Generalised extreme value statistics and sum of correlated variables.

and sum of correlated variables. Laboratoire Charles Coulomb CNRS & Université Montpellier 2 P.C.W. Holdsworth, É. Bertin (ENS Lyon), J.-Y. Fortin (LPS Nancy), S.T. Bramwell, S.T. Banks (UCL London). University of Warwick, May 17th 2012

Outline 1 Introduction Global quantities Surprising experimental fact 2 3 Model Volume fluctuations Physical interpretation

Outline Introduction Global quantities Surprising experimental fact 1 Introduction Global quantities Surprising experimental fact 2 3 Model Volume fluctuations Physical interpretation

global quantities Introduction Global quantities Surprising experimental fact Definition: global quantity Example Measurable quantity, sum of microscopic or local variables. Magnetization of a system of N spins : M = 1 N N k=1 σ k. Total dissipated power in a fluid : P = k p(r k)

Probleme Introduction Global quantities Surprising experimental fact Quantity of interest Example Question : Probability density function (PDF) of global quantities Gaussian distribution central limit theorem (CLT) Poisson distribution and a full zoo of other possibilities... What could we say in general of PDF of global quantities?

Outline Introduction Global quantities Surprising experimental fact 1 Introduction Global quantities Surprising experimental fact 2 3 Model Volume fluctuations Physical interpretation

Global quantities Surprising experimental fact Injected power fluctuations in a turbulent fluid [Labbe, Pinton, Fauve, J. Phys II (Paris), 6, 10099 (1996)] Ω L Ω Experience Contra-rotating disks Angular speed Ω constant Reynolds number Re = L 2 Ω/ν 5 10 5 : Turbulent regime. Injected power fluctuations

Global quantities Surprising experimental fact Injected power fluctuations in a turbulent fluid [Pinton, Holdsworth, Labbé, Phys. Rev. E, 60 R2452 (1999)] Figure: Measured injected power PDF for various Reynolds numbers.

Global quantities Surprising experimental fact Magnetisation fluctuations in the 2d XY model [P. Archambault et al., J. Appl. Phys. 83, 7234 (1998)] Figure: PDF obtained by Monte Carlo simulations.

A surprising similarity Global quantities Surprising experimental fact [Bramwell, Holdsworth, Pinton, Nature (1998)] Figure: The two previous results.

Density fluctuation in a tokamak Global quantities Surprising experimental fact [B. Ph. van Milligen et al., Phys. Plasmas 12, 052507 (2005)] Figure: Plasma density fluctuations in a tokamak.

Global quantities Surprising experimental fact Resistance of a disordered conductor [C. Pennetta et al., SPIE Proc. 5471 (2004)] Close to the critical point: 2d resistor network In a steady state UNIVERSITÁ degli STUDI di LECCE Monte Carlo simulations Fluctuations closed to the threshold. 12x50 =0.33 rks NxN: N=100, 125, 150 and j=i/n=const e PDF becomes independent of the size Longitudinal geometry

Probleme Introduction Global quantities Surprising experimental fact Experimental fact Similar distributions in various complex systems Turbulent flows, magnetic systems, SOC models, electroconvections, Freederick transitions, granular materials, spin glass, universe sheets... Question : Origine for this similarity? Limit theorem for complex systems? Similar physics behind these distributions?

Global quantities Surprising experimental fact BHP distribution and extreme value statistics Interesting observation : [ G a (µ) exp ab a (µ s a ) ae ba(µ sa)], a = π/2 1 1x10-1 BHP distribution Generalised Gumbel distribution 1x10-2 1x10-3 1x10-4 1x10-5 -8-6 -4-2 0 2

Extreme value statistics Global quantities Surprising experimental fact Extreme value statistics Example For a integer, G a is Gumbel distribution, an aymptotic distribution for extremes Let X n = {x 1,..., x n } be a set of n random variables independent and identically distributed. Let z n = Max (k) (X n ) be the k th largest element of X n. P n (z n ) n G k ( z) Note that here k is an integer by construction

Global quantities Surprising experimental fact A phenomenology based on extremes? Question : What is the meaning of a generalized Gumbel distribution (a R)? Physics controled by a hidden complex process? n X = x k Max[f (x 1...x n )]? k=1 Interpretation of generalised extreme value statistics?

Outline Introduction 1 Introduction Global quantities Surprising experimental fact 2 3 Model Volume fluctuations Physical interpretation

[É. Bertin and M. Clusel, J.Phys.A 39 7607 (2006)] k th largest element Gumbel G k, k integer Figure: Principle of the proof

Equivalence: ordering Ordering X N a set of N IID random variables x k. x 2 x 10 x 4 x 7 x 1 x 6 x 5 x 9 x 8 x 3 Increments u k z k z k+1, 1 k N 1, u N z N. z n [ ] N Max (n) X N = u k, k=n Interpretation

Equivalence: ordering Ordering X N a set of N IID random variables x k. z 10 z 9 z 8 z 7 z 6 z 5 z 4 z 3 z 2 z 1 z k = x σ(k), with σ ordering permutation z 1 z 2... z N. Increments u k z k z k+1, 1 k N 1, u N z N. z n [ ] N Max (n) X N = u k, k=n

Equivalence: ordering Ordering X N a set of N IID random variables x k. u 10 u 9 u 8 u 7 u 6 u 5 u 4 u 3 z 10 z 9 z 8 z 7 z 6 z 5 z 4 z 3 u 2 z 2 u 1 z 1 Increments u k z k z k+1, 1 k N 1, u N z N. z n [ ] N Max (n) X N = u k, k=n Interpretation

Equivalence: ordering Ordering X N a set of N IID random variables x k. Increments u k z k z k+1, 1 k N 1, u N z N. z n [ ] N Max (n) X N = u k, k=n Interpretation Equivalence between sums and extremes.

Equivalence: correlation Induced correlations Joint probability J k,n (u k,..., u N ) = N! Ordering process a priori non-iid. 0 dz N P(z N )... dz 1 P(z 1 ) z 2 N 1 δ(u N z N ) δ(u n z n + z n+1 ) n=k

Equivalence: correlation Integration Successive integration u dz k 1 P(z k 1 )... F(u) u dz P(z) Shift of indices. Final result J k,n (u k,..., u N ) = z 2 dz 1 P(z 1 ) = ( N ) k 1 N! (k 1)! F u i i=k 1 (k 1)! F(u)k 1, N n=k ( N ) P u i. i=n

Equivalence: correlation Final result J k,n (u k,..., u N ) = General case: ( N ) k 1 N! (k 1)! F u i i=k Not factorised : impossible to write J N (u 1,..., u N ) N n=1 Random variables U n are not IID. N n=k π n (u n ), ( N ) P u i. i=n

Introduction [É. Bertin and M. Clusel, J.Phys.A 39 7607 (2006)] equivalence k th largest element Sum of random variables J k Gumbel G k, k integer Figure: Principle of the proof

Outline Introduction 1 Introduction Global quantities Surprising experimental fact 2 3 Model Volume fluctuations Physical interpretation

Generalisation: extended joint probability Previous results J k,n (u k,..., u N ) = Generalisation ( N ) k 1 N! (k 1)! F u i i=k N n=k ( N ) P u i. i=n J N (u 1,..., u N ) = Γ(N) Z N [ ( N )] N ( N ) Ω F u n P u i n=1 n=1 i=n

Generalisation: extended joint probability Generalisation J N (u 1,..., u N ) = Γ(N) Z N [ ( N )] N ( N ) Ω F u n P u i n=1 n=1 i=n Functions Ω(F), (arbitrary) positive function of F Ω(F) F 0 F a 1, a R Z N = 1 0 dv Ω(v) (1 v)n 1.

Introduction [É. Bertin and M. Clusel, J.Phys.A 39 7607 (2006)] equivalence k th largest element Sum of random variables J k Gumbel G k, k integer Figure: Principle of the proof Sum of random variables jointed by J a

Outline Introduction 1 Introduction Global quantities Surprising experimental fact 2 3 Model Volume fluctuations Physical interpretation

The simple exponential case Factorisation conditions (x, y), P(x + y) = P(x)P(y) P(x) = κe κx (F 1, F 2 ), Ω(F 1 F 2 ) = Ω(F 1 )Ω(F 2 ) Ω(F) = F a 1 Factorised joint probability J N (u 1,..., u N ) = N n=1 π n (u n ), π n (u n ) = (n + a 1)κ e (n+a 1)κun. Variables u n Independent, but non-identically distributed.

The simple exponential case Definitions u n is distributed according to π n (u n ), a R. s N = N n=1 u n. Let Υ N be the distribution of s n. Fourier transform of Υ N F [Υ N ] (ω) = Question : N F [π n ] (ω) = n=1 N n=1 ( ) iω 1 1 +. κ(n + a 1) What is the asymptotic distribution lim N Υ N?

The simple exponential case First two moments of Υ N S N = N U n = 1 κ n=1 N n=1 N σn 2 = Var(U n ) = 1 κ 2 n=1 Breakdown of central limit theorem 1 n + a 1, N n=1 1 (n + a 1) 2. lim σ N < N Lindeberg condition not satisfied.

The simple exponential case: asymptotic distribution Rescaled variable µ = s S N, with σ = lim σ σ N. N Φ N (µ) = συ N (σµ + S N ).

The simple exponential case: asymptotic distribution Rescaled variable µ = s S N, with σ = lim σ σ N. N Φ N (µ) = συ N (σµ + S N ). Large N limit: Φ = lim N Φ N F [Φ ] (ω) = n=1 ( 1 + ) iω 1 ( exp σκ(n + a 1) iω σκ(n + a 1) ).

The simple exponential case: asymptotic distribution Rescaled variable µ = s S N, with σ = lim σ σ N. N Φ N (µ) = συ N (σµ + S N ). Large N limit: Φ = lim N Φ N Φ (µ) = G a (µ), G a, Gumbel distribution of parameter a R.

Introduction [É. Bertin and M. Clusel, J.Phys.A 39 7607 (2006)] equivalence k th largest element Sum of random variables J k Gumbel G k, k integer Gumbel G a, a real Sum of random variables jointed by J a Figure: Principle of the proof

Physical interpretation 1/f noise with cut-off 1D lattice model, continuous variable φ x on each site. ˆφ q = 1 L L x=1 φ xe iqx. Total energy : E = q ˆφ q 2 = q u q, P(u n ) = κe κun and Ω(F) = F a 1. Correlation ˆφ q 2 1 q +m, Correlation length ξ = P(E) = G a (E). m = 2π(a 1) L L 2π(a 1)

Physical interpretation Correlation ˆφ q 2 1 q +m, Correlation length ξ = Limit cases m = 2π(a 1) L L 2π(a 1) ξ : pure 1/f noise Gumbel a = 1 ξ 0: uncorrelated white noise Gaussian a

Interpretation of generalised extreme value statistics Summary Extreme value statistics sums of correlated random variables of the correlated sums, and not of the extreme problem. Conclusion Observation of generalised Gumbel distribution Correlation and not necessarly extremes

Outline Introduction Model Volume fluctuations Physical interpretation 1 Introduction Global quantities Surprising experimental fact 2 3 Model Volume fluctuations Physical interpretation

Model Volume fluctuations Physical interpretation A simple model for [É. Bertin et al., J.Stat.Mech P07019 (2008)] Piston Particle g Model N point particles Quasi 1d cylinder Equilibrium at T Reflective wall at z = 0 Piston at z p Potentials Piston: U p (z p ) Particle: U(z)

Model Volume fluctuations Physical interpretation A simple model for Joint probability P N (z 1,..., z N, z p ) = 1 Z e βup(zp) Volume fluctuations Integration on particles positions: N i=1 e βu(z i ) Θ(z p z i ), P(z p ) = 1 ( zp ) N Z e βup(zp) dz e βu(z), 0

Outline Introduction Model Volume fluctuations Physical interpretation 1 Introduction Global quantities Surprising experimental fact 2 3 Model Volume fluctuations Physical interpretation

Two simple cases Model Volume fluctuations Physical interpretation Free particles A particle as piston U(z) = 0 P(z p ) = 1 Z zn p e βup(zp) Gaussian fluctuations in the large N limit. U p (z p ) = U(z p ) P(z p ) = d dz p G(z p ) N+1 Standard extreme value statistics.

Case U(z) = U 0 z α, Model Volume fluctuations Physical interpretation U p (z p ) = U 0 zγ p Ordering z 1,..., z N satisfying 0 < z i < z p permutation: σ : z σ(1) z σ(2)... z σ(n) space interval: h i h i = z σ(i) z σ(i 1), i = 2,..., N. h i = u N+2 i Equilibrium probability distribution J N (h 1,..., h N+1 ) = K Ω [ F ( N+1 i=1 )] N+1 h i i=1 P i j=1 h i,

Case U(z) = U 0 z α, Model Volume fluctuations Physical interpretation U p (z p ) = U 0 zγ p Equilibrium probability distribution J N (h 1,..., h N+1 ) = K Ω Functions P(z) = λ e βu(z), F(z) = [ F ( N+1 i=1 z )] N+1 h i i=1 P(z )dz. P Ω(y) = 1 ] [βu(f λ exp 1 (y)) βu p (F 1 (y)), y ]0, 1[. i j=1 h i,

Case U(z) = U 0 z α, Model Volume fluctuations Physical interpretation U p (z p ) = U 0 zγ p Equilibrium probability distribution J N (h 1,..., h N+1 ) = K Ω Limit N : case α = γ [ F ( N+1 i=1 )] N+1 h i i=1 P i j=1 h i, Ω(y) Γ ( ) 1 a ( α a 1 y ln 1 ) (a 1)(1 1 α ) (y 0) α(βu 0 ) 1/α y with a = U 0 /U 0 generalised Gumbel distribution of parameter a

Case U(z) = U 0 z α, Model Volume fluctuations Physical interpretation U p (z p ) = U 0 zγ p Equilibrium probability distribution J N (h 1,..., h N+1 ) = K Ω Limit N [ F ( N+1 Depending of the behaviour of Ω i=1 γ > α: Gaussian distribution γ < α: Exponential distribution )] N+1 h i i=1 P γ = α: Generalised extreme value distribution i j=1 h i,

Outline Introduction Model Volume fluctuations Physical interpretation 1 Introduction Global quantities Surprising experimental fact 2 3 Model Volume fluctuations Physical interpretation

Model Volume fluctuations Physical interpretation Physical interpretation: gravity case α = γ = 1 Equation of states z p = N+1 j=1 h j k bt mg (1 ln + N a ( P a V = ak b T ln 1 + N a M Nm: P V = Nk b T M Nm: P a V = ak b T ln(n/a) ). ), P a = Mg/S = amg/s. Compressibility V P = 1 ( k b T V 2 V 2) κ = 1 V V P 1 ln N. Abnormally small fluctuations, logarithmic decay

Conclusion Introduction Model Volume fluctuations Physical interpretation So called generalised EVS : Associated with sum of correlated random variables A priori no simple extreme processes at play Misleading name! Any suggestion?

Bibliography Introduction Model Volume fluctuations Physical interpretation Labbe, Pinton, Fauve, J. Phys II (Paris), 6, 10099 (1996) J.-F. Pinton, P. Holdsworth, R. Labbe, Phys. Rev. E, 60 R2452 (1999) P. Archambault, S. T. Bramwell, J.-Y. Fortin, P. C. W. Holdsworth, S. Peysson, and J.-F. Pinton, J. Appl. Phys. 83, 7234 (1998) S.T. Bramwell, P. Holdsworth, J.-F. Pinton, Nature 396 552 (1998) B. Ph. van Milligen et al., Phys. Plasmas 12, 052507 (2005) C. Pennetta et al., SPIE Proc. 5471 (2004) K. Dahlstedt and H. J. Jensen, J Phys A:Math Gen 34, 11193 (2001) É. Bertin and M. Clusel, J.Phys.A 39 7607 (2006) É. Bertin, M. Clusel and P.C.W. Holdsworth, J.Stat.Mech P07019 (2008) M. Clusel and É. Bertin, Int.J.Mod.Phys B 22, 3311 (2008)

Model Volume fluctuations Physical interpretation Consider a set of realisations {u n } of N (correlated) random variables U n, with the joint probability (??). We then define as above the random variable S N = N n=1 U n, and let Υ N be the probability density of S N. Then Υ N is given by Υ N (s) = 0 = Γ(N) Z N du N...du 1 J N (u 1,..., u N ) δ P(s) Ω ( F(s) ) I N (s), ( s ) N u n. n=1 with I N (s) = 0 du N P(u N )... 0 du 1 δ ( s ) N u n. n=1

Model Volume fluctuations Physical interpretation To evaluate I N, let us start by integrating over u 1, using 0 du 1 δ ( s ) ( N u n = Θ s n=1 ) N u n, n=2 where Θ is the Heaviside distribution. This changes ( the upper bound of the integral over u 2 by u2 max = max 0, s ) N n=2 u n. Then the integration over u 2 leads to u max 2 0 [ ( N ) ] ( du 2 P(u 2 ) = F u n F(s) Θ s n=3 ) N u n, n=3

Model Volume fluctuations Physical interpretation By recurrence it is then possible to show that I N (s) = 1 ( N 1, 1 F(s)) Γ(N) finally yielding the following expression for Υ N : Υ N (s) = 1 P(s) Ω ( F(s) ) ( ) N 1. 1 F(s) (1) Z N In the following sections, we assume that Ω(F) behaves asymptotically as a power law Ω(F) Ω 0 F a 1 when F 0 (a > 0). Under this assumption, we deduce from Eq. (1) the different limit distributions associated with the different classes of asymptotic behaviours of P at large x.

Model Volume fluctuations Physical interpretation P(x) decays faster than any power law at large x. To that purpose, we define sn by F(s N ) = a/n. If a is an integer, this is nothing but the typical value of the a th largest value of s in a sample of size N. As P is unbounded we have lim N s N = +. Let us introduce g(s) = ln F(s) and, assuming g (s N ) 0, define the rescaled variable v by s = sn + v g (sn (2) ).

Model Volume fluctuations Physical interpretation For large N, series expansion of g around s N : g(s) = g(s N ) + v + n=1 1 g (n) (sn ) n! g (sn v n. )n For P in the Gumbel class, g (n) (s N )/g (s N )n is bounded as a function of n so that the series converges. In addition, one has lim N so that g(s) may be written as g (n) (sn ) g (sn = 0, n 2, )n g(s) = g(s N ) + v + ε N(v), with lim N ε N(v) = 0. (3)

Model Volume fluctuations Physical interpretation Given that P(s) = g (s) F(s), one gets using Eqs. (1) and (3) Φ N (v) = 1 g (s N )Υ N(s) = 1 g (s) Z N g (sn ) F(s) Ω( F(s) )( 1 F(s) ) N 1, where s is given by Eq. (2). For P in the Gumbel class, it can be checked that, for fixed v g ( sn lim + v/g (sn )) N g (sn ) = 1. Besides, F ( s N + v/g (s N )) 0 when N, so that one can use the small F expansion of Ω(F).

Model Volume fluctuations Physical interpretation Altogether, one finds Φ N (v) Ω ( 0 a ) a [ e av aε N (v) 1 a ] N 1 Z N N N e v ε N(v) Using a simple change of variable in Eq. (??), one can show that N a Z lim N N Ω 0 = Γ(a) It is then straightforward to take the asymptotic limit N, leading to Φ (v) = aa Γ(a) exp [ av ae v].

Model Volume fluctuations Physical interpretation In order to recover the usual expression for the generalised Gumbel distribution, one simply needs to introduce the reduced variable µ = v v σ v, with, Ψ being the digamma function, v = ln a Ψ(a), σ 2 v = Ψ (a). The variable µ is then distributed according to a generalised Gumbel distribution.

Model Volume fluctuations Physical interpretation To sum up, if one considers the sum S N of N 1 random variables linked by the joint probability (??), then the asymptotic distribution of the reduced variable µ defined by with S N = s N µ = s N S N σ N, ln a Ψ(a) Ψ + g (sn ), σ N = (a) g (sn ). is the generalised Gumbel distribution. back