Non Extensive Thermostatistics'

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• Non Extensive Thermostatistics.
• Motivation
• Variety of physical systems where Boltzmann-Gibbs
  thermostatistics fail
• Long range interactions.
• Long range memory- Non markovian evolution.

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B.G. Statistics - A reminder.

• Entropy
• Constraints
• Maximize the objective
• Yields distribution

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Postulate C. Tsallis J. Stat. Phys. 52 p479
(1988)

• Generalized entropy where q characterizes
  the extensivity of the statistics.
• Note For q1 regular B.G. Statistics is
  recovered

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Properties of Sq
Sq/k
q-1
q-0.5
q0
q0.5

• Positivity
• Concavity Sq is concave (convex) for qgt0
  (qlt0)
• Generalized AdditivitySq is super (sub)
  additive for qlt1 (qgt1)

q1
q2
qoo
p
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Generalized canonical ensemble

• We demand that at equilibrium, Sq have an
  extremum under constraints.
• We define a generalized average energy
• Maximization Objective
• Yields distribution

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Connection with ThermodynamicsE.M.F. Curado
C. Tsallis J. Phys. A. 24 pL69 (1991)

• Define
• Legendre transform
• Thermodynamic relations

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Generalized Thermodynamic relationships

• Thermo stability Heat capacityC. Tsallis
  Phys. Lett. A, 206 p389 (1995)
• H theorem.A. M. Mariz Phys. Lett. A, 165 p366
  (1995)
• Focker Plank Eq.

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• Problem
• ?i not invariant under uniform translation of
  energy spectrum.
• Non extensivity of Generalized EnergyUq(AB) ?
  Uq (A)Uq (B)
• Solution
• Normalization of Generalized Energy
• Previous results modified but still valid.
• C. Tsallis et al Physica A, 291 p534 (1998)

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Applications

• Heat capacity of Hydrogen Atom.L.S. Lucena et
  al, Phys. Rev. E, 51 p6247 (1995)
• Self gravitating systems in Astrophysics.A. R.
  Plastino A. Plastino, Phys. Lett. A,193 p251
  (1991)
• Cosmic background microwave radiation.C.
  Tsallis et al, Phys. Rev. E, 52 p1447 (1995)
• Long range interaction systems. P. Jund et al,
  Phys. Rev. B, 52 p50 (1995)
• Levy flights, systematic derivation.

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Levy Flights

• Albert Einstein 1905

• Paul Levy 1937

Brownian motion
Levy distributions
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Levy Flights

• Albert Einstein 1905
• one step distribution funct. central limit
  theorem
• N step distribution funct.

Paul Levy 1937 Levy-Gnedenko
Levy distribution
C. Tsallis et al. PRL 75 p3589 (1995)
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Summary

• A generalized non extensive thermodynamics can be
  formulated using a generalized entropy Sq.
• Many mathematical characteristics of B.G.
  Thermostatistics are preserved.
• Application to variety of systems unaccounted for
  by B.G. Statistics.