Download presentation

Presentation is loading. Please wait.

Published byKenneth Anthony Modified over 2 years ago

1
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

7
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

8
Large deviation theory and Statistical Mechanics Rare events: Tails of probability distributions Rates of convergence to equilibrium BG: lies on LDT NEXT: ¿ q-LDT ?

9
Large deviation theory and Statistical Mechanics G. Ruiz & C. Tsallis, Phys. Lett.A 376 (2012) 2451-2454. G. Ruiz & C. Tsallis, Phys. Lett. A 377 (2013) 491-495.

10
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)

11
: 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:

12
: 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

13
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) 1442-1445. 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) 033502.

14
Non-BG: N strongly correlated coins Histograms: A. Rodriguez, V. Schwammle, C. Tsallis, J. Stat. Mech (2008)P09006. Discretization: Suport:

15
Average number of heads per toss : : Large deviations in (Q, )-model

16
Average number of heads per toss : :

17
Large Deviation Principle in (Q, )-model Average number of heads per toss : :

18
Generalized q-rate function: What about q-generalized relative entropy? Large Deviation Principle in (Q, )-model Average number of heads per toss : :

19
Large Deviation Principle in (Q, )-model Asymptotic numerical behavior

20
Numericaly known calculation Large Deviation Principle in (Q, )-model Asymptotic expansion of q-exponential :

21
Bounding numerical results: Large Deviation Principle in (Q, )-model

22
Bounding numerical results: Large Deviation Principle in (Q, )-model

23
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

24
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

25
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

26
Kaniadakis’ logarithm and -exponential Kaniadakis’ logarithm and -exponential (back)

Similar presentations

OK

A scale invariant probabilistic model based on Leibniz-like pyramids Antonio Rodríguez 1,2 1 Dpto. Matemática Aplicada y Estadística. Universidad Politécnica.

A scale invariant probabilistic model based on Leibniz-like pyramids Antonio Rodríguez 1,2 1 Dpto. Matemática Aplicada y Estadística. Universidad Politécnica.

© 2018 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Ppt on do's and don'ts of group discussion clip Ppt on two point perspective definition Ppt on tungsten inert gas welding Ppt on brand equity management Ppt on home automation and security for mobile devices Ppt on teachers day download Ppt on 21st century skills map Ppt on indian union budget 2013-14 Download ppt on 3d printing technology Ppt on water the essence of life