Transient bi-fractional diffusion: space-time coupling inducing the coexistence of two fractional diffusions

Anomalous diffusion is researched within the framework of the coupled continuous time random walk model, in which the space-time coupling is considered through the correlated function g(t) ~ t ^γ , 0 ≤ γ< 2, and the probability density function ω(t) of a particle’s transition time t follows a power law for large t: ω(t) ~ t ^{? (1 + α)},1 <α< 2. The bi-fractional generalized master equation is derived analytically which can be applied to describe the transient bi-fractional diffusion phenomenon which is induced by the space-time coupling and the asymptotic behavior of ω(t). Numerical results show that for the transient bi-fractional diffusion, there is a transition from one fractional diffusion to another one in the diffusive process.     

Dielectric relaxations,magnetodielectric and magnetoelectric interactions in Bi0.6La0.4MnO3 ceramics

Complex permittivity ε*/ε₀ = ε′/ε₀–iε″/ε₀ of the bismuth–lanthanum manganite Bi_0.6La_0.4MnO₃ ceramics has been measured in the temperature range of 10–220 K at frequencies f = 20–10⁶ Hz and magnetic inductions B = 0–0.846 T. At a temperature of 80 K, the spectra ε′/ε₀(t) and ε″/ε₀(t) demonstrate the dielectric relaxation that is a superposition of contributions of several relaxation processes, each of which is dominant in its frequency range: I (f < 10³ Hz, II (10³ < f < 10⁵ Hz), and III (10⁵ < f < 10⁶ Hz). In the range of 10–120 K, anomalous behavior of ε′/ε₀(T) and ε″/ε₀(T) is observed near the temperature of the transition from the paramagnetic to ferromagnetic phase and is due to the Anderson localization of charge carrier on a spin disorder.     

Probing the sign-changeable interaction between dark energy and dark matter with current observations

We consider the models of vacuum energy interacting with cold dark matter in this study, in which the coupling can change sigh during the cosmological evolution. We parameterize the running coupling b by the form b(a) = b_0 a + b_e(1-a), where at the earlytime the coupling is given by a constant b_e and today the coupling is described by another constant b_0. We explore six specific models with(i) Q = b(a)H_0ρ_0,(ii) Q = b(a)H_0ρ_(de),(iii) Q = b(a)H_0ρ_c,(iv) Q = b(a)Hρ_0,(v) Q = b(a)Hρ_(de), and(vi) Q = b(a)Hρ_c.The current observational data sets we use to constrain the models include the JLA compilation of type Ia supernova data, the Planck 2015 distance priors data of cosmic microwave background observation, the baryon acoustic oscillations measurements,and the Hubble constant direct measurement. We find that, for all the models, we have b_0 0 and b_e 0 at around the 1σ level,and b_0 and b_e are in extremely strong anti-correlation. Our results show that the coupling changes sign during the evolution at about the 1σ level, i.e., the energy transfer is from dark matter to dark energy when dark matter dominates the universe and the energy transfer is from dark energy to dark matter when dark energy dominates the universe.     

Self-similar magnetic structures and magnetic flux “Giant” creep

The penetration of a magnetic flux into a type-II high-T _c superconductor occupying the half-space x > 0 is considered. At the superconductor surface, the magnetic field amplitude increases in accordance with the law b(0, t) = b ₀(1 + t)^m (in dimensionless coordinates), where m > 0. The velocity of penetration of vortices is determined in the regime of thermally activated magnetic flux flow: v = v ₀exp?ub;?(U _0/T )(1-b?b/?x)?ub;, where U ₀ is the effective pinning energy and T is the thermal energy of excited vortex filaments (or their bundles). magnetic flux “Giant” creep (for which U ₀/T? 1) is considered. The model Navier-Stokes equation is derived with nonlinear “viscosity” v ∝ U ₀/T and convection velocity v _f ∝ (1 ? U ₀/T). It is shown that motion of vortices is of the diffusion type for j → 0 (j is the current density). For finite current densities 0 < j < j _c, magnetic flux convection takes place, leading to an increase in the amplitude and depth of penetration of the magnetic field into the superconductor. It is shown that the solution to the model equation is finite at each instant (i.e., the magnetic flux penetrates to a finite depth). The penetration depth x _eff ^A (t) ∝ (1 + t)^{(1 + m/2)/2} of the magnetic field in the superconductor and the velocity of the wavefront, which increases linearly in exponent m, exponentially in temperature T, and decreases upon an increase in the effective pinning barrier, are determined. A distinguishing feature of the solutions is their self-similarity; i.e., dissipative magnetic structures emerging in the case of giant creep are invariant to transformations b(x, t) = β^m b(t/β, x/β^{(1 + m/2)/2}), where β > 0.     

Stabilizer of a function in the gage group

It is proved that, for the dimension d of the stabilizer of an analytic function z(x, y) in the gage pseudogroup G = {z(x, y) → c(z(a(x), b(y))}, there are precisely four possibilities: (1) d = ∞ and the complexity of z is zero, (2) d = 3 and the complexity of z is equal to one, (3) d = 1 and z is equivalent the function r(x + y) ? x of complexity two, (4) d = 0 in all remaining cases.     

Parabolic Anderson Model in a Dynamic Random Environment: Random Conductances

The parabolic Anderson model is defined as the partial differential equation ? u(x, t)/? t = κ Δ u(x, t) + ξ(x, t)u(x, t), x ∈ ? ^d , t ≥ 0, where κ ∈ [0, ∞) is the diffusion constant, Δ is the discrete Laplacian, and ξ is a dynamic random environment that drives the equation. The initial condition u(x, 0) = u ₀(x), x ∈ ? ^d , is typically taken to be non-negative and bounded. The solution of the parabolic Anderson equation describes the evolution of a field of particles performing independent simple random walks with binary branching: particles jump at rate 2d κ, split into two at rate ξ ∨ 0, and die at rate (?ξ) ∨ 0. In earlier work we looked at the Lyapunov exponents

$$ \lambda _{p}(\kappa ) = \lim\limits _{t\to \infty } \frac {1}{t} \log \mathbb {E} ([u(0,t)]^{p})^{1/p}, \quad p \in \mathbb{N} , \qquad \lambda _{0}(\kappa ) = \lim\limits _{t\to \infty } \frac {1}{t}\log u(0,t). $$

For the former we derived quantitative results on the κ-dependence for four choices of ξ : space-time white noise, independent simple random walks, the exclusion process and the voter model. For the latter we obtained qualitative results under certain space-time mixing conditions on ξ. In the present paper we investigate what happens when κΔ is replaced by Δ^??, where ?? = {??(x, y) : x, y ∈ ? _d , x ～ y} is a collection of random conductances between neighbouring sites replacing the constant conductances κ in the homogeneous model. We show that the associated annealed Lyapunov exponents λ _p (??), p ∈ ?, are given by the formula

$$ \lambda _{p}(\mathcal{K} ) = \text{sup} \{\lambda _{p}(\kappa ) : \, \kappa \in \text{Supp} (\mathcal{K} )\}, $$

where, for a fixed realisation of ??, Supp(??) is the set of values taken by the ??-field. We also show that for the associated quenched Lyapunov exponent λ ₀(??) this formula only provides a lower bound, and we conjecture that an upper bound holds when Supp(??) is replaced by its convex hull. Our proof is valid for three classes of reversible ξ, and for all ?? satisfying a certain clustering property, namely, there are arbitrarily large balls where ?? is almost constant and close to any value in Supp(??). What our result says is that the annealed Lyapunov exponents are controlled by those pockets of ?? where the conductances are close to the value that maximises the growth in the homogeneous setting. In contrast our conjecture says that the quenched Lyapunov exponent is controlled by a mixture of pockets of ?? where the conductances are nearly constant. Our proof is based on variational representations and confinement arguments.

     

Impurity transport in fractal media in the presence of a degrading diffusion barrier

We have analyzed the transport regimes and the asymptotic forms of the impurity concentration in a randomly inhomogeneous fractal medium in the case when an impurity source is surrounded by a weakly permeable degrading barrier. The systematization of transport regimes depends on the relation between the time t ₀ of emergence of impurity from the barrier and time t _* corresponding to the beginning of degradation. For t ₀ < t _*, degradation processes are immaterial. In the opposite situation, when t ₀ > t _*, the results on time intervals t < t _* can be formally reduced to the problem with a stationary barrier. The characteristics of regimes with t _* < t < t ₀ depend on the scenario of barrier degradation. For an exponentially fast scenario, the interval t _* < t < t ₀ is very narrow, and the transport regime occurring over time intervals t < t _* passes almost jumpwise to the regime of the problem without a barrier. In the slow power-law scenario, the transport over long time interval t _* < t < t ₀ occurs in a new regime, which is faster as compared to the problem with a stationary barrier, but slower than in the problem without a barrier. The asymptotic form of the concentration at large distances from the source over time intervals t < t ₀ has two steps, while for t > t ₀, it has only one step. The more remote step for t < t ₀ and the single step for t > t ₀ coincide with the asymptotic form in the problem without a barrier.     

On rational functions of first-class complexity

It is proved that, for every rational function of two variables P(x, y) of analytic complexity one, there is either a representation of the form f(a(x) + b(y)) or a representation of the form f(a(x)b(y)), where f(x), a(x), b(x) are nonconstant rational functions of a single variable. Here, if P(x, y) is a polynomial, then f(x), a(x), and b(x) are nonconstant polynomials of a single variable.