Entropy and Mutability for the <i>q</i>-State Clock Model in Small Systems
In this paper, we revisit the <i>q</i>-state clock model for small systems. We present results for the thermodynamics of the <i>q</i>-state clock model for values from <inline-formula> <math display="inline"> <semantics> <mrow> <mi>q<...
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| Format: | Article |
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MDPI AG
2018-12-01
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| Series: | Entropy |
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| Online Access: | https://www.mdpi.com/1099-4300/20/12/933 |
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| author | Oscar A. Negrete Patricio Vargas Francisco J. Peña Gonzalo Saravia Eugenio E. Vogel |
| author_facet | Oscar A. Negrete Patricio Vargas Francisco J. Peña Gonzalo Saravia Eugenio E. Vogel |
| author_sort | Oscar A. Negrete |
| collection | DOAJ |
| description | In this paper, we revisit the <i>q</i>-state clock model for small systems. We present results for the thermodynamics of the <i>q</i>-state clock model for values from <inline-formula> <math display="inline"> <semantics> <mrow> <mi>q</mi> <mo>=</mo> <mn>2</mn> </mrow> </semantics> </math> </inline-formula> to <inline-formula> <math display="inline"> <semantics> <mrow> <mi>q</mi> <mo>=</mo> <mn>20</mn> </mrow> </semantics> </math> </inline-formula> for small square lattices of <inline-formula> <math display="inline"> <semantics> <mrow> <mi>L</mi> <mo>×</mo> <mi>L</mi> </mrow> </semantics> </math> </inline-formula>, with L ranging from <inline-formula> <math display="inline"> <semantics> <mrow> <mi>L</mi> <mo>=</mo> <mn>3</mn> </mrow> </semantics> </math> </inline-formula> to <inline-formula> <math display="inline"> <semantics> <mrow> <mi>L</mi> <mo>=</mo> <mn>64</mn> </mrow> </semantics> </math> </inline-formula> with free-boundary conditions. Energy, specific heat, entropy, and magnetization were measured. We found that the Berezinskii⁻Kosterlitz⁻Thouless (BKT)-like transition appears for <inline-formula> <math display="inline"> <semantics> <mrow> <mi>q</mi> <mo>></mo> <mn>5</mn></mrow></semantics></math></inline-formula>, regardless of lattice size, while this transition at <inline-formula> <math display="inline"> <semantics> <mrow> <mi>q</mi> <mo>=</mo> <mn>5</mn> </mrow> </semantics> </math> </inline-formula> is lost for <inline-formula> <math display="inline"> <semantics> <mrow> <mi>L</mi> <mo><</mo> <mn>10</mn></mrow></semantics></math></inline-formula>; for <inline-formula> <math display="inline"> <semantics> <mrow> <mi>q</mi> <mo>≤</mo> <mn>4</mn></mrow></semantics></math></inline-formula>, the BKT transition is never present. We present the phase diagram in terms of <i>q</i> that shows the transition from the ferromagnetic (FM) to the paramagnetic (PM) phases at the critical temperature <inline-formula> <math display="inline"> <semantics> <msub> <mi>T</mi> <mn>1</mn> </msub> </semantics> </math> </inline-formula> for small systems, and the transition changes such that it is from the FM to the BKT phase for larger systems, while a second phase transition between the BKT and the PM phases occurs at <inline-formula> <math display="inline"> <semantics> <msub> <mi>T</mi> <mn>2</mn></msub></semantics></math></inline-formula>. We also show that the magnetic phases are well characterized by the two-dimensional (2D) distribution of the magnetization values. We made use of this opportunity to carry out an information theory analysis of the time series obtained from Monte Carlo simulations. In particular, we calculated the phenomenological mutability and diversity functions. Diversity characterizes the phase transitions, but the phases are less detectable as <i>q</i> increases. Free boundary conditions were used to better mimic the reality of small systems (far from any thermodynamic limit). The role of size is discussed. |
| first_indexed | 2024-04-14T02:01:22Z |
| format | Article |
| id | doaj.art-544591008d104491908e5be5a0f458df |
| institution | Directory Open Access Journal |
| issn | 1099-4300 |
| language | English |
| last_indexed | 2024-04-14T02:01:22Z |
| publishDate | 2018-12-01 |
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| series | Entropy |
| spelling | doaj.art-544591008d104491908e5be5a0f458df2022-12-22T02:18:49ZengMDPI AGEntropy1099-43002018-12-01201293310.3390/e20120933e20120933Entropy and Mutability for the <i>q</i>-State Clock Model in Small SystemsOscar A. Negrete0Patricio Vargas1Francisco J. Peña2Gonzalo Saravia3Eugenio E. Vogel4Departamento de Física, Universidad Técnica Federico Santa María, Valparaíso 2340000, ChileDepartamento de Física, Universidad Técnica Federico Santa María, Valparaíso 2340000, ChileDepartamento de Física, Universidad Técnica Federico Santa María, Valparaíso 2340000, ChileDepartamento de Ciencias Físicas, Universidad de La Frontera, Temuco 4811230, ChileCentro para el Desarrollo de la Nanociencia y la Nanotecnología, CEDENNA, Santiago 8320000, ChileIn this paper, we revisit the <i>q</i>-state clock model for small systems. We present results for the thermodynamics of the <i>q</i>-state clock model for values from <inline-formula> <math display="inline"> <semantics> <mrow> <mi>q</mi> <mo>=</mo> <mn>2</mn> </mrow> </semantics> </math> </inline-formula> to <inline-formula> <math display="inline"> <semantics> <mrow> <mi>q</mi> <mo>=</mo> <mn>20</mn> </mrow> </semantics> </math> </inline-formula> for small square lattices of <inline-formula> <math display="inline"> <semantics> <mrow> <mi>L</mi> <mo>×</mo> <mi>L</mi> </mrow> </semantics> </math> </inline-formula>, with L ranging from <inline-formula> <math display="inline"> <semantics> <mrow> <mi>L</mi> <mo>=</mo> <mn>3</mn> </mrow> </semantics> </math> </inline-formula> to <inline-formula> <math display="inline"> <semantics> <mrow> <mi>L</mi> <mo>=</mo> <mn>64</mn> </mrow> </semantics> </math> </inline-formula> with free-boundary conditions. Energy, specific heat, entropy, and magnetization were measured. We found that the Berezinskii⁻Kosterlitz⁻Thouless (BKT)-like transition appears for <inline-formula> <math display="inline"> <semantics> <mrow> <mi>q</mi> <mo>></mo> <mn>5</mn></mrow></semantics></math></inline-formula>, regardless of lattice size, while this transition at <inline-formula> <math display="inline"> <semantics> <mrow> <mi>q</mi> <mo>=</mo> <mn>5</mn> </mrow> </semantics> </math> </inline-formula> is lost for <inline-formula> <math display="inline"> <semantics> <mrow> <mi>L</mi> <mo><</mo> <mn>10</mn></mrow></semantics></math></inline-formula>; for <inline-formula> <math display="inline"> <semantics> <mrow> <mi>q</mi> <mo>≤</mo> <mn>4</mn></mrow></semantics></math></inline-formula>, the BKT transition is never present. We present the phase diagram in terms of <i>q</i> that shows the transition from the ferromagnetic (FM) to the paramagnetic (PM) phases at the critical temperature <inline-formula> <math display="inline"> <semantics> <msub> <mi>T</mi> <mn>1</mn> </msub> </semantics> </math> </inline-formula> for small systems, and the transition changes such that it is from the FM to the BKT phase for larger systems, while a second phase transition between the BKT and the PM phases occurs at <inline-formula> <math display="inline"> <semantics> <msub> <mi>T</mi> <mn>2</mn></msub></semantics></math></inline-formula>. We also show that the magnetic phases are well characterized by the two-dimensional (2D) distribution of the magnetization values. We made use of this opportunity to carry out an information theory analysis of the time series obtained from Monte Carlo simulations. In particular, we calculated the phenomenological mutability and diversity functions. Diversity characterizes the phase transitions, but the phases are less detectable as <i>q</i> increases. Free boundary conditions were used to better mimic the reality of small systems (far from any thermodynamic limit). The role of size is discussed.https://www.mdpi.com/1099-4300/20/12/933<i>q</i>-state clock modelentropyBerezinskii–Kosterlitz–Thouless transition |
| spellingShingle | Oscar A. Negrete Patricio Vargas Francisco J. Peña Gonzalo Saravia Eugenio E. Vogel Entropy and Mutability for the <i>q</i>-State Clock Model in Small Systems Entropy <i>q</i>-state clock model entropy Berezinskii–Kosterlitz–Thouless transition |
| title | Entropy and Mutability for the <i>q</i>-State Clock Model in Small Systems |
| title_full | Entropy and Mutability for the <i>q</i>-State Clock Model in Small Systems |
| title_fullStr | Entropy and Mutability for the <i>q</i>-State Clock Model in Small Systems |
| title_full_unstemmed | Entropy and Mutability for the <i>q</i>-State Clock Model in Small Systems |
| title_short | Entropy and Mutability for the <i>q</i>-State Clock Model in Small Systems |
| title_sort | entropy and mutability for the i q i state clock model in small systems |
| topic | <i>q</i>-state clock model entropy Berezinskii–Kosterlitz–Thouless transition |
| url | https://www.mdpi.com/1099-4300/20/12/933 |
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