NONEXTENSIVITY IN A DARK MAXIMUM ENTROPY LANDSCAPE

Transcription

1 BEYOND 010 Cape Town, SA NONEXTENSIVITY IN A DARK MAXIMUM ENTROPY LANDSCAPE M. P. LEUBNER Institute for Astro- and Particle Physics University of Innsbruck, Austria

2 TOPICS (1) FUNDAMENTALS OF NONEXTENSIVE STATISTICS () ENTROPY BIFURCATION AND DUALITY (3) CONSEQUENCES OF NONEXTENSIVITY (1) Non-thermal equilibria in thermostatistics - plasma () Gravitational equilibria in self-gravitating systems - DM (3) Nonextensive negative pressure domain DE (4) Discrete cosmic inhomogeneity scales (4) Summary Four fundamentally different domains in nonxtensive statistics depending on sign and value of the governing entropic index

3 NATURE IS NONEXTENSIVE ANY MEMBER OF AN ENSEMBLE OF PARTICLES gravitational / electromagnetic / strong / weak long-range interactions correlations, coupling TWO EXTREME CASES (1) crystal max order min entropy, exact geometry () therm gas min order max entropy, BGS statistics NATURE SOMEWHERE BETWEEN NATURE IS COMPLEX STATISTICS

4 EXTENSIVE FUNDAMENTALS Statistical Mechanics adoption of specific Entropy Functional S shortcut for vast, detailed microscopic information stored in system + connection to macroscopic behavior no systematic context determining S Boltzmann Gibbs proposed fundation of standard statistics Shannon: descrete form entropic form for ergodic systems S = - S p i ln p i Systems visiting with equal probability all allowed states Chaotic systems molecular chaos hypothesis Extensive systems: no correlations, members independent BUT: nonergodicity is the generic case for complex systems

5 NONEXTENSIVE FUNDAMENTALS Nonextensive Statistics what is S for non-ergodic systems S = ( p 1 1/ i 1) p i probability of i-th microstate S extremized for equiprobabilit p i = 1 Tsallis, 1988 q-entropy conjecture fl S = - S p i ln p i transformation = 1/(1-q) (1) symmetry in entropic index Leubner, 00 - k () link astrophysical nonext PDFs 1 S( A+ B) = S( A) + S( B) + S( A) S( B) > 0 superextensive < 0 subextensive DUALITY + less organized state fi entropy increase - higher organized state fi entropy decrease

6 Nonext statistics - applications Chemistry, Biology, Economics, Linguistics, Medicine, Geophysics, Social & Cognitive Science, Computer Science, Information Theory Physics: Ç Astrophysical plasma distributions in velocity space Ç PDF s of differences of fluctuations in plasma turbulence Ç DM and plasma density distributions of self-gravitating systems Ç Distribution of peculiar velocities of spiral galaxies Ç Cosmic microwave background radiation Ç Nonextensive dark energy model - chaotic scalar fields Ç Transverse momenta distribution in hadronic jets, e - e + annihilation Ç Diffusion of charm quarks in quark-gluon plasma Ç Hierarchy of discrete structure scales in the universe etc. etc..

7 EXTENSIVE / NONEXTENSIVE EXTENSIVE SYSTEMS standart BGS statistics particles independent feo no interactions/correlations Euclidian space time logarithmic entropy measure additive Maxwell DF f M (E, Φ) NONEXTENSIVE SYSTEMS particles not independent feo long-range forces / memory, correlations, fractal space time Nonextensive statistics generalized power-law entropy pseudoadditive power-law DF f (E, Φ)

8 DUALITY (A) Duality of NEXT equilibria TWO FAMILIES (, ) OF STATIONARY STATES (Karlin, 00) thermodynamic equilibrium () î maximum entropy state kinetic equilibrium ( ) î zeros of the collision integral constraint for the nonextensive equilibrium index fl nonextensive thermodynamic equilibria Κ > 0 nonextensive kinetic equilibria Κ < 0 with condition = - DUALITY limiting BGS state for = : SELF - DUAL ï extensivity the extensive = BGS state bifurcates for finite values of

9 (B) Duality of NEXT heat capacity TWO FAMILIES OF HEAT CAPACITIES (Almeida, 001) Virial theorem K + U = 0 E = K + U = - K K = 3/ Nk B T C = de/dt C = - 3/ Nk B < 0 d de 1 1 ( ) = β 1 = kt β related to energy derivative of heat bath d de 1 ( ) = β > 0... > = < 0... < 0 positive HC infinite HC negative HC Κ > 0..positive HC thermodynamic systems Κ = ideal heat bath, loss/gain energy without change of T Κ < 0..negative HC self-interacting systems The BGS self-dual state ( = ) bifurcates for finite IDENTIFICATION > 0 thermodynamic state plasma < 0 self-interacting state DM î

10 FROM ENTROPY GENERALIZATION TO PDFs S extremize entropy under conservation of mass and energy variational problem δs - αδm - βδe = 0 α,β Lagrage multiplieres POWER-LAW DISTRIBUTIONS, BIFURCATION kˆ 0 halo k> 0 core k< 0 f hc v = Bhc ( 1+ ) σ incorporate sign of in distribution B c = N 1/ π v th 1/ Γ( ) Γ( 1/ ) normalization kî k B c = N 1/ π v th Γ( + 3/ ) 1/ Γ( + 1) σ variance, mean energy

11 nd moments and constraints f c v = Bc ( 1+ ) σ 0 k f h = B h v ( 1 ) σ halo different generalized nd moments core p = 3/ Nm k > 0 p = v f dv k < 0 p B identical for temperatures p = + 3/ pb increased and 0 <k<3/ neg pressure as compared to BGS state decreased and v = σ cutoff

12 KAPPA - DEPENDENT CHARACTERISTICS f(v) [normalized] f(v) [normalized] nonextensive distribution family and Maxwellians = v [normalized] v [normalized] halo, k > 0 core, k < 0 BI KAPPA PDF core-halo superposition

13 (1) Electromagnetic Interactions Κ > 0, C > 0, p > 0 BEYOND: REPLACE STANDARD MAXWELLIANS BY NEXT DISTR. FAMILY a) Energy distributions in astrophysical plasmas e - VDF observed nonext theory Leubner, 00, 004 b) PDF s in astrophysical plasma turbulence δx = X ( t + τ ) X ( t) amplitude PDFs of differences of plasma or magnetic field fluctuations Leubner & Voros, 005, 006

14 () Grav. Equ. - DM/plasma density distributions STANDARD CONTEXT: GRAV. INFALL MODELS; PRESSURE BALANCE BEYOND: EXTREMIZE GENERALIZED ENTROPY OF NEXT STATISTICS hot gas ï DM halo ï elm. interacting, fully ionized plasma, thermodyn. equilibrium self grav., weakly interacting particles, dynamical equilibrium extremize S in grav. Potential Ψ conservation of mass and energy ï energy distribution bifurcation: > 0 < 0 S = ( ± ± 1 v / Ψ ( r ) = 1+ f E B p 1 1/ i σ 1) integration over v ρ ± 1 Ψ = ρ0 1 σ 3/ limit = : extensive exponential density profile ρ = ρ Ψ σ exp( / ) 0

15 Nonextensive spatial density variation Κ ˆ 0, C ˆ 0, p > 0 ΔΨ = 4π Gρ combine ρ ± 1 Ψ = ρ0 1 σ 3/ ( ) ( ) 1/ 3/ d ρ dρ 1 1 dρ 4πG 3/ ρ + 1 ρ 0 = dr r dr 3/ ρ dr σ ρ0 Ψ = Ψ(r) Leubner, 005, 006 ρ(r) radial density distribution of spherically symmetric hot plasma ( k > 0 ) and dark matter ( k < 0 ) = BGS selfduality, conventional isothermal sphere

16 Non-extensive family of density profiles Nonextensive family of density profiles ρ ± (r), = 3 10 Convergence to the selfdual BGS solution =

17 Testing Nonextensive Density Distributions on Simulations, Observations DM simulations Hydro simulations Observations Κ < 0 Κ > 0 Κ < observations log(ρ/ρ c ) NFW log(ρ) [normalized] 10 1 double beta 10 R log(r) [Mpc] log(r) [normalized] Kronberger, Leubner, van Kampen A&A, 006 Integrated mass profiles Pointecouteau et al., A&A 005

18 (3) Nonextensivity in a dark energy landscape Universe currently in accelerated expansion 73% DE - 3% DM - 4% ordinary matter STANDARD DARK ENERGY MODELS: (1) Vacuum energy of self-interacting scalar field potential energy generates a cosmological constant () Quintessence models with slowly evolving scalar fields and nontrivial equation of state (3) String theoretical candidates of scalar fields and consequences (4) Exotic forms of matter BEYOND: NEXT STAT + SCALAR FIELD POTENTIAL

19 Scale factor: dark energy, matter, radiation 3 components (j) in homogeneous, isotropic universe - adiabatic expansion:.. R R 4πG = ( ρde + 3pDE + ρm + ρr + 3p 3 r ) ρ...energy density, p pressure H =. ( R/ R) dρ dt j = 3H ( ρ j + p j ) conservation of energy Equation of state for dominant species j w = / ρ j j p j = const ρ 3(1 ) j ~ R +w j ρ m ~ R -3 w m = 0 ρ r ~ R -4 w r = 1/3 condition 3p ρ DE ~ const w DM = - 1, DE < 1 ρ DE + ρ m class. vacuum energy condition

20 NEXT chaotic scalar fields as DE model consider chaotic strongly self-interacting scalar fields associate chaotic behavior with vacuum fluctuations fl probability distribution of fluctuating field p( φ) = π 1 1 φ Beck, 004 associate generalized NEXT distributions identify E = mφ / and = 1/ 1 p( E) ~ (1 + βe) β -1 = m thermal energy coincides with mass satisfy property of NEXT DE gas: w = -1?

21 Nonextensive dark energy domain 0 < Κ < 3/, C > 0, p < 0 ρ nd moment î p ~ v f ( v) dv pde = ρ 3/ DE DE equation of state (Λ) w ~ p DE / ρ DE ~ - 1 density parameter Ω DE = ρ DE ρde + ρ m + ρ r ~ 0.75 entropic index specifies degree of correlations / heat capacity Ç dark energy behaves like ordinary gas ( C > 0 ) Ç subject to high degree of correlations ( ~ 1/ ) Ç satisfying equation of state w ~ p DE / ρ DE ~ - 1

22 ENTROPIC DOMAINS S = ( p 1 1/ i 1) Negative entropy: Entropy that system exports to keep own entropy low Schrödinger, 1943 Green: BGS extensive statistics, ergodic, S() = S M = const, = Red: nonextensive thermostat, non-ergodic, S() > S M, > 0, reduced order Blue: nonextensive kin. stat, DM, non-ergodic, S() < S M, < 0, increased order 0 < < 3/ : DE domain, chaotic scalar fields, S() < S M, ~ 1/, m ~ 10e-38 m e

23 (4) Discrete cosmic inhomogeneity scales STANDARD CONTEXT: NONE BEYOND: NEXT STAT + NETWORK SCIENCE DIMENSIONLESS emerging, highly interdisciplinary research area tradition in graph theory, discrete mathematics, sociology, information /communication theory, biology, physics undirected network of network ensembles with N nodes and L links Common context: node entropy, link entropy (1) Entropy per node in a network ensemble N i S i = 1/N 0 ln N i (Hartley information measure ident BGS) () Probability of given link (i) p i = L / ( N 0 (N 0-1)/ ) (3) Perform nonextensive generalization with N 0 >> 1

24 Nonextensive network generalization S = ( p 1 1/ i 1) Let members of cluster merge links within / between subclusters N 0 total number of links in entire original cluster network n total number of links within each subcluster network N. total number of links between all subclusters as closed units condition: subclusters obey closed network nodes correspond to particles - node conservation: N*n = N 0 HOW DOES ENSEMBLE RE-ARRANGE EXT. SUM OF LINKPROB. S = {[( N n + N )/N 0 ] 1-1/ -1 } Ext. LINK ENTROPY Leubner, M. P., 000, 00, 010 N = -1/3 N 0 /3 no dependence on - connected or not

25 Recursion context - network hierarchy (1) Hierarchical tree - nested generalization N i+1 = -1/3 N i /3 () Introducing physics (m, r) through natural invariant m ~ r α scale invariance ( α ~ ) m i+1 =(m i M) 1/3 r i+1 =( r i R) 1/3 (3) Network boundary conditions e.g. ρ c, H 0 or N 0, m 0

26 Radius mass relation hierarchical sequence of discrete cosmic structure scales mimic time evolution by stepping up the hierachical tree towards larger cluster mergers

27 Sequence of structure scales MASS RADIUS # MEMBERS physical scale 8.5e e e+004 superclusters 3.5e e e+006 galaxy clusters 9.1e e+03.8e+010 galaxies 3.8e+040.3e e+015 globular clusters 3.3e+03.8e e+03 stellar systems.7e e e+035 planetesimals 1.9e e e+054 condensed matter 1.e e-013.1e+081 particle scale 1.8e e e+1 quintessence key role in evolutionary scenarios: globular clusters building blocks of galaxies planetesimals building blocks of stellar systems condensed matter nature appears highly complex quintessence scalar field m F ~ 1e-66g

28 5 fundamental domains in NEXT statistics Κ = - Κ = 0 Κ = 3/ Κ = (1) (3) fi (4) fi () fi (1) C < 0 C > 0 C > 0 p > 0 p < 0 p > 0 ρ > 0 ρ < 0 ρ > 0 BGS self interaction highest correlations BGS (gravity) correlations.. (electromagnetic) DM DE gen. thermostat. C heat capacity, p pressure, ρ density (5) no dependence on entropic index discrete structure scales

29 SUMMARY nature appears nonlocal and nonlinear on all observable levels requiring a generalization of standard BGS statistics nonextensive theory accounts for long-range interactions and correlations and is naturally subject to entropy bifurcation BEYOND STANDARD BGS ENTROPY the specific physical domains covered by the entropic index are manifest in (1) astrophysical energy distributions and PDF s in plasma turbulence () DM and plasma density distributions in gravitationlly clustered structures (3) Repulsive DE landscape of negative equation of state (4) Hierarchy of fundamental structure scales

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