# Investigation of Finite-Size 2D Ising Model with a Noisy Matrix of Spin-Spin Interactions

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Essential Expressions, the Equation of State

#### 2.1. The Effect of the Finite Grid Dimension

#### 2.2. The Effect of Noise

#### 2.3. Evaluating the Spectral Density

## 3. The Experiment Description

## 4. Experimental Results

#### 4.1. The Free and Internal Energy

#### 4.2. The Energy Variance

#### 4.3. The Critical Temperature

#### 4.4. The Ground State

#### 4.5. The Entropy

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Free energy $f(\beta )$ at different noise amplitudes $\eta =0;0.4;0.8;1.2;1.6;2.0;2.5;3$. Lower curves correspond to greater values of $\eta $. The red marks indicate the values that are found by the n-vicinity method with the aid of Formulae (15)–(17) at zero noise amplitude. The grid dimension $L=400$.

**Figure 2.**(

**a**) Internal energy $U(\beta )$ at different noise amplitudes $\eta \in [0,1.7]$ spaced by 0.1 intervals. The red marks indicate the values that are found by the n-vicinity method with the aid of Formulae (15)–(17) at zero noise amplitude. (

**b**) $\eta \in [1.8,3.0]$ spaced by 0.1 intervals, the lower curves correspond to greater $\eta $. The grid dimension $L=400$.

**Figure 3.**The energy variance ${\sigma}^{2}(\beta )$ at different noise amplitudes $\eta $: (

**a**) $\eta \in [0,1.7]$ and (

**b**) it changes by 0.1 intervals in range $\eta \in [1.8,3.0]$. The red marks indicate values ${\sigma}^{2}$ produced by Formula (5). The grid dimension $L=400$.

**Figure 4.**(

**a**) The critical temperature ${\beta}_{\u0441}$ and (

**b**) energy variance at the critical temperature ${\sigma}_{c}^{2}$ as functions of noise amplitude $\eta $. The solid lines correspond to Formulae (24)–(25). $L=400$.

**Figure 5.**(

**a**) Energy ${E}_{0}$ and (

**b**) magnetization ${M}_{0}$ of the ground state of the system as a function of noise amplitude. $L=400$.

**Figure 6.**(

**a**) Spectral density $\mathrm{\Psi}(E)$ and (

**b**) its first derivative for some noise amplitudes $\eta =0;0.3;0.7;1.1;1.5;1.8;2.2;2.5;3$. The marks show the zero-noise curve. The grid dimension $L=400$.

**Figure 7.**The second derivative of spectral density $\ddot{\mathrm{\Psi}}(E)$ at (

**a**) $\eta =[0,1.7]$ and (

**b**) $\eta =[1.8,3]$, the reading spacing is 0.1. The marks denote the zero-noise curve (

**a**) and the curve for $\eta =1.8$ resulted from (27) (

**b**). The grid dimension $L=400$.

**Table 1.**The energy of ground state ${E}_{0}$ and its magnetization ${M}_{0}$, critical values ${\beta}_{c}$, ${f}_{c}$, ${U}_{c}$ and ${\sigma}_{c}^{2}$ for different noise amplitudes.

$\mathit{\eta}$ | ${\mathit{E}}_{\mathbf{0}}$ | ${\mathit{M}}_{\mathbf{0}}$ | ${\mathit{\beta}}_{\mathit{c}}$ | ${\mathit{f}}_{\mathit{c}}$ | ${\mathit{U}}_{\mathit{c}}$ | ${\mathit{\sigma}}_{\mathit{c}}^{\mathbf{2}}$ |
---|---|---|---|---|---|---|

0 | −1.995 | 1 | 0.442 | −0.6931 | −1.978 × 10^{5} | 12.958 |

0.1 | −1.995 | 1 | 0.443 | −0.6931 | −1.986 × 10^{5} | 11.427 |

0.2 | −1.995 | 1 | 0.444 | −0.6932 | −0.0101 | 12.566 |

0.3 | −1.995 | 1 | 0.445 | −0.6932 | −0.0103 | 11.627 |

0.4 | −1.996 | 1 | 0.452 | −0.6933 | −0.0211 | 11.476 |

0.5 | −1.994 | 1 | 0.454 | −0.6934 | −0.0324 | 10.666 |

0.6 | −1.993 | 1 | 0.459 | −0.6936 | −0.0447 | 9.719 |

0.7 | −1.994 | 1 | 0.465 | −0.6939 | −0.0581 | 8.328 |

0.8 | −1.996 | 1 | 0.476 | −0.6946 | −0.0849 | 7.642 |

0.9 | −1.996 | 1 | 0.484 | −0.6957 | −0.1143 | 6.518 |

1.0 | −1.993 | 1 | 0.503 | −0.6979 | −0.1599 | 5.603 |

1.1 | −1.996 | 0.9998 | 0.515 | −0.7010 | −0.2109 | 4.656 |

1.2 | −1.995 | 0.9987 | 0.536 | −0.7065 | −0.2815 | 3.629 |

1.3 | −1.994 | 0.9943 | 0.562 | −0.7156 | −0.3747 | 2.775 |

1.4 | −1.996 | 0.9839 | 0.591 | −0.7327 | −0.5107 | 1.998 |

1.5 | −2.002 | 0.9602 | 0.623 | −0.7527 | −0.6414 | 1.380 |

1.6 | −2.014 | 0.9060 | - | - | - | - |

1.7 | −2.033 | 0.2155 | - | - | - | - |

1.8 | −2.065 | 0.0312 | - | - | - | - |

1.9 | −2.098 | 0.0241 | - | - | - | - |

2.0 | −2.139 | 0.0058 | - | - | - | - |

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**MDPI and ACS Style**

Kryzhanovsky, B.; Malsagov, M.; Karandashev, I.
Investigation of Finite-Size 2D Ising Model with a Noisy Matrix of Spin-Spin Interactions. *Entropy* **2018**, *20*, 585.
https://doi.org/10.3390/e20080585

**AMA Style**

Kryzhanovsky B, Malsagov M, Karandashev I.
Investigation of Finite-Size 2D Ising Model with a Noisy Matrix of Spin-Spin Interactions. *Entropy*. 2018; 20(8):585.
https://doi.org/10.3390/e20080585

**Chicago/Turabian Style**

Kryzhanovsky, Boris, Magomed Malsagov, and Iakov Karandashev.
2018. "Investigation of Finite-Size 2D Ising Model with a Noisy Matrix of Spin-Spin Interactions" *Entropy* 20, no. 8: 585.
https://doi.org/10.3390/e20080585