Geodesics of Random Riemannian Metrics

We analyze the disordered Riemannian geometry resulting from random perturbations of the Euclidean metric. We focus on geodesics, the paths traced out by a particle traveling in this quenched random environment. By taking the point of the view of the particle, we show that the law of its observed environment is absolutely continuous with respect to the law of the random metric, and we provide an explicit form for its Radon–Nikodym derivative. We use this result to prove a “local Markov property” along an unbounded geodesic, demonstrating that it eventually encounters any type of geometric phenomenon. We also develop in this paper some general results on conditional Gaussian measures. Our Main Theorem states that a geodesic chosen with random initial conditions (chosen independently of the metric) is almost surely not minimizing. To demonstrate this, we show that a minimizing geodesic is guaranteed to eventually pass over a certain “bump surface,” which locally has constant positive curvature. By using Jacobi fields, we show that this is sufficient to destabilize the minimizing property.     

On the Material Invariant Formulation of Maxwell’s Displacement Current

Maxwell accounted for the apparent elastic behavior of the electromagnetic field by augmenting Ampere’s law with the so-called displacement current, in much the same way that he treated the viscoelasticity of gases. Maxwell’s original constitutive relations for both electrodynamics and fluid dynamics were not material invariant. In the theory of viscoelastic fluids, the situation was later corrected by Oldroyd, who introduced the upper-convective derivative. Assuming that the electromagnetic field should follow the general requirements for a material field, we show that if the upper convected derivative is used in place of the partial time derivative in the displacement current term, Maxwell’s electrodynamics becomes material invariant. Note, that the material invariance of Faraday’s law is automatically established if the Lorentz force is admitted as an integral part of the model. The new formulation ensures that the equation for conservation of charge is also material invariant in vacuo. The viscoelastic medium whose apparent manifestation are the known phenomena of electrodynamics is called here the metacontinuum.     

Charged black holes in the infinite derivative theory of gravity

The infinite derivative theory of gravity is a generalization of Einstein gravity with many interesting properties,but the black hole solutions in this theory are still not fully understood.In the paper,we concentrate on studying the charged black holes in such a theory.Adding the electromagnetic field part to the effective action,we show how the black hole solutions around the Reissner-Nordstrom metric can be solved perturbatively and iteratively.We further calculate the corresponding temperature,entropy and electrostatic potential of the black holes and verify the first law of thermodynamics.     

Force law in material media,hidden momentum and quantum phases

We address to the force law in classical electrodynamics of material media, paying attention on the force term due to time variation of hidden momentum of magnetic dipoles. We highlight that the emergence of this force component is required by the general theorem, deriving zero total momentum for any static configuration of charges/currents. At the same time, we disclose the impossibility to add this force term covariantly to the Lorentz force law in material media. We further show that the adoption of the Einstein–Laub force law does not resolve the issue, because for a small electric/magnetic dipole, the density of Einstein–Laub force integrates exactly to the same equation, like the Lorentz force with the inclusion of hidden momentum contribution. Thus, none of the available expressions for the force on a moving dipole is compatible with the relativistic transformation of force, and we support this statement with a number of particular examples. In this respect, we suggest applying the Lagrangian approach to the derivation of the force law in a magnetized/polarized medium. In the framework of this approach we obtain the novel expression for the force on a small electric/magnetic dipole, with the novel expression for its generalized momentum. The latter expression implies two novel quantum effects with non-topological phases, when an electric dipole is moving in an electric field, and when a magnetic dipole is moving in a magnetic field. These phases, in general, are not related to dynamical effects, because they are not equal to zero, when the classical force on a dipole is vanishing. The implications of the obtained results are discussed.     

One-loop induced topological term in QED3

We consider the 2+1 dimensional massive QED and discuss the induced topological term in the one-loop level by using the higher derivative regularization. We show that the higher derivative regularization, which manifestly preserves the parity invariance when the fermion mass vanishes, automatically leads to a nonvanishing topological term in low energies. Although this higher derivative regularization formally violates the local gauge invariance, we propose a systematic way to recover the gauge invariance by adding local counter terms dictated by Ward-Takahashi identities. In practical applications, this regularization is interesting in connection with the discussion of the dynamical origin of the quantum Hall effect.     

Non-exponential extinction of radiation by fractional calculus modelling

Possible deviations from exponential attenuation of radiation in a random medium have been recently studied in several works. These deviations from the classical Beer–Lambert law were justified from a stochastic point of view by Kostinski (2001) [1]. In his model he introduced the spatial correlation among the random variables, i.e. a space memory. In this note we introduce a different approach, including a memory formalism in the classical Beer–Lambert law through fractional calculus modelling. We find a generalized Beer–Lambert law in which the exponential memoryless extinction is only a special case of non-exponential extinction solutions described by Mittag–Leffler functions. We also justify this result from a stochastic point of view, using the space fractional Poisson process. Moreover, we discuss some concrete advantages of this approach from an experimental point of view, giving an estimate of the deviation from exponential extinction law, varying the optical depth. This is also an interesting model to understand the meaning of fractional derivative as an instrument to transmit randomness of microscopic dynamics to the macroscopic scale.     

On the process of filtration of fractional viscoelastic liquid food

In the process of filtration, fluid impurities precipitate/accumulate; this results in an uneven inner wall of the filter, consequently leading to non-uniform suction/injection. The Riemannian–Liouville fractional derivative model is used to investigate viscoelastic incompressible liquid food flowing through a permeable plate and to generalize Fick’s law. Moreover, we consider steady-state mass balance during ultrafiltration on a plate surface, and a fractional-order concentration boundary condition is established, thereby rendering the problem real and complex. The governing equation is numerically solved using the finite difference algorithm. The effects of the fractional constitutive models, generalized Reynolds number, generalized Schmidt number, and permeability parameter on the velocity and concentration fields are compared. The results show that an increase in fractionalorderαin the momentum equation leads to a decrease in the horizontal velocity. Anomalous diffusion described by the fractional derivative model weakens the mass transfer; therefore, the concentration decreases with increasing fractional derivativeγin the concentration equation.     

Motion of matter in metric-affine theory of gravitation

Based on the Lie derivative technique in a general space with affine connection (L₄, g), we show that in the metric-affine theory of gravitation, the law of conservation of the energy-momentum tensor for matter and consequently also the equations of motion for matter stemming from this law are (as in the general theory of relativity) a consequence of the gravitational field equations. We derive the hydrodynamic equation of motion for an ideal Weyssenhoff—Raabe spin fluid in Weyl space. We discuss the possibilities for observation of space—time nonmetricity.Moscow State Pedagogical University. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 1, pp. 76–82, January, 1994.     

Experimental evidence for resonant tunneling in a luttinger liquid

We have measured the low-temperature conductance of a one-dimensional island embedded in a single mode quantum wire. The quantum wire is fabricated using the cleaved edge overgrowth technique and the tunneling is through a single state of the island. Our results show that while the resonance line shape fits the derivative of the Fermi function the intrinsic linewidth decreases in a power law fashion as the temperature is reduced. This behavior agrees quantitatively with Furusaki's model for resonant tunneling in a Luttinger liquid.     

Fractional diffusion with two time scales

Moving particles that rest along their trajectory lead to time-fractional diffusion equations for the scaling limit distributions. For power law waiting times with infinite mean, the equation contains a fractional time derivative of order between 0 and 1. For finite mean waiting times, the most revealing approach is to employ two time scales, one for the mean and another for deviations from the mean. For finite mean power law waiting times, the resulting equation contains a first derivative as well as a derivative of order between 1 and 2. Finite variance waiting times lead to a second-order partial differential equation in time. In this article we investigate the various solutions with regard to moment growth and scaling properties, and show that even infinite mean waiting times do not necessarily induce subdiffusion, but can lead to super-diffusion if the jump distribution has non-zero mean.     

Series expansion solutions for the multi-term time and space fractional partial differential equations in two- and three-dimensions

Fractional partial differential equations with more than one fractional derivative in time describe some important physical phenomena, such as the telegraph equation, the power law wave equation, or the Szabo wave equation. In this paper, we consider two- and three-dimensional multi-term time and space fractional partial differential equations. The multi-term time-fractional derivative is defined in the Caputo sense, whose order belongs to the interval (1,2],(2,3],(3,4] or (0,m], and the space-fractional derivative is referred to as the fractional Laplacian form. We derive series expansion solutions based on a spectral representation of the Laplacian operator on a bounded region. Some applications are given for the two- and three-dimensional telegraph equation, power law wave equation and Szabo wave equation.     

The case of escape probability as linear in short time

We derive rigorously the short-time escape probability of a quantum particle from its compactly supported initial state, which has a discontinuous derivative at the boundary of the support. We show that this probability is linear in time, which seems to be a new result. The novelty of our calculation is the inclusion of the boundary layer of the propagated wave function formed outside the initial support. This result has applications to the decay law of the particle, to the Zeno behaviour, quantum absorption, time of arrival, quantum measurements, and more.     

Pressure dependence of the melting temperature of rare-gas solids

According to Lindemann's law and the Debye model and with the assumption that the volume derivative [Formula: see text] of the Grüineisen parameter [Formula: see text] is a constant depending on the material, we present a new expression for the analysis of the experimental data for the melting temperature of solids under a high pressure. The test on rare-gas solids (Ne, Ar, Kr and Xe) shows that the calculated results are in good agreement with the corresponding experimental data.     

First-order system least squares and the energetic variational approach for two-phase flow

This paper develops a first-order system least-squares (FOSLS) formulation for equations of two-phase flow. The main goal is to show that this discretization, along with numerical techniques such as nested iteration, algebraic multigrid, and adaptive local refinement, can be used to solve these types of complex fluid flow problems. In addition, from an energetic variational approach, it can be shown that an important quantity to preserve in a given simulation is the energy law. We discuss the energy law and inherent structure for two-phase flow using the Allen–Cahn interface model and indicate how it is related to other complex fluid models, such as magnetohydrodynamics. Finally, we show that, using the FOSLS framework, one can still satisfy the appropriate energy law globally while using well-known numerical techniques.     

Determination of true temperature of opaque materials via spectral distribution of thermal radiation intensity: application of Wien’s displacement law

The equation for the derivative connecting surface spectral emissivity, wavelength, and thermodynamic (true) temperature of an opaque heated body at the point of spectral maximum of thermal radiation was obtained. It is suggested to solve the problem of determining the true temperature of an opaque surface in two stages. At the first stage, the spectral range, most comfortable for approximation of body emissivity, is distinguished using a special function (relative emissivity), and the true temperature is determined. At the second stage, the true temperature is determined again using the resulting equation for the derivative. The dimensionless parameter that connects the radiative properties of material with the peak wavelength and characterizes deviation from Wien’s displacement law was found. If the absolute value of this parameter is low, the value of true temperature obtained at the first step can be specified at the second stage. This approach is illustrated by experimental data obtained at comparison of spectral radiance of the temperature lamps.     

Linear Response Function for Coupled Hyperbolic Attractors

The SRB measures of a hyperbolic system are widely accepted as the measures that are physically relevant. It has been shown by Ruelle that they depend smoothly on the system. Furthermore, Ruelle showed by a separate argument that the first derivative, i.e., the linear response function, admits a geometric interpretation. In this paper, we consider thermodynamic limits of SRB measures in lattices of coupled hyperbolic attractors. In a previous paper, using Markov partitions and thermodynamic formalism, we had established the smooth dependence of thermodynamic limits of SRB measures. Here, we establish that the linear response function admits a geometric interpretation. The formula is analogous to the one found by Ruelle for finite dimensional systems if one term is reinterpreted appropriately. We show that the limiting derivative is the thermodynamic limit of the derivatives in finite volume. We also obtain similar results for the derivatives of the entropy. Supported in part by NSF grants.     

A fractional order optimal 4D chaotic financial model with Mittag-Leffler law

In this paper, a fractional 4D chaotic financial model with optimal control is investigated. The fractional derivative used in this financial model is Atangana–Baleanu derivative. The existence and uniqueness conditions of solutions for the proposed model are derived based on Mittag-Leffler law. Optimal control is incorporated into the model to maximize output. The Adams–Moulton scheme of the Atangana–Baleanu derivative is utilized to obtain the numerical results which produce new attractors. Euler-Lagrange optimality conditions are determined for the fractional 4D chaotic financial model. The numerical results show that the memory factor has a great influences on the dynamics of the model.     

Rigorous Generalization of Young's Law for Heterogeneous and Rough Substrates

We consider a SOS type model of interfaces on a substrate which is both heterogeneous and rough. We first show that, for appropriate values of the parameters, the differential wall tension that governs wetting on such a substrate satisfies a generalized law which combines both Cassie and Wenzel laws. Then in the case of an homogeneous substrate, we show that this differential wall tension satisfies either the Wenzel's law or the Cassie's law, according to the values of the parameters.     

Plausible black hole solutions in higher-derivative gravity

Here we obtain explicit black hole solutions in Extension Gravity models with high-order derivative terms, while the Lichnerowicz-type theorem simplifies our analysis by vanishing Ricci's scalar curvature. We find out two explicit static, spherical solutions that satisfy the presented action: the first one is the same usual Schwarzschild solution and the other one is the new non-Schwarzschild solution. It means that Schwarzschild's solution following the no-hair theorem can describe any black hole object on each gravity theory. Without considering the first law of thermodynamics for it, we show that the non-Schwarzschild solution is depending on its set of constants, and then we consider its entropy and other thermodynamic parameters for specific values of the constants.     

Higher derivative correction to Kaluza–Klein black hole solution

We investigate the attractor mechanism in a Kaluza–Klein black hole solution in the presence of higher derivative terms. In particular, we discuss the attractor behavior of static black holes by using the effective potential approach as well as the entropy function formalism. We consider different higher derivative terms with a general coupling to the moduli field. For the R ² theory, we use an effective potential approach, looking for solutions which are analytic near the horizon and showing that they exist and enjoy attractor behavior. The attractor point is determined by extremization of the modified effective potential at the horizon. We study the effect of the general higher derivative corrections of R ⁿ terms. Using the entropy function we define the modified effective potential and we find the conditions to have the attractor solution. In particular for a single charged Kaluza–Klein black hole solution we show that a higher derivative correction dresses the singularity for an appropriate coupling, and we can find the attractor solution.