Multiscale Ansatz Research Articles

Modelling diffusive transport of particles interacting with slit nanopore walls: The case of fullerenes in toluene filled alumina pores

Accurate modeling of diffusive transport of nanoparticles across nanopores is a particularly challenging problem. The reason is that for such narrow pores the large surface-to-volume ratio amplifies the relevance of the nanoscopic details and of the effective interactions at the interface with pore walls. Close to the pore wall, there is no clear separation between the length scales associated with molecular interactions, layering of the solvent at the interface with the pore and the particle size. Therefore, the standard hydrodynamic arguments may not apply and alternative solutions to determining average transport coefficients need to be developed. We here address this problem by offering a multiscale ansatz that uses effective potentials determined from molecular dynamics simulations to parametrise a four state stochastic model for the positional configuration of the particle in the pore. This is in turn combined with diffusivities in the centre of the pore and at the pore wall to calculate the average diffusion constant. We apply this model to the diffusion of fullerenes in a toluene filled slit nanopore and calculate the mean diffusion coefficient as a function of the pore size. We show that the accuracy of our model is affected by the partial slip of the toluene on the pore wall.

An efficient reformulation of a multiscale method for the eddy current problem

PurposeThis paper aims to improve the efficiency of a numerical method to treat the eddy current problem on a laminated material, where using a mesh that resolves each individual laminate would be too computationally expensive.Design/methodology/approachThe domain is modeled using a coarse mesh that treats the laminated material as a bulk with averaged properties. The fine-structured behavior is recovered by introducing micro-shape functions in the ansatz space. One such method is analyzed to find further model restrictions.FindingsBy using a special reformulation, it is possible to eliminate the additional degrees of freedom introduced by the multiscale ansatz at the cost of an additional modeling error that decreases with the laminate thickness.Originality/valueThe paper gives a computationally more efficient approximate variant to a known multiscale method.

Levitron: multi-scale analysis of stability

The Levitron is a typical system for complex Hamiltonian dynamics. Using accurate integrators numerical stability studies were done, especially with respect to variations of the magnetic moment μ for different initial positions x and z. For large μ, equivalent to stronger magnetic fields, the region of stable trajectories is splitted into two parts, whereas for small μ, only one stability region is observed. A linear ansatz is not sufficient to explain this splitting of regions. Vertical and transverse stability conditions have to be combined to understand this behaviour using a multi-scale ansatz. For different hole sizes in the magnetic base plate, the same behaviour appears. For larger holes one has to use stronger magnetic fields. Physically, the stability limits can be identified as critical gradients (forces) of the underlying potential. In x-direction, the stability boundaries are determined by a maximum x-gradient of the potential, which is allowed to act on the top. This derivative of the potential determines a force acting on the top. If this force in x-direction gets too strong, the top deviates from a stable trajectory and gets unstable.

MULTISCALE BORN-OPPENHEIMER THEORY OF COLLECTIVE ELECTRON-NUCLEAR DYNAMICS IN NANOSYSTEMS

Born-Oppenheimer theory is based on the separation in timescales between the nuclear and electron dynamics implied by the electron-to-nuclear mass ratio. This makes it naturally fit into a multiscale analysis. It is shown that a fully dynamical Born-Oppenheimer theory follows from a multiscale ansatz on the wave function and a Taylor expansion in the mass ratio. Allowing for a larger spatial scale of electron motion yields an understanding of boson, fermion, and more complex excitations that involve quasi-particles with an effective mass not equal to that of the electron. The theory involves a unified asymptotic expansion in a mass and length scale ratio, and preserves all many-body effects via accounting for the full strength of the interparticle forces. A novel mean-field theory emerges based on the fact that long-scale migration allows each electron to interact with many others on the space-time scale relevant to the coarse-grained equation. Implications for computational methods and applications to quantum nanosystems such as quantum dots, nanowires, superconducting nanoparticles, and liquid He droplets are discussed.

Interaction of modulated pulses in scalar multidimensional nonlinear lattices

We investigate the macroscopic dynamics of sets of an arbitrary finite number of weakly amplitude-modulated pulses in a multidimensional lattice of particles. The latter are assumed to exhibit scalar displacement under pairwise nonlinear interaction potentials of arbitrary range and are embedded in a nonlinear background field. By an appropriate multiscale ansatz, we derive formally the explicit evolution equations for the macroscopic amplitudes up to an arbitrarily high-order of the scaling parameter, thereby deducing the resonance and nonresonance conditions on the fixed wave vectors and frequencies of the pulses, which are required for that. The derived equations are justified rigorously in time intervals of macroscopic length. Finally, for sets of up to three pulses we present a complete list of all possible interactions and discuss their ramifications for the corresponding, explicitly given macroscopic systems.

High order multi-scale wall-laws, Part I: The periodic case

In this work we present new wall-laws boundary conditions including microscopic oscillations. We consider a newtonian flow in domains with periodic rough boundaries that we simplify considering a Laplace operator with periodic inflow and outflow boundary conditions. Following the previous approaches, see [A. Mikelic, W. Jager, J. Diff. Eqs, 170, 96-122, (2001) ] and [Y. Achdou, O. Pironneau, F. Valentin, J. Comput. Phys, 147, 1, 187-218, (1998)], we construct high order boundary layer approximations and rigorously justify their rates of convergence with respect to epsilon (the roughness' thickness). We establish mathematically a poor convergence rate for averaged second-order wall-laws as it was illustrated numerically for instance in [Y. Achdou, O. Pironneau, F. Valentin, J. Comput. Phys, 147, 1, 187-218, (1998)]. In comparison, we establish exponential error estimates in the case of explicit multi-scale ansatz. This motivates our study to derive implicit first order multi-scale wall-laws and to show that its rate of convergence is at least of order epsilon to the three halves. We provide a numerical assessment of the claims as well as a counter-example that evidences the impossibility of an averaged second order wall-law. Our paper may be seen as the first stone to derive efficient high order wall-laws boundary conditions.

Dispersive evolution of pulses in oscillator chains with general interaction potentials

We study the dispersive evolution of modulated pulses in a nonlinear oscillator chain embedded in a background field. The atoms of the chain interact pairwise with an arbitrary but finite number of neighbors. The pulses are modeled as macroscopic modulations of the exact spatiotemporally periodic solutions of the linearized model. The scaling of amplitude, space and time is chosen in such a way that we can describe how the envelope changes in time due to dispersive effects. By this multiscale ansatz we find that the macroscopic evolution of the amplitude is given by the nonlinear Schrödinger equation. The main part of the work is focused on the justification of the formally derived equation: We show that solutions which have initially the form of the assumed ansatz preserve this form over time-intervals with a positive macroscopic length. The proof is based on a normal-form transformation constructed in Fourier space, and the results depend on the validity of suitable nonresonance conditions.