Two-time scale subordination in physical processes with long-term memory

We describe dynamical processes in continuous media with a long-term memory. Our consideration is based on a stochastic subordination idea and concerns two physical examples in detail. First we study a temporal evolution of the species concentration in a trapping reaction in which a diffusing reactant is surrounded by a sea of randomly moving traps. The analysis uses the random-variable formalism of anomalous diffusive processes. We find that the empirical trapping-reaction law, according to which the reactant concentration decreases in time as a product of an exponential and a stretched exponential function, can be explained by a two-time scale subordination of random processes. Another example is connected with a state equation for continuous media with memory. If the pressure and the density of a medium are subordinated in two different random processes, then the ordinary state equation becomes fractional with two-time scales. This allows one to arrive at the Bagley-Torvik type of state equation.     

Quantum hall conductivity in a Landau type model with a realistic geometry II

We use a mathematical framework that we introduced in a previous paper to study geometrical and quantum mechanical aspects of a Hall system with finite size and general boundary conditions. Geometrical structures control possibly the integral or fractional quantization of the Hall conductivity depending on the value of NB/2π (N is the number of charge carriers and B is the magnetic field). When NB/2π is irrational, we show that monovaluated wave functions can be constructed only on the graph of a free group with two generators. When NB/2π is rational, the relevant space becomes a punctured Riemann surface. We finally discuss our results from a phenomenological viewpoint.     

Incomplete normalization of probability on multifractals

This work is an extension of the incomplete probability theory from the simple case of monofractals previously studied to the more general case of multifractals that can occur in the phase space without equiprobable partition.     

An information-theoretic link between spacetime symmetries and quantum linearity

A nonlinear generalisation of Schrodinger’s equation is obtained using information-theoretic arguments. The nonlinearities are controlled by an intrinsic length scale and involve derivatives to all orders thus making the equation mildly nonlocal. The nonlinear equation is homogeneous, separable, conserves probability, but is not invariant under spacetime symmetries. Spacetime symmetries are recovered when a dimensionless parameter is tuned to vanish, whereby linearity is simultaneously established and the length scale becomes hidden. It is thus suggested that if, in the search for a more basic foundation for Nature’s Laws, an inference principle is given precedence over symmetry requirements, then the symmetries of spacetime and the linearity of quantum theory might both be emergent properties that are intrinsically linked. Supporting arguments are provided for this point of view and some testable phenomenological consequences are highlighted. The generalised Klien-Gordon and Dirac equations are also studied, leading to the suggestion that nonlinear quantum dynamics with intrinsically broken spacetime symmetries might be relevant to understanding the problem of neutrino mass (lessness) and oscillations: among other observations, this approach hints at the existence of a hidden discrete family symmetry in the Standard Model of particle physics.     

Quantum probability calculations using a first-principles quantum Monte Carlo method

A first-principles Monte Carlo code that exactly simulates quantum dynamics is presented which makes no use of amplitude calculations, only noise sources. The subtle question concerning how to map random choices in amplitude interferences is explained. In this formalism negative values of the Wigner function have clear logical meaning.     

Many worlds and the appearance of probability in quantum mechanics

The interpretation of the squared norm as probability and the apparent stochastic nature of observation in the quantum theory are derived from the law of large numbers and the algebraic properties of infinite sequences of simultaneous quantum observables. It is argued that this result validates the many-worlds view of quantum reality.     

Geometric algebra techniques for general relativity

Geometric (Clifford) algebra provides an efficient mathematical language for describing physical problems. We formulate general relativity in this language. The resulting formalism combines the efficiency of differential forms with the straightforwardness of coordinate methods. We focus our attention on orthonormal frames and the associated connection bivector, using them to find the Schwarzschild and Kerr solutions, along with a detailed exposition of the Petrov types for the Weyl tensor.     

Hidden variables in quantum mechanics: Generic models, set-theoretic forcing, and the appearance of probability

The hidden-variables premise is shown to be equivalent to the existence of generic filters for systems of commuting observables of a quantum system. The significance of this equivalence is interpreted in light of the theory of generic filters and boolean-valued models in set theory. The apparent stochastic nature of quantum observation is derived for these hidden-variables models.     

Duality, phase structures, and dilemmas in symmetric quantum games

Symmetric quantum games for 2-player, 2-qubit strategies are analyzed in detail by using a scheme in which all pure states in the 2-qubit Hilbert space are utilized for strategies. We consider two different types of symmetric games exemplified by the familiar games, the Battle of the Sexes (BoS) and the Prisoners’ Dilemma (PD). These two types of symmetric games are shown to be related by a duality map, which ensures that they share common phase structures with respect to the equilibria of the strategies. We find eight distinct phase structures possible for the symmetric games, which are determined by the classical payoff matrices from which the quantum games are defined. We also discuss the possibility of resolving the dilemmas in the classical BoS, PD, and the Stag Hunt (SH) game based on the phase structures obtained in the quantum games. It is observed that quantization cannot resolve the dilemma fully for the BoS, while it generically can for the PD and SH if appropriate correlations for the strategies of the players are provided.     

Noise-induced coupling and phase transition in initially homogeneous bistable system

We show that a colored spatial noise induces a heterogeneous behavior and coupling of initially uncoupled single bistable units. A formal approximation reduces a non-Markovian stochastic process described by the initial set of equations into Markovian process in terms of Langevin equation, for which a simple piecewise linear emulation was used to represent the nonlinear deterministic force. It turned out that the coupling leads to a phase transition due to the noise-induced diffusive term. As an example, a typical bistable noisy system with symmetric double-well potential was studied.     

General covariance of the non-abelian DBI-action: checks and balances

We perform three tests on our proposal to implement diffeomorphism invariance in the non-abelian D0-brane DBI action as a base-point independence constraint between matrix Riemann normal coordinate systems. First, we show that T-duality along an isometry correctly interchanges the potential and kinetic terms in the action. Second, we show that the method to impose base-point independence using an auxiliary dN²-dimensional nonlinear sigma model also works for metrics which are curved along the brane, provided a physical gauge choice is made at the end. Third, we show that without alteration this method is applicable to higher order in velocities. Testing specifically to order 4, we elucidate the range of validity of the symmetrized trace approximation to the non-abelian DBI action.     

Granger causality and contiguity between stochastic processes

Although according to many econometricians the definition of causality proposed by Granger differs from other definitions of causation in the philosophy of science, in this Letter we argue that it is not completely lacking in philosophical legitimacy. We attempt to shed new light on the nexus between Granger causality and the concept of contiguity. In particular, we prove that the existence of a Granger causal link between two stochastic processes requires that these be “contiguous” or that there exist a chain of processes, one contiguous to the next, which link the two processes.     

Generic two-phase coexistence, relaxation kinetics, and interface propagation in the quadratic contact process: Analytic studies

The quadratic contact process is implemented as an adsorption-desorption model on a two-dimensional square lattice. The model involves random adsorption at empty sites, and correlated desorption requiring diagonal pairs of empty neighbors. A simulation study of this model [D.-J. Liu, X. Guo, J.W. Evans, Phys. Rev. Lett. 98 (2007) 050601] revealed the existence of generic two-phase coexistence between a low-coverage active steady-state and a completely covered absorbing state. Here, an analytic treatment of model behavior is developed based on truncation approximations to the exact master equations. Applying this approach for spatially homogeneous states, we characterize steady-state behavior as well as the kinetics of relaxation to the steady-states. Extending consideration to spatially inhomogeneous states, we obtain discrete reaction-diffusion type equations characterizing evolution. These are employed to analyze an orientation-dependence of the propagation of planar interfaces between active and absorbing states which underlies the generic two-phase coexistence. We also describe the dynamics and critical forms of planar perturbations of the active state and of droplets of one phase embedded in the other.     

Analysis of the spatial and temporal distributions between successive earthquakes: Nonextensive statistical mechanics viewpoint

The spatial and temporal distributions between successive earthquakes are treated in the framework of nonextensive statistical mechanics. We find temporal distributions exhibit the power law behavior; q-exponential with q>1. It means the earthquakes are strongly correlated in time. The spatial distributions obey the q-exponential form with q<1. We also examine the dependence of the q exponent on magnitude range, covering period, time interval and size of the region where data are gathered. The conjecture of Abe et al. [S. Abe, N. Suzuki, Physica A 350 (2005) 588] has been examined for different categories of data. The results show a strange relation between q values of the spatial and temporal distributions.     

Schottky barrier properties of various metal (Zr, Ti, Cr, Pt) contact on p-GaN revealed from I-V-T measurement

Schottky barrier contact using three different metal (Zr, Ti, Cr and Pt) and Ohmic contact using Ni were made on same epitaxial growth layer of p-GaN. Measurements were carried out using current-voltage-temperature (I-V-T) in the range of 27-100°C. Under forward bias and room-temperature (RT), the ideality factors (η) were determined to be 2.38, 1.82, 1.51 and 2.63, respectively, for Zr, Ti, Cr and Pt. The Schottky barrier height (SBH) and effective Richardson coefficient A^** were measured through modified Norde plot as one of the analysis tools. Barrier heights of 0.84, 0.82, 0.77 and 0.41 eV for Zr, Ti, Cr and Pt, respectively, were obtained from the modified Norde plot. Schottky barrier heights of Zr, Ti, or Cr/p-GaN were also measured through activation energy plot, and determined to be in the same range (∼0.87 eV) and Pt at 0.49 eV. These results indicate that the Fermi level seems to be pinned due to the value of slope parameter (S) was very low (S = −0.25).     

Evidence for a gravitational Myers effect

An indication for the existence of a collective Myers solution in the non-abelian D0-brane Born-Infeld action is the presence of a tachyonic mode in fluctuations around the standard diagonal background. We show that this computation for non-abelian D0-branes in curved space has the geometric interpretation of computing the eigenvalues of the geodesic deviation operator for U(N)-valued coordinates. On general grounds one therefore expects a geometric Myers effect in regions of sufficiently negative curvature. We confirm this by explicit computations for non-abelian D0-branes on a sphere and a hyperboloid. For the former the diagonal solution is stable, but not so for the latter. We conclude by showing that near a Schwarzschild black hole one also finds a tachyonic mode in the fluctuation spectrum, signaling the possibility of a near-black-hole gravitationally induced Myers effect.     

Exact solutions of the Fokker-Planck equations with moving boundaries

By means of time-dependent similarity transformations, we derive exact solutions of the Fokker-Planck equations with moving boundaries in the presence of: (1) a time-dependent linear force and (2) a time-dependent nonlinear force. The method of similarity transformation is simple and can be easily applied to more general Fokker-Planck equations. Furthermore, the knowledge of the exact solutions in closed form can be useful as a benchmark to test approximate numerical or analytical procedures.     

A Bayesian account of quantum histories

We investigate whether quantum history theories can be consistent with Bayesian reasoning and whether such an analysis helps clarify the interpretation of such theories. First, we summarise and extend recent work categorising two different approaches to formalising multi-time measurements in quantum theory. The standard approach consists of describing an ordered series of measurements in terms of history propositions with non-additive ‘probabilities.’ The non-standard approach consists of defining multi-time measurements to consist of sets of exclusive and exhaustive history propositions and recovering the single-time exclusivity of results when discussing single-time history propositions. We analyse whether such history propositions can be consistent with Bayes’ rule. We show that certain class of histories are given a natural Bayesian interpretation, namely, the linearly positive histories originally introduced by Goldstein and Page. Thus, we argue that this gives a certain amount of interpretational clarity to the non-standard approach. We also attempt a justification of our analysis using Cox’s axioms of probability theory.     

Random, but not so much a parameterization for the returns and correlation matrix of financial time series

A parameterization that is a modified version of a previous work is proposed for the returns and correlation matrix of financial time series and its properties are studied. This parameterization allows easy introduction of non-stationarity and it shows several of the characteristics of the true, observed realizations, such as fat tails, volatility clustering, and a spectrum of eigenvalues of the correlation matrix that can be understood as an extension of Random Matrix Theory results. The predicted behavior of this parameterization for the eigenvalues is compared with the eigenvalues of Brazilian assets and it is shown that those predictions fit the data better than Random Matrix Theory.     

Duality in 2 + 1D quantum elasticity: superconductivity and quantum nematic order

Superfluidity and superconductivity are traditionally understood in terms of an adiabatic continuation from the Bose-gas limit. Here we demonstrate that at least in a 2 + 1D Bose system, superfluidity can arise in a strict quantum field-theoretic setting. Taking the theory of quantum elasticity (describing phonons) as a literal quantum field theory with a bosonic statistic, superfluidity and superconductivity (in the EM charged case) emerge automatically when the shear rigidity of the elastic state is destroyed by the proliferation of topological defects (quantum dislocations). Off-diagonal long range order in terms of the field operators of the constituent particles is not required. This is one of the outcomes of the broader pursuit presented in this paper. In essence, it amounts to the generalization of the well known theory of crystal melting in two dimensions by Nelson et al. [Phys. Rev. B 19 (1979) 2457; Phys. Rev. B 19 (1979) 1855], to the dynamical theory of bosonic states exhibiting quantum liquid-crystalline orders in 2 + 1 dimensions. We strongly rest on the field-theoretic formalism developed by Kleinert [Gauge fields in Condensed Matter, vol. II: Stresses and Defects, Differential Geometry, Crystal Defects, World Scientific, Singapore, 1989] for classical melting in 3D. Within this framework, the disordered states correspond to Bose condensates of the topological excitations, coupled to gauge fields describing the capacity of the elastic medium to propagate stresses. Our focus is primarily on the nematic states, corresponding with condensates of dislocations, under the topological condition that disclinations remain massive. The dislocations carry Burgers vectors as topological charges. Conventional nematic order, i.e., the breaking of space-rotations, corresponds in this field-theoretic duality framework with an ordering of the Burgers vectors. However, we also demonstrate that the Burgers vectors can quantum disorder despite the massive character of the disclinations. We identify the physical nature of the ‘Coulomb nematic’ suggested by Lammert et al. [Phys. Rev. Lett. 70 (1993) 1650; Phys. Rev. E 52 (1995) 1778] on gauge-theoretical grounds. The 2 + 1D quantum liquid crystals differ in fundamental regards from their 3D classical counterparts due to the presence of a dynamical constraint. This constraint is the glide principle, well known from metallurgy, which states that dislocations can only propagate in the direction of their Burgers vector. In the present framework this principle plays a central role. This constraint is necessary to decouple compression rigidity from the dislocation condensate. The shear rigidity is not protected, and as a result the shear modes acquire a Higgs mass in the dual condensate. This is the way the dictum that translational symmetry breaking goes hand in hand with shear rigidity emerges in the field theory. However, because of the glide principle compression stays massless, and the fluids are characterized by an isolated massless compression mode and are therefore superfluids. Glide also causes the shear Higgs mass to vanish at orientations perpendicular to the director in the ordered nematic, and the resulting state can be viewed as a quantum smectic of a novel kind. Our most spectacular result is a new hydrodynamical way of understanding the conventional electromagnetic Meissner state (superconducting state). Generalizing to the electromagnetically charged elastic medium (‘Wigner Crystal’) we find that the Higgs mass of the shear gauge fields, becoming finite in the nematic quantum fluids, automatically causes a Higgs mass in the electromagnetic sector by a novel mechanism.