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.

A TEM study of the influence of microstructure on the superplastic behavior of a U-5.8 wt.% Nb alloy

1986 ◽  
Vol 44 ◽  
pp. 568-569
Author(s):  
G. Mackiewicz Ludtka
Keyword(s):  
Grain Size ◽  
Heating Rate ◽  
Phase Field ◽  
Single Phase ◽  
Superplastic Behavior ◽  
Two Phase ◽  
Single Phase Field

Historically, metals exhibit superplasticity only while forming in a two-phase field because a two-phase microstructure helps ensure a fine, stable grain size. In the U-5.8 Nb alloy, superplastici ty exists for up to 2 h in the single phase field (γ1) at 670°C. This is above the equilibrium monotectoid temperature of 647°C. Utilizing dilatometry, the superplastic (SP) U-5.8 Nb alloy requires superheating to 658°C to initiate the α+γ2 → γ1 transformation at a heating rate of 1.5°C/s. Hence, the U-5.8 Nb alloy exhibits an anomolous superplastic behavior.

Frustration effect following correlated nonreinforcement.

1968 ◽  
Vol 78 (3, Pt.1) ◽  
pp. 396-400 ◽  
Author(s):  
Frank A. Logan
Keyword(s):  
Frustration Effect

Incomplete reduction of reward and the frustration effect with hunger constant.

1969 ◽  
Vol 80 (3, Pt.1) ◽  
pp. 493-497
Author(s):  
Lawrence A. Hall ◽  
John N. Marr
Keyword(s):  
Frustration Effect

Remarks on nonlinear elastic waves with null conditions

2020 ◽  
Vol 26 ◽  
pp. 121
Author(s):  
Dongbing Zha ◽  
Weimin Peng
Keyword(s):  
Initial Data ◽  
Global Existence ◽  
Elastic Waves ◽  
Wave Equations ◽  
Alternative Proof ◽  
Classical Solutions ◽  
Small Initial Data ◽  
Hyperelastic Materials ◽  
Variational Structure ◽  
Nonlinear Elastic

For the Cauchy problem of nonlinear elastic wave equations for 3D isotropic, homogeneous and hyperelastic materials with null conditions, global existence of classical solutions with small initial data was proved in R. Agemi (Invent. Math. 142 (2000) 225–250) and T. C. Sideris (Ann. Math. 151 (2000) 849–874) independently. In this paper, we will give some remarks and an alternative proof for it. First, we give the explicit variational structure of nonlinear elastic waves. Thus we can identify whether materials satisfy the null condition by checking the stored energy function directly. Furthermore, by some careful analyses on the nonlinear structure, we show that the Helmholtz projection, which is usually considered to be ill-suited for nonlinear analysis, can be in fact used to show the global existence result. We also improve the amount of Sobolev regularity of initial data, which seems optimal in the framework of classical solutions.

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