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Towards a large deviation theory for statistical mechanical complex systems

Towards a large deviation theory for statistical mechanical complex systems. 1 Centro Brasileiro de Pesquisas Fisicas . Brazil 2 Universidad Politécnica de Madrid. Spain . 3 Santa Fe Institute , USA. G. Ruiz López 1,2 , C. Tsallis 1,3.

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Towards a large deviation theory for statistical mechanical complex systems

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  1. Towards a large deviation theory for statistical mechanical complex systems 1Centro Brasileiro de Pesquisas Fisicas. Brazil 2Universidad Politécnica de Madrid. Spain. 3Santa Fe Institute, USA G. Ruiz López1,2, C. Tsallis1,3

  2. Towards a large deviation theory for statistical mechanical complex systems 1Centro Brasileiro de Pesquisas Fisicas. Brazil 2Universidad Politécnica de Madrid. Spain. 3Santa Fe Institute, USA G. Ruiz López1,2, C. Tsallis1,3

  3. Largedeviationtheory and StatisticalMechanics • Rare events: • Tails of probability distributions • Rates of convergence to equilibrium • BG: lies on LDT NEXT: ¿ q-LDT ?

  4. Largedeviationtheory and StatisticalMechanics G. Ruiz & C. Tsallis, Phys. Lett .A 376 (2012) 2451-2454. G. Ruiz & C. Tsallis, Phys. Lett. A 377 (2013) 491-495.

  5. Physicalscenario of a possible LDT generalization a) Standard many-bodyHamiltoniansystem in thermalequilibrium (T) BG weight: (short-range + ergodic = extensiveenergy) LDT probability: ( BG relativeentropyper particle) • LDT probability: b) d-dimensional classicalsystem: 2-body interactions • Largeranged ( ) ( intensive variable)

  6. LDT standardresults: Nuncorrelatedcoins • Outcomes: 2 (eachtoss) 2N(Ntosses) • Number of heads, n: Containingnheads: Probability of nheads: • Averagenumber of heads per toss in a range: WeakLaw of largenumbers: Rate at whichlimitisattained: LargeDeviationPrinciple (r1 : ratefunction)

  7. LDT standardresults: Nuncorrelatedcoins • Outcomes: 2 (eachtoss) 2N (N tosses) • Number of heads Containingnheads: Probability of nheads: • Averagenumber of heads per toss in a range: WeakLaw of largenumbers: Rate at whichlimitisattained: LargeDeviationPrinciple (r1 : ratefunction)

  8. Ratefunction and relativeentropy • a) Independent random variables Standard CLT Relative entropy: N uncorrelated coins (W=2, p1=x, p2=1-x): b) Strongly correlated random variables q-CLT q-Generalized relative entropy: S.Umarov, C. Tsallis, S. Steinberg, Milan J. Math. 76 (2008) 307. S. Umarov, C. Tsallis, M. Gell-Mann, S. Steinberg, J. Math. Phys. 51 (2010) 033502. C. Tsallis, Phys. Rev. E 58 (1998) 1442-1445.

  9. Non-BG: Nstronglycorrelatedcoins A. Rodriguez, V. Schwammle, C. Tsallis, J. Stat. Mech (2008)P09006. Discretization: Suport: Histograms:

  10. Largedeviations in (Q, g, d)-model Average number of heads per toss : :

  11. Largedeviations in (Q, g, d)-model Average number of heads per toss : :

  12. LargeDeviationPrinciple in (Q, g, d)-model Average number of heads per toss : :

  13. LargeDeviationPrinciple in (Q, g, d)-model Average number of heads per toss : : • Generalizedq-ratefunction: • Whataboutq-generalizedrelativeentropy?

  14. LargeDeviationPrinciple in (Q, g, d)-model Asymptotic numerical behavior

  15. LargeDeviationPrinciple in (Q, g, d)-model Asymptotic expansion of q-exponential : Numericalyknowncalculation

  16. LargeDeviationPrinciple in (Q, g, d)-model • Boundingnumericalresults:

  17. LargeDeviationPrinciple in (Q, g, d)-model • Boundingnumericalresults:

  18. LargeDeviationPrinciple in (Q, g, d)-model For all strongly correlated systems which have Q-Gaussians (Q>1) as attractors in the sense of the central limit theorem, a model-dependent set [q>1, B(x)>0,rq(low)(x)>0, rq(up)(x)>0] might exists such that P(N;n/N<x) satisfies these inequalities:

  19. LargeDeviationPrinciple in (Q, g, d)-model For all strongly correlated systems which have Q-Gaussians (Q>1) as attractors in the sense of the central limit theorem, a model-dependent set [q>1, B(x)>0,rq(low)(x)>0, rq(up)(x)>0] might exists such that P(N;n/N<x) satisfies these inequalities:

  20. Conclusions • Weaddress a family of models of stronglycorrelated variables of a certainclasswhoseattractors, in theprobabilityspace, are Q-Gaussians (Q>1). TheyillustratehowtheclassicalLargeDeviationTheory can begeneralized. • WeconjecturethatforallstronglycorrelatedsystemsthathaveQ-Gaussians (Q>1) as attractors in thesense of the central limittheorem, a model-dependent set [q>1, B(x)>0,rq(low)(x)>0, rq(up)(x)>0] mightexistssuchthat P(N;n/N<x)satisfies: • Theargument of theq-logarithmicdecay of largedeviationsremainsextensive in ourmodel. Thisreinforcesthefactthat, accordingto NEXT for a wideclass of systemswhoseelements are stronglycorrelated, a value of indexqexistssuchtharSq preserves extensivity. • Ourmodels open thedoorto a q-generalization of virtuallymany of theclassicalresults of thetheory of largedeviations. • Thepresentresults do suggestthemathematicalbasisfortheubiquity of q-exponentialenergydistributions in nature.

  21. (back) Kaniadakis’ k-logarithm and k-exponential

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