We investigated the impact of Kaniadakis statistics on thermodynamic properties for a square magnetic grid. We used the Ising model. We reported numerical results for a two-dimensional magnetic network in a thermal bath. We calculated the probabilities of transition between states using -statistics, the Metropolis dynamics for stochastic processes of the finite-sized magnetic network, considering that the system interacts with a thermal reservoir. We investigated the behavior of various thermodynamic properties. We observed typical measurements of magnetization, energy and specific heat. Increasing the  parameter reduces the critical temperature. We observed by the measurements of the fourth order Binder cumulative of magnetization, that for different network sizes and different values of the parameter , the system transition temperature magnetic decreases as ? increases.

Chagas, Maria Liduína das

Costa, Rafael Santos da

Moura, Thiago Rafael da Silva

Neyra, Jair Rodrigues

Nunes, Thiago Crisóstomo Carlos

Silva, Maria das Graças Dias da

Portuguese

Brazilian Journals

Brazilian Applied Science Review  Braz. Ap. Sci. Rev, Curitiba, v. 4, n. 3, p. 1535-1543 mai./jun. 2020           ISSN 2595-3621  1535  The relativistic effect of critical temperature reduction in the two dimensional Ising model  O efeito relativístico da redução crítica de temperatura no modelo bidimensional de Ising  DOI:10.34115/basrv4n3-063  Recebimento dos originais: 20/04/2020 Aceitação para publicação: 27/05/2020  Rafael Santos da Costa Graduando do Curso de Licenciatura em Matemática pela Universidade Federal do Pará Instituição: Universidade Federal do Pará, Faculdade de Matemática - UFPA Endereço: Rua Raimundo Santana Cruz, s/n, Bairro São Tomé, Salinópolis – PA, Brasil E-mail: rafael_santos723@outlook.com  Jair Rodrigues Neyra Graduando do Curso de Engenharia de Exploração e Produção de Petróleo pela Universidade Federal do Pará Instituição: Universidade Federal do Pará, Faculdade de Engenharia - UFPA Endereço: Rua Raimundo Santana Cruz, s/n, Bairro São Tomé, Salinópolis – PA, Brasil E-mail: jairneyra30@gmail.com  Maria Liduína das Chagas Doutora em Física Pela Universidade Federal do Rio Grande do Norte Instituição: Universidade Federal do Sul e Sudeste do Pará, Faculdade de Física - Instituto de Ciências Exatas Marabá – PA Endereço: Cidade Universitária Campus Do Tauarizinho, Marabá-PA, Brasil E-mail: liduina@unifesspa.edu.br  Maria das Graças Dias da Silva Doutora em Física Pela Universidade Federal do Rio Grande do Norte Instituição: Universidade do Estado do Rio Grande do Norte, Faculdade de Ciências Exatas e Naturais - UERN Mossoró – RN Endereço: Rua Almino, 478 – Centro – Mossoró/RN E-mail: gracinha_dias@hotmail.com  Thiago Crisóstomo Carlos Nunes Doutor em Física Pela Universidade Federal do Rio Grande do Norte Instituição: Universidade Federal do Rio Grande do Norte Endereço: Avenida Senador Salgado Filho, 3000, Bairro Lagoa Nova, Natal – RN, Brasil E-mail: thiago.cris@yahoo.com.br  Brazilian Applied Science Review  Braz. Ap. Sci. Rev, Curitiba, v. 4, n. 3, p. 1535-1543 mai./jun. 2020           ISSN 2595-3621  1536  Thiago Rafael da Silva Moura Doutor em Física Pela Universidade Federal do Rio Grande do Norte Instituição: Universidade Federal do Pará, Faculdade de Física - UFPA Endereço: Rua Raimundo Santana Cruz, s/n, Bairro São Tomé, Salinópolis – PA, Brasil E-mail: trsmoura@yahoo.com.br  ABSTRACT We investigated the impact of Kaniadakis statistics on thermodynamic properties for a square magnetic grid. We used the Ising model. We reported numerical results for a two-dimensional magnetic network in a thermal bath. We calculated the probabilities of transition between states using 𝜅-statistics, the Metropolis dynamics for stochastic processes of the finite-sized magnetic network, considering that the system interacts with a thermal reservoir. We investigated the behavior of various thermodynamic properties. We observed typical measurements of magnetization, energy and specific heat. Increasing the 𝜅 parameter reduces the critical temperature. We observed by the measurements of the fourth order Binder cumulative of magnetization, that for different network sizes and different values of the parameter 𝜅, the system transition temperature magnetic decreases as κ increases.  Keywords: 𝜅-exponential, Ising model, phase transitions.  RESUMO Investigamos o impacto das estatísticas Kaniadakis nas propriedades termodinâmicas de uma grade magnética quadrada. Utilizamos o modelo de Ising. Relatamos resultados numéricos para uma rede magnética bidimensional em banho termal. Calculamos as probabilidades de transição entre estados usando a estatística k, a dinâmica Metropolis para processos estocásticos da rede magnética de tamanho finito, considerando que o sistema interage com um reservatório térmico. Investigamos o comportamento de várias propriedades termodinâmicas. Observamos medidas típicas de magnetização, energia e calor específico. Aumentar o parâmetro κ reduz a temperatura crítica. Observamos pelas medições do Fichário de quarta ordem cumulativo de magnetização, que para diferentes tamanhos de rede e diferentes valores do parâmetro κ, a temperatura de transição do sistema magnético diminui à medida que κ aumenta.  Palavras-chave: κ-exponencial, modelo de Ising, transições de fase.   1 INTRODUCTION The area of phase transitions and critical phenomena is one of the central themes in Physics. This type of study is carried out in several physical systems: fluids, liquid crystals, metal alloys, spin glasses, magnetic materials, piezoelectric, ferroelectric materials, biological systems, etc. Magnetic systems are of particular interest for the potential of biomedical applications in magnetic hyperthermia and for the construction of new technological devices such as: magnetic memories, nano-antennas, magnetic tunneling junctions, etc. [1-8] The Ising model is a prototype model, proposed to provide an explanation of the phase transition of a magnetic system. The Ising model consists of 𝑁 spins, subjected to short-range interactions, spread over the sites of a two-dimensional network. Brazilian Applied Science Review  Braz. Ap. Sci. Rev, Curitiba, v. 4, n. 3, p. 1535-1543 mai./jun. 2020           ISSN 2595-3621  1537  Since Gibbs proposed the first generalization of Maxwell-Boltzmann's entropy, many other generalizations, whether classical or quantum, have emerged within the context of Statistical Mechanics, among these several we mentioned Tsalis [9] and Kaniadakis [10], [11]. A common point between these new entropies is the fact that these distributions, which form the basis of such generalizations, depend on some deforming parameter. In 2001, G. Kaniadakis proposed a new statistic, the κ-statistic, which generalizes the Maxwell-Boltzmann-Gibbs statistic by varying the deformation parameter 𝜅. The theory is guided by the principle of kinetic interaction (PKI). According to PKI, the temporal evolution of the function of distribution of identical particles subject to binary collisions leads us to a functional that is always increasing over time, satisfying the statement of the irreversibility of the second law of Thermodynamics. Kaniadakis concluded that such a function was related to a type of entropy defined as:                               𝑆𝜅 = −⟨𝑙𝑛𝜅[𝑓(𝑥)]⟩ = −∫𝑑𝑥 𝑓(𝑥)𝑙𝑛𝜅[𝑓(𝑥)]                            (1)  in which 𝑓(𝑥) is the velocity distribution of the particles and 𝑙𝑛𝜅 is the logarithm deformed by the parameter 𝜅. The 𝑙𝑛𝜅 is a real and decreasing function ∀𝑥 ∈ 𝑅 given by:                                               𝑙𝑛𝜅(𝑥) =𝑥𝜅−𝑥−𝜅2𝜅                                                 (2)  its inverse function is called 𝜅-exponential [12]. It is quantified by                                                        𝑒𝑥𝑝𝜅(±𝑥) = (√1 + 𝜅2𝑥2 ± 𝜅𝑥)1𝜅                                 (3)  which is set for the interval 0 ⩽ 𝑥 < 1. When we take the limit of 𝜅 → 0, the function that is 𝜅-exponential becomes an ordinary exponential.                                                         𝑙𝑖𝑚𝜅→0𝑒𝑥𝑝𝜅 (±𝑥) = 𝑒𝑥𝑝(±𝑥)                                           (4)  2 THE THEORETICAL MODEL The model deals with a network of short-range interactions between spins, described by the following Hamiltonian  Brazilian Applied Science Review  Braz. Ap. Sci. Rev, Curitiba, v. 4, n. 3, p. 1535-1543 mai./jun. 2020           ISSN 2595-3621  1538                       𝐸 = −𝐽∑ ∑ 𝜎𝑖 ∙ 𝜎𝑗𝑗𝑖 + ?⃗? . ∑ 𝜎𝑗𝑗                  (5)  𝐽 is the exchange constant, ?⃗?  the applied magnetic field and { 𝜎 = ±1 } is the set of spins in the network. The first term of equation (5) describes how the exchange interaction occurs between adjacent spins, the pair ⟨𝑖, 𝑗⟩. The second term of equation (5) Zeeman energy, where ?⃗? = 𝐻𝑥 ?̂? is the magnetic field applied along the axis 𝑥.   Magnetization is defined as the sum of all spins in the network,                        ⟨𝑀⟩ =1𝑁∑ 𝜎𝑖𝑁𝑖=1                            (6)   Deforming the Boltzmann distribution function in terms of 𝜅-statistics, the probability that a given specific state will occur in our model is      𝑊𝜅(𝛼) =1𝑍𝑒𝑥𝑝𝜅(−𝛽𝐸𝛼)                           (7)  in which 𝛽 =1𝐾𝐵𝑇 and 𝑍 is the system partition function, which can be written as                   𝑍 = ∑ 𝑒𝑥𝑝𝜅𝑁𝑖=1 (𝛽𝐸𝑖)                             (8)  in which 𝐾𝐵 is the Boltzmann constant and 𝑇 is the absolute temperature.  2.1 NUMERICAL EXPERIMENTATIONS  We proceeded with numerical experiments to obtain the measurements of the physical observables. We estimated the observables concerning the solutions of the master equation. The master equation is              𝑑𝑃(𝑖,𝑗)𝑑𝑡= ∑ {𝑊(𝑖, 𝑗)𝑃(𝑗, 𝑡) − 𝑊(𝑗, 𝑖)𝑃(𝑖, 𝑡)}𝑖≠𝑗      (9)  in which 𝑊(𝑖, 𝑗) is the state's probability transition rate 𝑖 for 𝑗 and 𝑃(𝑗, 𝑡) is the state probability 𝑗 in the instant of time: 𝑡. Given the transition rates, we found the stationary solution that satisfies the condition:     ∑ {𝑊(𝑖, 𝑗)𝑃(𝑗, 𝑡) − 𝑊(𝑗, 𝑖)𝑃(𝑖, 𝑡)}𝑖≠𝑗 = 0                  (10) Brazilian Applied Science Review  Braz. Ap. Sci. Rev, Curitiba, v. 4, n. 3, p. 1535-1543 mai./jun. 2020           ISSN 2595-3621  1539   The numerical solution of the master equation for the equilibrium state equation (10) served as the basis for the development of the Metropolis algorithm [13].    The Metropolis algorithm is summarized as follows:  1. Configure an initial 𝑗 configuration for the system; 2. Generate tentative configuration of the j-th simulation cell; 3. If 𝛥𝐸𝑗,𝑖 = 𝐸𝑗 − 𝐸𝑖 < 0, the configuration 𝑖 is accepted; 4. Otherwise, a random number is generated 𝛿 ∈ (0,1). If 𝛿 < 𝑃(𝛥𝐸𝑗,𝑖) ∝ 𝑒𝑥𝑝𝜅(𝛥𝐸𝑖,𝑗 𝐾𝐵⁄ 𝑇) the new configuration is accepted. 5. Otherwise, the 𝑗 configuration remains.  To obtain our numerical results we worked with reduced constants 𝐽 = 1 and 𝐾𝐵 = 1.  The exchange interaction occurs between the first neighbors of a variable on a network site. Naturally, the so-called edge effect arises, a problem that we overcome with the use of periodic boundary conditions, where a variable contained in one edge interacts, according to equation (5), with another located in an opposite edge. We defined the initial state of the magnetic grid with all spins aligned in the same direction, for instance, they are coupled ferromagnetically. This type of coupling is characteristic of materials in the ferromagnetic state. Starting from this initial network, we implemented the Metropolis algorithm, we will not describe the algorithm here, but the interested reader can consult the reference [14]. For the quantities ⟨𝐸⟩ and ⟨𝑚⟩ we have the following relationships:                        ⟨𝐸⟩ =1𝑁∑ 𝐸𝑖𝑁𝑖=1                           (11)                                  ⟨𝑚⟩ =1𝑁∑ 𝑀𝑖𝑁𝑖=1                             (12)   Specific heat at a constant volume, 𝐶𝑣, is proportional to the energy variance. According to reference [15], we have:            𝐶𝑣 =1𝑁𝐾𝐵𝑇2[⟨𝐸2⟩ − ⟨𝐸⟩2]               (13)  Brazilian Applied Science Review  Braz. Ap. Sci. Rev, Curitiba, v. 4, n. 3, p. 1535-1543 mai./jun. 2020           ISSN 2595-3621  1540  Besides, we measured the fourth order Binder cumulative of magnetization [14].                 𝑈4(𝑀) = 1 −⟨𝑀4⟩3⟨𝑀2⟩2                  (14)  3 RESULTS We carried out numerical experiments on networks with lateral dimensions of the order of equivalent to 𝐿 = 4,8,14,16 and 24, using 106 Monte Carlo steps. We discarded 2.5𝑥104, starting the accounting of physical observables after reaching statistical stability. In figure 1, we presented typical measurements of magnetization and Helmholtz free κenergy. We observed that the average magnetization reaches the transition temperature more quickly with increasing 𝜅. The expected value of energy reaches its saturation point, more quickly, the higher the quantitative values of 𝜅.   Figure 1 - In (a) we have the average magnetization per particle. In (b) the average energy per particle. In both, we observed that as 𝜅 increases, the phase transition occurs at an increasingly lower temperature. The curve for 𝜅 = 10−9 in the two graphs, confirm that at the limit: 𝜅 → 0 our model falls back to the classic Ising model.  In figure 2, we show the results of the specific heat for different values of 𝜅. We noted that as 𝜅 grows, the second derivative of energy, has maximums closer to 𝑇 = 0. The results shown in figures 1 and 2, show the shift in transition temperature 𝑇𝐶 as 𝜅 increases. Brazilian Applied Science Review  Braz. Ap. Sci. Rev, Curitiba, v. 4, n. 3, p. 1535-1543 mai./jun. 2020           ISSN 2595-3621  1541   Figure 2 - In (a) we observed the presence of peaks occurring around a region close to 𝑇 = 0. In (b) observed that in other region there are other minor peaks, which in turn are related to 𝑇𝐶  described by the Binder Cumulative 𝑈4(𝑀).   Figure 3 shows typical Binder Cumulative measurements for 𝜅 = 10−9 and 𝜅 = 0.9, with different network sizes, depending on the temperature. Square side nets were used 𝐿 = 4,8,14,16 and 24. Throughout the cumulant measurements, the region where the lines cross is the critical point measurement region, which is 𝑇𝐶 = 2.27 for 𝜅 = 10−9  and 𝑇𝐶 = 1.07 for 𝜅 = 0.9.   Figure 3 - Typical measurements of the fourth order Binder Cumulative for magnetization 𝑈4(𝑀) as a function of temperature T for different network sizes, in (a) with 𝜅 = 10−9, we observed that 𝑇𝐶 = 2.27, in (b) we observed that with the increase of the parameter, it has a reduction in 𝑇𝐶  , because taking 𝜅 = 0.9 we observed 𝑇𝐶 = 1.07.   4 CONCLUSION We studied a magnetic system, square network, in thermal bath, with the Hamiltonian of Ising. We used 𝜅-exponential distribution, generalization of Boltzmann statistics, to measure typical physical observables. We analyzed the effect of the disturbance caused by the parameter 𝜅 on the system, we Brazilian Applied Science Review  Braz. Ap. Sci. Rev, Curitiba, v. 4, n. 3, p. 1535-1543 mai./jun. 2020           ISSN 2595-3621  1542  observed that with the reduction of𝜅 our model tries to fall back on the classic Ising model. Still, we obtained that with the increase of the parameter𝜅, 𝑇𝐶 is reduced, leading to the transition between the ferromagnetic and paramagnetic phases occurring at lower temperatures. We estimated the critical point related to each value of the parameter𝜅, using the fourth order Binder Cumulative for magnetization 𝑈4(𝑀).   For future contributions we intended to investigate, through the study of critical exponents, the class of universality of our model.   REFERENCES  [1] Slonczewski, J.C. (1989). Phys. Rev. B 39, 6995.    [2] Mattis, D.C. (1981) “The Theory of Magnetism I.” Springer Verlag, Berlin.  [3] Gerlach, W. and Stern, O. (1992). Z. Phys. 9, 349.  [4] Lodder, J.C. (2006) “Patterned Nanomagnetic Films”, in: D.J. Sellmyer and R. Skomski (eds), Advanced Magnetic Materials Nanostructures, Springer, Berlin, pp.261-293.  [5] AJAYAN, P.M. Nanotubes from carbo. Chem. Rev., Washington DC, v.99, n.7, 1999.  [6] BAIBICH, M. N.; BROTO, J. M.; FERT, A.; VAN DAU, F. Nguyen; PETROFF, F.; EITENNE, P.; CREUZET, G.; FRIEDERICH, A.  and CHAZELAS, J. . Giant Magnetoresistance of (001)Fe/(001)Cr Magnetic Superlattices. Physical Review Letters. v.61, .21, 1988.  [7] BINASCH, G.; GRÜNBERG, P.; SAURENCH, F. and ZINN, W. . Enhanced magnetoresistance in layered magnetic structures with antiferomagnetic interlayer exchange. Physical Review B. v.39, n.7, 1989.  [8] SHINJO, T. Nanomagnetism and Spintronics. 1st ed. Amsterdã: Elsevier, 2009.  Brazilian Applied Science Review  Braz. Ap. Sci. Rev, Curitiba, v. 4, n. 3, p. 1535-1543 mai./jun. 2020           ISSN 2595-3621  1543  [9] TSALLIS, C. Possible generalization of Boltzmann-Gibbs statistics. J. Stat. Phys., v. 52, p. 470, 1988.  [10] KANIADAKIS, G. Non-linear kinetics underlying generalized statistics. Physica A, v. 256, p. 405–425, 2001.  [11] KANIADAKIS, G. Statistical mechanics in the context of special relativity. Phys. Rev.E, v. 66, p. 56125, 2002.  [12] KANIADAKIS, G. Theoretical foundations and mathematical formalism of the power-law tailed statistical distributions. Entropy, v. 15, n. 10, p. 3983–4010, 2013.  [13] N.Metropolis, A. Rosenbluth, M. Rosenbluth, A. Teller, E. Teller, Equation of State Calculations by Fast Computing Machines, Journal of Chemical Physics 21, 1087, 1953.  [14] NEWMAN, M. E. J.; BARKEMA, G. T. Monte Carlo Methods in Statistical Physics. [S.l.]: Oxford University Press Inc, New York, 1999.  [15] TSAI, S.; SALINAS, S. R. A. Fourth-order cumulants to characterize the phase transitions of a spin-1 ising model. Braz. J. Phys., v. 28, n. 1, 1998.  

The relativistic effect of critical temperature reduction in the two dimensional Ising model / O efeito relativístico da redução crítica de temperatura no modelo bidimensional de Ising

https://ojs.brazilianjournals.com.br/ojs/index.php/BASR/article/download/10733/9261

The relativistic effect of critical temperature reduction in the two dimensional Ising model / O efeito relativístico da redução crítica de temperatura no modelo bidimensional de Ising

Abstract

Similar works

Full text

Available Versions

Brazilian Journals