×

Non-linear kinetics underlying generalized statistics. (English) Zbl 0969.82537

Summary: The purpose of the present effort is threefold. Firstly, it is shown that there exists a principle, that we call kinetical interaction principle (KIP), underlying the nonlinear kinetics in particle systems, independently on the picture (Kramers, Boltzmann) used to describe their time evolution. Secondly, the KIP imposes the form of the generalized entropy associated to the system and permits to obtain the particle statistical distribution, both as stationary solution of the nonlinear evolution equation and as the state which maximizes the generalized entropy. Thirdly, the KIP allows, on one hand, to treat all the classical or quantum statistical distributions already known in the literature in a unifying scheme and, on the other hand, suggests how we can introduce naturally new distributions. Finally, as a working example of the approach to the nonlinear kinetics here presented, a new non-extensive statistics is constructed and studied starting from a one-parameter deformation of the exponential function holding the relation \(f(-x)f(x)=1\).

MSC:

82B40 Kinetic theory of gases in equilibrium statistical mechanics
PDF BibTeX XML Cite
Full Text: DOI arXiv

References:

[1] Druyvenstein, M., Physica (Eindhoven), 10, 6, (1930)
[2] Druyvenstein, M., Physica (Eindhoven), 1, 1003, (1934)
[3] Renyi, A., Probability theory, (1970), North-Holland Amsterdam · Zbl 0206.18002
[4] Sharma, B.D.; Mittal, D.P., J. math. sci., 1, 28, (1975)
[5] Tsallis, C., J. stat. phys., 52, 479, (1988)
[6] Abe, S., Phys. lett. A, 224, 326, (1997)
[7] Papa, A.R.R., J. phys. A, 31, 5271, (1998)
[8] Borges, E.P.; Roditi, I., Phys. lett. A, 246, 399, (1998)
[9] Landsberg, P.T.; Vedral, V., Phys. lett. A, 247, 211, (1998)
[10] Anteneodo, C.; Plastino, A.R., J. phys. A, 32, 1089, (1999)
[11] Frank, T.D.; Daffertshofer, A., Physica A, 272, 497, (1999)
[12] Gentile, G., Nuovo cimento, 17, 493, (1940)
[13] Green, H.S., Phys. rev., 90, 270, (1953)
[14] Greenberg, O.W.; Mohapatra, R.N., Phys. rev. D, 39, 2032, (1989)
[15] Biedenharn, L.C., J. phys. A, 22, L873, (1989) · Zbl 0708.17015
[16] Haldane, F.D.M., Phys. rev. lett., 66, 1529, (1991)
[17] Wu, Y.S., Phys. rev. lett., 73, 922, (1994)
[18] Acharya, R.; Narayana Swamy, P., J. phys. A, 27, 7247, (1994)
[19] Buyukkilic, F.; Demirhan, D., Phys. lett. A, 181, 24, (1993)
[20] Uehling, E.A.; Uhlenbeck, G.E., Phys. rev., 43, 552, (1933)
[21] Plastino, A.R.; Plastino, A., Physica A, 222, 347, (1995) · Zbl 1020.82502
[22] Tsallis, C.; Buckman, D.J., Phys. rev. E, 54, R2197, (1996)
[23] Compte, A.; Jou, D., J. phys. A, 29, 4321, (1996)
[24] Compte, A.; Jou, D.; Katayama, Y., J. phys. A, 30, 1023, (1997)
[25] Stariolo, D.A., Phys. rev. E, 55, 4806, (1997)
[26] Martinez, S.; Plastino, A.R.; Plastino, A., Physica A, 259, 183, (1998)
[27] Borland, L., Phys. rev. E, 57, 6634, (1998)
[28] Borland, L., Phys. lett. A, 245, 67, (1998)
[29] Borland, L.; Pennini, F.; Plastino, A.R.; Plastino, A., Eur. phys. J. B, 12, 285, (1999)
[30] Kaniadakis, G.; Lapenta, G., Phys. rev. E, 62, 3246, (2000)
[31] Kaniadakis, G.; Lavagno, A.; Quarati, P., Nucl. phys. B, 466, 527, (1996)
[32] Kaniadakis, G.; Lavagno, A.; Quarati, P., Phys. lett. A, 227, 227, (1997)
[33] Kaniadakis, G.; Quarati, P., Phys. rev. E, 49, 5103, (1994)
[34] Rossani, A.; Kaniadakis, G., Physica A, 277, 349, (2000)
[35] Frank, T.D., Phys. lett. A, 267, 298, (2000)
[36] Kaniadakis, G.; Quarati, P., Phys. rev. E, 48, 4263, (1993)
[37] Boghosian, B.M., Phys. rev. E, 53, 4754, (1996)
[38] Kaniadakis, G.; Lavagno, A.; Quarati, P., Phys. lett. B, 369, 308, (1996)
[39] Kaniadakis, G.; Lavagno, A.; Lissia, M.; Quarati, P., Physica A, 261, 359, (1998)
[40] Coraddu, M.; Kaniadakis, G.; Lavagno, A.; Lissia, M.; Mezzorani, G.; Quarati, P., Braz. J. phys., 29, 153, (1999)
[41] Tsallis, C.; de Souza, A.M.C., Phys. lett. A, 235, 444, (1994)
[42] Alemany, P.A.; Zanette, D.H., Phys. rev. E, 49, R956, (1994)
[43] Plastino, A.R.; Plastino, A., Physica A, 174, 384, (1993)
[44] Lavagno, A.; Kaniadakis, G.; Rego-Monteiro, M.; Quarati, P.; Tsallis, C., Astro. lett. comm., 35, 449, (1998)
[45] Clayton, D.D., Nature, 249, 131, (1974)
[46] Clayton, D.D.; Eliahu, E.; Newman, M.J.; Talbor, R.J., Astrophys. J., 199, 494, (1975)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.