes that each earthquake can trigger other earthquakes (“aftershocks”). An aftershock sequence results in this model from the cascade of aftershocks of each past earthquake. The triggering efficiency of earthquakes is assumed to vanish below a lower magnitude limit m₀, in order to ensure the convergence of the theory and may reflect the physics of state-and-velocity frictional rupture. We show that, to a good approximation, the statistical distribution of seismic rates of events with magnitudes above m_d generated by an ETAS model with branching ratio n is the same as that of events generated by another ETAS model with effective parameter n(m_d). Our present analysis thus confirms, for the full statistical (time-independent or large time-window approximation) properties, the results obtained previously by one of us and Werner, based solely on the average seismic rates (the first-order moment of the statistics). Our analysis also demonstrates that this correspondence is not exact, as there are small corrections which can be systematically calculated, in terms of additional contributions that can be mapped onto a different branching model. We also show that this approximate correspondence of the ETAS model onto itself obtained by changing m₀ into m_d, and n into n(m_d) holds only with respect to its statistical properties and not for all its space-time properties.     

Plane-wave approximation in theS-matrix and the construction of bound-state wave functions

Three inversion problem approaches — byGelfand-Levitan, Marchenko andPetrá? [5] —in both non-relativistic and relativistic (Klein-Gordon) variants are used in an approximation scheme selected to construct bound-state wave functions which are advantageous for purposes of model hadron physics. This family of wave functions is created exclusively by theS-matrix quantities and derived in the approximation which requires the Jost function to be equal to the unity throughout the continuous spectrum (the plane-wave approximation). As a consequence of the difference in boundary conditions of the mentioned approaches, the resulting approximate wave functions are not identical, but it is shown that there exists a parallelism as to the form among them. This parallelism is explained more extensively in the non-relativistic case, where the transformation properties of alternative sets of functions are treated. In the present paper it is demonstrated that in the relativistic variants of the above approaches the non-relativistic plane-wave-approximation form of the constructed wave functions for a given bound state is preserved.     

Influence of an external magnetic field on the threshold of multipactor onset on a dielectric surface

In this paper, we propose a theoretical model of the initial stage of development of a one-sided secondary-emission (multipactor) discharge on a dielectric surface. Consideration is based on a statistical method supported by an exact analytical solution for the transit-time distribution function of secondary electrons. The general integral equation allowing us to determine the stationary emission-phase distribution functions and the threshold of multiplicator onset in the presence of an external magnetic field is formulated. It is shown that the presence of an external magnetic field can significantly change the conditions of multipactor onset. It is found that the discharge-zone boundaries calculated within the framework of a statistical model are in qualitative agreement with the results obtained by the Monte-Carlo method. __________ Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 50, No. 2, pp. 118–133, February 2007.     

选择模型的理论扩展分析

选择模型为解释指数律、幂律的成因提供了基于偏好的概率分析方法.通过对选择试验的研究,在视野规律和选项分布之间导出了一对变换,称其为偏序变换.对偏序变换连续化,导出了对应的连续变换.仿照离散分布,定义了连续分布的比较函数,并沟通了离散和连续分布的伴随关系.通过选择模型,对幂律访问现象进行了仿真.选择理论对于复杂巨系统和社会行为的研究具有较重要的借鉴作用,可扩展统计物理学和概率论的理论基础. 关键词： 选择模型 幂律现象 偏序变换 比较函数     

The Multilevel Splitting algorithm for graph colouring with application to the Potts model

Calculating the partition function of the zero-temperature antiferromagnetic model is an important problem in statistical physics. However, an exact calculation is hard since it is strongly connected to a fundamental combinatorial problem of counting proper vertex colourings in undirected graphs, for which an efficient algorithm is not known to exist. Thus, one has to rely on approximation techniques. In this paper, we formulate the problem of the partition function approximation in terms of rare-event probability estimation and investigate the performance of a particle-based algorithm, called Multilevel Splitting, for handling this setting. The proposed method enjoys a provable probabilistic performance guarantee and our numerical study indicates that this algorithm is capable of delivering accurate results using a relatively modest amount of computational resources.     

Violation of the Porod law in a freely cooling granular gas in one dimension

We study a model of freely cooling inelastic granular gas in one dimension, with a restitution coefficient which approaches the elastic limit below a relative velocity scale delta. While at early times (tdelta;{-1}) it exhibits a new fluctuation-dominated phase ordering state. We find distinct scaling behavior for the (i) density distribution function, (ii) occupied and empty gap distribution functions, (iii) the density structure function, and (iv) the velocity structure function, as compared to the completely inelastic sticky gas. The spatial structure functions (iii) and (iv) violate the Porod law. Within a mean-field approximation, the exponents describing the structure functions are related to those describing the spatial gap distribution functions.     

Quantum kinetics and thermalization in a particle bath model

We study the dynamics of relaxation and thermalization in an exactly solvable model of a particle interacting with a harmonic oscillator bath. Our goal is to understand the effects of non-Markovian processes on the relaxational dynamics and to compare the exact evolution of the distribution function with approximate Markovian and non-Markovian quantum kinetics. There are two different cases that are studied in detail: (i) a quasiparticle (resonance) when the renormalized frequency of the particle is above the frequency threshold of the bath and (ii) a stable renormalized "particle" state below this threshold. The time evolution of the occupation number for the particle is evaluated exactly using different approaches that yield to complementary insights. The exact solution allows us to investigate the concept of the formation time of a quasiparticle and to study the difference between the relaxation of the distribution of bare particles and that of quasiparticles. For the case of quasiparticles, the exact occupation number asymptotically tends to a statistical equilibrium distribution that differs from a simple Bose-Einstein form as a result of off-shell processes whereas in the stable particle case, the distribution of particles does not thermalize with the bath. We derive a non-Markovian quantum kinetic equation which resums the perturbative series and includes off-shell effects. A Markovian approximation that includes off-shell contributions and the usual Boltzmann equation (energy conserving) are obtained from the quantum kinetic equation in the limit of wide separation of time scales upon different coarse-graining assumptions. The relaxational dynamics predicted by the non-Markovian, Markovian, and Boltzmann approximations are compared to the exact result. The Boltzmann approach is seen to fail in the case of wide resonances and when threshold and renormalization effects are important.     

Monostatic and bistatic statistical shadowing functions from a one-dimensional stationary randomly rough surface according to the observation length: I. Single scattering

When solving electromagnetic rough-surface scattering problems, the effect of shadowing by the surface roughness often needs to be considered, especially as the illumination angle approaches grazing incidence. This paper presents the Ricciardi-Sato, as well as the Wagner and the Smith formulations for calculating the monostatic and bistatic statistical shadowing functions from a one-dimensional rough stationary surface, which are valid for an uncorrelated Gaussian process with an infinite surface length. In this paper, these formulations are extended to include a finite surface length and any uncorrelated process. The inclusion of a finite surface length is needed to extend the single-reflection shadowing function to the more general multiple-reflection case, presented in the following companion paper. Comparisons of these shadowing functions with the exact numerical solution for the shadowing (using surfaces with Gaussian and Lorentzian autocorrelation functions for a Gaussian process) shows that the Smith formulation without correlation is a good approximation, and that including correlation only weakly improves the model. This paper also presents a method to include the shadowing effect in the electromagnetic scattering problem.     

Calculation of the Gluon Distribution Function Using Alternative Method for the Proton Structure Function

A calculation of the proton structure function F2(x,Q^2) is reported with an approximation method that relates the reduced cross section derivative and the F2(x, Q^2) scaling violation at low x by using quadratic form for the structure function. This quadratic form approximation method can be used to determine the structure function F2 (x, Q^2) from the HERA reduced cross section data taken at low x. This new approach can determine the structure functions F2(x,Q^2) with reasonable precision even for low x values which have not been investigated. We observe that the Q^2 dependence is quadratic over the full kinematic covered range. To test the validity of our new determined structure functions, we find the gluon distribution function in the leading order approximation with our new calculation for the structure functions and compare them with the QCD parton distribution functions.     

Extraction of Structure Function and Gluon Distribution Function at Low-x from Cross Section Derivative by Regge Behavior

An approximation method based on Regge behavior is presented. This new method relates the reduced cross section derivative and the structure function Regge behavior at low x. With the use of this approximation method, the C and λ parameters are calculated from the HERA reduced cross section data taken at low-x. Also, we calculate the structure functions F₂(x,Q²) even for low-x values, which have not been investigated. To test the validity of calculated structure functions, we find the gluon distribution function in the Leading order approximation based on Regge behaviour of structure function and compare to the NLO QCD fit to H1 data and NLO parton distribution function.     

Husimi Function of Excited Squeezed Vacuum State

The q-p phase-space distribution function is a popular tool to study semiclassical physics and to describe the quantum aspects of a system. In this paper by using the pure state density operator formula of the Husimi operator Δ_h(q,p;κ)=|p,q〉_κκ〈p,q| we deduce the Husimi function of the excited squeezed vacuum state. Then we study the behavior of Husimi distribution graphically.     

On the connection of the formulas for entropy and stationary distribution

As is well known in statistical physics, the stationary distribution can be obtained by maximizing entropy. We show how one can reconstruct the formula for entropy knowing the formula for the stationary distribution. A general case is discussed and some concrete physical examples are considered.     

Duality Between Spin Networks and the 2D Ising Model

The goal of this paper is to exhibit a deep relation between the partition function of the Ising model on a planar trivalent graph and the generating series of the spin network evaluations on the same graph. We provide respectively a fermionic and a bosonic Gaussian integral formulation for each of these functions and we show that they are the inverse of each other (up to some explicit constants) by exhibiting a supersymmetry relating the two formulations. We investigate three aspects and applications of this duality. First, we propose higher order supersymmetric theories that couple the geometry of the spin networks to the Ising model and for which supersymmetric localization still holds. Secondly, after interpreting the generating function of spin network evaluations as the projection of a coherent state of loop quantum gravity onto the flat connection state, we find the probability distribution induced by that coherent state on the edge spins and study its stationary phase approximation. It is found that the stationary points correspond to the critical values of the couplings of the 2D Ising model, at least for isoradial graphs. Third, we analyze the mapping of the correlations of the Ising model to spin network observables, and describe the phase transition on those observables on the hexagonal lattice. This opens the door to many new possibilities, especially for the study of the coarse-graining and continuum limit of spin networks in the context of quantum gravity.     

Thermodynamic functions of both simple (monomeric) and polymeric melts: MFA approach and Monte Carlo simulation

The influence of equilibrium polymerization on the thermodynamic properties of model systems consisting of building units with well-defined characteristics and interactions is investigated. The systems under study are thought to resemble more or less accurately undercooled melts, and the calculations performed give the configurational part of the thermodynamic functions of these melts. The whole investigation is performed by using two approaches. In the first one, the temperature courses of the entropies and the specific heat of the systems as well as the average flexibility and the mean chain length of the polymer molecules are obtained in the framework of a mean field approximation (MFA). In the second approach, the bulk characteristics and the configurational properties of the model systems are obtained by using Monte Carlo simulations (MCS). The correspondence and the differences between the thermodynamic properties for the same systems in the two approaches are analyzed and discussed. The model used in our MCS is a modified version of the so-called independent monomer state model proposed earliar by Jari? and Bennemann. The temperature course of the thermodynamic functions of the systems under investigation is analyzed and compared with existing experimental data. The existence of phase transition corresponding to melt-crystal or order-disorder transformation in the system is discussed based on the two approaches.     

Code Optimization,Frozen Glassy Phase and Improved Decoding Algorithms for Low-Density Parity-Check Codes

The statistical physics properties of low-density parity-check codes for the binary symmetric channel are investigated as a spin glass problem with multi-spin interactions and quenched random fields by the cavity method. By evaluating the entropy function at the Nishimori temperature, we find that irregular constructions with heterogeneous degree distribution of check(bit) nodes have higher decoding thresholds compared to regular counterparts with homogeneous degree distribution. We also show that the instability of the mean-field calculation takes place only after the entropy crisis, suggesting the presence of a frozen glassy phase at low temperatures. When no prior knowledge of channel noise is assumed(searching for the ground state), we find that a reinforced strategy on normal belief propagation will boost the decoding threshold to a higher value than the normal belief propagation. This value is close to the dynamical transition where all local search heuristics fail to identify the true message(codeword or the ferromagnetic state). After the dynamical transition, the number of metastable states with larger energy density(than the ferromagnetic state)becomes exponentially numerous. When the noise level of the transmission channel approaches the static transition point, there starts to exist exponentially numerous codewords sharing the identical ferromagnetic energy.     

Quantummechanical solutions of the laser masterequation I

The masterequation for the statistical operatorW of a Laser mode andA active two-level atoms 1 is solved by using the coherent state representation 2 of the lightfield. The ansatz forW represents the most general symmetrical coupling of the light mode to all atoms and therefore contains the full influence of quantum fluctuations of the atomic system on the light mode. The system of equations can be solved practically exactly in the stationary case and leads to a photon number distribution in the laser valid for arbitrary pumping. This distribution agrees with that found by a Fokker-Planck equation 3 for not too high pumping and approaches the Poisson distribution for very high pumping. The smooth transition of the inversion from σ₀ (below threshold) to σ (above threshold) can also be calculated.     

Statistical Theory of Two-Sided Multipactor

We develop a statistical theory of secondary-emission discharge (SED) taking the energy distribution of secondary electrons into account. The theory allows one to describe quantitatively the initial stage of development of a two-sided multipactor. For an arbitrary probability density of normal components of the ejection velocity and an arbitrary distance between the walls enclosing the microwave discharge plasma, we construct an analytical solution for the electron distribution function over transit times. The performed analysis is based on the results of a detailed study of conditions under which an electron reaches the opposite side. With allowance for the spread in thermal velocities, we derive a recurrence relation between the electron distribution functions over emission phases and formulate a general integral equation from which the resulting stationary distribution and the threshold of SED onset are determined.     

Non-linear dynamics of a driven two-level tunnelling defect

With increasing external-field amplitude the dynamics of a driven two-level system shows a cross-over from the weak-coupling or linear-response behaviour to a non-linear regime which is characterised by the appearance of Rabi oscillations, a field-dependent dynamical susceptibility, saturation of energy dissipation and additional frequencies in the correlation function. The system any initial distribution will relax towards a stationary state with a time-dependent polarisation vector. These features are derived from the propagator of the two-level system which is calculated using a perturbative scheme with a well-defined small parameter; the roatating-wave approximation of the Bloch equations is avoided. the dynamical behaviour is discussed by means of correlation and response functions.     

Local methods for constructing stationary distribution functions of systems of stochastic differential Langevin-type equations: Noise influence on simple bifurcation

The influence is considered of two additive correlated noise effects on a two-dimensional quadratic-nonlinear system describing the behavior of two hydrodynamic modes. Using the method of Gaussian approximation, local characteristics of the distribution function are calculated, which are used to construct the global distribution function with the aid of the method of fraction-rational approximations. It is shown that for a system at whose bifurcation point the asymptotic stability is lost, in an expanded space of parameters (bifurcation parameter in the absence of noise plus noise parameters) there appears an instability zone within which the stationary distribution function does not exist. The effect of noise correlation on the stationary characteristics of the system is studied.     

Odd-Excited Binomial States of the Radiation Field and Some of Their Statistical Properties

In this paper a new state called odd-excited binomial state (OEBS) is introduced. It interpolates between the odd number state and the odd-excited coherent state. We discuss some statistical properties, such as the Glauber second-order correlation function and squeezing phenomenon (normal and amplitude-squared squeezing) for this state. The quasiprobability distribution functions (Husimi Q-function and Wigner W-function) are also examined.