Diffusion problems on fractional nonlocal media

In this paper, the nonlocal diffusion in one-dimensional continua is investigated by means of a fractional calculus approach. The problem is set on finite spatial domains and it is faced numerically by means of fractional finite differences, both for what concerns the transient and the steady-state regimes. Nonlinear deviations from classical solutions are observed. Furthermore, it is shown that fractional operators possess a clear physical-mechanical meaning, representing conductors, whose conductance decays as a power-law of the distance, connecting non-adjacent points of the body.     

Phenomenological approach to classical and quantum relaxation of the harmonic oscillator

A phenomenological approach for construction of the Markovian master equation describing the relaxation of the harmonic oscillator to the equilibrium at temperatureT is presented. A superoperator formalism built up by analogy with the second quantization method of quantum mechanics is proposed for the formulation of both the classical and quantum master equations. The quantum-field-theoretical methods are used for the calculation of the mean values of the physical quantities and for the solution of the master equations.     

A fractional order diffusion-wave equation for time-dispersion media

Electromagnetic fields in time-dispersion media with a power-law dependence on time are analyzed. It is shown that these media are fractal and their fractal dimension is determined. Equations for scalar and vector potentials are derived using analogues of Maxwell??s equations for these types of media with the use of Caputo fractional derivatives. Electromagnetic fields in a bounded domain are numerically calculated for arbitrary functions of charge and current.     

Competing classical and quantum effects in shape relaxation of a metallic island

Pb islands grown on a Si substrate transform at room temperature from a flattop facet geometry into an unusual ring shape. The volume-preserving mass transport is catalyzed by the electrical field from the tip of a scanning tunneling microscope. The ring morphology results from the competing classical and quantum effects in the shape relaxation. The latter is enhanced by the large anisotropy of the effective mass, and leads to a sequential strip-flow growth on alternating strips of the same facet defined by substrate steps, showing its dynamical impact on the stability of a nanostructure.     

Magnetic long-range order induced by quantum relaxation in single-molecule magnets

Can magnetic interactions between single-molecule magnets (SMMs) in a crystal establish long-range magnetic order at low temperatures deep in the quantum regime, where the only electron spin fluctuations are due to incoherent magnetic quantum tunneling (MQT)? Put inversely: can MQT provide the temperature dependent fluctuations needed to destroy the ordered state above some finite T(c), although it should basically itself be a T-independent process? Our experiments on two novel Mn4 SMMs provide a positive answer to the above, showing at the same time that MQT in the SMMs has to involve spin-lattice coupling at a relaxation rate equaling that predicted and observed recently for nuclear-spin-mediated quantum relaxation.     

Universal nuclear spin relaxation and long-range order in nematics strongly confined in mass fractal silica gels

We show how the low-frequency dependence of the proton spin-lattice relaxation time T1(nu) of octylcyanobiphenyl liquid crystals confined in high-density silica gels evidences a long-range order nematic phase in spite of the strong confinement and random disorder of the gels. The universal value and frequency dependence observed, T1(nu) proportional, variant nu(2/3), is interpreted within a relaxation model due to director fluctuations in nematic liquid crystals confined to mass fractal porous media. The model provides a relation T1(nu) proportional, variant nu(2-d/2), giving a reliable value of the structural fractal dimension d(f)=2.67 for all the host silica gels.     

Quantum and classical relaxation in the proton glass

The hydrogen-bond network formed from a crystalline solution of ferroelectric RbH2PO4 and antiferroelectric NH4H2PO4 demonstrates glassy behavior, with proton tunneling the dominant mechanism for relaxation at low temperature. We characterize the dielectric response over seven decades of frequency and quantitatively fit the long-time relaxation by directly measuring the local potential energy landscape via neutron Compton scattering. The collective motion of protons rearranges the hydrogen bonds in the network. By analogy with vortex tunneling in superconductors, we relate the logarithmic decay of the polarization to the quantum-mechanical action.     

Separating the classical and quantum information via quantum cloning

An application of quantum cloning to optimally interface a quantum system with a classical observer is presented; in particular, we describe a procedure to perform a minimal disturbance measurement on a single qubit by adopting a 1-->2 cloning machine followed by a generalized measurement on a single clone and the anticlone or on the two clones. Such a scheme can be applied to enhance the transmission fidelity over a lossy quantum channel.     

Connecting the quantum and classical worlds

By considering (non‐relativistic) quantum mechanics as it is done in practice in particular in condensed‐matter physics, it is argued that a deterministic, unitary time evolution within a chosen Hilbert space always has a limited scope, leaving a lot of room for embedding the quantum‐classical transition into our current theories without recurring to difficult‐to‐accept interpretations of quantum mechanics. Nonunitary projections to initial and final states, the breaking of time‐reversal symmetry, a change of Hilbert space, and the introduction of classical concepts such as external potentials or localized atomic nuclei are widespread in quantum mechanical calculations. Furthermore, quantum systems require classical environments that enable the symmetry breaking that is necessary for creating the atomic configurations of molecules and crystals. This paper argues that such classical environments are provided by finite‐temperature macroscopic systems in which the range of quantum correlations and entanglement is limited. This leads to classical behavior on larger scales, and to collapse‐like events in all dynamical processes that become coupled to the thermalized degrees of freedom.

     

Transfer processes in fractal media

An irreversible process in fractal media involves coupling relation between the space and the time. The present note displays how the fractional derivation has to be introduced to describe this effect. As a result the law of the chemical diffusion to a fractal is given.     

Superposition in quantum and classical mechanics

Using the mathematical notion of an entity to represent states in quantum and classical mechanics, we show that, in a strict sense, proper superpositions are possible in classical mechanics.Dedicated to the Memory of Charles H. Randall.     

Comparison of a fractal analysis and debye relaxation for the structural relaxation in amorphous metals

Abstract

Magnetic After-Effect isotherms have been measured on a metallic glass pre-annealed for different times. The results have been analyzed simultaneously by Debye relaxation functions leading to spectra of activation enthalpies and by extended exponential functions leading to a unique effective activation enthalpy. The thermodynamic activation parameters issued from both analyses compare well. The average values of the enthalpy spectra determined by assuming Debye relaxations are equal to the effective enthalpies resulting from the non-Debye relaxation.

Magnetische Nachwirkungs-Isothermen wurden an amorphen Legierungen gemessen, die mit verschiedenen Anlasszeiten vorbehandelt wurden. Die Ergebnisse wurden sowohl unter der Annahme von Debye-Relaxationsprozessen analysiert, die mit Aktivierungsspektren beschrieben wurden, als auch mit gestreckten Exponentialfunktionen, die zu einer einheitlichen effektiven Aktivierungsenthalpie führt. Die thermodynamischen Parameter von beiden Analysen sind gut vergleichbar. Die Mittelwerte der Aktivierungsenthalpie-Spektren, welche sich bei Zugrundelegung der Debye-Relaxationsprozesse ergeben, entsprechen den effektiven Enthalpien, die sich bei Anwendung der gestreckten Exponentialfunktionen ergeben.

Des isothermes de trainage magnétique ont été réalisées sur des echantillons de verres métalliques ayant subi des traitements thermiques de différentes durées. Les résultats ont été simultanément analysés par des processus de relaxation de Debye engendrant des distributions d'enthalpies d'activation, et par des exponentielles etirées possédant des enthalpies effectives. Les paramètres thermodynamiques issus des deux analyses sont cohérents et comparables. Les valeurs moyennes des distributions d'enthalpies résultant de la relaxation de Debye sont identiques aux enthalpies effectives qui découlent des exponentielles etirées.     

Diffusion of overdamped classical solitons

We study the diffusion and deformation of classical solitons coupled to thermal noise. The diffusion coefficient for kinds in the gf⁴ theory is predicted up to the second order in kT. The prediction is verified by numerical simulations. Multiskyrmions in the vector O(3) sigma model are studied within the same formalism. Thermal noise results in a diffusion on the multisoliton collective coordinate space (moduli space). There are entropic forces which tend, for example, to bind pairs of solitons into bi-solitonic molecules.