Equivalence of Partition Functions Leads to Classification of Entropies and Means
We derive a two-parameter family of generalized entropies, <em>S<sub>pq</sub></em>, and means <em>m<sub>pq</sub></em>. To this end, assume that we want to calculate an entropy and a mean f...
| Main Authors: | , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2012-07-01
|
| Series: | Entropy |
| Subjects: | |
| Online Access: | http://www.mdpi.com/1099-4300/14/8/1317 |
| _version_ | 1811262734672592896 |
|---|---|
| author | Michel S. Elnaggar Achim Kempf |
| author_facet | Michel S. Elnaggar Achim Kempf |
| author_sort | Michel S. Elnaggar |
| collection | DOAJ |
| description | We derive a two-parameter family of generalized entropies, <em>S<sub>pq</sub></em>, and means <em>m<sub>pq</sub></em>. To this end, assume that we want to calculate an entropy and a mean for <em>n</em> non-negative real numbers {<em>x<sub>1</sub></em>,<em>…</em>,<em>x<sub>n</sub></em>}. For comparison, we consider {<em>m<sub>1</sub></em>,<em>…</em>,<em>m<sub>k</sub></em>} where <em>m<sub>i</sub> = m</em> for all <em>i = 1</em>,<em>…</em>,<em>k</em> and where <em>m</em> and <em>k</em> are chosen such that the <em>l<sup>p</sup></em> and <em>l<sup>q</sup></em> norms of {<em>x<sub>1</sub></em>,<em>…</em>,<em>x<sub>n</sub></em>} and {<em>m<sub>1</sub></em>,<em>…</em>,<em>m<sub>k</sub></em>} coincide. We formally allow <em>k</em> to be real. Then, we define <em>k</em>, log<em> k</em>, and <em>m</em> to be a generalized cardinality <em>k<sub>pq</sub></em>, a generalized entropy <em>S<sub>pq</sub></em>, and a generalized mean <em>m<sub>pq</sub></em> respectively. We show that this family of entropies includes the Shannon and Rényi entropies and that the family of generalized means includes the power means (such as arithmetic, harmonic, geometric, root-mean-square, maximum, and minimum) as well as novel means of Shannon-like and Rényi-like forms. A thermodynamic interpretation arises from the fact that the <em>l<sup>p</sup></em> norm is closely related to the partition function at inverse temperature <em>β</em><em> = p</em>. Namely, two systems possess the same generalized entropy and generalized mean energy if and only if their partition functions agree at two temperatures, which is also equivalent to the condition that their Helmholtz free energies agree at these two temperatures. |
| first_indexed | 2024-04-12T19:30:45Z |
| format | Article |
| id | doaj.art-95002d7368c14b4f83668072b397b7ae |
| institution | Directory Open Access Journal |
| issn | 1099-4300 |
| language | English |
| last_indexed | 2024-04-12T19:30:45Z |
| publishDate | 2012-07-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Entropy |
| spelling | doaj.art-95002d7368c14b4f83668072b397b7ae2022-12-22T03:19:20ZengMDPI AGEntropy1099-43002012-07-011481317134210.3390/e14081317Equivalence of Partition Functions Leads to Classification of Entropies and MeansMichel S. ElnaggarAchim KempfWe derive a two-parameter family of generalized entropies, <em>S<sub>pq</sub></em>, and means <em>m<sub>pq</sub></em>. To this end, assume that we want to calculate an entropy and a mean for <em>n</em> non-negative real numbers {<em>x<sub>1</sub></em>,<em>…</em>,<em>x<sub>n</sub></em>}. For comparison, we consider {<em>m<sub>1</sub></em>,<em>…</em>,<em>m<sub>k</sub></em>} where <em>m<sub>i</sub> = m</em> for all <em>i = 1</em>,<em>…</em>,<em>k</em> and where <em>m</em> and <em>k</em> are chosen such that the <em>l<sup>p</sup></em> and <em>l<sup>q</sup></em> norms of {<em>x<sub>1</sub></em>,<em>…</em>,<em>x<sub>n</sub></em>} and {<em>m<sub>1</sub></em>,<em>…</em>,<em>m<sub>k</sub></em>} coincide. We formally allow <em>k</em> to be real. Then, we define <em>k</em>, log<em> k</em>, and <em>m</em> to be a generalized cardinality <em>k<sub>pq</sub></em>, a generalized entropy <em>S<sub>pq</sub></em>, and a generalized mean <em>m<sub>pq</sub></em> respectively. We show that this family of entropies includes the Shannon and Rényi entropies and that the family of generalized means includes the power means (such as arithmetic, harmonic, geometric, root-mean-square, maximum, and minimum) as well as novel means of Shannon-like and Rényi-like forms. A thermodynamic interpretation arises from the fact that the <em>l<sup>p</sup></em> norm is closely related to the partition function at inverse temperature <em>β</em><em> = p</em>. Namely, two systems possess the same generalized entropy and generalized mean energy if and only if their partition functions agree at two temperatures, which is also equivalent to the condition that their Helmholtz free energies agree at these two temperatures.http://www.mdpi.com/1099-4300/14/8/1317cardinalitydimensionalityentropyequivalencefree energyinformation and thermodynamicsnormmeanpartition functionShannon and Rényi axioms |
| spellingShingle | Michel S. Elnaggar Achim Kempf Equivalence of Partition Functions Leads to Classification of Entropies and Means Entropy cardinality dimensionality entropy equivalence free energy information and thermodynamics norm mean partition function Shannon and Rényi axioms |
| title | Equivalence of Partition Functions Leads to Classification of Entropies and Means |
| title_full | Equivalence of Partition Functions Leads to Classification of Entropies and Means |
| title_fullStr | Equivalence of Partition Functions Leads to Classification of Entropies and Means |
| title_full_unstemmed | Equivalence of Partition Functions Leads to Classification of Entropies and Means |
| title_short | Equivalence of Partition Functions Leads to Classification of Entropies and Means |
| title_sort | equivalence of partition functions leads to classification of entropies and means |
| topic | cardinality dimensionality entropy equivalence free energy information and thermodynamics norm mean partition function Shannon and Rényi axioms |
| url | http://www.mdpi.com/1099-4300/14/8/1317 |
| work_keys_str_mv | AT michelselnaggar equivalenceofpartitionfunctionsleadstoclassificationofentropiesandmeans AT achimkempf equivalenceofpartitionfunctionsleadstoclassificationofentropiesandmeans |