A new fractional derivative involving the normalized sinc function without singular kernel

In this paper, a new fractional derivative involving the normalized sinc function without singular kernel is proposed. The Laplace transform is used to find the analytical solution of the anomalous heat-diffusion problems. The comparative results between classical and fractional-order operators are presented. The results are significant in the analysis of one-dimensional anomalous heat-transfer problems.     

Polarization of plane wave propagating inside elastic hexagonal system solids

Based on the reported physical parameters for hexagonal system solids,we have calculated the effects of anisotropy on polarization of plane P-wave propagation.Herein we report the results of calculations and the newly observed physical phenomena.It is found that,for a given propagation,if the polarization is parallel to the wave vector,so also to the Poynting vector.In such a case,the phase velocity is identical to the energy velocity;the quasi P-wave degenerates to a pure P-wave along the propagation.It is also noted that if the polarization is parallel to the Poynting vector but not to the wave vector,the propagating wave cannot be a pure P-wave.Furthermore,the polarization in a quasi P-wave may deviate from the wave vector for more than 45°,but the deviation from the Poynting vector is always less than 45°.The energy velocity of a quasi SV-wave can be larger than that of the quasi P-wave in some propagation directions,even though the phase velocity of a quasi SV-wave may never be larger than either the phase velocity or energy velocity of the quasi P-wave.Finally,in case of parameters ε=0 and δ*≠0,the polarization of a quasi P-wave has an observed symmetry at a 45°phase angle.The anisotropy of a hexagonal system solid determines if a pure P-wave can be created and what the propagation direction is for a plane wave propagating inside such a hexagonal system solid.     

Thermodynamic properties of the mixtures of some ionic liquids with alcohols using a simple equation of state

In the present work, we have used a simple equation of state called the GMA EoS to calculate the density of three ionic liquid mixtures including 1-butyl-3-methylimidazolum hexafluorophosphate, [BMIM] [PF₆] + methanol, 1-butyl-3-methylimidazolum tetrafluoroborate, [BMIM] [BF₄] + methanol, and [BMIM] [BF₄] + ethanol at different temperatures, pressures, and compositions. The isothermal compressibility, excess molar volumes, and excess Gibbs molar energy of these mixtures have been computed using this equation of state. The values of statistical parameters show that the GMA EoS can predict these thermodynamic properties very well within the experimental errors. The results show that isothermal compressibility of ionic liquids is lower than alcohols and the effect of temperature and pressure on the isothermal compressibility of ionic liquids is lower than alcohols. The excess molar volumes and excess molar Gibbs energy for these ionic liquid mixtures with alcohols are all negative at various temperatures and pressures over the whole composition range. The results have been interpreted in terms of intermolecular interactions and structural factors of the ionic liquids and alcohols.     

Ideal Mixture Approximation of Cluster Size Distributions at Low Density

We consider an interacting particle system in continuous configuration space. The pair interaction has an attractive part. We show that, at low density, the system behaves approximately like an ideal mixture of clusters (droplets): we prove rigorous bounds (a) for the constrained free energy associated with a given cluster size distribution, considered as an order parameter, (b) for the free energy, obtained by minimising over the order parameter, and (c) for the minimising cluster size distributions. It is known that, under suitable assumptions, the ideal mixture has a transition from a gas phase to a condensed phase as the density is varied; our bounds hold both in the gas phase and in the coexistence region of the ideal mixture. The present paper improves our earlier results by taking into account the mixing entropy.     

Further Implications of the Bessis–Moussa–Villani Conjecture

We find further implications of the BMV conjecture, which states that for hermitian matrices B≥0 and A, the function is the Laplace transform of a positive measure supported on [0,∞].     

Ergodicity of a Single Particle Confined in a Nanopore

We analyze the dynamics of a gas particle moving through a nanopore of adjustable width with particular emphasis on ergodicity. We give a measure of the portion of phase space that is characterized by quasiperiodic trajectories which break ergodicity. The interactions between particle and wall atoms are mediated by a Lennard-Jones potential, so that an analytical treatment of the dynamics is not feasible, but making the system more physically realistic. In view of recent studies, which proved non-ergodicity for systems with scatterers interacting via smooth potentials, we find that the non-ergodic component of the phase space for energy levels typical of experiments, is surprisingly small, i.e. we conclude that the ergodic hypothesis is a reasonable approximation even for a single particle trapped in a nanopore. Due to the numerical scope of this work, our focus will be the onset of ergodic behavior which is evident on time scales accessible to simulations and experimental observations rather than ergodicity in the infinite time limit.