Padé approximations and the critical exponents in the two- and three-dimensional Ising models

1975 ◽  
Vol 12 (1) ◽  
pp. 243-252 ◽  
Author(s):  
Paul H. E. Meijer ◽  
Richard A. Farrell
Keyword(s):  
Critical Exponents ◽  
Three Dimensional ◽  
Ising Models ◽  
Padé Approximations ◽  
Pade Approximations

scholarly journals Three-dimensional integral dry friction model for the motion of a rectangular body

2021 ◽  
Vol 21 (1) ◽  
pp. 14-21
Author(s):  
M. S. Salimov ◽  
I. V. Merkuriev
Keyword(s):  
Angular Velocity ◽  
Dry Friction ◽  
Three Dimensional ◽  
Friction Model ◽  
The Body ◽  
Horizontal Surface ◽  
Padé Approximations ◽  
Pade Approximations ◽  
Analytical Expressions ◽  
Rectangular Body

Introduction. A three-dimensional dry friction model in the interaction of a rectangular body and a horizontal rough surface is considered. It is assumed that there is no separation of the body from the horizontal surface. The body motion occurs under the conditions of combined dynamics when, in addition to the longitudinal movement, the body participates in twisting.Materials and Methods. Linear fractional Pade approximations are proposed, which replaced the cumbersome analytical expressions that most accurately describe the motion of bodies on rough surfaces. New mathematical models describing sliding and twisting of bodies with a rectangular base are proposed.Results. Analytical expressions of the principal vector and moment of friction for rectangular contact areas are developed and scientifically established. A friction model that takes into account the relationship between sliding and twisting speeds, which provides finding solutions for Pade dependences, is developed. After numerical solution to the equations of motion, the dependences of the sliding speed and angular velocity on time were obtained and constructed. Graphs of the dependences of the friction forces and their moment on two parameters (angular velocity and slip velocity) were constructed, which enabled to compare the integral and normalized models of friction. The comparison results showed good agreement of the integral model and the model based on Pade approximations.Discussion and Conclusions. The results obtained provide considering the dynamic coupling of components, which determines the force interaction of a rectangular body and a horizontal surface. These results can be used in mobile robotics. The analyzed motion of the body occurs through the motion control of a material point inside the body. Such mobile robots can be used when solving a wide class of problems: when creating autonomous robots for the exploration of outer space and planets; in the diagnosis and treatment in case of passing through complex structures of veins and arteries; in research under water, in places of large differential temperature; in underground operations.

scholarly journals Connecting Complex Electronic Pattern Formation to Critical Exponents

2021 ◽  
Vol 6 (4) ◽  
pp. 39
Author(s):  
Shuo Liu ◽  
Erica W. Carlson ◽  
Karin A. Dahmen
Keyword(s):  
Cross Sections ◽  
Critical Exponents ◽  
Nearest Neighbor ◽  
Condensed Matter ◽  
Three Dimensional ◽  
Spin Correlation ◽  
Ising Models ◽  
Connected Sets ◽  
Universality Classes ◽  
Self Similar

Scanning probes reveal complex, inhomogeneous patterns on the surface of many condensed matter systems. In some cases, the patterns form self-similar, fractal geometric clusters. In this paper, we advance the theory of criticality as it pertains to those geometric clusters (defined as connected sets of nearest-neighbor aligned spins) in the context of Ising models. We show how data from surface probes can be used to distinguish whether electronic patterns observed at the surface of a material are confined to the surface, or whether the patterns originate in the bulk. Whereas thermodynamic critical exponents are derived from the behavior of Fortuin–Kasteleyn (FK) clusters, critical exponents can be similarly defined for geometric clusters. We find that these geometric critical exponents are not only distinct numerically from the thermodynamic and uncorrelated percolation exponents, but that they separately satisfy scaling relations at the critical fixed points discussed in the text. We furthermore find that the two-dimensional (2D) cross-sections of geometric clusters in the three-dimensional (3D) Ising model display critical scaling behavior at the bulk phase transition temperature. In particular, we show that when considered on a 2D slice of a 3D system, the pair connectivity function familiar from percolation theory displays more robust critical behavior than the spin-spin correlation function, and we calculate the corresponding critical exponent. We discuss the implications of these two distinct length scales in Ising models. We also calculate the pair connectivity exponent in the clean 2D case. These results extend the theory of geometric criticality in the clean Ising universality classes, and facilitate the broad application of geometric cluster analysis techniques to maximize the information that can be extracted from scanning image probe data in condensed matter systems.

scholarly journals Quantum critical exponents for a disordered three-dimensional Weyl node

2015 ◽  
Vol 92 (11) ◽  
Author(s):  
Björn Sbierski ◽  
Emil J. Bergholtz ◽  
Piet W. Brouwer
Keyword(s):  
Critical Exponents ◽  
Three Dimensional ◽  
Quantum Critical

Padé approximations to the logarithm II: Identities, recurrences, and symbolic computation

2006 ◽  
Vol 11 (2) ◽  
pp. 139-158 ◽  
Author(s):  
Kathy Driver ◽  
Helmut Prodinger ◽  
Carsten Schneider ◽  
J. A. C. Weideman
Keyword(s):  
Symbolic Computation ◽  
Padé Approximations ◽  
Pade Approximations

Nucleation and metastability in three-dimensional Ising models

1984 ◽  
Vol 36 (3-4) ◽  
pp. 447-470 ◽  
Author(s):  
Dieter W. Heermann ◽  
Antonio Coniglio ◽  
W. Klein ◽  
Dietrich Stauffer
Keyword(s):  
Three Dimensional ◽  
Ising Models
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