Noise-induced multilevel Landau–Zener transitions: Density matrix investigation

The generalised multilevel Landau–Zener problem is solved by applying the density matrix technique within the framework of nonstationary perturbation theory. The exact survival probability is achieved as a proof of the Brundobler–Elzer hypothesis (Brundobler and Elzer (1993) [38]). The effect of classical Gaussian noise is investigated by averaging the solution over the noise realisation. A generalised formula for slow noise-induced transition probability is obtained and found to agree exactly with all known results. Exact results are reported for the Demkov–Osherov model in the slow and fast noise limits. Thermal transition probabilities are obtained via the activation Arrhenius law and observed to tailor a qubit from thermal decoherence.     

Verhulst model with Lévy white noise excitation

The transient dynamics of the Verhulst model perturbed by arbitrary non-Gaussian white noise is investigated. Based on the infinitely divisible distribution of the Lévy process we study the nonlinear relaxation of the population density for three cases of white non-Gaussian noise: (i) shot noise; (ii) noise with a probability density of increments expressed in terms of Gamma function; and (iii) Cauchy stable noise. We obtain exact results for the probability distribution of the population density in all cases, and for Cauchy stable noise the exact expression of the nonlinear relaxation time is derived. Moreover starting from an initial delta function distribution, we find a transition induced by the multiplicative Lévy noise, from a trimodal probability distribution to a bimodal probability distribution in asymptotics. Finally we find a nonmonotonic behavior of the nonlinear relaxation time as a function of the Cauchy stable noise intensity.     

Effect of the illumination length on the statistical distribution of the field scattered from one-dimensional random rough surfaces: analytical formulae derived from the small perturbation method

We consider a dielectric plane surface with a local cylindrical perturbation illuminated by a monochromatic plane wave. The perturbation is represented by a random function assuming values with a Gaussian probability density with zero mean value. Outside the perturbation zone, the scattered field can be represented by a superposition of a continuous spectrum of outgoing plane waves. The stationary phase method leads to the asymptotic field, the angular dependence of which is given by the scattering amplitudes of the propagating plane waves. The small perturbation method applied to the Rayleigh integral and the boundary conditions gives a first-order approximation of the scattering amplitudes. We show that the real part and the imaginary part of the scattering amplitudes are Gaussian stochastic variables with zero mean values and unequal variances. The values of variances depend on the length of the perturbation zone. In most cases, the probability density function for the amplitude is a Hoyt distribution and the phase is not uniformly distributed between -π and π. The standard Rayleigh and uniform distributions are obtained for special values of the length and in the case of an infinite illumination length.     

Remarks concerning the derivation and the expansion of the master equation

The present work consist of two parts: In the first part we apply the method of quasilinearization to the differential equation describing the time development of the quantum-mechanical probability density. In this way we derive the master equation without resorting to perturbation theory. In the second part of the paper, for a general form of the master equation which is an integro-differential equation, we test the accuracy of the Fokker-Planck approximation with the help of a solvable model. Then we study an alternative way of reducing the integro-differential equation to a partial differential equation. By expanding the transition probability W(q, q′), and the distribution function in terms of a complete set of functions, we show that for certain forms of W(q, q′), the master equation can be transformed exactly to partial differential equations of finite order.     

Random Matrix Theory and the Anderson Model

This paper is devoted to a discussion of possible strategies to prove rigorously the existence of a metal-insulator Anderson transition for the Anderson model in dimension d≥3. The possible criterions used to define such a transition are presented. It is argued that at low disorder the lowest order in perturbation theory is described by a random matrix model. Various simplified versions for which rigorous results have been obtained in the past are discussed. It includes a free probability approach, the Wegner n-orbital model and a class of models proposed by Disertori, Pinson, and Spencer, Comm. Math. Phys. 232:83–124 (2002). At last a recent work by Magnen, Rivasseau, and the author, Markov Process and Related Fields 9:261–278 (2003) is summarized: it gives a toy modeldescribing the lowest order approximation of Anderson model and it is proved that, for d=2, its density of states is given by the semicircle distribution. A short discussion of its extension to d≥3 follows.     

Random anisotropy in lattice gas model

We consider a lattice-gas model with infinite-range interaction with site dependent random anisotropy distributed with a Gaussian distribution. The random anisotropy lattice-gas analogous of the random field Ising model is solved exactly using a replica theory. We show that, at finite temperature, the introduction of disorder eliminates completely the phase transition, and destroy the equivalence between real gases and Ising magnets. Whereas at T = 0, the density of occupied sites has a step-like behavior as function of the random anisotropy.     

Effect of the illumination length on the statistical distribution of the field scattered from one-dimensional random rough surfaces: analytical formulae derived from the small perturbation method

We consider a dielectric plane surface with a local cylindrical perturbation illuminated by a monochromatic plane wave. The perturbation is represented by a random function assuming values with a Gaussian probability density with zero mean value. Outside the perturbation zone, the scattered field can be represented by a superposition of a continuous spectrum of outgoing plane waves. The stationary phase method leads to the asymptotic field, the angular dependence of which is given by the scattering amplitudes of the propagating plane waves. The small perturbation method applied to the Rayleigh integral and the boundary conditions gives a first-order approximation of the scattering amplitudes. We show that the real part and the imaginary part of the scattering amplitudes are Gaussian stochastic variables with zero mean values and unequal variances. The values of variances depend on the length of the perturbation zone. In most cases, the probability density function for the amplitude is a Hoyt distribution and the phase is not uniformly distributed between –π and π. The standard Rayleigh and uniform distributions are obtained for special values of the length and in the case of an infinite illumination length.     

A prediction theory of random level distribution based on a generalized difference type of Fokker-Planck equation and its application to environmental noise and vibration,II: Experimental study

In a companion paper [1], a procedure for solving the short time prediction problem in terms of the transition probability distribution has been theoretically derived, for discrete time-sampled data. Explicit algorithms for estimating the non-stationary moment statistics of arbitrary order also have been derived, based on a generalized difference equation of Fokker-Planck type for the conditional probability distributed function, which is central to the theory. In this paper, evidence for the validity and effectiveness of the proposed method is presented, as obtained not only by means of digital simulation but also by using road traffic noise data obtained experimentally in Hiroshima. For several non-stationary random processes simulated by means of random numbers, the theoretical and experimental conditional probability functions are compared. For non-stationary road traffic noise data the theoretically predicted and experimentally determined confidence intervals are compared; in these comparisons several types of conditional probability function and various values of weighting parameter are used in the algorithm. All of the theoretical results show good agreement with the experimental results.     

Nearly linear mappings and their applications

A simple 2-dimensional mapping is considered, both analytically and numerically, for which all nonlinear effects are of the same order as the perturbations and of the same origin. Properties of the stochastic instability are investigated, taking the beam-beam interaction in a storage ring as an important particular example of a dynamic system that can be modelled with such a mapping. The special case of time-dependent mappings is discussed. It is shown that low-frequency time dependence sharply decreases the critical perturbation strength for the stochastic transition.     

Statistical properties of N random sinusoidal waves in additive Gaussian noise

The first order probability density functions of the sums of N independent sinusoidal waves having random amplitudes and phases in additive Gaussian noise are studied for the cases where N is fixed and where N in Poisson distributed. The conditional moments about the origin are obtained in closed form or both situations. The corresponding probability density functions for the envelope are also studied. The even conditional moments about the origin are also obtained in closed form. Representative numerical results are presented.     

Perturbative calculation of critical exponents for the Bose–Hubbard model

We develop a strategy for calculating critical exponents for the Mott insulator-to-superfluid transition shown by the Bose–Hubbard model. Our approach is based on the field-theoretic concept of the effective potential, which provides a natural extension of the Landau theory of phase transitions to quantum critical phenomena. The coefficients of the Landau expansion of that effective potential are obtained by high-order perturbation theory. We counteract the divergency of the weak-coupling perturbation series by including the seldom considered Landau coefficient a ₆ into our analysis. Our preliminary results indicate that the critical exponents for both the condensate density and the superfluid density, as derived from the two-dimensional Bose–Hubbard model, deviate by less than 1 % from the best known estimates computed so far for the three-dimensional XY universality class.     

Statistical damage identification of structures with frequency changes

Model updating methods based on structural vibration data have being rapidly developed and applied to detect structural damage in civil engineering. But uncertainties existing in the structural model and measured vibration data might lead to unreliable damage detection. In this paper a statistical damage identification algorithm based on frequency changes is developed to account for the effects of random noise in both the vibration data and finite element model. The structural stiffness parameters in the intact state and damaged state are, respectively, derived with a two-stage model updating process. The statistics of the parameters are estimated by the perturbation method and verified by Monte Carlo technique. The probability of damage existence is then estimated based on the probability density functions of the parameters in the two states. A higher probability statistically implies a more likelihood of damage occurrence. The presented technique is applied to detect damages in a numerical cantilever beam and a laboratory tested steel cantilever plate. The effects of using different number of modal frequencies, noise level and damage level on damage identification results are also discussed.     

Stationary response of multi-degree-of-freedom vibro-impact systems to Poisson white noises

The stationary response of multi-degree-of-freedom (MDOF) vibro-impact (VI) systems to random pulse trains is studied. The system is formulated as a stochastically excited and dissipated Hamiltonian system. The constraints are modeled as non-linear springs according to the Hertz contact law. The random pulse trains are modeled as Poisson white noises. The approximate stationary probability density function (PDF) for the response of MDOF dissipated Hamiltonian systems to Poisson white noises is obtained by solving the fourth-order generalized Fokker-Planck-Kolmogorov (FPK) equation using perturbation approach. As examples, two-degree-of-freedom (2DOF) VI systems under external and parametric Poisson white noise excitations, respectively, are investigated. The validity of the proposed approach is confirmed by using the results obtained from Monte Carlo simulation. It is shown that the non-Gaussian behaviour depends on the product of the mean arrival rate of the impulses and the relaxation time of the oscillator.     

Theoretical analysis on phase behaviour of a liquid crystalline material – effect of molecular motions

Molecular structure, and phase behaviour of 2-Cyano-N-[4-(4-n-pentyloxybenzoyloxy)-benzylidene] aniline (CPBBA) has been reported with respect to translational and orientational motions. The atomic net charge and dipole moment components at each atomic centre have been evaluated using the complete neglect differential overlap (CNDO/2) method. The modified Rayleigh–Schrodinger perturbation theory along with multicentered–multipole expansion method has been employed to evaluate the long-range intermolecular interactions, while a ‘6-exp’ potential function has been assumed for short-range interactions. The interaction energy values obtained through these computations have been used as input to calculate the configurational probability at room temperature (300 K), and nematic–isotropic transition temperature (396.5 K). On the basis of stacking, in-plane, and terminal interaction energy calculations, all possible geometrical arrangements between the molecular pairs have been considered. Molecular arrangements inside a bulk of materials have been discussed in terms of their relative order. Further, translational rigidity parameter has been estimated as a function of temperature to understand the phase behaviour of the compound. The present model is helpful to understand the effect of molecular motions on ordering, and phase behaviour of the mesogenic compounds.     

Qualitative change of fluctuation observed in real traffic flow

We studied the nature of fluctuations around the phase transition of vehicular traffic by analyzing a time series of successive variations of velocity, obtained from single-vehicle data measured by an onboard apparatus. We found that the probability density function calculated from the time series of variation of velocity is transformed irreversibly in the critical region, where a Gaussian distribution changes into a Lévy stable symmetrical distribution. The power-law tail in the Lévy distribution indicated that the time series of velocity variation exhibits the nature of the critical fluctuations generally observed in phase transitions driven far from equilibrium. Furthermore, single-vehicle data enabled us to calculate the time evolution of the local flux–density relation, which suggested that the vehicular traffic system spontaneously approaches a delicate balance between metastable states and congested-flow states. The nature of fluctuations enables us to understand mechanisms behind the spontaneous decay of the metastable branch at the phase transition. The power-law tail in the probability density function suggests that dynamical processes of vehicular traffic in the critical region are related to a time-discrete stochastic process driven by random amplification with additive external noise.     

Ein Beitrag zur Theorie der inneren Feldemission

The internal field emission (Zener-tunneling) is examined from two directions, using a onedimensional model. In section I the transition probability is calculated using the Houston expansion, but without perturbation theory, as has been used in previous papers by other authors. The calculation permits an examination of the various perturbation approximations. It is found, that the validity of the first approximation depends critically upon the residue of the interband coupling function. This residue contains the Keldysh factor, which is determined here. In section II the transition probability is calculated using free electron functions, again without perturbation theory. The same transition probability is obtained in both cases.     

A modified pitchfork bifurcation as a model of the TI–TII transition in liquid helium counterflow with external noise

An amended pitchfork bifurcation is introduced to model recent experiments by Griswold and Tough on superfluid turbulence in liquid helium counterflow subject to strong external noise. We adopt the generalized white noise limit of Blankenship and Papanicolaou to take a short-correlation-time limit of the nonlinear noise which enters into the model, and we implement this limit by means of the wideband perturbation expansion. Novel boundary conditions are applied to the resultant diffusion process in order to obtain behavior in qualitative agreement with the observations at low vortex line density. We are able to account for the sharp peak in probability observed experimentally at a small positive line density. The drift and diffusion of our diffusion process may be estimated experimentally; we describe how to do this.     

Statistical study of the radiation losses due to surface roughness for a dilectric slab deposited on a metal substrate

This paper presents an original method of analyzing radiation loss from dielectric slab with random wall imperfections. The method is based on Maxwell’s equations under their covariant form written in a non-orthogonal coordinate system. The solution is found by using a perturbation method applied to the smooth surface problem. The statistical characteristics of the radiation intensity, the average value and the probability density function, are analytically determined.     

Newtonian Kinetic Theory and the Ergodic-Nonergodic Transition

In a recent work we have discussed how kinetic theory, the statistics of classical particles obeying Newtonian dynamics, can be formulated as a field theory. The field theory can be organized to produce a self-consistent perturbation theory expansion in an effective interaction potential. In the present work we use this development for investigating ergodic-nonergodic (ENE) transitions in dense fluids. The theory is developed in terms of a core problem spanned by the variables ρ, the number density, and B, a response density. We set up the perturbation theory expansion for studying the self-consistent model which gives rise to a ENE transition. Our main result is that the low-frequency dynamics near the ENE transition is the same for Smoluchowski and Newtonian dynamics. This is true despite the fact that term by term in a density expansion the results for the two dynamics are fundamentally different.