Λ-parameters for asymmetric lattice gauge theories with fermions

Dimensional quantities obtained from Monte Carlo simulations on the lattice depend on the lattice mass parameter, Λ_L. To make a connection with continuum physics, a relationship is needed between Λ_L and the Λ-parameters of the continuum theory. This has been done for the euclidean symmetric lattice by others. However, in order to incorporate finite temperature into Monte Carlo studies, or to study the transition from the euclidean formulation to the hamiltonian formulation of gauge theories, asymmetric lattices (a_s≠a_t) may be used. In this paper, the assymetric calculations are extended and the ratio $\text{Λ}{}_{\mathrm{<mtext>min</mtext>}}\text{Λ}{}_{\mathrm{<mtext>L</mtext>}}$, where Λ_min is the continuum mass parameter in the minimal subtraction scheme, is given to one loop for n_f flavors of Wilson and Susskind massless fermions on an asymmetric four-dimensional lattice for two different asymmetric lattice actions.     

Semiclassical expanding discrete space-times

Given the close ties between general relativity and geometry one might reasonably expect that quantum effects associated with gravitation might also be tied to the geometry of space-time, namely, to some sort of discreteness in space-time itself. In particular we suppose space-time to consist of a discrete lattice of points rather than the usual continuum. Since astronomical evidence seems to suggest that the universe is expanding, we also demand that the lattice is expanding. Some of the implications of such a model are that the proton should presently be stable, and the universe should be closed although the mechanism for closure is quantum mechanical.     

Dynamical mass generation in the continuum thirring model

We study the fundamental—adjoint SU(2) lattice gauge theory in four dimensions by Monte Carlo simulations. We show that the string tension at the endpoint of the first-order line is finite. The endpoint is therefore not a second-order critical point of the traditional variety. We also measure the lattice Λ parameter from simulations along a line close to this endpoint and find $\text{Λ = (3.0 ± 0.3) × 10}{}^{\mathrm{?3}}\text{σ}$. This disagrees from previous measurements for the pure fundamental Wilson theory by a factor of 4. Our results cast serious doubt on all continuum limit estimates made near quasi-critical and crossover regions.     

Diffusion-Limited Reactions and Mortal Random Walkers in Confined Geometries

Motivated by the diffusion-reaction kinetics on interstellar dust grains, we study a first-passage problem of mortal random walkers in a confined two-dimensional geometry. We provide an exact expression for the encounter probability of two walkers, which is evaluated in limiting cases and checked against extensive kinetic Monte Carlo simulations. We analyze the continuum limit which is approached very slowly, with corrections that vanish logarithmically with the lattice size. We then examine the influence of the shape of the lattice on the first-passage probability, where we focus on the aspect ratio dependence: Distorting the lattice always reduces the encounter probability of two walkers and can exhibit a crossover to the behavior of a genuinely one-dimensional random walk. The nature of this transition is also explained qualitatively.     

Universality of the continuum limit of lattice quantum theories

The lattice approximation of the naïve continuum action in quantum mechanics or in field theory is not uniquely determined. We investigate to what extent corrections to the lattice action, which vanish in the naïve continuum limit, affect the continuum limit when taking quantum fluctuations into account. In the quantum mechanical case, modifications of the lattice action may induce non-trivial corrections to the potential of the system and thereby change the structure of the theory completely. We verify this statement analytically as well as numerically by performing a Monte Carlo simulation. In the field theoretical case we argue that the lattice corrections considered do not affect the physics of the continuum limit, at least not for asymptotically free gauge field theories. In four dimensions, one might encounter finite renormalization of CP violating terms.     

An isotropic dynamically consistent gradient elasticity model derived from a 2D lattice

This paper presents a derivation of a second-order isotropic continuum from a 2D lattice. The derived continuum is isotropic and dynamically consistent in the sense that it is unconditionally stable and prohibits the infinite speed of energy propagation. The Lagrangian density of the continuum is obtained from the Lagrange function of the underlying lattice. This density is used to obtain the expressions for standard and higher-order stresses in direct correspondence with the equations of the continuum motion. The derived continuum is characterized by two additional parameters relative to the classical elastic continuum. These are the characteristic lengthscale and a dimensionless continualization parameter, which characterizes indirectly the timescale of the derived continuum. The margins for the latter parameter are found from the stability analysis. It is envisaged that the continualization parameter could be measured employing a high-frequency pulse propagating along the surface of the continuum. Excitation and propagation of such pulse is studied theoretically in this paper.     

The homology of defective crystal lattices and the continuum limit

The problem of extending fields that are defined on lattices to fields defined on the continua that they become in the continuum limit is basically one of continuous extension from the 0‐skeleton of a simplicial complex to its higher‐dimensional skeletons. If the lattice in question has defects, as well as the order parameter space of the field, then this process might be obstructed by characteristic cohomology classes on the lattice with values in the homotopy groups of the order parameter space. The examples from solid‐state physics that are discussed are quantum spin fields on planar lattices with point defects or orientable space lattices, vorticial flows or director fields on lattices with dislocations or disclinations, and monopole fields on lattices with point defects.     

A model for the self-organization of microtubules driven by molecular motors

We propose a two-dimensional model for the organization of stabilized microtubules driven by molecular motors in an unconfined geometry. In this model two kinds of dynamics are competing. The first one is purely diffusive, with an interaction between the rotational degrees of freedom, while the second one is a local drive, dependent on microtubule polarity. As a result, there is a configuration dependent driving field. Applying a molecular field approximation, we are able to derive continuum equations. A study on the solutions of these equations shows non-equilibrium inhomogeneous steady states in various regions of the parameter space. The presence and stability of such self-organized states are investigated in terms of entropy production. Numerical simulations confirm our analytic results. Received 4 August 1999 and Received in final form 24 November 1999     

Finite size effects in euclidean lattice thermodynamics for non-interacting bose and fermi systems

In the Monte Carlo simulation of QCD, the euclidean form of the partition function is evaluated on a finite lattice. We use this method to calculate the partition function for non-interacting Bose and Fermi fields. Here the expressions on the lattice can be evaluated in closed form and the continuum limit is well-known; this provides us with a measure for finite lattice size effects in such approaches.     

Long Time Behavior of the Continuum Limit¶of the Toda Lattice, and the Generation¶of Infinitely Many Gaps from C∞ Initial Data

We analyze a continuum limit of the finite non-periodic Toda lattice through an associated constrained maximization problem over spectral density functions. The maximization problem was derived by Deift and McLaughlin using the Lax–Levermore approach, initially developed for the zero dispersion limit of the KdV equation. It encodes the evolution of the continuum limit for all times, including evolution through shocks. The formation of gaps in the support of the maximizer is indicative of oscillations in the Toda lattice and the lack of strong convergence of the continuum limit. For large times, the maximizer tends to have zero gaps, which is the continuum analogue of the sorting property of the finite lattice. Using methods from logarithmic potential theory, we show that this behavior depends crucially on the initial data. We exhibit initial data for which the zero gap ansatz holds uniformly in the spatial parameter (at large times), and other initial data for which this uniformity fails at all times. We then construct an example of C ^∞ smooth initial data generating, at a later time, infinitely many gaps in the support of the maximizer, while for even larger times, all gaps have closed. Received: 8 May 2000 / Accepted: 27 March 2001     

Nonideal effects in quantum field-effect directional coupler

The nonideal effects in a quantum field-effect directional coupler where two quantum wires are coupled through a finite potential barrier are studied by adopting the lattice Green function method. The results show that the electron energy distribution, asymmetric geometry and finite temperature all have obvious influence on the electron transfer of the coupler. Only for the electrons with energies in a certain region, can the complete periodic transfer between two quantum wires take place. The conductance of these electrons as a function of the barrier length and potential height exhibits a fine periodic or quasi-periodic pattern. For the electrons with energies beyond the region, however, the complete periodic transfer does not hold any more since many irregular oscillations are superimposed on the conductance profile. In addition, the finite temperature and asymmetric geometry both can reduce the electron transfer efficiency.     

The effective geometry Monte Carlo algorithm: Applications to molecular communication

In this work, we address the systematic biases and random errors stemming from finite step sizes encountered in diffusion simulations. We introduce the Effective Geometry Monte Carlo (EG-MC) simulation algorithm which modifies the geometry of the receiver. We motivate our approach in a 1D toy model and then apply our findings to a spherical absorbing receiver in a 3D unbounded environment. We show that with minimal computational cost the impulse response of this receiver can be precisely simulated using EG-MC. Afterwards, we demonstrate the accuracy of our simulations and give tight constraints on the single free parameter in EG-MC. Finally, we comment on the range of applicability of our results. While we present the EG-MC algorithm for the specific case of molecular diffusion, we believe that analogous methods with effective geometry manipulations can be utilized to approach a variety of problems in other branches of physics such as condensed matter physics and cosmological large scale structure simulations.     

Landau–Ginzburg theory for ‘star’-shaped droplets

ABSTRACT

Molecular simulations have shown that when a nano-drop comprising a single spherical central ion and a dielectric solvent is charged above a well-defined threshold, it acquires a stable star morphology. A linear continuum model of the ‘star’-shapes comprised electrostatic and surface energy is not sufficient to describe these shapes. We employ combined molecular dynamics, continuum electrostatics and macroscopic modelling in order to construct a unified free energy functional that describes the observed star-shaped droplets. We demonstrate that the Landau free energy coupled to the third-order Steinhardt invariant mimics the shapes of droplets detected in molecular simulations. Using the maximum likelihood technique we build a universal free energy functional that describes droplets for a range of Rayleigh fissility parameter. The analysis of the macroscopic free energy demonstrates the origin of the finite amplitude perturbations just above the Rayleigh limit. We argue that the presence of the finite amplitude perturbations precludes the use of the small parameter perturbation method for the analysis of the shapes above the Rayleigh limit of the corresponding spherical shape.     

Critical film properties for epitaxy

The conditions for overlayer coherency were investigated, by means of molecular dynamic simulations, for two-dimensional triangular Lennard-Jones solids. Up to a certain critical misfit, i.e. difference in the film-substrate lattice parameters, the film remains coherent and in a state of homogeneous strain parallel to the interface, in agreement with continuum calculations based on energy criteria. However, contrary to the predictions of such theories, the critical misfit quickly saturates as a function of film thickness. Further, the generation of misfit dislocations was favored by reducing the film-substrate bonding parameter but inhibited by an increase in this parameter. These factors indicate that in our overlayer-substrate system the occurrence of misfit dislocations is better described by a critical local force rather than a critical energy criterion.     

A channel drop filter in hetero-woodpile-structure

By finite difference time domain method, a channel drop filter with four-port is designed, analyzed, and theoretically simulated in the hetero-woodpile-structure. Hetero-woodpile-structure includes three parts namely, along the propagation direction, the constant lattice in the core woodpile is different from two cladding woodpiles. This channel drop filter is comprised of two straight waveguides separated by an air-cavity in the same layer. Through simulations, we found that adjustment of the resonant-mode in heterostructure can be achieved in various ways, such as only changing the lattice constants, or changing both lattice constants, and the cavity size. The results also show that this structure can realize the energy transfer between bus and drop waveguides.     

Multiscale simulation from atomistic to continuum – coupling molecular dynamics (MD) with the material point method (MPM)

A new multiscale simulation approach is introduced that couples atomistic-scale simulations using molecular dynamics (MD) with continuum-scale simulations using the recently developed material point method (MPM). In MPM, material continuum is represented by a finite collection of material points carrying all relevant physical characteristics, such as mass, acceleration, velocity, strain and stress. The use of material points at the continuum level provides a natural connection with the atoms in the lattice at the atomistic scale. A hierarchical mesh refinement technique in MPM is presented to scale down the continuum level to the atomistic level, so that material points at the fine level in MPM are allowed to directly couple with the atoms in MD. A one-to-one correspondence of MD atoms and MPM points is used in the transition region and non-local elastic theory is used to assure compatibility between MD and MPM regions, so that seamless coupling between MD and MPM can be accomplished. A silicon single crystal under uniaxial tension is used in demonstrating the viability of the technique. A Tersoff-type, three-body potential was used in the MD simulations. The coupled MD/MPM simulations show that silicon under nanometric tension experiences, with increasing elongation in elasticity, dislocation generation and plasticity by slip, void formation and propagation, formation of amorphous structure, necking, and final rupture. Results are presented in terms of stress–strain relationships at several strain rates, as well as the rate dependence of uniaxial material properties. This new multiscale computational method has potential for use in cases where a detailed atomistic-level analysis is necessary in localized spatially separated regions whereas continuum mechanics is adequate in the rest of the material.     

IMPROVEMENT OF THERMODYNAMICAL MONTE CARLO SIMULATION IN SU(2) PURE GAUGE FIELD THEORY ——AN APPROACH TO CONTINUUM PHYSICS LIMIT

The thermodynamical quantities of SU(2) pure lattice gauge field have been simulated first time on the asymmetric lattice (ξ>1).The finite size effect and continuum physics limits have also been studied.The results show that the use of asymmetric lattice is of benefit to calculate the thermodynamical quantities and study the behavior of continuum physics limits.In addition,it is explained that the efficiency of the whole Monte Carlo simulation and the calculation of heat capacity will be improved quite a lot by using bias sampling technique.     

QCD sum rules,the spontaneous breakdown of chiral symmetry and short-distance behaviour in lattice gauge theories

We analyze the behaviour that correlation functions ought to have on the lattice in order to reproduce QCD sum rules in the continuum limit. We formulate a set of relations between lattice correlation functions of meson operators at small time separation and the quark condensates responsible for spontaneous breakdown of chiral symmetery. We suggest that the degree to which such relations are satisfied will provide a set of consistency checks on the ability of lattice Monte Carlo simulations to reproduce the correct spontaneous chiral symmetry breaking of the continuum theory.     

On the thermodynamics of an ideal fermi gas on the lattice at finite density

We study the ideal gas of fermions on a lattice at finite density for both naive and Wilson fermions. Comparing the thermodynamical quantities thus calculated with the known results in the continuum theory, we are led to propose a modification of the naive form of the lattice action, which is same for both the naive and the Wilson fermions. The thermodynamical quantities, calculated by using this form, are shown to have the correct continuum limit.