1 Non-equilibrium evolution of Long-range Interacting Systems and Polytrope Kyoto University...
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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.)
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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)
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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
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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)
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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
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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
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n=6 Energy-density contrastrelation for stellar polytrope
stable unstable
unstable state appears at n>5(gravothermal instability)
Stellar polytrope as quasi-equilibrium state
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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.
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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.
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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.
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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
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15
Antoni and Ruffo,PRE 52(1995)2361
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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
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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
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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
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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
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Time evolution of M and Tphys (1) 20
N=10000, 10 samples of simulations
Initial cond. : Water Bag
69.0/ NEU
Spatially homogeneous
0M
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Time evolution of M and Tphys (2) 21
N=10000, 10 samples of simulations
Initial cond. : Water Bag
69.0/ NEU
Spatially inhomogeneous
0M
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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.
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23N=1000, single sample
62.0/ NEULong term behavior
early stage behavior
Theoretical prediction for thermal equilibrium. n
Tphys – U relation at equilibrium
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62.0/ NEULong term behavior
early stage behavior
Tphys at early stage of qss
N=1000, single sample Tphys – U relation at qss
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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
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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
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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
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69.0/ NEU
600800t
69.2n
Evolutionary track on the polytrope sequence
69.2n
N=10000, 10 samples of simulations
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69.0/ NEU95.3n
700300t
Evolutionary track on the polytrope sequenceN=10000, 10 samples of simulations
95.3n
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69.0/ NEU
800800t
46.6n
Evolutionary track on the polytrope sequenceN=10000, 10 samples of simulations
46.6n
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1000800t
9.13n69.0/ NEU
Evolutionary track on the polytrope sequenceN=10000, 10 samples of simulations
9.13n
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1800800t
6.58n69.0/ NEU
Evolutionary track on the polytrope sequenceN=10000, 10 samples of simulations
6.58n
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2500100t
4.66n69.0/ NEU
Evolutionary track on the polytrope sequenceN=10000, 10 samples of simulations
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69.0/ NEU
3000700t
597n
Evolutionary track on the polytrope sequenceN=10000, 10 samples of simulations
597n
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35
Failure of single polytrope description due to Thermodynamical instability (negative specific heat).
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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
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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.
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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
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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
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40
5000t
A description by double polytropes might work.
5000t
double polytrope single polytrope
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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
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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
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43How to determine polytropic index n.
)1()32()2()2(4
12 2222
22
MTMnTnn
nphysphys
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44
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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
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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
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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
![Page 46: 1 Non-equilibrium evolution of Long-range Interacting Systems and Polytrope Kyoto University Masa-aki SAKAGAMI Collaboration with T.Kaneyama ( Kyoto U.](https://reader034.fdocuments.net/reader034/viewer/2022051401/5697bfa71a28abf838c99054/html5/thumbnails/46.jpg)
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
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50
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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.
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Initial : polytrope U=1.9 52
t (logarithmic)
Mag
neti
zati
on
dist. func.
p
Negative specific heat
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Initial: polytrope U=1.7 53
t (logarithmic)
Mag
neti
zati
on
dist. func.
p
positive specific heat
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Initial WB U=1.9 54
t (logarithmic)
Mag
neti
zati
on
dist. func.
p
Negative specific heat
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Initial WB U=1.7 55
t (logarithmic)
Mag
neti
zati
on
dist. func.
p
positive specific heat
![Page 53: 1 Non-equilibrium evolution of Long-range Interacting Systems and Polytrope Kyoto University Masa-aki SAKAGAMI Collaboration with T.Kaneyama ( Kyoto U.](https://reader034.fdocuments.net/reader034/viewer/2022051401/5697bfa71a28abf838c99054/html5/thumbnails/53.jpg)
56
How to derive the evolution equation for polytropic index, q or n.
self-gravitating systems
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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
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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
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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 ,,
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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.
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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. ?
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62
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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
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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?