Tsallis entropy

In physics, the Tsallis entropy is a generalization of the standard Boltzmann–Gibbs entropy. It was introduced in 1988 by Constantino Tsallis [1] as a basis for generalizing the standard statistical mechanics. In the scientific literature, the physical relevance of the Tsallis entropy was occasionally debated. However, from the years 2000 on, an increasingly wide spectrum of natural, artificial and social complex systems have been identified which confirm the predictions and consequences that are derived from this nonadditive entropy, such as nonextensive statistical mechanics,[2] which generalizes the Boltzmann-Gibbs theory.

Among the various experimental verifications and applications presently available in the literature, the following ones deserve a special mention:

1. The distribution characterizing the motion of cold atoms in dissipative optical lattices, predicted in 2003 [3] and observed in 2006.[4]
2. The fluctuations of the magnetic field in the solar wind enabled the calculation of the q-triplet (or Tsallis triplet).[5]
3. The velocity distributions in driven dissipative dusty plasma.[6]
4. Spin glass relaxation.[7]
5. Trapped ion interacting with a classical buffer gas.[8]
6. High energy collisional experiments at LHC/CERN (CMS, ATLAS and ALICE detectors) [9][10] and RHIC/Brookhaven (STAR and PHENIX detectors).[11]

Among the various available theoretical results which clarify the physical conditions under which Tsallis entropy and associated statistics apply, the following ones can be selected:

1. Anomalous diffusion.[12][13]
2. Uniqueness theorem.[14]
3. Sensitivity to initial conditions and entropy production at the edge of chaos.[15][16]
4. Probability sets which make the nonadditive Tsallis entropy to be extensive in the thermodynamical sense.[17]
5. Strongly quantum entangled systems and thermodynamics.[18]
6. Thermostatistics of overdamped motion of interacting particles.[19][20]
7. Nonlinear generalizations of the Schroedinger, Klein-Gordon and Dirac equations.[21]

For further details a bibliography is available at http://tsallis.cat.cbpf.br/biblio.htm

Given a discrete set of probabilities $\{p_i\}$ with the condition $\sum_i p_i=1$, and $q$ any real number, the Tsallis entropy is defined as

$S_q(p_i) = {1 \over q - 1} \left( 1 - \sum_i p_i^q \right),$

where $q$ is a real parameter sometimes called entropic-index. In the limit as $q \to 1$, the usual Boltzmann-Gibbs entropy is recovered, namely

$S_{BG} = S_1(p) = -k \sum_i p_i \ln p_i .$

For continuous probability distributions, we define the entropy as

$S_q(p) = {1 \over q - 1} \left( 1 - \int (p(x))^q\, dx \right),$

where $p(x)$ is a probability density function.

The Tsallis Entropy has been used along with the Principle of maximum entropy to derive the Tsallis distribution.

Various relationships

The discrete Tsallis entropy satisfies

$S_q = -\lim_{x\rightarrow 1}D_q \sum_i p_i^x$

where Dq is the q-derivative with respect to x. This may be compared to the standard entropy formula:

$S = -\lim_{x\rightarrow 1}\frac{d}{dx} \sum_i p_i^x$

Given two independent systems A and B, for which the joint probability density satisfies

$p(A, B) = p(A) p(B),\,$

the Tsallis entropy of this system satisfies

$S_q(A,B) = S_q(A) + S_q(B) + (1-q)S_q(A) S_q(B).\,$

From this result, it is evident that the parameter $|1-q|$ is a measure of the departure from additivity. In the limit when q = 1,

$S(A,B) = S(A) + S(B),\,$

which is what is expected for an additive system. This property is sometimes referred to as "pseudo-additivity".

Exponential families

Many common distributions like the normal distribution belongs to the statistical exponential families. Tsallis entropy for an exponential family can be written (Nielsen, Nock, 2011) as

$H^T_q(p_F(x;\theta)) = \frac{1}{1-q} \left((e^{F(q\theta)-q F(\theta)}) E_p[e^{(q-1)k(x)}]-1 \right)$

where F is log-normalizer and k the term indicating the carrier measure. For multivariate normal, term k is zero, and therefore the Tsallis entropy is in closed-form.

References

1. ^ Tsallis, C. (1988). "Possible generalization of Boltzmann-Gibbs statistics". Journal of Statistical Physics 52: 479–487. doi:10.1007/BF01016429. edit
2. ^ Tsallis, Constantino (2009). Introduction to nonextensive statistical mechanics : approaching a complex world (Online-Ausg. ed.). New York: Springer. ISBN 978-0-387-85358-1.
3. ^ Lutz, E. (2003). "Anomalous diffusion and Tsallis statistics in an optical lattice". Physical Review A 67 (5). doi:10.1103/PhysRevA.67.051402. edit
4. ^ Douglas, P.; Bergamini, S.; Renzoni, F. (2006). "Tunable Tsallis Distributions in Dissipative Optical Lattices". Physical Review Letters 96 (11). doi:10.1103/PhysRevLett.96.110601. edit
5. ^ Burlaga, L. F.; - Viñas, A. F. (2005). "Triangle for the entropic index q of non-extensive statistical mechanics observed by Voyager 1 in the distant heliosphere". Physica A: Statistical Mechanics and its Applications 356 (2–4): 375. doi:10.1016/j.physa.2005.06.065. edit
6. ^ Liu, B.; Goree, J. (2008). "Superdiffusion and Non-Gaussian Statistics in a Driven-Dissipative 2D Dusty Plasma". Physical Review Letters 100 (5). doi:10.1103/PhysRevLett.100.055003. edit
7. ^ Pickup, R.; Cywinski, R.; Pappas, C.; Farago, B.; Fouquet, P. (2009). "Generalized Spin-Glass Relaxation". Physical Review Letters 102 (9). doi:10.1103/PhysRevLett.102.097202. edit
8. ^ Devoe, R. (2009). "Power-Law Distributions for a Trapped Ion Interacting with a Classical Buffer Gas". Physical Review Letters 102 (6). doi:10.1103/PhysRevLett.102.063001. edit
9. ^ Khachatryan, V.; Sirunyan, A.; Tumasyan, A.; Adam, W.; Bergauer, T.; Dragicevic, M.; Erö, J.; Fabjan, C. et al. (2010). "Transverse-Momentum and Pseudorapidity Distributions of Charged Hadrons in pp Collisions at sqrt\s]=7  TeV". Physical Review Letters 105 (2). doi:10.1103/PhysRevLett.105.022002. edit
10. ^ Chatrchyan, S.; Khachatryan, V.; Sirunyan, A. M.; Tumasyan, A.; Adam, W.; Bergauer, T.; Dragicevic, M.; Erö, J. et al. (2011). "Charged particle transverse momentum spectra in pp collisions at $\sqrt {s} = 0.9$ and 7 TeV". Journal of High Energy Physics 2011 (8). doi:10.1007/JHEP08(2011)086. edit
11. ^ Adare, A.; Afanasiev, S.; Aidala, C.; Ajitanand, N.; Akiba, Y.; Al-Bataineh, H.; Alexander, J.; Aoki, K. et al. (2011). "Measurement of neutral mesons in p+p collisions at sqrt\s]=200  GeV and scaling properties of hadron production". Physical Review D 83 (5). doi:10.1103/PhysRevD.83.052004. edit
12. ^ Plastino, A. R.; Plastino, A. (1995). "Non-extensive statistical mechanics and generalized Fokker-Planck equation". Physica A: Statistical Mechanics and its Applications 222: 347. doi:10.1016/0378-4371(95)00211-1. edit
13. ^ Tsallis, C.; Bukman, D. (1996). "Anomalous diffusion in the presence of external forces: Exact time-dependent solutions and their thermostatistical basis". Physical Review E 54 (3): R2197. doi:10.1103/PhysRevE.54.R2197. edit
14. ^ Abe, S. (2000). "Axioms and uniqueness theorem for Tsallis entropy". Physics Letters A 271: 74–79. doi:10.1016/S0375-9601(00)00337-6. edit
15. ^ Lyra, M.; Tsallis, C. (1998). "Nonextensivity and Multifractality in Low-Dimensional Dissipative Systems". Physical Review Letters 80: 53. doi:10.1103/PhysRevLett.80.53. edit
16. ^ Baldovin, F.; Robledo, A. (2004). "Nonextensive Pesin identity: Exact renormalization group analytical results for the dynamics at the edge of chaos of the logistic map". Physical Review E 69 (4). doi:10.1103/PhysRevE.69.045202. edit
17. ^ Tsallis, C.; Gell-Mann, M.; Sato, Y. (2005). "Asymptotically scale-invariant occupancy of phase space makes the entropy Sq extensive". Proceedings of the National Academy of Sciences 102 (43): 15377. doi:10.1073/pnas.0503807102. edit
18. ^ Caruso, F.; Tsallis, C. (2008). "Nonadditive entropy reconciles the area law in quantum systems with classical thermodynamics". Physical Review E 78 (2). doi:10.1103/PhysRevE.78.021102. edit
19. ^ Andrade, J.; Da Silva, G.; Moreira, A.; Nobre, F.; Curado, E. (2010). "Thermostatistics of Overdamped Motion of Interacting Particles". Physical Review Letters 105 (26). doi:10.1103/PhysRevLett.105.260601. edit
20. ^ Ribeiro, M.; Nobre, F.; Curado, E. M. (2012). "Time evolution of interacting vortices under overdamped motion". Physical Review E 85 (2). doi:10.1103/PhysRevE.85.021146. edit
21. ^ Nobre, F.; Rego-Monteiro, M.; Tsallis, C. (2011). "Nonlinear Relativistic and Quantum Equations with a Common Type of Solution". Physical Review Letters 106 (14). doi:10.1103/PhysRevLett.106.140601. edit