# Padé and Post-Padé Approximations for Critical Phenomena

## Abstract

**:**

## 1. Introduction

## 2. Ruina-Dieterich Model for Landslides and Factor Approximants. Critical Point

#### 2.1. Factor Approximants

#### 2.2. Ruina-Dieterich Model

#### 2.2.1. Case of Relaxation Phenomena

#### 2.2.2. Case of Critical Behavior

## 3. Critical Index. Direct Methods of Calculation the Index

#### 3.1. Some Basic Definitions

#### Root Approximants

#### 3.2. Examples of Exact Critical Index Reconstruction with Corrected Approximants. General Ansatz

#### 3.3. Approximate Methods for Critical Index Computation

## 4. Additive Self-Similar Approximants and DLog Additive Recursive Approximants

#### 4.1. $DLog$ Additive Recursive Approximants. Finite Critical Point

#### Conductivity of Regular Composite. Hexagonal Array

#### 4.2. $DLog$ Additive Recursive Approximants. Critical Behavior at Infinity

#### 4.2.1. Anharmonic Partition Integral. Interpolation

#### 4.2.2. Quartic Anharmonic Oscillator. Interpolation

#### 4.3. Thermal Entropy in N = 4 Super Yang-Mills Theory

#### 4.3.1. Schwinger Model. Interpolation

**Ground state.**We test the technique of $DLog$ additive approximants first on the case previously explored in [38], and calculate the ground state of the model, given as a function of the dimensionless variable $x=m/g$. Here m stands for electron mass and g is the coupling parameter. It also has the dimension of mass. The energy is $E=M-2m$, corresponding to a vector boson of mass $M(x)$.

**Excited state.**Consider now the first excited state of the model, corresponding to a scalar boson. The small-x expansion for the energy of the excited “scalar” state, can be extracted from [48,49,50,51] is given as follows,

#### 4.3.2. Correlation Energy of One-Dimensional Electron Gas. Interpolation

## 5. Concluding Remarks

## Funding

## Conflicts of Interest

## References

- Bender, C.M.; Boettcher, S. Determination of f(∞) from the asymptotic series for f(x) about x = 0. J. Math. Phys.
**1994**, 35, 1914–1921. [Google Scholar] [CrossRef] - Bogoliubov, N.N.; Shirkov, D.V. Quantum Fields; Benjamin-Cummings Pub. Co.: San Francisco, CA, USA, 1982. [Google Scholar]
- Nayfeh, A.H. Perturbation Methods; John Wiley: New York, NY, USA, 1981. [Google Scholar]
- Baker, G.A.; Graves-Moris, P. Padé Approximants; Cambridge University: Cambridge, UK, 1996. [Google Scholar]
- Gluzman, S.; Yukalov, V.I. Self-similar extrapolation from weak to strong coupling. J. Math. Chem.
**2010**, 48, 883–913. [Google Scholar] [CrossRef] [Green Version] - Gluzman, S.; Yukalov, V.I. Self-similarly corrected Padè approximants for indeterminate problem. Eur. Phys. J. Plus
**2016**, 131, 340–361. [Google Scholar] [CrossRef] [Green Version] - Gluzman, S.; Mityushev, V.; Nawalaniec, W. Computational Analysis of Structured Media; Academic Press (Elsevier): Cambridge, MA, USA, 2017. [Google Scholar]
- Dryga’s, P.; Gluzman, S.; Mityushev, V.; Nawalaniec, W. Applied Analysis of Composite Media; Woodhead Publishing (Elsevier): Sawston, UK, 2020. [Google Scholar]
- Bender, C.M.; Orszag, S.A. Advanced Mathematical Methods for Scientists and Engineers. Asymptotic Methods and Perturbation Theory); Springer: New York, NY, USA, 1999. [Google Scholar]
- Baker, G.A.; Gammel, J.L. The Padé approximant. J. Math. Anal. Appl.
**1961**, 2, 21–30. [Google Scholar] [CrossRef] [Green Version] - Yukalov, V.I. Theory of perturbations with a strong interaction. Moscow Univ. Phys. Bull.
**1976**, 51, 10–15. [Google Scholar] - Yukalov, V.I. Model of a hybrid crystal. Theor. Math. Phys.
**1976**, 28, 652–660. [Google Scholar] [CrossRef] - Kleinert, H. Path Integrals in Quantum Mechanics, Statistics, Polymer Physics and Financial Markets; World Scientific: Singapore, 2006. [Google Scholar]
- Kröger, H. Fractal geometry in quantum mechanics, field theory and spin systems. Phys. Rep.
**2000**, 323, 81–181. [Google Scholar] [CrossRef] - Ma, S. Theory of Critical Phenomena; Benjamin: London, UK, 1976. [Google Scholar]
- Yukalov, V.I.; Gluzman, S. Self-similar exponential approximants. Phys. Rev. E
**1998**, 58, 1359–1382. [Google Scholar] [CrossRef] [Green Version] - Gluzman, S.; Yukalov, V.I. Unified approach to crossover phenomena. Phys. Rev. E
**1998**, 58, 4197–4209. [Google Scholar] [CrossRef] [Green Version] - Andrianov, I.; Awrejcewicz, J.; Danishevs’kyy, V.; Ivankov, S. Asymptotic Methods in the Theory of Plates with Mixed Boundary Conditions; John Wiley & Sons: Hoboken, NJ, USA, 2014. [Google Scholar]
- Andrianov, I.; Shatrov, A. Padé Approximation to Solve the Problems of Aerodynamics and Heat Transfer in the Boundary Layer; IntechOpen: London, UK, 2020. [Google Scholar] [CrossRef]
- Gluzman, S.; Yukalov, V.I.; Sornette, D. Self-similar factor approximants. Phys. Rev. E
**2003**, 67, 026109. [Google Scholar] [CrossRef] [Green Version] - Yukalov, V.I.; Gluzman, S.; Sornette, D. Summation of Power Series by Self-Similar Factor Approximants. Physica A
**2003**, 328, 409–438. [Google Scholar] [CrossRef] [Green Version] - Yukalova, E.P.; Yukalov, V.I.; Gluzman, S. Solution of differential equations by self-similar factor approximants. Ann. Phys.
**2008**, 323, 3074–3090. [Google Scholar] [CrossRef] [Green Version] - Helmstetter, A.; Sornette, D.; Grasso, J.-R.; Andersen, J.V.; Gluzman, S.; Pisarenko, V. Slider-block friction model for landslides: Implication for prediction of mountain collapse. J. Geophys. Res.
**2004**, 109, B02409. [Google Scholar] [CrossRef] - Sornette, D.; Helmstetter, A.; Andersen, J.V.; Gluzman, S.; Grasso, J.-R.; Pisarenko, V.F. Towards landslide predictions: Two case studies. Phys. A
**2004**, 338, 605–632. [Google Scholar] [CrossRef] [Green Version] - Scholz, C.H. Earthquakes and friction laws. Nature
**1998**, 391, 37–42. [Google Scholar] [CrossRef] - Gluzman, A.; Sornette, D.; Yukalov, V.I. Generalized exponential laws by self-similar exponential approximants. Int. J. Mod. Phys. C
**2003**, 14, 509–527. [Google Scholar] [CrossRef] [Green Version] - He, H.X.; Hamer, C.J.; Oitmaa, J. High-temperature series expansions for the (2 + 1)-dimensional Ising model. J. Phys. A
**1990**, 23, 1775–1787. [Google Scholar] [CrossRef] - Gluzman, S. Nonlinear Approximations to Critical and Relaxation Processes. Available online: https://www.researchgate.net/publication/344189697_Nonlinear_approximations_to_critical_and_relaxation_processes (accessed on 2 September 2020). [CrossRef]
- Gluzman, S.; Yukalov, V.I. Self-similarly corrected Pade approximants for nonlinear equations. Int. J. Mod. Phys. B
**2019**, 33, 1950353. [Google Scholar] [CrossRef] - Gluzman, S.; Yukalov, V.I. Critical indices from self-similar root approximants. Eur. Phys. J. Plus
**2017**, 132, 535. [Google Scholar] [CrossRef] [Green Version] - Baker, G.A., Jr.; Gutierrez, G.; de Llano, M. Classical and quantum hard sphere fluids: Theory and experiment. Ann. Phys.
**1984**, 153, 283–300. [Google Scholar] [CrossRef] - Gluzman, S.; Yukalov, V.I. Effective summation and interpolation of series by self-similar root approximants. Mathematics
**2015**, 3, 510–526. [Google Scholar] [CrossRef] [Green Version] - Oller, J.A. Unitarization technics in hadron physics with historical remarks. Symmetry
**2020**, 12, 1114. [Google Scholar] [CrossRef] - Kastening, B. Fluctuation pressure of a fluid membrane between walls through six loops. Phys. Rev. E
**2006**, 73, 011101. [Google Scholar] [CrossRef] [Green Version] - Grosberg, A.Y.; Khokhlov, A.R. Statistical Physics of Macromolecules; AIP Press: Woodbury, NY, USA, 1994. [Google Scholar]
- Bender, C.M.; Wu, T.T. Anharmonic oscillator. Phys. Rev.
**1969**, 184, 1231–1260. [Google Scholar] [CrossRef] - Banks, T.; Torres, T.J. Two point Padè approximants and duality. arXiv
**2013**, arXiv:1307.3689v2. [Google Scholar] - Yukalov, V.I.; Gluzman, S. Self-similar interpolation in high-energy physics. Phys. Rev. D
**2015**, 91, 125023. [Google Scholar] [CrossRef] [Green Version] - Gluzman, S.; Yukalov, V.I. Additive self-similar approximants. J. Math. Chem.
**2017**, 55, 607–622. [Google Scholar] [CrossRef] [Green Version] - Czaplinski, T.; Drygas, P.; Gluzman, S.; Mityushev, V.; Nawalaniec, W.; Zietek, G. Elastic properties of a unidirectional composite reinforced with hexagonal array of fibers. Arch. Mech.
**2018**, 70, 1–33. [Google Scholar] - Sommerfeld, A. Integrazione asintotica dell equazione differenziale di Thomas-Fermi. Rend R Accad Lincei
**1932**, 15, 293–308. [Google Scholar] - Perrins, W.T.; McKenzie, S.R.; McPhedran, R.C. Transport properties of regular array of cylinders. Proc. R. Soc. A
**1979**, 369, 207–225. [Google Scholar] - Drygaś, P.; Filishtinski, L.A.; Gluzman, S.; Mityushev, V. Conductivity and elasticity of graphene-type composites. In 2D and Quasi-2D Composite and Nano Composite Materials, Properties and Photonic Applications; McPhedran, R., Gluzman, S., Mityushev, V., Rylko, N., Eds.; Elsevier: Amsterdam, The Netherlands, 2020; Chapter 8; pp. 193–231. [Google Scholar]
- Hioe, F.T.; Montroll, E.W. Quantum theory of anharmonic oscillators. I. Energy levels of oscillators with positive quartic anharmonicity. J. Math. Phys.
**1975**, 16, 1945–1955. [Google Scholar] [CrossRef] - Yukalov, V.I.; Yukalova, L.P.; Gluzman, S. Self-Similar Interpolation in Quantum Mechanics. Phys. Rev. A
**1998**, 58, 96–115. [Google Scholar] [CrossRef] [Green Version] - Schwinger, J. Gauge invariance and mass. Phys. Rev.
**1962**, 128, 2425–2428. [Google Scholar] [CrossRef] - Banks, T.; Susskind, L.; Kogut, J. Strong-coupling calculations of lattice gauge theories: (1 + 1)-dimensional exercises. Phys. Rev. D
**1976**, 13, 1043–1053. [Google Scholar] [CrossRef] - Carrol, A.; Kogut, J.; Sinclair, D.K.; Susskind, L. Lattice gauge theory calculations in 1 + 1 dimensions and the approach to the continuum limit. Phys. Rev. D
**1976**, 13, 2270–2277. [Google Scholar] [CrossRef] - Vary, J.P.; Fields, T.J.; Pirner, H.J. Chiral perturbation theory in the Schwinger model. Phys. Rev. D
**1996**, 53, 7231–7238. [Google Scholar] [CrossRef] [PubMed] - Adam, C. The Schwinger mass in the massive Schwinger model. Phys. Lett. B
**1996**, 382, 383–388. [Google Scholar] [CrossRef] [Green Version] - Striganesh, P.; Hamer, C.J.; Bursill, R.J. A new finite-lattice study of the massive Schwinger model. Phys. Rev. D
**2000**, 62, 034508. [Google Scholar] [CrossRef] [Green Version] - Hamer, C.J.; Weihong, Z.; Oitmaa, J. Series expansions for the massive Schwinger model in Hamiltonian lattice theory. Phys. Rev. D
**1997**, 56, 55–67. [Google Scholar] [CrossRef] [Green Version] - Coleman, S. More about the massive Schwinger model. Ann. Phys. (N. Y.)
**1976**, 101, 239–267. [Google Scholar] [CrossRef] - Hamer, C.J. Lattice model calculations for SU(2) Yang-Mills theory in 1 + 1 dimensions. Nucl. Phys. B
**1977**, 121, 159–175. [Google Scholar] [CrossRef] - Byrnes, T.M.R.; Striganesh, P.; Bursill, R.J.; Hamer, C.J. Density matrix renormalization group approach to the massive Schwinger model. Phys. Rev. D
**2002**, 66, 013002. [Google Scholar] [CrossRef] [Green Version] - Kröger H, H.; Scheu, N. The massive Schwinger model—A Hamiltonian lattice study in a fast moving frame. Phys. Lett. B
**1998**, 121, 58–63. [Google Scholar] [CrossRef] [Green Version] - Byrnes, T.M.R.; Hamer, C.J.; Weihong, Z.; Morrison, S. Application of Feynman-Kleinert approximants to the massive Schwinger model on a lattice. Phys. Rev. D
**2003**, 68, 016002. [Google Scholar] [CrossRef] [Green Version] - Loos, P.F. High-density correlation energy expansion of the one-dimensional uniform electron gas. J. Chem. Phys.
**2013**, 138, 064108. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Cioslowski, J. Robust interpolation between weak-and strong-correlation regimes of quantum systems. J. Chem. Phys.
**2012**, 136, 044109. [Google Scholar] [CrossRef] [PubMed]

**Figure 1.**Numerical solution to (5) is shown with solid line. The factor approximant ${F}_{f}^{*}(t)$ is shown with dashed line. The Padé approximant ${F}_{p}^{*}(t)$ is shown with dotted line.

**Figure 2.**Numerical solution to (5) is shown with solid line; factor approximant ${\Phi}_{4}^{*}(t)$ is shown with dashed line; factor approximant ${\Phi}_{6}^{*}(t)$ is shown with dotted line; factor approximant ${\Phi}_{8}^{*}(t)$ is shown with dot-dashed line.

**Figure 3.**Numerical solution to (5) is shown with solid line. Factor approximant ${\Phi}_{8}^{*}(t)$ is shown with dashed line. Padé approximant ${P}_{4,4}(t)$ is shown with dotted line.

**Figure 4.**Approximate reconstruction, Debye-Hukel function. The relative percentage error $\u03f5$ for the critical index, with varying n, is shown. The results are obtained from the formula (31).

**Figure 5.**Approximate reconstruction for quartic anharmonic oscillator. The relative percentage error $\u03f5$ for the critical index with varying n, is shown. The results are obtained from formula (31).

**Figure 6.**Approximate calculation of the critical index, for the Wilson loop. The relative percentage error $\u03f5$ for the critical index with varying n, is shown. The results are computed by applying the technique of corrected approximants in slightly modified form (34).

**Figure 7.**Approximate calculation of the critical index for the anharmonic partition integral. The error $\u03f5$ expressed in percentages, is presented. The dependence of the error is shown on the approximation order. It is obtained by applying the $Log$ Padé approximants according to the formula (40).

**Table 1.**Ground-state energy (vector state) of Schwinger model, dependent on the nondimensional parameter x. Different approximations are compared, such as Density matrix renormalization group, ${E}_{DMRG}$; $DLog$-additive approximants, ${E}_{DLog}$; fast moving frame estimates, ${E}_{FMFE}$; the self-similar additive approximant ${E}_{ad}^{*}$; and self-similar root approximant ${R}_{5}^{*}$ from [38].

x | ${\mathit{E}}_{\mathit{DMRG}}$ | ${\mathit{E}}_{\mathit{DLog}}$ | ${\mathit{E}}_{\mathit{FMFE}}$ | ${\mathit{R}}_{5}^{*}$ | ${\mathit{E}}_{\mathit{ad}}^{*}$ |
---|---|---|---|---|---|

0 | 0.56419 | 0.564 | N.A | 0.564 | 0.564 |

0.125 | 0.5395 | 0.540 | 0.528 | 0.54 | 0.539 |

0.25 | 0.51918 | 0.520 | 0.511 | 0.519 | 0.519 |

0.5 | 0.48747 | 0.49 | 0.489 | 0.487 | 0.487 |

1 | 0.4444 | 0.444 | 0.455 | 0.444 | 0.442 |

2 | 0.398 | 0.397 | 0.394 | 0.392 | 0.389 |

4 | 0.340 | 0.342 | 0.339 | 0.337 | 0.334 |

8 | 0.287 | 0.287 | 0.285 | 0.284 | 0.282 |

16 | 0.238 | 0.237 | 0.235 | 0.235 | 0.235 |

32 | 0.194 | 0.193 | 0.191 | 0.192 | 0.192 |

**Table 2.**The energy of scalar state, or of the first excited state of Schwinger model, dependent on the dimensionless parameter x, in different approximations: Finite lattice estimates, ${E}_{FLE}$; $DLog$-additive approximant, ${E}_{DLog}$; fast moving frame estimates, ${E}_{FMFE}$; the self-similar additive approximant ${E}_{ad}^{*}$; and the Padé approximant ${P}_{5,6}$.

x | ${\mathit{E}}_{\mathit{FLE}}$ | ${\mathit{E}}_{\mathit{DLog}}$ | ${\mathit{E}}_{\mathit{FMFE}}$ | ${\mathit{E}}_{\mathit{ad}}^{*}$ | ${\mathit{P}}_{5,6}$ |
---|---|---|---|---|---|

0 | 1.11 | 1.13 | N.A | 1.13 | 1.13 |

0.125 | 1.22 | 1.22 | 1.314 | 1.21 | 1.24 |

0.25 | 1.24 | 1.23 | 1.279 | 1.23 | 1.26 |

0.5 | 1.20 | 1.19 | 1.227 | 1.22 | 1.24 |

1 | 1.12 | 1.1 | 1.128 | 1.15 | 1.15 |

2 | 1.00 | 0.97 | 0.991 | 1.01 | 1.01 |

4 | 0.85 | 0.83 | 0. 837 | 0.85 | 0.85 |

8 | 0.68 | 0.69 | 0. 690 | 0.7 | 0.7 |

16 | 0.56 | 0.56 | 0.559 | 0.57 | 0.56 |

32 | 0.45 | 0.45 | 0.447 | 0.45 | 0.45 |

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**MDPI and ACS Style**

Gluzman, S.
Padé and Post-Padé Approximations for Critical Phenomena. *Symmetry* **2020**, *12*, 1600.
https://doi.org/10.3390/sym12101600

**AMA Style**

Gluzman S.
Padé and Post-Padé Approximations for Critical Phenomena. *Symmetry*. 2020; 12(10):1600.
https://doi.org/10.3390/sym12101600

**Chicago/Turabian Style**

Gluzman, Simon.
2020. "Padé and Post-Padé Approximations for Critical Phenomena" *Symmetry* 12, no. 10: 1600.
https://doi.org/10.3390/sym12101600