The Physical Meaning of the Rényi Relative Entropy

Year 2017, Volume: 21 Issue: 1, 292 - 295, 31.03.2017

Gökhan Barış Bağcı

Abstract

The Boltzmann-Gibbs relative entropy provides the difference between the off-equilibrium and equilibrium free energy terms associated with Boltzmann-Gibbs entropy. In this work, we studied whether this physical meaning can be given to R\'{e}nyi relative entropy definition. We find that this is possible only in the limit as $q$ approaches to 1. This shows that R\'{e}nyi relative entropy has a physical meaning only when the system can already be explained by ordinary Boltzmann-Gibbs entropy. We also note that this result is independent of the internal energy constraint employed.

Keywords

Rényi relative entropy, Free energy; Entropy maximization

References

• [1] Rényi A. 1961. On measures of entropy and information. Proceedings of the Fourth Berkeley Symposium on Mathematics, Statistics and Probability, California, 547-561.
• [2] Lenzi, E. K., Mendes, R. S., da Silva, L. R. 2000. Statistical mechanics based on Renyi entropy. Physica A, 280(2000), 337-345.
• [3] Bashkirov, A. G. 2004. On maximum entropy principle, superstatistics, power-law distribution and Renyi parameter. Physica A, 340(2004), 153-162.
• [4] Wilde, M. M., Winter A., Yang D. 2014. Strong Converse for the Classical Capacity of Entanglement-Breaking and Hadamard Channels via a Sandwiched Rényi Relative Entropy. Comm. Math. Phys., 331(2014), 593-622.
• [5] Seshadreesan, K. P., Berta M., Wilde M. M. 2015. Rényi squashed entanglement, discord, and relative entropy differences. J. Phys. A, 48(2015), 395303 (1)-395303 (42).
• [6] Lashkari, N. 2014. Relative Entropies in Conformal Field Theory. Phys. Rev. Lett., 113(2014), 051602(1)-051602 (4).
• [7] Coles, P. J., Colbeck R., Yu L., Zwolak M. 2012. Uncertainty Relations from Simple Entropic Properties. Phys. Rev. Lett., 108(2012), 210405 (1)-210405 (42).
• [8] Lostaglio, M., Jennings D., Rudolph T. 2015. Description of quantum coherence in thermodynamic processes requires constraints beyond free energy. Nature Communications, 6(2015), 1-9.
• [9] Gray, R. 2011. Entropy and Information Entropy. 2nd, Springer Verlag GmbH and Co. New York, 65p.
• [10] Uffink, J. 1995. Can the maximum entropy principle be explained as a consistency requirement?. Stud. in Hist. and Phil. of Mod. Phys., 26(1995), 223-261.
• [11] Qian, H. 2001. Relative entropy:Free energy associated with equilibrium fluctuations and nonequilibrium deviations. Phys. Rev. E, 63(2001), 042103(1)-042103(4).
• [12] Harremoes, P. 2006. Interpretations of Rényi entropies and divergences. Physica A, 365(2006), 57-62.
• [13] Campisi, C., Ba˘gcı, G. B. 2007. Tsallis Ensemble as an Exact Orthode. Phys. Lett. A, 362(2007), 11-15.
• [14] Marino, M. 2007. A generalized thermodynamics for power-law statistics. Physica A, 386(2007), 135-154.
• [15] Misra, A., Singh, U., Bera, M. N., Rajagopal A. K. 2015. Quantum Rényi relative entropies affirm universality of thermodynamics. Phys. Rev. E, 92(2015), 042161(1)-042103(8).