Mathematical Physics (MP)
http://hdl.handle.net/20.500.11824/16
Sat, 16 Feb 2019 07:01:15 GMT2019-02-16T07:01:15ZModeling anomalous heat diffusion: Comparing fractional derivative and non-linear diffusivity treatments
http://hdl.handle.net/20.500.11824/931
Modeling anomalous heat diffusion: Comparing fractional derivative and non-linear diffusivity treatments
Falcini F.; Garra R.; Voller V.
In the Fourier heat conduction equation, when the flux definition is expressed as the product of a constant diffusivity and the temperature gradient, the characteristic length scale evolves as the square root of time. However, if we replace the 1 st order transient and gradient terms in the Fourier equation with fractional derivatives and/or define a non-linear spatially dependent diffusivity, it is possible to generate an anomalous space-time scaling, i.e., a scaling where the time exponent differs from the expected value of 1/2 . To compare and contrast the possible consequences of using fractional calculus along with a non-linear flux, we investigate a space-time fractional heat diffusion equation that involves a non-linear diffusivity. Following presentation of the governing non-linear fractional equation, we arrive at a space-time scaling that accounts for the combined anomalous contributions of memory (fractional derivative in time), non-locality (fractional derivative in space), and a non-linear diffusivity. We demonstrate how this scaling can manifest in a physical setting by considering the analytical solution of a non-linear fractional space-time diffusion equation, a limit case Stefan problem related to moisture infiltration into a porous media. A direct physically realizable simulation of this process shows how the anomalous space-time scaling is explicitly related to measures of both the memory and non-linearity in the system. Overall, the findings from this work clearly show how the definition of a non-linear diffusivity might contribute to anomalous diffusion behavior and suggests that, in modeling a particular observation, the roles of fractional derivatives and a suitably defined non-linear diffusivity are interchangeable.
Thu, 01 Nov 2018 00:00:00 GMThttp://hdl.handle.net/20.500.11824/9312018-11-01T00:00:00ZIsotropic Bipolaron-Fermion-Exchange Theory and Unconventional Pairing in Cuprate Superconductors
http://hdl.handle.net/20.500.11824/930
Isotropic Bipolaron-Fermion-Exchange Theory and Unconventional Pairing in Cuprate Superconductors
Bru J.B.; de Siqueira Pedra W.; Delgado de Pasquale A.
The discovery of high-temperature superconductors in 1986 represented a major experimental breakthrough (Nobel Prize 1987), but their theoretical explanation is still a subject of much debate. These materials have many exotic properties, such as d- and p-wave pairing and density waves. The appearance of unconventional pairing is examined from a microscopic model, taking into account important properties of hole-doped copper oxides. Weconsider an exchange interaction between fermions and dominantly inter-site bipolarons to be the mechanism which leads to the pairing. We connect its momentum dependency to the well-established fermion-phonon anomalies in cuprate superconductors. Since charge carriers in these materials are strongly correlated, we add a screened Coulomb repulsion to this exchange term. We avoid any ad hoc assumptions like anisotropy, but rather provide a microscopic explanation of unconventional pairing for coupling strengths that are in accordance with experimental facts. One important outcome is a mathematically rigorous elucidation of the role of Coulomb repulsion in unconventional pairing, which is shown to be concomitant with a strong depletion of superconducting pairs. Our theory, applied to the special case
of LaSr 214, predicts at optimal doping (i) a coherence length of 21A, which is the same as that obtained from the Ginzburg-Landau critical magnetic field measured for this material, and (ii) d-wave pair formation in the pseudogap regime, i.e., at temperatures much higher thanthe superconducting transition temperature. We think that the understanding of pairing symmetry and the pseudogap phase are central issues in the theoretical comprehension of high-temperature superconductivity, with
possible technological applications like s-, d-, and p-wave Josephson junctions used nowadays in quantum computers.
Mon, 10 Dec 2018 00:00:00 GMThttp://hdl.handle.net/20.500.11824/9302018-12-10T00:00:00ZAccuracy of Classical Conductivity Theory at Atomic Scales for Free Fermions in Disordered Media
http://hdl.handle.net/20.500.11824/929
Accuracy of Classical Conductivity Theory at Atomic Scales for Free Fermions in Disordered Media
Aza N.J.B.; Bru J.B.; de Siqueira Pedra W.; Ratsimanetrimanana A.
The growing need for smaller electronic components has recently sparked the interest in the breakdown of the classical conductivity theory near the atomic scale, at which quantum effects should dominate. In 2012, experimental measurements of electric resistance of nanowires in Si doped with phosphorus atoms demonstrate that quantum effects on charge transport
almost disappear for nanowires of lengths larger than a few nanometers, even at very low temperature (4.2K). We mathematically prove, for non-interacting lattice fermions with disorder, that quantum uncertainty of microscopic electric current density around their (classical) macroscopic values is suppressed, exponentially fast with respect to the volume of the region of the lattice where an external electric field is applied. This is in accordance with the above experimental observation. Disorder is modeled
by a random external potential along with random, complex-valued, hopping amplitudes. The celebrated tight-binding Anderson model is one particular example of the general case considered here. Our mathematical analysis is based on Combes-Thomas estimates, the Akcoglu-Krengel ergodic theorem, and the large deviation formalism, in particular the GĂ¤rtner-Ellis theorem.
Tue, 22 Jan 2019 00:00:00 GMThttp://hdl.handle.net/20.500.11824/9292019-01-22T00:00:00ZDecay of Complex-time Determinantal and Pfaffian\ Correlation Functionals in Lattices
http://hdl.handle.net/20.500.11824/928
Decay of Complex-time Determinantal and Pfaffian\ Correlation Functionals in Lattices
Aza N.J.B.; Bru J.B.; de Siqueira Pedra W.
We supplement the determinantal and Pfaffian bounds of Sims and Warzel (Commun Math Phys 347:903--931, 2016) for many-body localization of quasi-free fermions, by considering the high dimensional case and complex-time correlations. Our proof uses the analyticity of correlation functions via the Hadamard three-line theorem. We show that the dynamical localization for the one-particle system yields the dynamical localization for the many-point fermionic correlation functions, with respect to the Hausdorff distance in the determinantal case. In Sims and Warzel (2016), a stronger notion of decay for many-particle configurations was used but only at dimension one and for real times. Considering determinantal and Pfaffian correlation functionals for complex times is important in the study of weakly interacting fermions.
Wed, 24 Jan 2018 00:00:00 GMThttp://hdl.handle.net/20.500.11824/9282018-01-24T00:00:00Z