Non-extensive statistical theory of Non-extensive statistical theory of dark matter and plasma density distributionsdark matter and plasma density distributions

in clustered structuresin clustered structures

DARK 2007, SYDNEY

M. P. LEUBNERM. P. LEUBNER

Institute for Astro- and Particle PhysicsInstitute for Astro- and Particle PhysicsUniversity of Innsbruck, AustriaUniversity of Innsbruck, Austria

c o r ec o r e – – h a l oh a l o leptokurtic long-leptokurtic long-tailedtailed

PERSISTENT FEATURE PERSISTENT FEATURE OF DOF DIFFERENTIFFERENTASTROPHYSICAL ENVIRONMENTSASTROPHYSICAL ENVIRONMENTS

standard Boltzmann-Gibbs statistics not applicablestandard Boltzmann-Gibbs statistics not applicable

thermo-statisticalthermo-statistical properties of interplanetary mediumproperties of interplanetary medium PDFs ofPDFs of turbulenturbulentt fluctuations of astrophysical plasmasfluctuations of astrophysical plasmas

sself – organized criticality ( SOC ) - Per Bak, 1985 elf – organized criticality ( SOC ) - Per Bak, 1985

PRONOUNCEDNON-GAUSSIANDISTRIBUTIONS

GRAVITATIONAL EQUILIBRIA of CLUSTERED SRUCTURESGRAVITATIONAL EQUILIBRIA of CLUSTERED SRUCTURES

Empirical fitting relations –Empirical fitting relations – DM density profiles DM density profiles

(3 )

1~( / ) (1 / )DM

s sr r r r

2 2

1~(1 / )(1 / )DM

s sr r r r

Burkert, 95 / Salucci, 00non-singular

Navarro, Frenk & White, 96, 97NFW, singular

Fukushige 97, Moore 98, Moore 99…

Zhao, 1996singular

Ricotti, 2003: good fits on all scales: dwarf galaxies clusters

2

1~( / )(1 / )DM

s sr r r r

Empirical fitting relations – Empirical fitting relations – GAS density profilesGAS density profiles

3/ 2~ (1 / )GAS cr r Cavaliere, 1976: single β-model

Generalization

convolution of two β-models double β-model

Aim: resolving β-discrepancy: Bahcall & Lubin, 1994

good representation of hot plasma density distribution

galaxies / clusters

Xu & Wu, 2000, Ota & Mitsuda, 2004

β ~ 2/3 ...kinetic DM energy / thermal gas energy

Dark Matter - Hot GasDark Matter - Hot Gas

DM halo DM halo self gravitating system of weakly interacting

particles in dynamical equilibrium

hot gas electromagnetic interacting high temperature

plasma in thermodynamical equilibrium

any astrophysical system

long-range gravitational / electromagnetic interactions

develop theory…

FROM EXPONENTIAL DEPENDENCEFROM EXPONENTIAL DEPENDENCETO TO POWER - LAW DISTRIBUTIONSPOWER - LAW DISTRIBUTIONS

not applicable accounting for long-range interactionsnot applicable accounting for long-range interactions

THUSTHUS

introduce correlations viaintroduce correlations via “NON-EXTENSIVE STATISTICS” “NON-EXTENSIVE STATISTICS” derivederive corresponding power-law distribution corresponding power-law distribution

iiBB ppkS lnStandard Boltzmann-Gibbs statisticsbased on extensive entropy measure

pi…probability of the ith microstate, S extremized for equiprobability

Assumtion: particles independent from e.o. no correlations

isotropy of velocity directions “extensivity“

Consequence: entropy of subsystems additive Maxwell PD

NON - EXTENSIVE STATISTICS NON - EXTENSIVE STATISTICS

Subsystems A, B:Subsystems A, B: EXTENSIVE EXTENSIVE

non-extensive statistics non-extensive statistics Renyi, 1955; Tsallis,85 Renyi, 1955; Tsallis,85

NON-EXTENSIVE ENTROPY BIFURKATIONNON-EXTENSIVE ENTROPY BIFURKATION

Dual nature + tendency to less organized state, entropy Dual nature + tendency to less organized state, entropy increaseincrease

- - tendency to higher organized state, entropy tendency to higher organized state, entropy decreasedecrease

generalized entropy (kgeneralized entropy (kBB = 1, = 1, - ∞ ≤ - ∞ ≤ κκ ≤≤ + + ∞∞))

1/1/κκ long long – – rangerange interactionsinteractions / / mixing mixing quantifies degree of non-extensivity /couplingsquantifies degree of non-extensivity /couplings accounts for non-localityaccounts for non-locality / correlations / correlations

)1( /11 ipS

)1/(1 q

1( ) ( ) ( ) ( ) ( )q q q q qS A B S A S B S A S B

2

21ch ch

vf B

Bifurcation manifest in

Equilibrium power-law velocity distributions, bifurcation 0

restriction

max thv v

thermal cutoff

HALO CORE

3/ 2h thv

3/ 2c thv

different normalizationand different

generalized higher moments

> 0 < 0

FROM ENTROPY GENERALIZATION TO PDFsNO GRAVITY

Sκ … extremizing entropy under conservation of mass and energy

3/ 2 Leubner, ApJ 2004

Leubner & Vörös, ApJ 2005

STANDARD EQUILIBRIUM OF N-BODY SYSTEM STANDARD EQUILIBRIUM OF N-BODY SYSTEM NO CORRELATIONS but GRAVITYNO CORRELATIONS but GRAVITY

spherical symmetric, self-gravitating, collisionlessspherical symmetric, self-gravitating, collisionless

Equilibrium via Poisson’s equationEquilibrium via Poisson’s equation

f(vf(v22 + + Φ) = f(E) … energy distribution) = f(E) … energy distribution

2 314 ( )

2G f v d v

Available by extremizing BGS entropy, conservation of mass and energy

exponential energy distributionextensive, independent particles

(relative potential Ψ = - Φ + Φ0 , vanishes at systems boundary)

After integrating over all velocities:

202 3/ 2 2

/ 2( ) exp( )

(2 )r

vf E

isothermal, self-gravitating sphere of gas == phase-space density distribution of collisionless system of particles

20 exp( / ) 4 G

GRAVITATIONAL EQUILIBRIUM OF N-BODY GRAVITATIONAL EQUILIBRIUM OF N-BODY SYSTEM; NON-EXTENSIVE CORRELATIONSSYSTEM; NON-EXTENSIVE CORRELATIONS

long-range interactions long-range interactions non-extensive systems

extremize non-extensive entropy,conservation of mass and energyin gravitational potential Ψ: equilibrium distribution

02 3/ 2 3/ 2

( )

(2 ) ( 3 / 2)B

3/ 2

0 2

11

2

2

1 / 2( ) 1r

vf E B

02 3/ 2 3/ 2

( 5 / 2)

(2 ) ( 1)B

02 3/ 2 3/ 2

( )

(2 ) ( 3 / 2)B

integration over v

limit κ = ∞∞ : expo – form of extensive statistics

20 exp( / )

BIFURCATION

> 0 < 0

Ψ = Ψ(r)

NON-EXTENSIVE NON-EXTENSIVE SPATIAL DENSITY VARIATIONSPATIAL DENSITY VARIATION

1/(3/ 2 )

22 2

0

1 41

d d Gr

r dr dr

1/ 3/ 2222

2 20

4 3/ 22 1 11 03/ 2

Gd d d

dr r dr dr

3/ 2

0 2

11

combine

ρ(r) … radial density distribution of spherically symmetric hot plasma ( > 0 ) and dark matter ( < 0 )

κ = = ∞∞ … BGS selfduality, conventional isothermal sphere … BGS selfduality, conventional isothermal sphere

4 G

Leubner, ApJ, 2005, 2006

physics of physics of σσ and and κκ

generally variance σ = σ(r)

(1) DM: σ(r) … velocity dispersion of members of cluster

(2) GAS: σ(r) … thermal speed of plasma v 2th= 2kBT/m

keep radial dependence σ = σ(r) relation κκ, , σ, ρ and κκ, T, T, ρ

ρ(r) … radial density distribution of spherically symmetric hot plasma ( > 0 ) and dark matter ( < 0 ) density distribution with spatially varying variance σ

κ = ∞, = ∞, σ = const … BGS selfdual isothermal sphere … BGS selfdual isothermal sphere solutionsolution

ΚΚ(r)(r) Du, 2007

DUALITY OF EQUILIBRIA AND HEAT CAPACITY DUALITY OF EQUILIBRIA AND HEAT CAPACITY IN NON-EXTENSIVE STATISTICSIN NON-EXTENSIVE STATISTICS

(A) two families ((A) two families (κ’,κ) of STATIONARY STATES (Karlin et al., 2002) of STATIONARY STATES (Karlin et al., 2002)

non-extensive thermodynamic equilibria, non-extensive thermodynamic equilibria, Κ > 0

non-extensive kinetic equilibria, non-extensive kinetic equilibria, Κ’ < 0

related by related by κ’ = - - κ

limiting BGS state for limiting BGS state for κ = ∞ = ∞ self-duality extensivity

(B) two families of HEAT CAPACITY ((B) two families of HEAT CAPACITY (Almeida, 2001)

Κ > 0 … finite positive … thermodynamic systemsΚ < 0 … finite negative … self-gravitating systems

non-extensive bifurcation of the BGS κ = = ∞∞,, self-dual staterequires to identify Κ > 0 … thermodynamic state of gas

Κ < 0 … self-gravitating state of DM

Non-extensive family of density profilesNon-extensive family of density profiles

Non-extensive family of density profiles ρ± (r) , κ = 3 … 10 = 3 … 10

Convergence to the selfdual BGS solution κ = = ∞∞

Non-extensive DM and GAS density profiles -Non-extensive DM and GAS density profiles -comparison with favored empirical modelscomparison with favored empirical models

Non-extensive GAS and DM densityNon-extensive GAS and DM density

profiles, profiles, κ = ± 7 as compared to = ± 7 as compared to

Burkert and NFW DM modelsBurkert and NFW DM models

and single/double and single/double ββ-models-models

Integrated mass of non-on-extensiveextensive

GAS and DM components, GAS and DM components, κ = = ± 7± 7

as compared toas compared to Burkert and NFW DM modelsBurkert and NFW DM modelsand single/double and single/double ββ-models-models

Non-extensive DM and GAS density profiles -Non-extensive DM and GAS density profiles -comparison with DM simulations and comparison with DM simulations and

observationsobservationsDM simulations

KronbergerLeubnervan Kampen

A&A, 2006

hydrodynamic simulations

Mair and Leubner

Integrated mass profile

A1413

Pointecouteau

et al.,

A&A 2005

SUMMARYSUMMARY

Non-extensive entropy generalization generates a bifurcationNon-extensive entropy generalization generates a bifurcationof the isothermal sphere solution into two power-law profilesof the isothermal sphere solution into two power-law profiles

The self-gravitating DM component as lower entropy state resides The self-gravitating DM component as lower entropy state resides beside the thermodynamic gas component of higher entropybeside the thermodynamic gas component of higher entropy

The bifurcation into the kinetic DM and thermodynamic gas branch The bifurcation into the kinetic DM and thermodynamic gas branch isis

controlled by a single parameter accounting for nonlocal controlled by a single parameter accounting for nonlocal correlationscorrelations

It is proposed to favor the family of non-extensive distributions,It is proposed to favor the family of non-extensive distributions,derived from the fundamental context of entropy generalization,derived from the fundamental context of entropy generalization,over empirical approaches when fitting observed density profilesover empirical approaches when fitting observed density profiles

of astrophysical structuresof astrophysical structures

Hot Plasma Simulation, M. Mair (2005)

Dark Matter Simulation, E. van Kampen T. Kronberger (2005)

Theory: M. P. Leubner, ApJL 632, L1, 2005

Comparison with simulationsComparison with simulations

DM popular phenomenological: Burkert, NFWDM popular phenomenological: Burkert, NFW GAS popular phenomenological: single / double GAS popular phenomenological: single / double ββ-models-modelsSolid: simulation (Solid: simulation (11, , 22 ... relaxation times), dashed: non- ... relaxation times), dashed: non-

extensiveextensive

dark matter (N – body) gas (hydro)

Kronberger, T. & van Kampen, E. Mair, M. & Domainko, W.