# 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

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**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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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