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We show that the uncertainty principle product for the position and momentum operators for a system described by the Hamiltonian $ \hat H= \frac{\hat{p}^2}{2m} +\frac{1}{2} m \omega^2 \hat{x}^2+\frac{\mu}{2}(\hat x \hat p+ \hat p \hat x)$ where $\mu<\omega$ reads $\Delta x \Delta p\ge\frac{\hbar \omega}{2\sqrt{\omega^2-\mu^2}}$. All the values bellow this lower bound are thus quantum-mechanically forbidden. We construct the annihilation and creation operators for this system and we calculate the expectation values of the operators $\hat p$ and $\hat x$ with respect to the corresponding coherent states.

We analytically derive the diffusion coefficients that drive a system of $N$ coupled harmonic oscillators to an equilibrium state exhibiting persistent correlations. It is shown that the main effect of the latter consists in a renormalization of the natural frequencies and the friction coefficients of the oscillators. We find that the Einstein relation may be satisfied at low temperatures with frequency-dependent effective friction coefficients, provided that the physical constraints are fulfilled. We also investigate the entanglement evolution in a bipartite bosonic Bogoliubov system initially prepared in a thermal squeezed state. It is found that, in contrast to what one may expect, strong coupling slows down the entanglement sudden death, and for initially separable states, entanglement generation may occur.

The mean field approximation is used to investigate the general features of the dynamics of a two-level atom in a ferromagnetic lattice close to the Curie temperature. Various analytical and numerical results are obtained. We first linearize the lattice Hamiltonian, and we derive the self-consistency equation for the order parameter of the phase transition for arbitrary direction of the magnetic field. The reduced dynamics is deduced by tracing out the degrees of freedom of the lattice, which results in the reduction of the dynamics to that of an atom in an effective spin bath whose size is equal to the size of a unit cell of the lattice. It is found that the dephasing and the excited state occupation probability may be enhanced by applying the magnetic field along some specific directions. The dependence on the change of the temperature and the magnitude of spin is also investigated. It turns out that the increase of thermal fluctuations may reduce the occupation probability of the excited state. The entanglement of two such atoms that occupy non-adjacent cells is studied and its variation in time is found to be not much sensitive to the direction of the magnetic field. Entanglement sudden death and revival is shown to occur close to the critical temperature.

A group theoretic analysis of the compressible Navier-Stokes equations of an ideal gas are carried out. The 12-dimensional Lie symmetry group is computed. The commutation table and the Levi decomposition of its Lie algebra are presented. The equations are reduced and self-similar one-, two- and three-dimensional solutions are computed. Many of them are graphically illustrated.

We report on the derivation of the heat transport equation for nonmetals using a quantum Markovian master equation in Lindblad form. We first establish the equations of motion describing the time variation of the on-site energy of atoms in a one dimensional periodic chain that is coupled to a heat reservoir. In the continuum limit, the Fourier law of heat conduction naturally emerges, and the heat conductivity is explicitly obtained. It is found that the effect of the heat reservoir on the lattice is described by a heat source density that depends on the diffusion coefficients of the atoms. We show that the Markovian dynamics is equivalent to the long wavelength approximation for phonons, which is typical for the case of elastic solids. The high temperature limit is shown to reproduce the classical heat conduction equation.

In this paper we make an overview of results relating the recent "discoveries" in differential geometry, such as higher structures and differential graded manifolds with some natural problems coming from mechanics. We explain that a lot of classical differential geometric constructions in the context can be conveniently described using the language of Q-structures, and thus Q-structure preserving integrators are potentially of great use in mechanics. We give some hints how the latter can be constructed, and formulate some open problems. Since the text is intended both to mathematics and mechanics communities, we tried to make it accessible to non-geometers as well., Comment: 20 pages, 2 figures

A stability analysis of the Borel-Laplace series summation technique, used as explicit time integrator, is carried out. Its numerical performance on stiff and non-stiff problems is analyzed. Applications to ordinary and partial differential equations are presented. The results are compared with those of many popular schemes designed for stiff and non-stiff equations.

This article introduces a new and general construction of discrete Hodge operator in the context of Discrete Exterior Calculus (DEC). This discrete Hodge operator enables to circumvent the well-centeredness limitation on the mesh with the popular diagonal Hodge. It allows a dual mesh based on any interior point, such as the incenter or the barycenter. It opens the way towards mesh-optimized discrete Hodge operators. In the particular case of a well-centered triangulation, it reduces to the diagonal Hodge if the dual mesh is circumcentric. Based on an analytical development, this discrete Hodge does not make use of Whitney forms, and is exact on piecewise constant forms, whichever interior point is chosen for the construction of the dual mesh. Numerical tests oriented to the resolution of fluid mechanics problems and thermal transfer are carried out. Convergence on various types of mesh is investigated.

The decay rate of an interstitial two-level spin impurity, located in the center of a unit cell of an anisotropic ferromagnetic lattice subjected to an external magnetic field is derived. The impurity is coupled to nearest-neighbor spins through Heisenberg $XY$ interaction. By mapping the lattice spin operators using the Holstein-Primakoff transformation, we establish the similarity with the Fano-Anderson model at low temperatures, and we calculate the retarded Green's function in one and two dimensions analytically for arbitrary coupling strength. It is shown that the reduced density matrix of the impurity satisfies an exact master equation in Lindblad form, from which the decay rate and the Lamb shift are deduced. The evolution in time of the latter together with the excited state occupation probability is investigated and its dependence on the applied magnetic field is discussed. It is found that there exists a critical resonance-like value of the magnetic field around which the behavior of the decay rate and the density matrix changes drastically. The Markovian decay law, as given by the Fermi golden rule, does not hold in the weak-coupling regime unless the magnetic field is weak, typically less than the critical value. The weak-coupling regime is further treated perturbatively up to second order, and the obtained results are compared with the exact solution. We also discuss the Zeno regime of the dynamics, where it is shown that at short times, the effective decay rate is twice as small as the exact decay rate, and that when the impurity energy lies outside the lattice continuum, the measurement speeds up the decay of the survival probability., Comment: 18 pages, close to the published version

The interpolation on Grassmann manifolds in the framework of parametric evolution partial differential equations is presented. Interpolation points on the Grassmann manifold are the subspaces spanned by the POD bases of the available solutions corresponding to the chosen parameter values. The well-known Neville-Aitken's algorithm is extended to Grassmann manifold, where interpolation is performed in a recursive way via the geodesic barycenter of two points. The performances of the proposed method are illustrated through three independent CFD applications, namely: the Von Karman vortex shedding street, the lid-driven cavity with inflow and the flow induced by a rotating solid. The obtained numerical simulations are pertinent both in terms of the accuracy of results and the time computation., Comment: Preprint submitted to Journal of Computational Physics

We derive power series expansions for the magnetization, the internal energy, and the specific heat of the Heisenberg XX chain that are valid at low temperatures. The coefficients of the series obtained depend logarithmically on the fugacity. It is shown that depending on whether the magnetic field exceeds or not the critical point, the effects of either the coupling of the spins and the magnetic field can have different characters, as indicated by the different power laws established.

This article reviews some integrators particularly suitable for the numerical resolution of differential equations on a large time interval. Symplectic integrators are presented. Their stability on exponentially large time is shown through numerical examples. Next, Dirac integrators for constrained systems are exposed. An application on chaotic dynamics is presented. Lastly, for systems having no exploitable geometric structure, the Borel-Laplace integrator is presented. Numerical experiments on Hamiltonian and non-Hamiltonian systems are carried out, as well as on a partial differential equation. Keywords: Symplectic integrators, Dirac integrators, long-time stability, Borel summation, divergent series., Comment: 33 pages, 18 figures

In this note we describe how some objects from generalized geometry appear in the qualitative analysis and numerical simulation of mechanical systems. In particular we discuss double vector bundles and Dirac structures. It turns out that those objects can be naturally associated to systems with constraints -- we recall the mathematical construction in the context of so called implicit Lagrangian systems. We explain how they can be used to produce new numerical methods, that we call Dirac integrators. On a test example of a simple pendulum in a gravity field we compare the Dirac integrators with classical explicit and implicit methods, we pay special attention to conservation of constrains. Then, on a more advanced example of the Ziegler column we show that the choice of numerical methods can indeed affect the conclusions of qualitative analysis of the dynamics of mechanical systems. We also tell why we think that Dirac integrators are appropriate for this kind of systems by explaining the relation with the notions of geometric degree of non-conservativity and kinematic structural stability., Comment: 19 pages, 6 figures

The study of the decoherence of qubits in spin systems is almost restricted to environments whose constituents are spin-$\frac{1}{2}$ particles. In this paper we consider environments that are composed of particles of higher spin, and we investigate the consequences on the dynamics of a qubit coupled to such baths via Heisenberg $XY$ and Ising interactions. It is shown that while the short time decay in both cases gets faster as the magnitude of the spin increases, the asymptotic behavior exhibits an improvement of the suppression of the decoherence when the coupling is through Heisenberg $XY$ interactions. In the case of a transverse Ising model, we find that the mean field approximation breaks down for high values of the spin., Comment: Preprint; 27 pages, 8 figures

These short notes present to the reader (students, in particular) a concise approach to the derivation of the propagator of Hamiltonians with position-dependent kinetic energy. The formalism is applied to the von Roos Hamiltonian with arbitrary ordering ambiguity parameters, and a simple scheme to convert the problem to a constant-mass motion is presented. The motion in curved spaces is treated along the same lines, where a phase space formulation is used to derive the propagator for arbitrary discretization choices., Comment: 20 pages

We show that the probability distribution corresponding to a fully random tracial state of a system of spin-S particles satisfies a diffusion-like equation. The diffusion coefficient turns out to be equal to $S(S+1)/6$, where $S$ is the magnitude of the spin of each particle. We also present a bosonization scheme for the lowering and raising total spin operators., Comment: 10 pages

We report on a nontrivial bosonization scheme for spin operators. It is shown that in the large $N$ limit, at infinite temperature, the operators $\sum_{k=1}^N \hat s_{k\pm}/\sqrt{N}$ behave like the creation and annihilation operators, $a^\dag$ and $a$, corresponding to a harmonic oscillator in thermal equilibrium, whose temperature and frequency are related by $\hbar\omega/k_B T=\ln 3$. The $z$ component is found to be equivalent to the position variable of another harmonic oscillator occupying its ground Gaussian state at zero temperature. The obtained results are applied to the Heisenberg XY Hamiltonian at finite temperature., Comment: 12 pages, preprint, we have included a brief discussion of the antiferromagnetic case

The object of this paper is to investigate, classically and quantum mechanically, the relation existing between the position-dependent effective mass and damping-antidamping dynamics. The quantization of the equations of motion is carried out using the geometric interpretation of the motion, and we compare it with the one based on the ordering ambiguity scheme. Furthermore, we apply the obtained results to a Fermi gas of damped-antidamped particles, and we solve the Schr\"odinger equation for an exponentially increasing (decreasing) mass in the presence of the Morse potential., Comment: Preprint, 24 pages, 2 figures. We now derive the correct condition that should be satisfied by the derivative of the wave function at the discontinuity point and show that the reflexion coefficient is independent of $\eta$. Figure 1 and the corresponding discussion have been removed. Two references added. To be published in Journal of Physics A: Mathematical and Theoretical

The elements of the quantum mechanical diffusion matrix, leading to a Gibbs equilibrium state for a set of $N$ coupled quantum harmonic oscillators are derived within Lindblad's axiomatic approach. Consequences of the fundamental constraints on the quantum friction coefficients are discussed. We derive the equations of motion for the expectation values and variances, and we solve them analytically. We apply our results to the description of the charge and mass asymmetry coordinates in heavy-ion collisions, and we investigate the effect of dissipation on tunneling in sub-barrier processes., Comment: 29 pages, 10 figures

Exact analytical results for the dynamics of two interacting qubits each of which is embedded in its own spin star bath are presented. The time evolution of the concurrence and the purity of the two-qubit system is investigated for finite and infinite numbers of environmental spins. The effect of qubit-qubit interactions on the steady state of the central system is investigated, Comment: 24 pages, 7 figures; comments welcome

We derive analytical properties for the degeneracy $\nu(N,j)$ occurring in the decomposition $\bigoplus\limits_{j}^\frac{N}{2}\nu(N,j)\mathbb C^{2j+1}$ of the state space $\mathbb C^{2\otimes N}$. We also investigate the dynamics of two qubits coupled via Ising interactions to separate spin baths, and we study the thermodynamic limit., Comment: 14 pages, 6 figures, minor typographical errors fixed

The time evolution of a spin-1/2 particle under the influence of a locally applied external magnetic field, and interacting with anisotropic spin environment in thermal equilibrium at temperature $T$ is studied. The exact analytical form of the reduced density matrix of the central spin is calculated explicitly for finite number of bath spins. The case of an infinite number of environmental spins is investigated using the convergence of the rescaled bath operators to normal Gaussian random variables. In this limit, we derive the analytical form of the components of the Bloch vector for antiferromagnetic interactions within the bath, and we investigate the short-time and long-time behavior of reduced dynamics. The effect of the external magnetic field, the anisotropy and the temperature of the bath on the decoherence of the central spin is discussed., Comment: 17 pages, 13 figures (compressed), one table. To appear in Phys. Rev. B

Since they represent fundamental physical properties in turbulence (conservation laws, wall laws, Kolmogorov energy spectrum, ...), symmetries are used to analyse common turbulence models. A class of symmetry preserving turbulence models is proposed. This class is refined such that the models respect the second law of thermodynamics. Finally, an example of model belonging to the class is numerically tested., Comment: Published in SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) at http://www.emis.de/journals/SIGMA/