Coarsening in Two-Dimensional Soap Froths and the Large-Q Potts Model

1991 ◽  
Vol 237 ◽  
Author(s):  
Gary S. Grest ◽  
James A. Glaziers ◽  
Michael P. Anderson ◽  
Elizabeth A. Holm ◽  
David J. Srolovitz
Keyword(s):  
Potts Model ◽  
Initial Conditions ◽  
Domain Size ◽  
Detailed Comparison ◽  
Distribution Functions ◽  
Domain Growth ◽  
Vertex Angle ◽  
Two Dimensional ◽  
Lattice Simulation ◽  
Universal Domain

ABSTRACTA detailed comparison between the experimental evolution of a two-dimensional soap froth and the large Q state Potts model is presented. The pattern evolution starting from identical initial conditions will be compared as well as a variety of distribution functions and correlations of the two systems. Simulations on different lattices show that the discrete lattice of the Potts model causes deviations from universal domain growth by weakening the vertex angle boundary conditions that form the basis of von Neumann's law. We show that the anisotropy inherent in a discrete lattice simulation, which masks the underlying ‘universal’ grain growth, can be overcome by increasing the range of the interaction between spins or increasing the temperature. Excellent overall agreement between the kinetics, topological distributions and domain size distributions between the low lattice anisotropy Potts-model simulations and the soap froth suggests that the Potts model is useful for studying domain growth in a wide variety of physical systems.

Effects of lattice anisotropy and temperature on domain growth in the two-dimensional Potts model

1991 ◽  
Vol 43 (6) ◽  
pp. 2662-2668 ◽  
Author(s):  
Elizabeth A. Holm ◽  
James A. Glazier ◽  
David J. Srolovitz ◽  
Gary S. Grest
Keyword(s):  
Potts Model ◽  
Domain Growth ◽  
Two Dimensional ◽  
Lattice Anisotropy

Light front field theory: An advanced Primer

2007 ◽  
Vol 57 (3) ◽  
Author(s):  
L'ubomír Martinovič
Keyword(s):  
Field Theory ◽  
Detailed Comparison ◽  
Light Front ◽  
Two Dimensional ◽  
Schwinger Model ◽  
Initial Space ◽  
Model Hamiltonian ◽  
Spatial Volume ◽  
Hamiltonian Perturbation Theory ◽  
Light Front Quantization

Light front field theory: An advanced PrimerWe present an elementary introduction to quantum field theory formulated in terms of Dirac's light front variables. In addition to general principles and methods, a few more specific topics and approaches based on the author's work will be discussed. Most of the discussion deals with massive two-dimensional models formulated in a finite spatial volume starting with a detailed comparison between quantization of massive free fields in the usual field theory and the light front (LF) quantization. We discuss basic properties such as relativistic invariance and causality. After the LF treatment of the soluble Federbush model, a LF approach to spontaneous symmetry breaking is explained and a simple gauge theory - the massive Schwinger model in various gauges is studied. A LF version of bosonization and the massive Thirring model are also discussed. A special chapter is devoted to the method of discretized light cone quantization and its application to calculations of the properties of quantum solitons. The problem of LF zero modes is illustrated with the example of the two-dimensional Yukawa model. Hamiltonian perturbation theory in the LF formulation is derived and applied to a few simple processes to demonstrate its advantages. As a byproduct, it is shown that the LF theory cannot be obtained as a "light-like" limit of the usual field theory quantized on an initial space-like surface. A simple LF formulation of the Higgs mechanism is then given. Since our intention was to provide a treatment of the light front quantization accessible to postgradual students, an effort was made to discuss most of the topics pedagogically and a number of technical details and derivations are contained in the appendices.

Extended two-dimensional belief function based on divergence measurement

2020 ◽  
pp. 1-8
Author(s):  
Jianping Fan ◽  
Jing Wang ◽  
Meiqin Wu
Keyword(s):  
Belief Function ◽  
Distribution Functions ◽  
Divergence Measure ◽  
Uncertain Information ◽  
Belief Functions ◽  
Two Dimensional ◽  
Probability Distribution Functions ◽  
Basic Probability ◽  
The Difference ◽  
Supporting Degree

The two-dimensional belief function (TDBF = (mA, mB)) uses a pair of ordered basic probability distribution functions to describe and process uncertain information. Among them, mB includes support degree, non-support degree and reliability unmeasured degree of mA. So it is more abundant and reasonable than the traditional discount coefficient and expresses the evaluation value of experts. However, only considering that the expert’s assessment is single and one-sided, we also need to consider the influence between the belief function itself. The difference in belief function can measure the difference between two belief functions, based on which the supporting degree, non-supporting degree and unmeasured degree of reliability of the evidence are calculated. Based on the divergence measure of belief function, this paper proposes an extended two-dimensional belief function, which can solve some evidence conflict problems and is more objective and better solve a class of problems that TDBF cannot handle. Finally, numerical examples illustrate its effectiveness and rationality.

scholarly journals Geometrical four-point functions in the two-dimensional critical Q-state Potts model: the interchiral conformal bootstrap

2020 ◽  
Vol 2020 (12) ◽  
Author(s):  
Yifei He ◽  
Jesper Lykke Jacobsen ◽  
Hubert Saleur
Keyword(s):  
Potts Model ◽  
Correlation Functions ◽  
Liouville Theory ◽  
Analytical Determination ◽  
Conformal Weight ◽  
Two Dimensional ◽  
Conformal Blocks ◽  
Conformal Bootstrap ◽  
Point Correlation

Abstract Based on the spectrum identified in our earlier work [1], we numerically solve the bootstrap to determine four-point correlation functions of the geometrical connectivities in the Q-state Potts model. Crucial in our approach is the existence of “interchiral conformal blocks”, which arise from the degeneracy of fields with conformal weight hr,1, with r ∈ ℕ*, and are related to the underlying presence of the “interchiral algebra” introduced in [2]. We also find evidence for the existence of “renormalized” recursions, replacing those that follow from the degeneracy of the field $$ {\Phi}_{12}^D $$ Φ 12 D in Liouville theory, and obtain the first few such recursions in closed form. This hints at the possibility of the full analytical determination of correlation functions in this model.

Waves and turbulence on a beta-plane

1975 ◽  
Vol 69 (3) ◽  
pp. 417-443 ◽  
Author(s):  
Peter B. Rhines
Keyword(s):  
Weak Interaction ◽  
Initial Conditions ◽  
Computer Experiments ◽  
Zonal Flow ◽  
Similarity Solutions ◽  
Large Reynolds Number ◽  
Two Dimensional ◽  
Homogeneous Fluid ◽  
Restoring Force ◽  
Model Equations

Two-dimensional eddies in a homogeneous fluid at large Reynolds number, if closely packed, are known to evolve towards larger scales. In the presence of a restoring force, the geophysical beta-effect, this cascade produces a field of waves without loss of energy, and the turbulent migration of the dominant scale nearly ceases at a wavenumber kβ = (β/2U)½ independent of the initial conditions other than U, the r.m.s. particle speed, and β, the northward gradient of the Coriolis frequency.The conversion of turbulence into waves yields, in addition, more narrowly peaked wavenumber spectra and less fine-structure in the spatial maps, while smoothly distributing the energy about physical space.The theory is discussed, using known integral constraints and similarity solutions, model equations, weak-interaction wave theory (which provides the terminus for the cascade) and other linearized instability theory. Computer experiments with both finite-difference and spectral codes are reported. The central quantity is the cascade rate, defined as \[ T = 2\int_0^{\infty} kF(k)dk/U^3\langle k\rangle , \] where F is the nonlinear transfer spectrum and 〈k〉 the mean wavenumber of the energy spectrum. (In unforced inviscid flow T is simply U−1d〈k〉−1/dt, or the rate at which the dominant scale expands in time t.) T is shown to have a mean value of 3·0 × 10−2 for pure two-dimensional turbulence, but this decreases by a factor of five at the transition to wave motion. We infer from weak-interaction theory even smaller values for k [Lt ] kβ.After passing through a state of propagating waves, the homogeneous cascade tends towards a flow of alternating zonal jets which, we suggest, are almost perfectly steady. When the energy is intermittent in space, however, model equations show that the cascade is halted simply by the spreading of energy about space, and then the end state of a zonal flow is probably not achieved.The geophysical application is that the cascade of pure turbulence to large scales is defeated by wave propagation, helping to explain why the energy-containing eddies in the ocean and atmosphere, though significantly nonlinear, fail to reach the size of their respective domains, and are much smaller. For typical ocean flows, $k_{\beta}^{-1} = 70\,{\rm km} $, while for the atmosphere, $k_{\beta}^{-1} = 1000\,{\rm km}$. In addition the cascade generates, by itself, zonal flow (or more generally, flow along geostrophic contours).

Phase transition of a two-dimensional, multiplicatively coupledXY–Potts model

2009 ◽  
Vol 42 (22) ◽  
pp. 225001 ◽  
Author(s):  
Marcel Hellmann ◽  
Youjin Deng ◽  
Matthias Weiss ◽  
Dieter W Heermann
Keyword(s):  
Phase Transition ◽  
Potts Model ◽  
Two Dimensional
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