On the Thermodynamics of the <i>q</i>-Particles
Since the grand partition function <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>Z</mi><mi>q</mi></msub></semantics></math></inline-formula> for the so...
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| Format: | Article |
| Language: | English |
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MDPI AG
2022-01-01
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| Series: | Entropy |
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| Online Access: | https://www.mdpi.com/1099-4300/24/2/159 |
| _version_ | 1797480588793348096 |
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| author | Fabio Ciolli Francesco Fidaleo |
| author_facet | Fabio Ciolli Francesco Fidaleo |
| author_sort | Fabio Ciolli |
| collection | DOAJ |
| description | Since the grand partition function <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>Z</mi><mi>q</mi></msub></semantics></math></inline-formula> for the so-called <i>q</i>-particles (i.e., quons), <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>q</mi><mo>∈</mo><mo>(</mo><mo>−</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>)</mo></mrow></semantics></math></inline-formula>, cannot be computed by using the standard 2nd quantisation technique involving the full Fock space construction for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>q</mi><mo>=</mo><mn>0</mn></mrow></semantics></math></inline-formula>, and its <i>q</i>-deformations for the remaining cases, we determine such grand partition functions in order to obtain the natural generalisation of the Plank distribution to <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>q</mi><mo>∈</mo><mo>[</mo><mo>−</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>]</mo></mrow></semantics></math></inline-formula>. We also note the (non) surprising fact that the right grand partition function concerning the Boltzmann case (i.e., <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>q</mi><mo>=</mo><mn>0</mn></mrow></semantics></math></inline-formula>) can be easily obtained by using the full Fock space 2nd quantisation, by considering the appropriate correction by the Gibbs factor <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>1</mn><mo>/</mo><mi>n</mi><mo>!</mo></mrow></semantics></math></inline-formula> in the <i>n</i> term of the power series expansion with respect to the fugacity <i>z</i>. As an application, we briefly discuss the equations of the state for a gas of free quons or the condensation phenomenon into the ground state, also occurring for the Bose-like quons <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>q</mi><mo>∈</mo><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo></mrow></semantics></math></inline-formula>. |
| first_indexed | 2024-03-09T22:02:15Z |
| format | Article |
| id | doaj.art-118e331232d9472abcafddbab3bb824d |
| institution | Directory Open Access Journal |
| issn | 1099-4300 |
| language | English |
| last_indexed | 2024-03-09T22:02:15Z |
| publishDate | 2022-01-01 |
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| record_format | Article |
| series | Entropy |
| spelling | doaj.art-118e331232d9472abcafddbab3bb824d2023-11-23T19:46:56ZengMDPI AGEntropy1099-43002022-01-0124215910.3390/e24020159On the Thermodynamics of the <i>q</i>-ParticlesFabio Ciolli0Francesco Fidaleo1Dipartimento di Matematica, Università di Roma Tor Vergata, Via della Ricerca Scientifica 1, I-00133 Roma, ItalyInstitute of Mathematics, Würzburg University, Emil-Fischer-Strße 40, 97074 Würzburg, GermanySince the grand partition function <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>Z</mi><mi>q</mi></msub></semantics></math></inline-formula> for the so-called <i>q</i>-particles (i.e., quons), <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>q</mi><mo>∈</mo><mo>(</mo><mo>−</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>)</mo></mrow></semantics></math></inline-formula>, cannot be computed by using the standard 2nd quantisation technique involving the full Fock space construction for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>q</mi><mo>=</mo><mn>0</mn></mrow></semantics></math></inline-formula>, and its <i>q</i>-deformations for the remaining cases, we determine such grand partition functions in order to obtain the natural generalisation of the Plank distribution to <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>q</mi><mo>∈</mo><mo>[</mo><mo>−</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>]</mo></mrow></semantics></math></inline-formula>. We also note the (non) surprising fact that the right grand partition function concerning the Boltzmann case (i.e., <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>q</mi><mo>=</mo><mn>0</mn></mrow></semantics></math></inline-formula>) can be easily obtained by using the full Fock space 2nd quantisation, by considering the appropriate correction by the Gibbs factor <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>1</mn><mo>/</mo><mi>n</mi><mo>!</mo></mrow></semantics></math></inline-formula> in the <i>n</i> term of the power series expansion with respect to the fugacity <i>z</i>. As an application, we briefly discuss the equations of the state for a gas of free quons or the condensation phenomenon into the ground state, also occurring for the Bose-like quons <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>q</mi><mo>∈</mo><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo></mrow></semantics></math></inline-formula>.https://www.mdpi.com/1099-4300/24/2/159thermodynamics of <i>q</i>-particlesquonsgrand canonical ensemblegrand partition functionBose Einstein Condensation |
| spellingShingle | Fabio Ciolli Francesco Fidaleo On the Thermodynamics of the <i>q</i>-Particles Entropy thermodynamics of <i>q</i>-particles quons grand canonical ensemble grand partition function Bose Einstein Condensation |
| title | On the Thermodynamics of the <i>q</i>-Particles |
| title_full | On the Thermodynamics of the <i>q</i>-Particles |
| title_fullStr | On the Thermodynamics of the <i>q</i>-Particles |
| title_full_unstemmed | On the Thermodynamics of the <i>q</i>-Particles |
| title_short | On the Thermodynamics of the <i>q</i>-Particles |
| title_sort | on the thermodynamics of the i q i particles |
| topic | thermodynamics of <i>q</i>-particles quons grand canonical ensemble grand partition function Bose Einstein Condensation |
| url | https://www.mdpi.com/1099-4300/24/2/159 |
| work_keys_str_mv | AT fabiociolli onthethermodynamicsoftheiqiparticles AT francescofidaleo onthethermodynamicsoftheiqiparticles |