# Nonadditive entropy: The concept and its use

• C. Tsallis
Regular Article - Theoretical Physics

## Abstract

The thermodynamical concept of entropy was introduced by Clausius in 1865 in order to construct the exact differential dS = $$\delta$$ Q/T , where $$\delta$$ Q is the heat transfer and the absolute temperature T its integrating factor. A few years later, in the period 1872-1877, it was shown by Boltzmann that this quantity can be expressed in terms of the probabilities associated with the microscopic configurations of the system. We refer to this fundamental connection as the Boltzmann-Gibbs (BG) entropy, namely (in its discrete form) $$\ensuremath S_{BG}=-k\sum_{i=1}^W p_i \ln p_i$$ , where k is the Boltzmann constant, and {p i} the probabilities corresponding to the W microscopic configurations (hence ∑W i=1 p i = 1 . This entropic form, further discussed by Gibbs, von Neumann and Shannon, and constituting the basis of the celebrated BG statistical mechanics, is additive. Indeed, if we consider a system composed by any two probabilistically independent subsystems A and B (i.e., $$\ensuremath p_{ij}^{A+B}=p_i^A p_j^B, \forall(i,j)$$ , we verify that $$\ensuremath S_{BG}(A+B)=S_{BG}(A)+S_{BG}(B)$$ . If a system is constituted by N equal elements which are either independent or quasi-independent (i.e., not too strongly correlated, in some specific nonlocal sense), this additivity guarantees SBG to be extensive in the thermodynamical sense, i.e., that $$\ensuremath S_{BG}(N) \propto N$$ in the N ≫ 1 limit. If, on the contrary, the correlations between the N elements are strong enough, then the extensivity of SBG is lost, being therefore incompatible with classical thermodynamics. In such a case, the many and precious relations described in textbooks of thermodynamics become invalid. Along a line which will be shown to overcome this difficulty, and which consistently enables the generalization of BG statistical mechanics, it was proposed in 1988 the entropy $$\ensuremath S_q=k [1-\sum_{i=1}^W p_i^q]/(q-1) (q\in{R}; S_1=S_{BG})$$ . In the context of cybernetics and information theory, this and similar forms have in fact been repeatedly introduced before 1988. The entropic form Sq is, for any q $$\neq$$ 1 , nonadditive. Indeed, for two probabilistically independent subsystems, it satisfies $$\ensuremath S_q(A+B)/k=[S_q(A)/k]+ [S_q(B)/k]+(1-q)[S_q(A)/k][S_q(B)/k] \neq S_q(A)/k+S_q(B)/k$$ . This form will turn out to be extensive for an important class of nonlocal correlations, if q is set equal to a special value different from unity, noted qent (where ent stands for entropy . In other words, for such systems, we verify that $$\ensuremath S_{q_{ent}}(N) \propto N (N \gg 1)$$ , thus legitimating the use of the classical thermodynamical relations. Standard systems, for which SBG is extensive, obviously correspond to q ent = 1 . Quite complex systems exist in the sense that, for them, no value of q exists such that Sq is extensive. Such systems are out of the present scope: they might need forms of entropy different from Sq, or perhaps --more plainly-- they are just not susceptible at all for some sort of thermostatistical approach. Consistently with the results associated with Sq, the q -generalizations of the Central Limit Theorem and of its extended Lévy-Gnedenko form have been achieved. These recent theorems could of course be the cause of the ubiquity of q -exponentials, q -Gaussians and related mathematical forms in natural, artificial and social systems. All of the above, as well as presently available experimental, observational and computational confirmations --in high-energy physics and elsewhere-- are briefly reviewed. Finally, we address a confusion which is quite common in the literature, namely referring to distinct physical mechanisms versus distinct regimes of a single physical mechanism.

## PACS

05.20.-y Classical statistical mechanics 02.50.Cw Probability theory 05.90.+m Other topics in statistical physics, thermodynamics, and nonlinear dynamical systems 05.70.-a Thermodynamics

## References

1. 1.
R. Clausius, Ann. Phys. (Leipzig) 125, 353 (1865)Google Scholar
2. 2.
L. Boltzmann, Weitere Studien \.uber das W\.armegleich\-gewicht unter Gas molek\.ulen (English translation: Further Studies on Thermal Equilibrium Between Gas Molecules), Wien Ber. 66, 275 (1872).Google Scholar
3. 3.
L. Boltzmann, Uber die Beziehung eines allgemeine mechanischen Satzes zum zweiten Haupsatze der Warmetheorie, Sitzungsber. K. Akad. Wiss. Wien, Math.-Naturwiss. 75, 67 (1877)Google Scholar
4. 4.
J.W. Gibbs, Elementary Principles in Statistical Mechanics -- Developed with Especial Reference to the Rational Foundation of Thermodynamics (C. Scribner’s Sons, New York, 1902Google Scholar
5. 5.
J. von Neumann, Thermodynamik quantummechanischer Gesamheiter, Gött. Nachr. 1, 273 (1927)Google Scholar
6. 6.
C.E. Shannon, A Mathematical Theory of Communication, Bell Syst. Tech. J. 27, 379Google Scholar
7. 7.
O. Penrose, Foundations of Statistical Mechanics: A Deductive Treatment (Pergamon, Oxford, 1970) p. 167.Google Scholar
8. 8.
C. Tsallis, J. Stat. Phys. 52, 479 (1988).Google Scholar
9. 9.
E.M.F. Curado, C. Tsallis, J. Phys. A 24, L69 (1991)Google Scholar
10. 10.
C. Tsallis, R.S. Mendes, A.R. Plastino, Physica A 261, 534 (1998).Google Scholar
11. 11.
S. Abe, Phys. Lett. A 271, 74 (2000).Google Scholar
12. 12.
C. Tsallis, M. Gell-Mann, Y. Sato, Proc. Natl. Acad. Sci. U.S.A. 102, 15377 (2005).Google Scholar
13. 13.
F. Caruso, C. Tsallis, Phys. Rev. E 78, 021101 (2008).Google Scholar
14. 14.
P. Calabrese, J. Cardy, J. Stat. P06002 (2004).Google Scholar
15. 15.
H.P. de Oliveira, I.D. Soares, E.V. Tonini, Phys. Rev. D 78, 044016 (2008).Google Scholar
16. 16.
P. Ginsparg, G. Moore, Lectures on $2D$ String Theory (Cambridge University Press, Cambridge, 1993).Google Scholar
17. 17.
J. Maddox, Nature 365, 103 (1993).Google Scholar
18. 18.
A. Pluchino, A. Rapisarda, C. Tsallis, EPL 80, 26002 (2007).Google Scholar
19. 19.
A. Pluchino, A. Rapisarda, C. Tsallis, Physica A 387, 3121 (2008).Google Scholar
20. 20.
A regularly updated bibliography is available at http:// tsallis.cat.cbpf.br/biblio.htm.Google Scholar
21. 21.
M. Gell-Mann, C. Tsallis (Editors), Nonextensive Entropy - Interdisciplinary Applications (Oxford University Press, New York, 2004).Google Scholar
22. 22.
J.P. Boon, C. Tsallis (Editors), Nonextensive Statistical Mechanics: New Trends, New perspectives, Europhys. News 36, no. 6 (2005).Google Scholar
23. 23.
C. Tsallis, Entropy, in Encyclopedia of Complexity and Systems Science (Springer, Berlin, 2009).Google Scholar
24. 24.
C. Tsallis, Introduction to Nonextensive Statistical Mechanics - Approaching a Complex World (Springer, New York, 2009).Google Scholar
25. 25.
A. Upadhyaya, J.-P. Rieu, J.A. Glazier, Y. Sawada, Physica A 293, 549 (2001).Google Scholar
26. 26.
S. Thurner, N. Wick, R. Hanel, R. Sedivy, L. Huber, Physica A 320, 475 (2003).Google Scholar
27. 27.
L. Diambra, L.C. Cintra, Q. Chen, D. Schubert, L. da F. Costa, Physica A 365, 481 (2006).Google Scholar
28. 28.
K.E. Daniels, C. Beck, E. Bodenschatz, Physica D 193, 208 (2004).Google Scholar
29. 29.
L.F. Burlaga, A.F.-Vinas, Physica A 356, 375 (2005).Google Scholar
30. 30.
L.F. Burlaga, A.F. Vinas, N.F. Ness, M.H. Acuna, Astrophys. J. 644, L83 (2006).Google Scholar
31. 31.
L.F. Burlaga, A. F-Vinas, C. Wang, J. Geophys. Res.-Space Phys. 112, A07206 (2007).Google Scholar
32. 32.
L.F. Burlaga, A.F.-Vinas, in Complexity, Metastability and Nonextensivity, edited by S. Abe, H.J. Herrmann, P. Quarati, A. Rapisarda, C. Tsallis, AIP Conf. Proc. 965, 259 (2007).Google Scholar
33. 33.
P. Douglas, S. Bergamini, F. Renzoni, Phys. Rev. Lett. 96, 110601 (2006).Google Scholar
34. 34.
B. Liu, J. Goree, Phys. Rev. Lett. 100, 055003 (2008).Google Scholar
35. 35.
R. Arevalo, A. Garcimartin, D. Maza, Eur. Phys. J. E 23, 191 (2007).Google Scholar
36. 36.
R. Arevalo, A. Garcimartin, D. Maza, Eur. Phys. J. ST 143, 191 (2007).Google Scholar
37. 37.
I. Bediaga, E.M.F. Curado, J. Miranda, Physica A 286, 156 (2000).Google Scholar
38. 38.
C. Beck, Physica A 286, 164 (2000).Google Scholar
39. 39.
C. Tsallis, J.C. Anjos, E.P. Borges, Phys. Lett. A 310, 372 (2003).Google Scholar
40. 40.
J.M. Conroy, H.G. Miller, Phys. Rev. D 78, 054010 (2008).Google Scholar
41. 41.
See the articles of the present topical issue and references therein.Google Scholar
42. 42.
C. Tsallis, A.R. Plastino, W.-M. Zheng, Chaos, Solitons Fractals 8, 885 (1997).Google Scholar
43. 43.
U. Tirnakli, C. Tsallis, M.L. Lyra, Eur. Phys. J. B 11, 309 (1999).Google Scholar
44. 44.
U. Tirnakli, Phys. Rev. E 62, 7857 (2000).Google Scholar
45. 45.
V. Latora, M. Baranger, A. Rapisarda, C. Tsallis, Phys. Lett. A 273, 97 (2000).Google Scholar
46. 46.
F. Baldovin, A. Robledo, Phys. Rev. E 66, R045104 (2002).Google Scholar
47. 47.
F. Baldovin, A. Robledo, Phys. Rev. E 69, 045202(R) (2004).Google Scholar
48. 48.
E. Mayoral, A. Robledo, Physica A 340, 219 (2004).Google Scholar
49. 49.
A. Robledo, Pramana J. Phys. 64, 947 (2005).Google Scholar
50. 50.
E. Mayoral, A. Robledo, Phys. Rev. E 72, 026209 (2005).Google Scholar
51. 51.
U. Tirnakli, C. Beck, C. Tsallis, Phys. Rev. E 75, 040106(R) (2007).Google Scholar
52. 52.
U. Tirnakli, C. Tsallis, C. Beck, arXiv: 0802.1138 [cond-mat.stat-mech].Google Scholar
53. 53.
C. Anteneodo, C. Tsallis, Phys. Rev. Lett. 80, 5313 (1998).Google Scholar
54. 54.
V. Latora, A. Rapisarda, C. Tsallis, Phys. Rev. E 64, 056134 (2001).Google Scholar
55. 55.
D.J.B. Soares, C. Tsallis, A.M. Mariz, L.R. da Silva, Europhys. Lett. 70, 70 (2005).Google Scholar
56. 56.
S. Thurner, C. Tsallis, Europhys. Lett. 72, 197 (2005).Google Scholar
57. 57.
S. Thurner, Europhys. News 36, 218 (2005).Google Scholar
58. 58.
D.R. White, N. Kejzar, C. Tsallis, D. Farmer, S. White, Phys. Rev. E 73, 016119 (2006).Google Scholar
59. 59.
M.D.S. de Meneses, S.D. da Cunha, D.J.B. Soares, L.R. da Silva, Prog. Theor. Phys. Suppl. 162, 131 (2006).Google Scholar
60. 60.
H. Hasegawa, Physica A 365, 383 (2006).Google Scholar
61. 61.
W. Li, Q.A. Wang, L. Nivanen, A. Le Mehaute, Physica A 368, 262 (2006).Google Scholar
62. 62.
S. Thurner, F. Kyriakopoulos, C. Tsallis, Phys. Rev. E 76, 036111 (2007).Google Scholar
63. 63.
C. Tsallis, Eur. Phys. J. ST 161, 175 (2008).Google Scholar
64. 64.
M.A. Fuentes, M.O. Caceres, Phys. Lett. A 372, 1236 (2008).Google Scholar
65. 65.
C. Anteneodo, C. Tsallis, J. Math. Phys. 44, 5194 (2003).Google Scholar
66. 66.
A.R. Plastino, A. Plastino, Physica A 222, 347 (1995).Google Scholar
67. 67.
C. Tsallis, D.J. Bukman, Phys. Rev. E 54, R2197 (1996).Google Scholar
68. 68.
V. Schwammle, E.M.F. Curado, F.D. Nobre, Eur. Phys. J. B 58, 159 (2007).Google Scholar
69. 69.
V. Schwammle, F.D. Nobre, E.M.F. Curado, Phys. Rev. E 76, 041123 (2007).Google Scholar
70. 70.
W. Thistleton, J.A. Marsh, K. Nelson, C. Tsallis, IEEE Trans. Inf. Theory 53, 4805 (2007).Google Scholar
71. 71.
S. Umarov, C. Tsallis, S. Steinberg, Milan J. Math. 76, 307 (2008).Google Scholar
72. 72.
C. Tsallis, S.M.D. Queiros, in Complexity, Metastability and Nonextensivity, edited by S. Abe, H.J. Herrmann, P. Quarati, A. Rapisarda, C. Tsallis, AIP Conf. Proc. 965, 8 (2007).Google Scholar
73. 73.
S.M.D. Queiros, C. Tsallis, in Complexity, Metastability and Nonextensivity, edited by S. Abe, H.J. Herrmann, P. Quarati, A. Rapisarda, C. Tsallis, AIP Conf. Proc. 965, 21 (2007).Google Scholar
74. 74.
S. Umarov, C. Tsallis, M. Gell-Mann, S. Steinberg, arXiv: cond-mat/0606038v2Google Scholar
75. 75.
A. Rodriguez, V. Schwammle, C. Tsallis, J. Stat. P09006 (2008).Google Scholar
76. 76.
K.P. Nelson, S. Umarov, arXiv: 0811.3777 [cs.IT] (2008).Google Scholar
77. 77.
G. Ruiz, C. Tsallis, Eur. Phys. J. B 67, 577 (2009).Google Scholar
78. 78.
M. Antoni, S. Ruffo, Phys. Rev. E 52, 2361 (1995).Google Scholar
79. 79.
A. Rapisarda, A. Pluchino, Europhys. News 36, 202 (2005)Google Scholar
80. 80.
A. Figueiredo, T.M. Rocha Filho, M.A. Amato, EPL 83, 30011 (2008).Google Scholar
81. 81.
A. Pluchino, A. Rapisarda, C. Tsallis, EPL 85, 60006 (2009).Google Scholar