Post on 17-Jan-2016
1
Non-equilibrium evolution of Long-range Interacting Systems and Polytrope
Kyoto University
Masa-aki SAKAGAMI
Collaboration with T.Kaneyama （ Kyoto U. ）A.Taruya (RESCEU, Tokyo U.)
2Outlines of this talk
(1) Self-gravitating N-body system
(2) 1-dim. Hamiltonian Mean-Field model (HMF model)
A short review of our previous work.
Extremal of generalized entropy by Tsallis
)1/(0 )()v,x( qqAf polytrope generalization of B-G dist.
It well describes non-equilibrium evolution of the system.
Two systems with long-range interaction
)(221 xv
Taruya and Sakagami, PRL90(2003)181101, Physica A322(2003)285
We show another example where non-equilibrium evolution ofthe system moves along a sequence of polytrope.
Thermodynamical instability implies a superposition of polytrope.(negative specific heat)
3
Self-gravitating N-body System
Typical example of a system under long-range force
i ij ji
ji
i i
i
xx
mGm
m
pH
||2
2
A System with N Particles (stars) (N>>1)
Particles interact with Newtonian gravity each other
Key word Negative Specific Heat
Gravothermal instabilityGravothermal instability Antonov 1962Lynden-Bell & Wood 1968
Long-term (thermodynamic) instability )( relaxtt
6
D= c e
–
reE
/GM
2 Largere c
e
e/D c
Maxwell-Boltzmann distribution as
Equilibrium
Maxwell-Boltzmann distributionMaxwell-Boltzmann distribution
;)exp()v,( xf )x(2/v2
crit
critD
= 0.335
= 709
)vx(ln)vx(vx 33BG ,f,fddS Boltzmann-Boltzmann-
Gibbs entropyGibbs entropy
ExtremizationEMS BG0
Adiabatic wall(perfectly reflecting boundary)
7A naïve generalization of BG statistics
Tsallis, J.Stat.Phys.52 (1988) 479
Power-law distribution
)1/(0 )()v,x( qqAf )x(2/v2
ppp
Thermostatistical treatment by generalized entropy
q-entropy )v,x()v,x(vx1
1 33q ppdd
qS q
BG limit q→1
q
q
pdd
pMf
)}v,x({vx
)}v,x({)v,x(
33
One-particle distribution function identified with escort distribution
EMS q0
9
n=6
Energy-density contrast relation for stellar polytrope
Polytrope index
n→∞BG limit
)1/(0 )()v,x( qqAf
Polytropic equation of state
(e.g., Binney & Tremaine 1987)
2
1
1
1
qn
)()( /11 rKrP nn
10
n=6 Energy-density contrastrelation for stellar polytrope
stable unstable
unstable state appears at n>5(gravothermal instability)
Stellar polytrope as quasi-equilibrium state
11Survey results of group (A)
The evolutionary track keeps the direction increasing the polytrope index “n”.
Once exceeding the critical value “Dcrit“, central density rapidly increases toward the core collapse.
12Run n3A : N-body simulation
Initial cond:Stellar polytrope
((n=3,D=10 ) )44
Density profileDensity profile One-particle distribution functionOne-particle distribution function
)(v2
1 2 x
Fitting to stellar polytropes is quite good until t ~ 30 trh,i.
13Overview of the results in sel-gravitating system
However, focusing on their transients, we found :
Stellar polytropes are not stable in timescale of two-body relaxation.
Quasi-equilibrium propertyQuasi-equilibrium property
Transient states approximately follow a sequence of stellar polytropes with gradually changing polytrope index “n”.
Quasi-attractive behaviorQuasi-attractive behavior
Even starting from non-polytropic states, system soon settles into a sequence of stellar polytropes.
14
Application of generalized entropy and polytrope to another long-range interacting system,
1-dimensional Hamiltonian mean-field model 1D-HMF model
Antoni and Ruffo,PRE 52(1995)2361
15
Antoni and Ruffo,PRE 52(1995)2361
16A naïve generalization of BG statistics
Tsallis, J.Stat.Phys.52 (1988) 479
Power-law distribution(polytrope)
)1/(0 )(),( qqApf )(2/2 p
hhh
Thermostatistical treatment by generalized entropy
q-entropy ),(),(1
1q phphdpd
qS q
BG limit q→1
q
q
phpdd
phNpf
)},(
)},({),(
One-particle distribution function identified with escort distribution
Physical quantities by 1-particle distribution 17
),( pfdpdN
),()(221 pfpdpdE
),(sin,cos1
),( pfdpdN
MM yx M
)','())'cos(1''1
)( pfdpdN
)cos(1)( M
number
energy
magnetization
potential
Polytropic distribution function 18
2/10),( nApf
2
1
1
q
qn cos1)( M
20 1)1( MTn phys
n
n
Md
MdM
)cos1(
)cos1(cos
0
0
),(/2 21 pfpdpdNKT Nphys
221 1/ MTNEU phys
polytrope index
)(221 p
BG limit q→1 n →∞
For given U and n, this eq. self-consistently determines magnitization M.
Chavanis and Campa, Eur.Phys.J. B76(2010)581Taruya and Sakagami, unpublished
As for defenition of Tphys,Abe, Phys.Lett. A281(2001)126
19Thermal equilibrium, ＢＧ Ｌｉｍｉｔ n→∞
generalized to seqences of polytrope, describing non-eqilibrium (?)
n=1000n=10n=4n=2n=0.5
n=1000n=10n=4n=2n=0.5
Antoni and Ruffo,PRE 52(1995)2361
homogeneous state
inhomogeneousstate 0M
Time evolution of M and Tphys (1) 20
N=10000, 10 samples of simulations
Initial cond. : Water Bag
69.0/ NEU
Spatially homogeneous
0M
Time evolution of M and Tphys (2) 21
N=10000, 10 samples of simulations
Initial cond. : Water Bag
69.0/ NEU
Spatially inhomogeneous
0M
Three stages of evolution of M and Tphys 22
N=10000, 10 samples of simulations69.0/ NEU
0Mquasi-stationary state (qss)
transient state
nearly equilibrium
The evolution over three stages are totally well described by sequences of polytropes.
23N=1000, single sample
62.0/ NEULong term behavior
early stage behavior
Theoretical prediction for thermal equilibrium. n
Tphys – U relation at equilibrium
24
62.0/ NEULong term behavior
early stage behavior
Tphys at early stage of qss
N=1000, single sample Tphys – U relation at qss
25
62.0/ NEULong term behavior
early stage behavior
N=1000, single sample Tphys – U relation at qss
Tphys at early stage of qssare explained by polytropewith n=0.5.
polytropen=0.5
26
69.0/ NEU
60000t
90.0n
Evolutionary track on the polytrope sequence
90.0n
N=10000, 10 samples of simulations69.0/ NEU
69.0U
f by simulation
prediction by polytrope
27
69.0/ NEU
300300t
58.1n
Evolutionary track on the polytrope sequence
58.1n
Even in qss, polytrope index n and distribution f change.
N=10000, 10 samples of simulations
28
69.0/ NEU
600800t
69.2n
Evolutionary track on the polytrope sequence
69.2n
N=10000, 10 samples of simulations
29
69.0/ NEU95.3n
700300t
Evolutionary track on the polytrope sequenceN=10000, 10 samples of simulations
95.3n
30
69.0/ NEU
800800t
46.6n
Evolutionary track on the polytrope sequenceN=10000, 10 samples of simulations
46.6n
31
1000800t
9.13n69.0/ NEU
Evolutionary track on the polytrope sequenceN=10000, 10 samples of simulations
9.13n
32
1800800t
6.58n69.0/ NEU
Evolutionary track on the polytrope sequenceN=10000, 10 samples of simulations
6.58n
33
2500100t
4.66n69.0/ NEU
Evolutionary track on the polytrope sequenceN=10000, 10 samples of simulations
34
69.0/ NEU
3000700t
597n
Evolutionary track on the polytrope sequenceN=10000, 10 samples of simulations
597n
35
Failure of single polytrope description due to Thermodynamical instability (negative specific heat).
36
n=6 Energy-density contrastrelation for stellar polytrope
stable unstable
unstable state appears at n>5(gravothermal instability)
Stellar polytrope as quasi-equilibrium state
37
H.Cohn Ap.J 242 p.765 (1980)
Heat flowHeat flow
corecore halohalo
0CV
0CV
Sel-similar core-collapse in Fokker-Planck eq.
self-gravitating system
When self-similar core collapse takes place,polytrope could not fit distribution function.
Halo could not catch up with core collapse.
38fitting of self-similar sol. with double polytrope
)(f
r
2
01
0)(n
cn
Af
Black dots: Numerical Self-Similar sol. by Heggie and Stevenson
Magenta lines: fitting by double polytrope
59.3
73.16
538.9
2
1
c
n
n
39Thermodynamical Instability due to negative specific heat
5000t
After a short time, single polytrope fails to describe the simulated distribution.
We prepare the initial state as Polytrope with U=0.6, n=1.
1n
40
5000t
A description by double polytropes might work.
5000t
double polytrope single polytrope
41
5000t
A description by double polytropes might work.
double polytrope parameters
62.0,2.1,95.0 121 nn
coexistence conditions (preliminary)
211
21 UU
)1( 2112
112
11 MTU
221
221
2 TU
2211 TTT
Summary and Discussion42
(1) Polytrope (Extremal of Generalized Entropy)
Self-gravitating system, 1D-HMF
(2) Break down of single polytrope description
(due to negative specific heat)
(Long-range interaction)
implies superposition of polytropes.
describes evolution along quasi-stationary and transient states to thermal equilibrium
43How to determine polytropic index n.
)1()32()2()2(4
12 2222
22
MTMnTnn
nphysphys
44
45
46Polytrope による準定常状態の記述の限界
H.Cohn Ap.J 242 p.765 (1980)
Heat flowHeat flow
corecore halohalo
0CV
0CV
Fokker-Planck eq. による Core-Collapse の解析 self-similar evolution
self-similar core collapse が始まると polytrope で fit できない
47
Self-similar sol. of F-P eq.Heggie and Stevenson, MN 230 p.223 (1988)
ln
rln
isothermal core
power law envelope
21.2 r
7.9n
48fitting of self-similar sol. with double polytrope
)(f
r
2
01
0)(n
cn
Af
Black dots: Numerical Self-Similar sol. by Heggie and Stevenson
Magenta lines: fitting by double polytrope
59.3
73.16
538.9
2
1
c
n
n
2D ＨＭＦモデル49
N
jijijijiji
iy
N
iix
yyxxyyxxN
ppH
,
2,
1
2,
)cos()cos()cos()cos(32
1
)(2
1
Antoni&Torcini PRE 57(1998) R6233Antoni, Ruffo&Torcini, PRE 66(2003) 025103R
2D HMF have the effect of energy transfer due to 2-body scattering process.
Interaction by Mean-field ： Long-range interacting system
Negative specific heat in some range of energy
50
51
T
U (energy)
95.1U
Mag
neti
zati
on
t / N
T-U curve Boltzmann case
Transient stpolytrope ?
Thermal equilibrium
44 109,10 NN
Vlasov phase ?
xv xv
dist. func.
dist. func.
Initial : polytrope U=1.9 52
t (logarithmic)
Mag
neti
zati
on
dist. func.
p
Negative specific heat
Initial: polytrope U=1.7 53
t (logarithmic)
Mag
neti
zati
on
dist. func.
p
positive specific heat
Initial WB U=1.9 54
t (logarithmic)
Mag
neti
zati
on
dist. func.
p
Negative specific heat
Initial WB U=1.7 55
t (logarithmic)
Mag
neti
zati
on
dist. func.
p
positive specific heat
56
How to derive the evolution equation for polytropic index, q or n.
self-gravitating systems
57Kinetic-theory approach
Fokker-Planck (F-P) model for stellar dynamicsFokker-Planck (F-P) model for stellar dynamics
orbit-averaged F-P eq.
tt
f
)()( ln16; 222 mG
2/322
)]([23
16)( rrdr
)(ln)(ln)](),(min[)()()(
ffffd
For a better understanding of the quasi-equilibrium states,
Complicated, but helpful for semi-analytic understanding
phase space volume
58Generalized Variational Principle for F-P eq.
Local potential
ft
fdff ln),( 0
0
2
0000
lnln),min(
4
ff
ffdd
Variation w.r.t. f F-P equation for0),(
0
0 ff
fff
Absolute minimum at a solution 0f
0),(),( 000 ffff
Application: Takahashi & Inagaki (1992); Takahashi (1993ab)
Glansdorff & Prigogine (1971)
Inagaki & Lynden-Bell (1990)
0f
0f fixed
59The evolution eq. for “n” from generalized variational principle
Assuming stellar polytropes with time-varying polytrope index as transient state,
2/3)(0 ])()[()( tnttAf
0),(0
0 ff
ffn
)offunctionstheareand( 0 nA
2
lnln)(
ln)(
)(
nene
nene
ne f
nn
ffd
f
nnd
tn
ee
ee
trial function
function of MEn ,,
60
Time evolution of polytrope index “n” fitted to N-body simulations
Semi-analytic prediction: evolution of “n”
Time-scale of quasi-equilibrium states is successfully reproduced from semi-analytic approach based on variational method.
Summary and Discussion61
(1) Polytrope (Extremum of Generalized Entropy) Transient states to thermal equilibrium
Self-gravitating system, 2D-HMF Negative specific heat
(2) Generalized variational princple for F-P eq.
Evolution eq. for polytropic index
Short-range attracting interaction
negative specific heat
Long-range interaction
Works in Progress:
Polytrope ? Superposition of Boltzmann dist. ?
62
Summary63
（１） 重力多体系長距離力（引力）比熱が負small system
非平衡進化準定常状態が存在
（２） 準定常状態 ポリトロープ状態の系列で記述できる
)()( /11 rKrP nn
2
1
1
1
qn
(3) ポリトロープ指数 n の時間発展
一般化された変分原理Fokker-Planck eq. ポリトロープ状態 : Trial func
指数 n の時間発展方程式が導出できる
Work in Progress64
(1) ポリトロープ 準定常状態： 他の例はあるか
(2) ポリトロープ 準定常状態： 長距離相互作用が本質？
(3) ポリトロープによる準定常状態の記述の限界
２次元ＨＭＦモデルの解析
Yukawa 型相互作用での解析
ポリトロープは core collapse 前しか適用できない？