NON-EXTENSIVE THEORY OFNON-EXTENSIVE THEORY OFDARK MATTER AND GAS DARK MATTER AND GAS

DENSITY DISTRIBUTIONS IN DENSITY DISTRIBUTIONS IN GALAXIES AND CLUSTERSGALAXIES AND CLUSTERS

M. P. LEUBNERM. P. LEUBNER

Institute for AstrophysicsInstitute for Astrophysics

University of Innsbruck, AustriaUniversity of Innsbruck, Austria

COSMO-05, BONN 2005

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

NON-GAUSSIANDISTRIBUTIONS

stellar stellar gravitationalgravitational equilibrium equilibrium

Empirical fitting relations - DMEmpirical fitting relations - DM

(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 - GASEmpirical fitting relations - GAS

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 - PlasmaDark Matter - Plasma

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

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

Hypothesis: isotropy of velocity directions extensivity

Consequence: entropy of subsystems additive Maxwell PDF

microscopic interactions short ranged, Euclidean space time

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

PSEUDOADDITIVE NON-EXTENSIVE ENTROPY BIFURKATIONPSEUDOADDITIVE NON-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, - ))

1/1/ 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

thh v

NB

2/1

thc v

NB

2/1

)2/1(

)(2/1

)1(

)2/3(2/1

normalization

power-law distributions, bifurcation 0

restriction

max thv v

thermal cutoff

HALO CORE

3/ 2h thv

3/ 2c thv

different

generalized 2nd moments

> 0 < 0

FROM ENTROPY GENERALIZATION TO PDFs

S … extremizing entropy under conservation of mass and energy

3/ 2 Leubner, ApJ 2004

Leubner & Vörös, ApJ 2005

EQUILIBRIUM OF N-BODY SYSTEMEQUILIBRIUM OF N-BODY SYSTEM NO CORRELATIONS NO CORRELATIONS

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

Equilibrium via Poisson’s equationEquilibrium via Poisson’s equation

f(r,v) = f(E) … mass distributionf(r,v) = f(E) … mass distribution

2 314 ( )

2G f v d v

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

Er = -v2/2 + Ψ and ΔΨ = - 4π G ρ

(2) exponential mass distribution extensive, independent

f(Er)… extremizing BGS entropy, conservation of mass and energy

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

EQUILIBRIUM OF N-BODY SYSTEM EQUILIBRIUM OF N-BODY SYSTEM CORRELATIONSCORRELATIONS

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

extremize non-extensive entropy,conservation of mass and energy corresponding distribution

02 3/ 2 3/ 2

( )

(2 ) ( 3 / 2)B

negative κ again energy cutoff v2/2 ≤ κ σ2 – Ψ, integration limit

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 = ∞∞ 20 exp( / )

bifurcation

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 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 and dark matter

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

4 G

Leubner, ApJ, 2005

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 profilesNon-extensive DM and GAS density profiles

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-extensiveon-extensive 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

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.

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