How to measure subdiffusion parameters

We propose a method to measure the subdiffusion parameter alpha and subdiffusion coefficient Dalpha which are defined by means of the relation chi2 = 2Dalpha / Gamma(1+alpha)(t alpha), where chi2 denotes a mean square displacement of a random walker starting from x = 0 at the initial time t = 0. The method exploits a membrane system where a substance of interest is transported in a solvent from one vessel to another across a thin membrane which plays here only an auxiliary role. We experimentally study a diffusion of glucose and sucrose in a gel solvent, and we precisely determine the parameters alpha and Dalpha, using a fully analytic solution of the fractional subdiffusion equation.     

Migration and proliferation dichotomy in tumor-cell invasion

We propose a two-component reaction-transport model for the migration-proliferation dichotomy in the spreading of tumor cells. By using a continuous time random walk (CTRW), we formulate a system of the balance equations for the cancer cells of two phenotypes with random switching between cell proliferation and migration. The transport process is formulated in terms of the CTRW with an arbitrary waiting-time distribution law. Proliferation is modeled by a standard logistic growth. We apply hyperbolic scaling and Hamilton-Jacobi formalism to determine the overall rate of tumor cell invasion. In particular, we take into account both normal diffusion and anomalous transport (subdiffusion) in order to show that the standard diffusion approximation for migration leads to overestimation of the overall cancer spreading rate.     

Nuclear matter equation of state and σ-meson parameters

We try to determine phenomenologically the extent of in-medium modification of σ-meson parameters so that the saturation observables of the nuclear matter equation of state (EOS) are reproduced. To calculate the EOS we have used Brueckner-Bethe-Goldstone formalism with Bonn potential as two-body interaction. We find that it is possible to understand all the saturation observables, namely, saturation density, energy per nucleon and incompressibility, by incorporating in-medium modification of σ-meson-nucleon coupling constant and σ-meson mass by a few per cent.     

On the Generalized Langevin Equation for a Rouse Bead in a Nonequilibrium Bath

We present the reduced dynamics of a bead in a Rouse chain which is submerged in a bath containing a driving agent that renders it out-of-equilibrium. We first review the generalized Langevin equation of the middle bead in an equilibrated bath. Thereafter, we introduce two driving forces. Firstly, we add a constant force that is applied to the first bead of the chain. We investigate how the generalized Langevin equation changes due to this perturbation for which the system evolves towards a steady state after some time. Secondly, we consider the case of stochastic active forces which will drive the system to a nonequilibrium state. Including these active forces results in an extra contribution to the second fluctuation–dissipation relation. The form of this active contribution is analysed for the specific case of Gaussian, exponentially correlated active forces. We also discuss the resulting rich dynamics of the middle bead in which various regimes of normal diffusion, subdiffusion and superdiffusion can be present.     

Subdiffusion of mixed origin with chemical reactions

We derive a reaction-subdiffusion equation that takes into account two microscopic mechanisms responsible for subdiffusion in real media. We show that the concentration profiles in media with identical subdiffusion exponents but with different microscopic structures can differ significantly at the same chemical kinetics.     

Finite temperature dynamics of vortices in the two dimensional anisotropic Heisenberg model

We study the effects of finite temperature on the dynamics of non-planar vortices in the classical, two-dimensional anisotropic Heisenberg model with XY- or easy-plane symmetry. To this end, we analyze a generalized Landau-Lifshitz equation including additive white noise and Gilbert damping. Using a collective variable theory with no adjustable parameters we derive an equation of motion for the vortices with stochastic forces which are shown to represent white noise with an effective diffusion constant linearly dependent on temperature. We solve these stochastic equations of motion by means of a Green's function formalism and obtain the mean vortex trajectory and its variance. We find a non-standard time dependence for the variance of the components perpendicular to the driving force. We compare the analytical results with Langevin dynamics simulations and find a good agreement up to temperatures of the order of 25% of the Kosterlitz-Thouless transition temperature. Finally, we discuss the reasons why our approach is not appropriate for higher temperatures as well as the discreteness effects observed in the numerical simulations. Received: 27 April 1998 / Revised: 2 September 1998 / Accepted: 10 September 1998     

Transitions between modulated phases in an Ising model with third-neighbour interactions (mean-field approximation)

We study order–disorder transitions in a three-dimensional Ising lattice in which all the spins belonging to the same xy plane have the same degree of disorder, so that the structure can be effectively reduced to a chain of layers. The layers interact with each other up to third neighbours. Employing the mean-field approximation, we find the different configurations that undergo the transition to total disorder in terms of the interaction constants and work out a diagram displaying the possible sequences of modulated phases that can be found when the temperature goes from 0 to the order–disorder transition point. At intermediate temperatures the average values of the spins of the layers for periodic structures are found by solving an equation system. Substitution of these values into the expression of the free energy allows one to determine the most stable structure for each set of interaction constants and for each temperature. The model predicts a transition between two modulated structures with the same wavelength but different unit cells, for suitable values of the interaction constants. The formalism is also applied to substances like UNi₂Si₂, with only a partial agreement with experiment.     

Perfect fluid quantum Universe in the presence of negative cosmological constant

We present perfect fluid Friedmann–Robertson–Walker quantum cosmological models in the presence of negative cosmological constant. In this work the Schutz’s variational formalism is applied for radiation, dust, cosmic string, and domain wall dominated Universes with positive, negative, and zero constant spatial curvature. In this approach the notion of time can be recovered. These give rise to Wheeler–DeWitt equations for the scale factor. We find their eigenvalues and eigenfunctions by using Spectral Method. After that, we use the eigenfunctions in order to construct wave packets for each case and evaluate the time-dependent expectation value of the scale factors, which are found to oscillate between finite maximum and minimum values. Since the expectation values of the scale factors never tends to the singular point, we have an initial indication that these models may not have singularities at the quantum level.     

Particle creation,classicality and related issues in quantum field theory: I. Formalism and toy models

The quantum theory of a harmonic oscillator with a time dependent frequency arises in several important physical problems, especially in the study of quantum field theory in an external background. While the mathematics of this system is straightforward, several conceptual issues arise in such a study. We present a general formalism to address some of the conceptual issues like the emergence of classicality, definition of particle content, back reaction etc. In particular, we parameterize the wave function in terms of a complex number (which we call excitation parameter) and express all physically relevant quantities in terms it. Many of the notions—like those of particle number density, effective Lagrangian etc., which are usually defined using asymptotic in–out states—are generalized as time-dependent concepts and we show that these generalized definitions lead to useful and reasonable results. Having developed the general formalism we apply it to several examples. Exact analytic expressions are found for a particular toy model and approximate analytic solutions are obtained in the extreme cases of adiabatic and highly non-adiabatic evolution. We then work out the exact results numerically for a variety of models and compare them with the analytic results and approximations. The formalism is useful in addressing the question of emergence of classicality of the quantum state, its relation to particle production and to clarify several conceptual issues related to this. In Paper II which is a sequel to this, the formalism will be applied to analyze the corresponding issues in the context of quantum field theory in background cosmological models and electric fields.     

Algebraic approaches in nuclear physics

In these lectures I discuss algebraic approaches in nuclear physics. The emphasis is on aspects that can not be found in the same form in other publications of the subject. We discuss the relation between bosons and fermions, as well as some of the standard formalism underlying all algebraic models in nuclear physics. We then contrast fermionic and bosonic models. We end with a case study on supersymmetry in superdeformed nuclei. Dedicated to the memory of Prof. P.W.M. Glaudemans, 1931–1998. Lecture given at the 10th Indian-Summer School of Nuclear Physics: “Theory of Many-Fermion Systems”, Prague, September 8–12, 1997.     

Generalized Langevin equation with fractional Gaussian noise: subdiffusion within a single protein molecule

By introducing fractional Gaussian noise into the generalized Langevin equation, the subdiffusion of a particle can be described as a stationary Gaussian process with analytical tractability. This model is capable of explaining the equilibrium fluctuation of the distance between an electron transfer donor and acceptor pair within a protein that spans a broad range of time scales, and is in excellent agreement with a single-molecule experiment.     

Aging in subdiffusion generated by a deterministic dynamical system

We investigate aging behavior in a simple dynamical system: a nonlinear map which generates subdiffusion deterministically. Asymptotic behaviors of the diffusion process are described using aging continuous time random walks. We show how these processes are described by an aging diffusion equation which is of fractional order. Our work demonstrates that aging behavior can be found in deterministic low dimensional dynamical systems.     

Subdiffusive axial transport of granular materials in a long drum mixer

Granular mixtures segregate radially by size when tumbled in a partially filled horizontal drum. The smaller component moves toward the axis of rotation and forms a buried core, which then splits into axial bands. Models have generally assumed that the axial segregation is opposed by diffusion. Using narrow pulses of the smaller component as initial conditions, we have characterized axial transport in the core. We find that the axial advance of the segregated core is well described by a self-similar concentration profile whose width scales as talpha, with alpha approximately 0.3<1/2. Thus, the process is subdiffusive rather than diffusive as previously assumed. We compare our results to two one-dimensional model equations which contain self-similarity and subdiffusion: a linear fractional diffusion model and the nonlinear porous medium equation.     

On the quantum-mechanical Fokker-Planck and Kramers-Chandrasekhar equation

In the classical theory of Brownian motion we can consider the Langevin equation as an infinitesimal transformation between the coordinates and momenta of a Brownian particle, given probabilistically, since the impulse appearing is characterized by a Gaussian random process. This probabilistic infinitesimal transformation generates a streaming on the distribution function, expressed by the classical Fokker-Planck and Kramers-Chandrasekhar equations. If the laws obeyed by the Brownian particle are quantum mechanical, we can reinterpret the Langevin equation as an operator relation expressing an infinitesimal transformation of these operators. Since the impulses are independent of the coordinates and momenta we can think of them as c numbers described by a Gaussian random process. The so resulting infinitesimal operator transformation induces a streaming on the density matrix. We may associate, according to Weyl functions with operators. The function associated with the density matrix is the Wigner function. Expressing, then, these operator relations in terms of these functions we can express the streaming as a continuity equation of the Wigner function. We find that in this parametrization the extra terms which appear are the same as in the classical theory, augmenting the usual Wigner equation.     

Boundary conditions for the subdiffusion equation

The boundary conditions for the subdiffusion equations are formulated using the continuous-time random walk model, as well as several versions of the random walk model on an irregular lattice. It is shown that the boundary conditions for the same equation in different models have different forms, and this difference considerably affects the solutions of this equation.     

Nonlinear mean field Fokker-Planck equations. Application to the chemotaxis of biological populations

We study a general class of nonlinear mean field Fokker-Planck equations in relation with an effective generalized thermodynamical (E.G.T.) formalism. We show that these equations describe several physical systems such as: chemotaxis of bacterial populations, Bose-Einstein condensation in the canonical ensemble, porous media, generalized Cahn-Hilliard equations, Kuramoto model, BMF model, Burgers equation, Smoluchowski-Poisson system for self-gravitating Brownian particles, Debye-Hückel theory of electrolytes, two-dimensional turbulence... In particular, we show that nonlinear mean field Fokker-Planck equations can provide generalized Keller-Segel models for the chemotaxis of biological populations. As an example, we introduce a new model of chemotaxis incorporating both effects of anomalous diffusion and exclusion principle (volume filling). Therefore, the notion of generalized thermodynamics can have applications for concrete physical systems. We also consider nonlinear mean field Fokker-Planck equations in phase space and show the passage from the generalized Kramers equation to the generalized Smoluchowski equation in a strong friction limit. Our formalism is simple and illustrated by several explicit examples corresponding to Boltzmann, Tsallis, Fermi-Dirac and Bose-Einstein entropies among others.     

Analytical results for periodically-driven two-level models in relation to Heun functions

Considering a homogeneous and isotropic Universe characterised by the Friedmann–Lemaître–Robertson–Walker line element, in this work, we have prescribed a general formalism for the cosmological solutions when the equation of state of the cosmic substance follows the general structure \(\phi (p, \rho ) = 0\), where \(p,\,\rho \) are respectively the pressure and the energy density of the cosmic substance. Using the general formalism we recover some well-known solutions, namely, when the cosmic substance obeys the linear equation of state, a Chaplygin-type equation of state, or a nonlinear equation of state. Thus, the current work offers a new technique to solve the cosmological solutions without any prior relation between p and \(\rho \).     

A multidimensional subdiffusion model： An arbitrage-free market

<正>To capture the subdiffusive characteristics of financial markets,the subordinated process,directed by the inverse Q-stale subordinator S_α(t) for 0 <α< 1,has been employed as the model of asset prices.In this article,we introduce a multidimensional subdiffusion model that has a bond and K correlated stocks.The stock price process is a multidimensional subdiffusion process directed by the inverse Q-stable subordinator.This model describes the period of stagnation for each stock and the behavior of the dependency between multiple stocks.Moreover,we derive the multidimensional fractional backward Kolmogorov equation for the subordinated process using the Laplace transform technique.Finally, using a martingale approach,we prove that the multidimensional subdiffusion model is arbitrage-free,and also gives an arbitrage-free pricing rule for contingent claims associated with the martingale measure.     

Limits on decaying dark energy density models from the CMB temperature–redshift relation

The nature of the dark energy is still a mystery and several models have been proposed to explain it. Here we consider a phenomenological model for dark energy decay into photons and particles as proposed by Lima (Phys Rev D 54:2571, 1996). He studied the thermodynamic aspects of decaying dark energy models in particular in the case of a continuous photon creation and/or disruption. Following his approach, we derive a temperature redshift relation for the cosmic microwave background (CMB) which depends on the effective equation of state w _eff and on the “adiabatic index” γ. Comparing our relation with the data on the CMB temperature as a function of the redshift obtained from Sunyaev–Zel’dovich observations and at higher redshift from quasar absorption line spectra, we find w _eff = −0.97 ± 0.03, adopting for the adiabatic index γ = 4/3, in good agreement with current estimates and still compatible with w _eff = −1, implying that the dark energy content being constant in time.