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.