The multidimensional formulation of the quantum lattice Boltzmann (QLB) scheme is validated against analytical solutions of the time-dependent nonrelativistic Schrödinger equation in two and three spatial dimensions. As a result, we... more
The multidimensional formulation of the quantum lattice Boltzmann (QLB) scheme is validated against analytical solutions of the time-dependent nonrelativistic Schrödinger equation in two and three spatial dimensions. As a result, we demonstrate the viability of the quantum lattice Boltzmann scheme for the numerical solution of the time-dependent Schrödinger equation in multiple spatial dimensions. In addition, the links between the QLB scheme and sequential splitting methods for partial differential equations are also clarified.
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The phenomenon of Anderson localization in expanding one-dimensional Bose-Einstein condensates is investigated by numerically solving the Gross-Pitaevskii equation with a random speckle potential. To this purpose, a quantum lattice... more
The phenomenon of Anderson localization in expanding one-dimensional Bose-Einstein condensates is investigated by numerically solving the Gross-Pitaevskii equation with a random speckle potential. To this purpose, a quantum lattice Boltzmann (QLB) method is used, and compared with a standard Crank-Nicolson scheme. The QLB simulations show evidence of Anderson localization even for relatively low-energy condensates, with a healing length as large as one-tenth of the Thomas-Fermi length. Moreover, very long-time simulations, lasting up to 15 000 optical confinement periods, indicate that the Anderson localization degrades in time, although at a very slow pace. In particular, the inverse localization length is found to decay according to a t;{-1/3} law. This lends support to the idea that localized wave functions, although not strictly ground states, represent extremely long-lived metastable states of the expanding condensate.
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The multidimensional formulation of the quantum lattice Boltzmann (qLB) scheme is extended to the case of nonlinear quantum wave equations. More specifically, imaginary-time formulations of the qLB scheme are developed and applied to the... more
The multidimensional formulation of the quantum lattice Boltzmann (qLB) scheme is extended to the case of nonlinear quantum wave equations. More specifically, imaginary-time formulations of the qLB scheme are developed and applied to the numerical computation of the ground state of the Gross-Pitaevskii equation in one and two spatial dimensions. The calculation is validated through detailed comparison with other numerical methods, as well as with analytical results based on the Thomas-Fermi approximation. The linear scaling of the time-step size with the spatial mesh spacing, a distinctive feature of the present quantum kinetic approach, is also numerically demonstrated.
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The derivation,of the quantum,lattice Boltzmann,model,is reviewed,with special emphasis on recent developments of the model, namely, the extension to a multi-dimensional formulation,and the application to the computation,of the ground... more
The derivation,of the quantum,lattice Boltzmann,model,is reviewed,with special emphasis on recent developments of the model, namely, the extension to a multi-dimensional formulation,and the application to the computation,of the ground state of the Gross-Pitaevskii equation,(GPE). Numerical,results for the linear and non- linear Schr¨odinger,equation,and,for the ground,state solution of the GPE are also presented,and validated,against analytical results or other classical schemes,such as Crank-Nicholson. PACS: 02.70.-c, 03.65-w, 03.67.Lx Key words: Quantum lattice Boltzmann, multi-dimensions, imaginary-time model, linear and
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ABSTRACT In this article, a lattice Boltzmann model for two immiscible fluids with same density but a large viscosity difference is developed. The effect of surface tension is included. An indicator function is used to model the interface... more
ABSTRACT In this article, a lattice Boltzmann model for two immiscible fluids with same density but a large viscosity difference is developed. The effect of surface tension is included. An indicator function is used to model the interface motion and is updated by a lattice Boltzmann scheme. The macroscopic equation for this function is derived using multi-scale analysis. Numerical results are compared with theoretical and experimental ones for the problem of drop deformation under steady linear flows. A qualitative comparison with results given by a level contour reconstruction method for drop collision in three dimensions is also shown. Other numerical applications, such as flow past obstacles and a falling drop, are presented.
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ABSTRACT Following an idea first proposed by Penrose in 1996 to explain the problem of quantum state reduction as a gravitational effect, Moroz, Penrose and Tod1 have shown that quantum state reduction due to gravitational interactions... more
ABSTRACT Following an idea first proposed by Penrose in 1996 to explain the problem of quantum state reduction as a gravitational effect, Moroz, Penrose and Tod1 have shown that quantum state reduction due to gravitational interactions could take place in about one second for the case of 1011 nucleons. However, keeping 1011 nucleons together in a quantum macroscopic state does not appear to be feasible as yet. The closest physical system to such a situation is provided by Bose-Einstein condensates (BEC) with attractive interactions. We present numerical simulations of the Schrödinger-Newton equations, which show that an attractive BEC with 103 atoms would yield a decorrelation time of the order of 10-2 seconds. Hence, a "Penrose-like" reduction, induced by BEC attractive interaction instead of gravity, might be observable and possibly monitored in current BEC experiments with attractive interactions.
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ABSTRACT Formal analogies between the Car-Parrinello (CP) ab-initio molecular dynamics for quantum many-body systems, and the Lattice Boltzmann (LB) method for classical and quantum fluids, are pointed out. A theoretical scenario, whereby... more
ABSTRACT Formal analogies between the Car-Parrinello (CP) ab-initio molecular dynamics for quantum many-body systems, and the Lattice Boltzmann (LB) method for classical and quantum fluids, are pointed out. A theoretical scenario, whereby the quantum LB would be coupled to the CP framework to speed-up many-body quantum simulations, is also discussed, together with accompanying considerations on the computational efficiency of the prospective CP-LB scheme.
Food engineers need to know the amount of time it takes to freeze, chill, or thaw their products and how much energy is required for that process. They also need to know the effect of storage, transport, or display conditions on the... more
Food engineers need to know the amount of time it takes to freeze, chill, or thaw their products and how much energy is required for that process. They also need to know the effect of storage, transport, or display conditions on the product temperatures. Currently experience plays a large role in identifying the time and energy needed, especially in situations where the food products can change from one day to the next. A reliable mathematical model can be of considerable help in optimizing a process and investigating the consequences of design changes.