Entropy, Age and Time Operator
The time operator and internal age are intrinsic features of entropy producing innovation processes. The innovation spaces at each stage are the eigenspaces of the time operator. The internal age is the average innovation time, analogous to lifetime computation. Time operators were originally introd...
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| Format: | Article |
| Language: | English |
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MDPI AG
2015-01-01
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| Series: | Entropy |
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| Online Access: | http://www.mdpi.com/1099-4300/17/1/407 |
| _version_ | 1818001794858483712 |
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| author | Ilias Gialampoukidis Ioannis Antoniou |
| author_facet | Ilias Gialampoukidis Ioannis Antoniou |
| author_sort | Ilias Gialampoukidis |
| collection | DOAJ |
| description | The time operator and internal age are intrinsic features of entropy producing innovation processes. The innovation spaces at each stage are the eigenspaces of the time operator. The internal age is the average innovation time, analogous to lifetime computation. Time operators were originally introduced for quantum systems and highly unstable dynamical systems. Extending the time operator theory to regular Markov chains allows one to relate internal age with norm distances from equilibrium. The goal of this work is to express the evolution of internal age in terms of Lyapunov functionals constructed from entropies. We selected the Boltzmann–Gibbs–Shannon entropy and more general entropy functions, namely the Tsallis entropies and the Kaniadakis entropies. Moreover, we compare the evolution of the distance of initial distributions from equilibrium to the evolution of the Lyapunov functionals constructed from norms with the evolution of Lyapunov functionals constructed from entropies. It is remarkable that the entropy functionals evolve, violating the second law of thermodynamics, while the norm functionals evolve thermodynamically. |
| first_indexed | 2024-04-14T03:37:53Z |
| format | Article |
| id | doaj.art-b4237052742d4bf0892a3755eff82672 |
| institution | Directory Open Access Journal |
| issn | 1099-4300 |
| language | English |
| last_indexed | 2024-04-14T03:37:53Z |
| publishDate | 2015-01-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Entropy |
| spelling | doaj.art-b4237052742d4bf0892a3755eff826722022-12-22T02:14:40ZengMDPI AGEntropy1099-43002015-01-0117140742410.3390/e17010407e17010407Entropy, Age and Time OperatorIlias Gialampoukidis0Ioannis Antoniou1School of Mathematics, Aristotle University of Thessaloniki, 54124 Thessaloniki, GreeceSchool of Mathematics, Aristotle University of Thessaloniki, 54124 Thessaloniki, GreeceThe time operator and internal age are intrinsic features of entropy producing innovation processes. The innovation spaces at each stage are the eigenspaces of the time operator. The internal age is the average innovation time, analogous to lifetime computation. Time operators were originally introduced for quantum systems and highly unstable dynamical systems. Extending the time operator theory to regular Markov chains allows one to relate internal age with norm distances from equilibrium. The goal of this work is to express the evolution of internal age in terms of Lyapunov functionals constructed from entropies. We selected the Boltzmann–Gibbs–Shannon entropy and more general entropy functions, namely the Tsallis entropies and the Kaniadakis entropies. Moreover, we compare the evolution of the distance of initial distributions from equilibrium to the evolution of the Lyapunov functionals constructed from norms with the evolution of Lyapunov functionals constructed from entropies. It is remarkable that the entropy functionals evolve, violating the second law of thermodynamics, while the norm functionals evolve thermodynamically.http://www.mdpi.com/1099-4300/17/1/407time operatorinternal ageMarkov chainsmixing timeTsallis entropyKaniadakis entropy |
| spellingShingle | Ilias Gialampoukidis Ioannis Antoniou Entropy, Age and Time Operator Entropy time operator internal age Markov chains mixing time Tsallis entropy Kaniadakis entropy |
| title | Entropy, Age and Time Operator |
| title_full | Entropy, Age and Time Operator |
| title_fullStr | Entropy, Age and Time Operator |
| title_full_unstemmed | Entropy, Age and Time Operator |
| title_short | Entropy, Age and Time Operator |
| title_sort | entropy age and time operator |
| topic | time operator internal age Markov chains mixing time Tsallis entropy Kaniadakis entropy |
| url | http://www.mdpi.com/1099-4300/17/1/407 |
| work_keys_str_mv | AT iliasgialampoukidis entropyageandtimeoperator AT ioannisantoniou entropyageandtimeoperator |