Linear Chain of Coupled Fractional Oscillators: Response Dynamics and Its Continuum Limit

2007 ◽  
Author(s):  
B. N. Narahari Achar ◽  
Tanya Prozny ◽  
John W. Hanneken
Keyword(s):  
Continuum Limit ◽  
Equations Of Motion ◽  
Linear Chain ◽  
Harmonic Oscillators ◽  
Fractional Dynamics ◽  
Response Dynamics ◽  
Numerical Application ◽  
Diffusion Wave ◽  
A Chain ◽  
The Continuum

The standard model of a chain of simple harmonic oscillators of Condensed Matter Physics is generalized to a model of linear chain of coupled fractional oscillators in fractional dynamics. The set of integral equations of motion pertaining to the chain of harmonic oscillators is generalized by taking the integrals to be of arbitrary order according to the methods of fractional calculus to yield the equations of motion of a chain of coupled fractional oscillators. The solution is obtained by using Laplace transforms. The continuum limit of the equations is shown to yield the fractional diffusion-wave equation in one dimension. The solution and numerical application of the set of equations and the continuum limit there of are discussed.

Response Dynamics in the Continuum Limit of the Lattice Dynamical Theory of Viscoelasticity (Fractional Calculus Approach)

2009 ◽  
Author(s):  
B. N. Narahari Achar ◽  
John W. Hanneken
Keyword(s):  
Integral Equations ◽  
Continuum Limit ◽  
Equations Of Motion ◽  
Linear Chain ◽  
Dynamical Theory ◽  
Harmonic Oscillators ◽  
Fractional Integrals ◽  
Response Dynamics ◽  
Dynamical Equations ◽  
The Continuum

A fractional diffusion-wave equation is derived in the continuum limit of the lattice dynamical equations of motion of a chain of coupled fractional oscillators obtained from the integral equations of motion of a linear chain of simple harmonic oscillators by generalization of the ordinary integrals into ones involving fractional integrals. The set of integral equations of motion pertaining to the chain of coupled fractional oscillators in the continuum limit is solved by using Laplace transforms. The response of the system to impulse and sinusoidal forcing is studied. Numerical applications are discussed with particular reference to energy flow and dissipation.

NONLINEAR FIELD ANALYSIS OF THE HUBBARD MODEL IN ONE AND TWO DIMENSIONS IN THE CONTINUUM LIMIT

2000 ◽  
Vol 14 (18) ◽  
pp. 1859-1890
Author(s):  
J. M. DIXON ◽  
J. A. TUSZYŃSKI ◽  
M. L. A. NIP ◽  
D. SEPT ◽  
K. J. E. VOS
Keyword(s):  
Continuum Limit ◽  
Dimensional Space ◽  
Equations Of Motion ◽  
Two Dimensions ◽  
Field Analysis ◽  
Field Equations ◽  
Phase Dynamics ◽  
Hartree Fock ◽  
Spin Fields ◽  
The Continuum

We investigate the Hubbard Hamiltonian's properties in the continuum limit by implementing the procedures of the Method of Coherent Structures (MCS). We obtain field equations of motion and analyse the phase dynamics of the resultant classical spin fields. We have performed analytical and numerical calculations to find appropriate physically acceptable solutions to the equations of motion in one-dimensional space. In two-dimensional space, among other types, we have found several different spin phases of vortex type, spiral patterns and parabolic spin arrangements. Our results are consistent with earlier Hartree–Fock finite-grid numerical simulations.

scholarly journals RISE OF THE CENTRIST: FROM BINARY TO CONTINUOUS OPINION DYNAMICS

2008 ◽  
Vol 19 (09) ◽  
pp. 1459-1475 ◽  
Author(s):  
GEORGE A. BAKER ◽  
JAMES P. HAGUE
Keyword(s):  
Monte Carlo ◽  
Monte Carlo Simulations ◽  
Continuum Limit ◽  
Linear Chain ◽  
Opinion Dynamics ◽  
Memory Effects ◽  
Time Step ◽  
Binary Model ◽  
Long Time ◽  
The Continuum

We propose a model that extends the binary "united we stand, divided we fall" opinion dynamics of Sznajd-Weron to handle continuous and multi-state discrete opinions on a linear chain. Disagreement dynamics are often ignored in continuous extensions of the binary rules, so we make the most symmetric continuum extension of the binary model that can treat the consequences of agreement (debate) and disagreement (confrontation) within a population of agents. We use the continuum extension as an opportunity to develop rules for persistence of opinion (memory). Rules governing the propagation of centrist views are also examined. Monte Carlo simulations are carried out. We find that both memory effects and the type of centrist significantly modify the variance of average opinions in the large timescale limits of the models. Finally, we describe the limit of applicability for Sznajd-Weron's model of binary opinions as the continuum limit is approached. By comparing Monte Carlo results and long time-step limits, we find that the opinion dynamics of binary models are significantly different to those where agents are permitted more than 3 opinions.

Instability and Collapse in Discrete Wave Equations

2005 ◽  
Vol 5 (3) ◽  
pp. 223-241
Author(s):  
A. Carpio ◽  
G. Duro
Keyword(s):  
Continuum Limit ◽  
Wave Equations ◽  
Numerical Solutions ◽  
Blow Up ◽  
Theoretical Predictions ◽  
Initial States ◽  
The Impact ◽  
The Continuum

AbstractUnstable growth phenomena in spatially discrete wave equations are studied. We characterize sets of initial states leading to instability and collapse and obtain analytical predictions for the blow-up time. The theoretical predictions are con- trasted with the numerical solutions computed by a variety of schemes. The behavior of the systems in the continuum limit and the impact of discreteness and friction are discussed.

Coupled dynamics of populations supported by discrete sites and their continuum limit

2010 ◽  
Vol 466 (2121) ◽  
pp. 2561-2586 ◽  
Author(s):  
Timothy R. Field ◽  
Robert J. A. Tough
Keyword(s):  
Evolution Equation ◽  
Closed Form ◽  
Generating Function ◽  
Rate Equation ◽  
Continuum Limit ◽  
Infinite Chain ◽  
Migration Process ◽  
The Mean ◽  
The Continuum ◽  
Birth Death

The illumination of single population behaviour subject to the processes of birth, death and immigration has provided a basis for the discussion of the non-Gaussian statistical and temporal correlation properties of scattered radiation. As a first step towards the modelling of its spatial correlations, we consider the populations supported by an infinite chain of discrete sites, each subject to birth, death and immigration and coupled by migration between adjacent sites. To provide some motivation, and illustrate the techniques we will use, the migration process for a single particle on an infinite chain of sites is introduced and its diffusion dynamics derived. A certain continuum limit is identified and its properties studied via asymptotic analysis. This forms the basis of the multi-particle model of a coupled population subject to single site birth, death and immigration processes, in addition to inter-site migration. A discrete rate equation is formulated and its generating function dynamics derived. This facilitates derivation of the equations of motion for the first- and second-order cumulants, thus generalizing the earlier results of Bailey through the incorporation of immigration at each site. We present a novel matrix formalism operating in the time domain that enables solution of these equations yielding the mean occupancy and inter-site variances in the closed form. The results for the first two moments at a single time are used to derive expressions for the asymptotic time-delayed correlation functions, which relates to Glauber’s analysis of an Ising model. The paper concludes with an analysis of the continuum limit of the birth–death–immigration–migration process in terms of a path integral formalism. The continuum rate equation and evolution equation for the generating function are developed, from which the evolution equation of the mean occupancy is derived, in this limit. Its solution is provided in closed form.

scholarly journals Effective Action and Measure in Matrix Model of IIB Superstrings

1997 ◽  
Vol 12 (31) ◽  
pp. 2331-2340 ◽  
Author(s):  
L. Chekhov ◽  
K. Zarembo
Keyword(s):  
Matrix Model ◽  
Effective Action ◽  
Continuum Limit ◽  
Auxiliary Field ◽  
Large N ◽  
The Matrix ◽  
Reparametrization Invariance ◽  
The Continuum ◽  
Induced Measure

We calculate an effective action and measure induced by the integration over the auxiliary field in the matrix model recently proposed to describe IIB superstrings. It is shown that the measure of integration over the auxiliary matrix is uniquely determined by locality and reparametrization invariance of the resulting effective action. The large-N limit of the induced measure for string coordinates is discussed in detail. It is found to be ultralocal and, thus, is possibly irrelevant in the continuum limit. The model of the GKM type is considered in relation to the effective action problem.

scholarly journals BOUNDARY CONDITIONS IN THE DIRAC APPROACH TO GRAPHENE DEVICES

2012 ◽  
Vol 14 ◽  
pp. 240-249 ◽  
Author(s):  
C.G. BENEVENTANO ◽  
E.M. SANTANGELO
Keyword(s):  
Boundary Conditions ◽  
Zeta Function ◽  
Continuum Limit ◽  
Casimir Energy ◽  
Local Boundary ◽  
Graphene Devices ◽  
The Continuum

We study a family of local boundary conditions for the Dirac problem corresponding to the continuum limit of graphene, both for nanoribbons and nanodots. We show that, among the members of such family, MIT bag boundary conditions are the ones which are in closest agreement with available experiments. For nanotubes of arbitrary chirality satisfying these last boundary conditions, we evaluate the Casimir energy via zeta function regularization, in such a way that the limit of nanoribbons is clearly determined.

A Hybrid Highly Parallelizable Algorithm for the Dynamics of Rigid Body Chain Systems

1999 ◽  
Author(s):  
Shanzhong Duan ◽  
Kurt S. Anderson
Keyword(s):  
Rigid Body ◽  
Degrees Of Freedom ◽  
High Efficiency ◽  
Equations Of Motion ◽  
Constraint Forces ◽  
Dynamics Of Rigid Body ◽  
A Chain ◽  
Explicit Determination ◽  
Order Algorithm

Abstract The paper presents a new hybrid parallelizable low order algorithm for modeling the dynamic behavior of multi-rigid-body chain systems. The method is based on cutting certain system interbody joints so that largely independent multibody subchain systems are formed. These subchains interact with one another through associated unknown constraint forces f¯c at the cut joints. The increased parallelism is obtainable through cutting the joints and the explicit determination of associated constraint loads combined with a sequential O(n) procedure. In other words, sequential O(n) procedures are performed to form and solve equations of motion within subchains and parallel strategies are used to form and solve constraint equations between subchains in parallel. The algorithm can easily accommodate the available number of processors while maintaining high efficiency. An O[(n+m)Np+m(1+γ)Np+mγlog2Np](0<γ<1) performance will be achieved with Np processors for a chain system with n degrees of freedom and m constraints due to cutting of interbody joints.

High-resolution spectroscopy near the continuum limit: the microwave spectrum of trans-3-bromo-1,1,1,2,2-pentafluoropropane

2018 ◽  
Vol 117 (9-12) ◽  
pp. 1351-1359 ◽  
Author(s):  
Frank E. Marshall ◽  
Nicole Moon ◽  
Thomas D. Persinger ◽  
David J. Gillcrist ◽  
Nelson E. Shreve ◽  
...  
Keyword(s):  
High Resolution ◽  
Continuum Limit ◽  
Microwave Spectrum ◽  
High Resolution Spectroscopy ◽  
The Continuum
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