Uniform-in-Time Continuum Limit of the Winfree Model on an Infinite Cylinder and Emergent Dynamics

2021 ◽  
Vol 20 (2) ◽  
pp. 1104-1134
Author(s):  
Seung-Yeal Ha ◽  
Myeongju Kang ◽  
Bora Moon
Keyword(s):  
Continuum Limit ◽  
Infinite Cylinder ◽  
Winfree Model ◽  
Time Continuum ◽  
Emergent Dynamics

scholarly journals Uniform-in-time continuum limit of the lattice Winfree model and emergent dynamics

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Seung-Yeal Ha ◽  
Myeongju Kang ◽  
Bora Moon
Keyword(s):  
Phase Field ◽  
Continuum Limit ◽  
Strength Function ◽  
Frustration Effect ◽  
Frequency Function ◽  
Classical Solutions ◽  
Compact Region ◽  
Winfree Model ◽  
Synchronous Behavior ◽  
Time Continuum

<p style='text-indent:20px;'>We study a uniform-in-time continuum limit of the lattice Winfree model(LWM) and its asymptotic dynamics which depends on system functions such as natural frequency function and coupling strength function. The continuum Winfree model(CWM) is an integro-differential equation for the temporal evolution of Winfree phase field. The LWM describes synchronous behavior of weakly coupled Winfree oscillators on a lattice lying in a compact region. For bounded measurable initial phase field, we establish a global well-posedness of classical solutions to the CWM under suitable assumptions on coupling function, and we also show that a classical solution to the CWM can be obtained as a <inline-formula><tex-math id="M1">\begin{document}$ L^1 $\end{document}</tex-math></inline-formula>-limit of a sequence of lattice solutions. Moreover, in the presence of frustration effect, we show that stationary states and bump states can emerge from some admissible class of initial data in a large and intermediate coupling regimes, respectively. We also provide several numerical examples and compare them with analytical results.</p>

scholarly journals Differentiable-path integrals in quantum mechanics

2015 ◽  
Vol 12 (09) ◽  
pp. 1550100 ◽  
Author(s):  
Benjamin Koch ◽  
Ignacio Reyes
Keyword(s):  
Quantum Mechanics ◽  
Continuum Limit ◽  
Energy Levels ◽  
Path Integrals ◽  
Time Intervals ◽  
Space Of Paths ◽  
Usual Quantum ◽  
The Mean ◽  
Time Continuum ◽  
Physical Quantities

A method is presented which restricts the space of paths entering the path integral of quantum mechanics to subspaces of Cα, by only allowing paths which possess at least α derivatives. The method introduces two external parameters, and induces the appearance of a particular time scale ϵD such that for time intervals longer than ϵD the model behaves as usual quantum mechanics. However, for time scales smaller than ϵD, modifications to standard formulation of quantum theory occur. This restriction renders convergent some quantities which are usually divergent in the time-continuum limit ϵ → 0. We illustrate the model by computing several meaningful physical quantities such as the mean square velocity 〈v2〉, the canonical commutator, the Schrödinger equation and the energy levels of the harmonic oscillator. It is shown that an adequate choice of the parameters introduced makes the evolution unitary.

scholarly journals Functional Integral and Stochastic Representations for Ensembles of Identical Bosons on a Lattice

2021 ◽  
Author(s):  
Manfred Salmhofer
Keyword(s):  
Continuum Limit ◽  
Canonical Ensemble ◽  
Functional Integral ◽  
Stochastic Representation ◽  
Functional Integrals ◽  
Uniform Bounds ◽  
Interacting Random Walks ◽  
Stochastic Representations ◽  
Time Continuum ◽  
Grand Canonical

AbstractRegularized coherent-state functional integrals are derived for ensembles of identical bosons on a lattice, the regularization being a discretization of Euclidian time. Convergence of the time-continuum limit is proven for various discretized actions. The focus is on the integral representation for the partition function and expectation values in the canonical ensemble. The connection to the grand-canonical integral is exhibited and some important differences are discussed. Uniform bounds for covariances are proven, which simplify the analysis of the time-continuum limit and can also be used to analyze the thermodynamic limit. The relation to a stochastic representation by an ensemble of interacting random walks is made explicit, and its modifications in presence of a condensate are discussed.

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.

scholarly journals More on the flavor dependence of mϱ/fπ

2021 ◽  
Vol 2021 (7) ◽  
Author(s):  
Andrey Yu. Kotov ◽  
Daniel Nogradi ◽  
Kalman K. Szabo ◽  
Lorinc Szikszai
Keyword(s):  
Gauge Theory ◽  
Symmetry Breaking ◽  
Finite Volume ◽  
Continuum Limit ◽  
Low Energy ◽  
Translational Symmetry ◽  
Fundamental Representation ◽  
Staggered Fermions ◽  
Link Type ◽  
Volume Effects

Abstract In previous work, [arXiv:1905.01909], we have calculated the mϱ/fπ ratio in the chiral and continuum limit for SU(3) gauge theory coupled to Nf = 2, 3, 4, 5, 6 fermions in the fundamental representation. The main result was that this ratio displays no statistically significant Nf-dependence. In the present work we continue the study of the Nf-dependence by extending the simulations to Nf = 7, 8, 9, 10. Along the way we also study in detail the Nf-dependence of finite volume effects on low energy observables and a particular translational symmetry breaking unphysical, lattice artefact phase specific to staggered fermions.

Active motion of multiphase oil droplets: emergent dynamics of squirmers with evolving internal structure

Soft Matter ◽  
2021 ◽  
Author(s):  
Xin Wang ◽  
Rui Zhang ◽  
Ali Mozaffari ◽  
Juan J. de Pablo ◽  
Nicholas L. Abbott
Keyword(s):  
Internal Structure ◽  
Oil Droplets ◽  
Active Motion ◽  
Over Time ◽  
Emergent Dynamics

Self-propelled motions of active droplets can be programmed by transforming their internal morphologies over time.

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