arXiv subjects

Mathieu Beau

Publications and source records attributed to Mathieu Beau.

29 records · Page 2Linked to original sources

arXiv2014

Théorie des champs des contraintes et déformations en relativité générale et expansion cosmologique: Theory of stress and strain fields in general relativity and cosmological expansion

In this article we propose to add stress-energy tensor to the Einstein equations, assuming that the matter-energy and the metric space-time is nothing but a continuous medium with some elastic properties. We first give a general expression of the stress tensor which is linearly related to the strain tensor. Then, we give the particular expression of the stress tensor for a spatially homnogeneous and isotropic cosmological medium. After that we derive the modified Friedmann equations. In first approximation, we end up with the usual term $Λg_{μν}$ , where the cosmological constant $Λ=K\varepsilon$ is related with the bulk modulus K and the relative variation of volume (dilatation). Then we derive corrections to the standard model in second approximation, which depend on these two new parameters.

gr-qc↗

arXiv2014

Rigorous investigation of the reduced density matrix for the ideal Bose gas in harmonic traps by a loop-gas-like approach

In this paper, we rigorously investigate the reduced density matrix (RDM) associated to the ideal Bose gas in harmonic traps. We present a method based on a sum-decomposition of the RDM allowing to treat not only the isotropic trap, but also general anisotropic traps. When focusing on the isotropic trap, the method is analogous to the loop-gas approach developed by W.J. Mullin in [38]. Turning to the case of anisotropic traps, we examine the RDM for some anisotropic trap models corresponding to some quasi-1D and quasi-2D regimes. For such models, we bring out an additional contribution in the local density of particles which arises from the mesoscopic loops. The close connection with the occurrence of generalized-BEC is discussed. Our loop-gas-like approach provides relevant information which can help guide numerical investigations on highly anisotropic systems based on the Path Integral Monte Carlo (PIMC) method.

math-ph↗

arXiv2014

Gaussian decay for a difference of traces of the Schrödinger semigroup associated to the isotropic harmonic oscillator

This paper deals with the derivation of a sharp estimate on the difference of traces of the one-parameter Schrödinger semigroup associated to the quantum isotropic harmonic oscillator. Denoting by $H_{\infty,κ}$ the self-adjoint realization in $L^{2}(\mathbb{R}^{d})$, $d \in \{1,2,3\}$ of the Schrödinger operator $-\frac{1}{2} Δ+ \frac{1}{2} κ^{2}\vert \bold{x}\vert^{2}$, $κ>0$ and by $H_{L,κ}$, $L>0$ the Dirichlet realization in $L^{2}(Λ_{L}^{d})$ where $Λ_{L}^{d}:=\{\bold{x} \in \mathbb{R}^{d}:- \frac{L}{2} < x_{l} < \frac{L}{2},\,l=1,\ldots,d\}$, we prove that the difference of traces $\mathrm{Tr}_{L^{2}(\mathbb{R}^{d})} \mathrm{e}^{-t H_{\infty,κ}} - \mathrm{Tr}_{L^{2}(Λ_{L}^{d})}\mathrm{e}^{-t H_{L,κ}}$, $t>0$ has for $L$ sufficiently large a Gaussian decay in $L$. Furthermore, the estimate that we derive is sharp in the two following senses: its behavior when $t \downarrow 0$ is similar to the one given by $\mathrm{Tr}_{L^{2}(\mathbb{R}^{d})}\ mathrm{e}^{-t H_{\infty,κ}} = (2\sinh( \fracκ{2}t))^{-d}$ and the exponential decay in $t$ arising from $\mathrm{Tr}_{L^{2}(\mathbb{R}^{d})}\mathrm{e}^{-t H_{\infty,κ}}$ when $t\uparrow \infty$ is preserved. For illustrative purposes, we give a simple application within the framework of quantum statistical mechanics.

math-ph↗

arXiv2014

Three-dimensional Quantum Slit Diffraction and Diffraction in Time

We study the quantum slit diffraction problem in three dimensions. In the treatment of diffraction of particles by a slit, it is usually assumed that the motion perpendicular to the slit is classical. Here we take into account the effect of the quantum nature of the motion perpendicular to the slit using the Green function approach [18]. We treat the diffraction of a Gaussian wave packet for general boundary conditions on the shutter. The difference between the standard and our three-dimensional slit diffraction models is analogous to the diffraction in time phenomenon introduced in [16]. We derive corrections to the standard formula for the diffraction pattern, and we point out situations in which this might be observable. In particular, we discuss the diffraction in space and time in the presence of gravity.

quant-ph↗

arXiv2013

Comment on higher derivative Lagrangians in relativistic theory

We discuss the consequences of higher derivative Lagrangians of the form $α_1 A_μ(x)\dot{x}^μ$, $α_2 G_μ(x)\ddot{x}^μ$, $α_3 B_μ(x)\dddot{x}^μ$, $α_4 K_μ(x)\ddddot{x}^μ$, $\cdots$, $U_{(n)μ}(x)x^{(n)μ}$ in relativistic theory. After establishing the equations of the motion of particles in these fields, we introduce the concept of the generalized induction principle assuming the coupling between the higher fields $U_{(n),μ}(x),\ n\geq1$ with the higher currents $j^{(n)μ}=ρ(x)x^{(n)μ}$, where $ρ(x)$ is the spatial density of mass or of electric charge. In addition, we discuss the analogy of the field $G_μ(x)$ with the gravitational field and its inclusion in the general relativity framework in the last section. This letter is an invitation to reflect on a generalisation of the concept of inertia and we also discuss this problem in the last section.

gr-qc↗

arXiv2012

Discrete-Time Path Distributions on Hilbert Space

We construct a path distribution representing the kinetic part of the Feynman path integral at discrete times similar to that defined by Thomas [1], but on a Hilbert space of paths rather than a nuclear sequence space. We also consider different boundary conditions and show that the discrete-time Feynman path integral is well-defined for suitably smooth potentials.

math-ph↗

arXiv2012

H2 molecule in strong magnetic fields

The Pauli-Hamiltonian of a molecule with fixed nuclei in a strong constant magnetic field is asymptotic, in norm-resolvent sense, to an effective Hamiltonian which has the form of a multi-particle Schrödinger operator with interactions given by one-dimensional δ-potentials. We study this effective Hamiltonian in the case of the H2 -molecule and establish existence of the ground state. We also show that the inter-nuclear equilibrium distance tends to 0 as the field-strength tends to infinity.

math-ph↗

arXiv2012

Feynman Path Integral approach to electron diffraction for one and two slits, analytical results

In this article we present an analytic solution of the famous problem of diffraction and interference of electrons through one and two slits (for simplicity, only the one-dimensional case is considered). In addition to exact formulas, we exhibit various approximations of the electron distribution which facilitate the interpretation of the results. Our derivation is based on the Feynman path integral formula and this work could therefore also serve as an interesting pedagogical introduction to Feynman's formulation of quantum mechanics for university students dealing with the foundations of quantum mechanics.

quant-ph↗

arXiv2010

The second critical point for the Perfect Bose gas in quasi-one-dimensional traps

We present a new model of quasi-one-dimensional trap with some unknown physical predictions about a second transition, including about a change in fractions of condensed coherence lengths due to the existence of a second critical temperature Tm < Tc. If this physical model is acceptable, we want to challenge experimental physicists in this regard.

cond-mat.quant-gas↗

arXiv2010

Quasi-one- and quasi-two-dimensional perfect Bose gas: the second critical density and generalised condensation

In this letter we discuss a relevance of the 3D Perfect Bose gas (PBG) condensation in extremely elongated vessels for the study of anisotropic condensate coherence and the "quasi-condensate". To this end we analyze the case of exponentially anisotropic (van den Berg) boxes, when there are two critical densities $ρ_c < ρ_m$ for a generalised Bose-Einstein Condensation (BEC). Here $ρ_c$ is the standard critical density for the PBG. We consider three examples of anisotropic geometry: slabs, squared beams and "cigars" to demonstrate that the "quasi-condensate" which exists in domain $ρ_c < ρ< ρ_m$ is in fact the van den Berg-Lewis-Pulé generalised condensation (vdBLP-GC) of the type III with no macroscopic occupation of any mode. We show that for the slab geometry the second critical density $ρ_m$ is a threshold between quasi- two-dimensional (quasi-2D) condensate and the three dimensional (3D) regime when there is a coexistence of the "quasi-condensate" with the standard one-mode BEC. On the other hand, in the case of squared beams and "cigars" geometries critical density $ρ_m$ separates quasi-1D and 3D regimes. We calculate the value of difference between $ρ_c, ρ_m$ (and between corresponding critical temperatures $T_m, T_c$) to show that observed space anisotropy of the condensate coherence can be described by a critical exponent $γ(T)$ related to the anisotropic ODLRO. We compare our calculations with physical results for extremely elongated traps that manifest "quasi-condensate".

cond-mat.quant-gas↗

arXiv2009

Scaling approach to existence of long cycles in Casimir boxes

We analyse the concept of generalized Bose-Einstein condensation (g-BEC), known since 1982 for the perfect Bose gas (PBG) in the Casimir (or anisotropic) boxes. Our aim is to establish a relation between this phenomenon and two concepts: the concept of long cycles and the Off-Diagonal-Long-Range-Order (ODLRO), which are usually considered as some adequate way to describe the standard BEC on the ground state for the cubic boxes. First we show that these three criterions are equivalent in this latter case. Then, basing on a scaling approach, we revise formu- lation of these concepts to prove that the classification of the g-BEC into three types I,II,III, implies a hierarchy of long cycles (depending on their size scale) as well as a hierarchy of ODLRO which depends on the coherence length of the condensate.

math-ph↗