### Abstract

In order to investigate the role of the parameter space on the third law of thermodynamics, we have checked the validity intervals of the Borges-Roditi, Abe and Sharma-Mittal entropies in the framework of the third law of thermodynamics. The two-parameter Borges-Roditi entropy conforms to the third law for a < 1, 0 < b < 1 or b < 1, 0 < a < 1 due to its a ? b symmetry, while the Abe entropy satisfies the third law in the interval 1 2 < σ < 1 or 1 < σ < 2. Since the validity interval of the single-parameter Abe entropy can be fully recovered from the two-parameter Borges-Roditi entropy, we note that the third law is immune to the reduction of the parameters in this particular case. The Sharma-Mittal entropy, in this same context, is valid in the interval 0 < q ≤ 1, whereas its second parameter r can take any real values without restriction. The interval 0 < q ≤ 1 is the intersection of the Sharma-Mittal entropy with its reduced expressions, i.e., the Tsallis and Rényi entropies from the third law perspective. This implies that the additional parameter r of the Sharma-Mittal entropy indeed renders it more generalized compared to the one-parameter Rényi and Tsallis entropies when validated by the third law of thermodynamics. Therefore, based on these observations, we deduce that an additional parameter can extend the interval of validity of the third law for a particular generalized entropy only when this additional parameter is not confined by reduction to some other generalized entropy definitions. We also check our theoretical results by using the one-dimensional Ising model with periodic boundary conditions under zero external field.

Original language | English |
---|---|

Article number | 1850274 |

Journal | International Journal of Modern Physics B |

Volume | 32 |

Issue number | 24 |

DOIs | |

Publication status | Published - Sep 30 2018 |

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### Keywords

- 3rd law of thermodynamics
- Abe and Sharma-Mittal entropies
- Borges-Roditi
- generalized entropies

### ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Condensed Matter Physics

### Cite this

*International Journal of Modern Physics B*,

*32*(24), [1850274]. https://doi.org/10.1142/S0217979218502740

**The parameter space and third law of thermodynamics for the Borges-Roditi, Abe and Sharma-Mittal entropies.** / Canturk, Bilal; Oikonomou, Thomas; Baris Bagci, G.

Research output: Contribution to journal › Article

*International Journal of Modern Physics B*, vol. 32, no. 24, 1850274. https://doi.org/10.1142/S0217979218502740

}

TY - JOUR

T1 - The parameter space and third law of thermodynamics for the Borges-Roditi, Abe and Sharma-Mittal entropies

AU - Canturk, Bilal

AU - Oikonomou, Thomas

AU - Baris Bagci, G.

PY - 2018/9/30

Y1 - 2018/9/30

N2 - In order to investigate the role of the parameter space on the third law of thermodynamics, we have checked the validity intervals of the Borges-Roditi, Abe and Sharma-Mittal entropies in the framework of the third law of thermodynamics. The two-parameter Borges-Roditi entropy conforms to the third law for a < 1, 0 < b < 1 or b < 1, 0 < a < 1 due to its a ? b symmetry, while the Abe entropy satisfies the third law in the interval 1 2 < σ < 1 or 1 < σ < 2. Since the validity interval of the single-parameter Abe entropy can be fully recovered from the two-parameter Borges-Roditi entropy, we note that the third law is immune to the reduction of the parameters in this particular case. The Sharma-Mittal entropy, in this same context, is valid in the interval 0 < q ≤ 1, whereas its second parameter r can take any real values without restriction. The interval 0 < q ≤ 1 is the intersection of the Sharma-Mittal entropy with its reduced expressions, i.e., the Tsallis and Rényi entropies from the third law perspective. This implies that the additional parameter r of the Sharma-Mittal entropy indeed renders it more generalized compared to the one-parameter Rényi and Tsallis entropies when validated by the third law of thermodynamics. Therefore, based on these observations, we deduce that an additional parameter can extend the interval of validity of the third law for a particular generalized entropy only when this additional parameter is not confined by reduction to some other generalized entropy definitions. We also check our theoretical results by using the one-dimensional Ising model with periodic boundary conditions under zero external field.

AB - In order to investigate the role of the parameter space on the third law of thermodynamics, we have checked the validity intervals of the Borges-Roditi, Abe and Sharma-Mittal entropies in the framework of the third law of thermodynamics. The two-parameter Borges-Roditi entropy conforms to the third law for a < 1, 0 < b < 1 or b < 1, 0 < a < 1 due to its a ? b symmetry, while the Abe entropy satisfies the third law in the interval 1 2 < σ < 1 or 1 < σ < 2. Since the validity interval of the single-parameter Abe entropy can be fully recovered from the two-parameter Borges-Roditi entropy, we note that the third law is immune to the reduction of the parameters in this particular case. The Sharma-Mittal entropy, in this same context, is valid in the interval 0 < q ≤ 1, whereas its second parameter r can take any real values without restriction. The interval 0 < q ≤ 1 is the intersection of the Sharma-Mittal entropy with its reduced expressions, i.e., the Tsallis and Rényi entropies from the third law perspective. This implies that the additional parameter r of the Sharma-Mittal entropy indeed renders it more generalized compared to the one-parameter Rényi and Tsallis entropies when validated by the third law of thermodynamics. Therefore, based on these observations, we deduce that an additional parameter can extend the interval of validity of the third law for a particular generalized entropy only when this additional parameter is not confined by reduction to some other generalized entropy definitions. We also check our theoretical results by using the one-dimensional Ising model with periodic boundary conditions under zero external field.

KW - 3rd law of thermodynamics

KW - Abe and Sharma-Mittal entropies

KW - Borges-Roditi

KW - generalized entropies

UR - http://www.scopus.com/inward/record.url?scp=85052904507&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85052904507&partnerID=8YFLogxK

U2 - 10.1142/S0217979218502740

DO - 10.1142/S0217979218502740

M3 - Article

VL - 32

JO - International Journal of Modern Physics B

JF - International Journal of Modern Physics B

SN - 0217-9792

IS - 24

M1 - 1850274

ER -