On the Three-Dimensional Central Moment Lattice Boltzmann Method

A three-dimensional (3D) lattice Boltzmann method based on central moments is derived. Two main elements are the local attractors in the collision term and the source terms representing the effect of external and/or self-consistent internal forces. For suitable choices of the orthogonal moment basis for the three-dimensional, twenty seven velocity (D3Q27), and, its subset, fifteen velocity (D3Q15) lattice models, attractors are expressed in terms of factorization of lower order moments as suggested in an earlier work; the corresponding source terms are specified to correctly influence lower order hydrodynamic fields, while avoiding aliasing effects for higher order moments. These are achieved by successively matching the corresponding continuous and discrete central moments at various orders, with the final expressions written in terms of raw moments via a transformation based on the binomial theorem. Furthermore, to alleviate the discrete effects with the source terms, they are treated to be temporally semi-implicit and second-order, with the implicitness subsequently removed by means of a transformation. As a result, the approach is frame-invariant by construction and its emergent dynamics describing fully 3D fluid motion in the presence of force fields is Galilean invariant. Numerical experiments for a set of benchmark problems demonstrate its accuracy.     

Correlation dimension of the sheared hard-disk Lorentz gas

The correlation dimension for the isokinetic Lorentz gas is calculated for hard disks using nonequilibrium molecular dynamics simulation. The trajectories are confined to a strange attractor embedded in a four-dimensional phase space—the additional degree of freedom having not been included properly until this work. This degree of freedom accounts for the explicit time dependence of the system (as quantified by the moving periodic cells under shear) and is significant because the collisions tend to synchronize with the periodic change of symmetry of the lattice at high shear rates.     

Symmetry-relations and dynamics of molecular crystals in the long wave limit

Symmetry relations for macroscopic constants are derived. Especially it is shown that the Voigt-symmetry holds for the indices of the elastic constants. This has been doubted several times for lattices of particles with additional degrees of freedom. In the second part a simple model for Br₂- and J₂-lattices is discussed. The intermolecular forces are assumed to be van-der-Waals-forces. The influence of the internal degrees of freedom on lattice dynamics is calculated especially for the elastic constants. Further the limiting frequencies forq?0 are given and compared with experimental values.     

Directed transport of coupled systems in symmetric periodic potentials

In this paper, we discuss the damped unidirectional motions of a coupled lattice in a periodic potential. Each particle in the lattice is subject to a time-periodic ac force. Our studies reveal that a directed transport process can be observed when the ac forces acting on the coupled lattice have a phase shift (mismatch). This directed motion is a collaboration of the coupling, the substrate potential, and the periodic force, which are all symmetric. The absence of any one of these three factors will not give rise to a directed current. We discuss the complex relations between the directed current and parameters in the system. Results in this paper can be accomplished in experiments. Moreover,our results can be generalized to the studies of directed transport processes in more complicated spatially extended systems.     

Dependence of the binding energy of the metal crystal lattice on the average number of conduction electrons

We establish the dependence of the electron binding energy in a separate isolated Wigner–Seitz cell of the metal crystal lattice on the average number of electrons located in this cell. The calculation is made using the modified Hellmann–Feynman theorem, which allows relating the eigenvalue of the steady-state Hamiltonian to the variation in its parameters that do not affect the degree of freedom of a system. As one of these parameters, we choose the average number of electrons in the cell. According to the calculated data, removal of 10–30% of electrons in monovalent metals leads to the crystal lattice fracture. The results obtained using the Hellmann–Feynman theorem are directly compared with the data of the jellium model.     

Effective actions from block diagonalization in lattice gauge theories

It is shown that the first step in Wilson's real space renormalization program for lattice gauge theories — namely the integration of internal degrees of freedom within one block — can be performed for “block-diagonalized” actions, which possess the proper continuum limit, just like the Wilson action. As a result, we obtain, on a lattice with double lattice spacing, an effective action which contains in the weak-coupling limit, in addition to the familiar plaquette terms, planar Wilson loops with six and eight links.     

Passage of a monomer through a nonrigid periodical substrate formed by noninteracting particles

The paper examines a classical system in one degree of freedom: a particle (monomer) interacting with a periodic lattice of independent, separated oscillators. The monomer can interact with oscillators via a short-range attractive potential force. The periodic lattice of oscillators may absorb the energy of the monomer launched at some initial velocity, but it does so in a very peculiar manner. The monomer velocity gradually decreases, approaching near some nonzero limit value. The limiting monomer velocities can assume discrete values only. This behavior of the monomer is accounted for by the existing resistance force that completely vanishes at certain monomer velocities.     

Rabi Spectroscopy and Sensitivity of a Floquet Engineered Optical Lattice Clock

We periodically modulate the lattice trapping potential of a87 Sr optical clock to Floquet engineer the clock transition. In the context of atomic gases in lattices, Floquet engineering has been used to shape the dispersion and topology of Bloch quasi-energy bands. Differently from these previous works manipulating the external(spatial) quasi-energies, we target the internal atomic degrees of freedom. We shape Floquet spin quasi-energies and measure their resonance profiles with Rabi spectroscopy. We provide the spectroscopic sensitivity of each band by measuring the Fisher information and show that this is not depleted by the Floquet dynamical modulation. The demonstration that the internal degrees of freedom can be selectively engineered by manipulating the external degrees of freedom inaugurates a novel device with potential applications in metrology, sensing and quantum simulations.     

Anharmonic effects of periodic molecular systems in the vibron model approximation

The vibron model approximation to take into account anharmonic effects in periodic systems is discussed. This is achieved by considering a simple one-dimensional molecular crystal with four degrees of freedom. In this case the lattice dynamical treatment is separated into two sets of modes, the modes associated to the molecular degrees of freedom and those corresponding to the motions of the centre of mass of the molecules. The non-interacting molecular modes are studied in detail, and analytical expressions for the energies and wave functions for the two-phonon manifold are obtained. A local-normal mode transition as a function of the interaction parameters is identified, similar to that in isolated molecular systems.     

Charges and Symmetries in Quantum Theories without Locality

We review some rigorous results (and include some new ones) on charges, symmetry breaking and related concepts in quantum theories without locality (micro-causality), relevant examples of which are quantum lattice systems, (nonrelativistic) many-body and lattice gauge theories. In particular, Goldstone's theorem and its generalizations (involving long-range forces) and Swieca's theorem on the connection between the absence of charged states and the existence of a mass gap are discussed.     

Exergy flows as bases of constructal law

How do macroscopic systems react when imposed to external forces? A recent analysis of Carnot’s theorem has pointed out that the systems do not convert all the inflow energy to work and dissipation, but that some energy will be incorporated to internal processes. These exergy flows appear as the heat exchanged with a second thermal reservoir of a thermodynamic general engine. Flows of energy from surroundings are driven by internal forces to the systemic process. Also, when the system has evolved to reach a stationary state, internal flows manifest themselves as internal processes. Here, the entropy generation extrema theorem is used to prove how the macroscopic system will react upon forces that are imposed by the external surroundings. This provides the link between the entropy generation extrema approach and the constructal law.     

Flexural vibration band gaps in a thin plate containing a periodic array of hemmed discs

Using a periodic expansion by means of the Bloch theorem, the flexural vibration band gaps are studied in a thin plate with two-dimensional ternary locally resonant structures, i.e. a thin epoxy plate containing a periodic square array of lead discs hemmed around by rubber. The full band gaps of flexural vibration in the thin plate are obtained within which sound and vibration will be forbidden. The numerical results are used to show how the width of the first full band gap depends on the radius ratio of lead disc to hemmed disc, filling fraction, lattice constant (distance between the centers of the nearest lead discs) and thickness of the thin plate. It is observed that the gap width can be changed a lot by modulating these physical parameters.     

Thermostating by Deterministic Scattering: The Periodic Lorentz Gas

We present a novel mechanism for thermalizing a system of particles in equilibrium and nonequilibrium situations, based on specifically modeling energy transfer at the boundaries via a microscopic collision process. We apply our method to the periodic Lorentz gas, where a point particle moves diffusively through an ensemble of hard disks arranged on a triangular lattice. First, collision rules are defined for this system in thermal equilibrium. They determine the velocity of the moving particle such that the system is deterministic, time-reversible, and microcanonical. These collision rules can systematically be adapted to the case where one associates arbitrarily many degrees of freedom to the disk, which here acts as a boundary. Subsequently, the system is investigated in nonequilibrium situations by applying an external field. We show that in the limit where the disk is endowed by infinitely many degrees of freedom it acts as a thermal reservoir yielding a well-defined nonequilibrium steady state. The characteristic properties of this state, as obtained from computer simulations, are finally compared to those of the so-called Gaussian thermostated driven Lorentz gas.     

A case study of the MSA approach to quantized polarizable media

The mean spherical approximation (MSA) has proved to be a very general and flexible method to analyze equilibrium statistical mechanical systems. In this note we test its accuracy against a simple one-dimensional model, i.e., a lattice gas of polarizable molecules where the internal degree of freedom is treated as quantized harmonic oscillators which interact via harmonic forces. This model can be solved exactly. We find a very good agreement between the MSA and exact solutions.2 The corresponding classical problem of polarizable particles was first solved in a mean spherical approximation (MSA) by M. Wertheim [J. Chem. Phys. 26:1425 (1973)]. He considered the model with nonfluctuating dipole moments. Later L. Pratt [Mol. Phys. 40:347 (1980)] and J. S. Høye and G. Stell [J. Chem. Phys. 73:461 (1980)] solved the corresponding classical problem in the MSA for particles with fluctuating dipole moments.     

The packing of soft materials: Molecular asymmetry,geometric frustration and optimal lattices in block copolymer melts

Block copolymer systems are well known for their ability to self-assemble into a wide array of periodic structures. Due to the abundance and adaptability of physical theories describing polymers, this system is ideal for the development of robust and testible predictions about amphiphilic self-assembly phenomena at large. We review the results of field-theoretic treatments of block copolymer melts, with the aim of understanding how self-assembly in this system can be understood in terms of optimal lattice geometry. The self-consistent (mean) field theory of block copolymer melts as well as its low temperature limit, strong-segregation theory, are presented in detail, highlighting the special role played by asymmetry in the copolymer architecture. Special attention is paid to micellar configurations, where a well-defined and simple notion of optimal lattice geometry emerges from a particular asymptotic limit of the full self-consistent field theory. In this limit, the stability of competing arrangements of copolymer micelles can be assessed in terms of two discrete measures of the lattice geometry, emphasizing the non-trivial coupling between the internal configurations of the fundamentally soft micelles and the periodic symmetry of the lattice.     

Exact wave-based Bloch analysis procedure for investigating wave propagation in two-dimensional periodic lattices

This paper presents an exact, wave-based approach for determining Bloch waves in two-dimensional periodic lattices. This is in contrast to existing methods which employ approximate approaches (e.g., finite difference, Ritz, finite element, or plane wave expansion methods) to compute Bloch waves in general two-dimensional lattices. The analysis combines the recently introduced wave-based vibration analysis technique with specialized Bloch boundary conditions developed herein. Timoshenko beams with axial extension are used in modeling the lattice members. The Bloch boundary conditions incorporate a propagation constant capturing Bloch wave propagation in a single direction, but applied to all wave directions propagating in the lattice members. This results in a unique and properly posed Bloch analysis. Results are generated for the simple problem of a periodic bi-material beam, and then for the more complex examples of square, diamond, and hexagonal honeycomb lattices. The bi-material beam clearly introduces the concepts, but also allows the Bloch wave mode to be explored using insight from the technique. The square, diamond, and hexagonal honeycomb lattices illustrate application of the developed technique to two-dimensional periodic lattices, and allow comparison to a finite element approach. Differences are noted in the predicted dispersion curves, and therefore band gaps, which are attributed to the exact procedure more-faithfully modeling the finite nature of lattice connection points. The exact method also differs from approximate methods in that the same number of solution degrees of freedom is needed to resolve low frequency, and arbitrarily high frequency, dispersion branches. These advantageous features may make the method attractive to researchers studying dispersion characteristics, band gap behavior, and energy propagation in two-dimensional periodic lattices.     

驻波激光场中囚禁离子内外自由度的周期纠缠

在Lamb-Dicke极限下,利用幺正变换,将处于驻波激光场中任意位置的囚禁离子哈密顿量变换为离子裸态基中的Jaynes-Commings模型哈密顿量,研究了其内外自由度的量子熵和纠缠.结果表明,在非共振条件下,囚禁离子系统内外自由度之间存在周期纠缠 关键词： 驻波激光场 囚禁离子 内外自由度 周期纠缠     

Nonperturbative solution for BLOCH electrons in constant magnetic fields

A general theoretical approach for the nonperturbative Bloch solution of Schr?dinger's equation in the presence of a constant magnetic field is presented. Using a singular gauge transformation based on a lattice of magnetic flux lines, an equivalent quantum system with a periodic vector potential is obtained. For rational magnetic fields this system forms a magnetic superlattice for which Bloch's theorem then applies. Extensions of the approach to particles with spin and many-body systems and connections to the theory of magnetic translation groups are discussed.     

Influence of internal degrees of freedom on the structure of one-dimensional systems

Summary The influence of internal degrees of freedom on the behaviour of one-dimensional systems is discussed. For systems with half-filled bands the coupling to internalviz. lattice coordinates decides whether Peierls distortion is caused by intramonomer coordinates or by a lattice coordinate. Thereby the various intramonomer degrees of freedom act cooperatively. We show that there is a small regime of parameters where both kinds of distortion exist simultaneously. For increasing temperature we find that distortions can also move from the lattice coordinate to the intramonomer coordinate.     

Simulations of Sisyphus cooling including multiple excited states

We extend the theory for laser cooling in a near-resonant optical lattice to include multiple excited hyperfine states. Simulations are performed treating the external degrees of freedom of the atom, i.e., position and momentum, classically, while the internal atomic states are treated quantum mechanically, allowing for arbitrary superpositions. Whereas theoretical treatments including only a single excited hyperfine state predict that the temperature should be a function of lattice depth only, except close to resonance, experiments have shown that the minimum temperature achieved depends also on the detuning from resonance of the lattice light. Our results resolve this discrepancy.