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

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

Large deviation theory and Statistical Mechanics Rare events:  Tails of probability distributions  Rates of convergence to equilibrium BG: lies on LDT NEXT: ¿ q-LDT ?

Large deviation theory and Statistical Mechanics G. Ruiz & C. Tsallis, Phys. Lett.A 376 (2012) G. Ruiz & C. Tsallis, Phys. Lett. A 377 (2013)

Physical scenario of a possible LDT generalization a) Standard many-body Hamiltonian system in thermal equilibrium (T) BG weight: BG weight: (short-range + ergodic = extensive energy) (short-range + ergodic = extensive energy) LDT probability: LDT probability: ( BG relative entropy per particle) ( BG relative entropy per particle) LDT probability: LDT probability: b) d-dimensional classical system: 2-body interactions Large ranged ( ) Large ranged ( ) ( intensive variable) ( intensive variable)

: Outcomes: 2 (each toss) 2 (each toss) 2 N (N tosses) 2 N (N tosses) Number of heads, n: Containing n heads: Containing n heads: Probability of n heads: Probability of n heads: LDT standard results: N uncorrelated coins Weak Law of large numbers: Rate at which limit is attained: Large Deviation Principle (r 1 : rate function) Large Deviation Principle (r 1 : rate function) Average number of heads per toss in a range:

: Outcomes: 2 (each toss) 2 N (N tosses) Number of heads Containing n heads: Containing n heads: Probability of n heads: Probability of n heads: Weak Law of large numbers: Rate at which limit is attained: Large Deviation Principle (r 1 : rate function) Large Deviation Principle (r 1 : rate function) Average number of heads per toss in a range: LDT standard results: N uncorrelated coins

a) Independent random variables Standard CLT Rate function and relative entropy Relative entropy: N uncorrelated coins (W=2, p 1 =x, p 2 =1-x): q-Generalized relative entropy: C. Tsallis, Phys. Rev. E 58 (1998) b) Strongly correlated random variables q-CLT 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)

Non-BG: N strongly correlated coins Histograms: A. Rodriguez, V. Schwammle, C. Tsallis, J. Stat. Mech (2008)P Discretization: Suport:

Average number of heads per toss : : Large deviations in (Q,  )-model

Average number of heads per toss : :

Large Deviation Principle in (Q,  )-model Average number of heads per toss : :

Generalized q-rate function: What about q-generalized relative entropy? Large Deviation Principle in (Q,  )-model Average number of heads per toss : :

Large Deviation Principle in (Q,  )-model Asymptotic numerical behavior

Numericaly known calculation Large Deviation Principle in (Q,  )-model Asymptotic expansion of q-exponential :

Bounding numerical results: Large Deviation Principle in (Q,  )-model

Bounding numerical results: Large Deviation Principle in (Q,  )-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,r q (low) (x)>0, r q (up) (x)>0] might exists such that P(N;n/N<x) satisfies these inequalities:

Large Deviation Principle in (Q,  )-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,r q (low) (x)>0, r q (up) (x)>0] might exists such that P(N;n/N<x) satisfies these inequalities:

Conclusions We address a family of models of strongly correlated variables of a certain class whose attractors, in the probability space, are Q-Gaussians (Q>1). They illustrate how the classical Large Deviation Theory can be generalized. We address a family of models of strongly correlated variables of a certain class whose attractors, in the probability space, are Q-Gaussians (Q>1). They illustrate how the classical Large Deviation Theory can be generalized. We conjecture that for all strongly correlated systems that have Q-Gaussians (Q>1) as attractors in the sense of the central limit theorem, a model-dependent set [q>1, B(x)>0,r q (low) (x)>0, r q (up) (x)>0] might exists such that P(N;n/N 1) as attractors in the sense of the central limit theorem, a model-dependent set [q>1, B(x)>0,r q (low) (x)>0, r q (up) (x)>0] might exists such that P(N;n/N<x) satisfies: The argument of the q-logarithmic decay of large deviations remains extensive in our model. This reinforces the fact that, according to NEXT for a wide class of systems whose elements are strongly correlated, a value of index q exists such thar S q preserves extensivity. The argument of the q-logarithmic decay of large deviations remains extensive in our model. This reinforces the fact that, according to NEXT for a wide class of systems whose elements are strongly correlated, a value of index q exists such thar S q preserves extensivity. Our models open the door to a q-generalization of virtually many of the classical results of the theory of large deviations. Our models open the door to a q-generalization of virtually many of the classical results of the theory of large deviations. The present results do suggest the mathematical basis for the ubiquity of q-exponential energy distributions in nature. The present results do suggest the mathematical basis for the ubiquity of q-exponential energy distributions in nature.

Kaniadakis’  logarithm and  -exponential Kaniadakis’  logarithm and  -exponential (back)