Transcript 長距離相互作用系の準安定状態(PPTファイル)

1
長距離相互作用系の
準安定状態
京都大学大学院人間・環境学研究科
阪上雅昭
Collaboration with
熱海合宿08/04/01
樽家篤史 (RESCEU, 東大)
岡村 隆 （関学）
立川崇之 （工学院大学）
D.C. Heggie (Univ. of Edinburgh)
Self-gravitating Stellar System
A System with N Particles (stars) (N>>1)
Particles interact with Newtonian gravity each other
2
H 

i
pi
2mi

i

ji
Gm i m j
| xi  x j |
Typical example of a system
under long-range force
Key word
Negative Specific Heat
Long-term (thermodynamic) instability ( t  trelax )
Gravothermal instability
Antonov 1962
Lynden-Bell & Wood 1968
2
Two typical examples of self-gravitating stellar systems
Globular cluster
N  10
6
collisional system (discussed in this lecture)
relaxation time << age of universe
Elliptical galaxy
collisionless system
N  10
11
(NOT discussed )
relaxation time >> age of universe
3
Time scales of Self-gravitating System
free-fall time
T free 
4
Motion driven
by mean
gravitational
potential
1
G
Dynamical time scale

Mean mass density
two-body relaxation time
T relax  0 . 1
N
ln N
T free
time scale for approaching
to thermal equilibrium
Time scale for
loss of memory
by two-body
collision
relaxation
Intuitive explanation of Negative Specific Heat
Circular Motion under Gravity
Eq. of Motion
m
K 
mathematica
Virial Theorem
Total Energy
v
2
R
Kinetic Energy
m
v

GmM
R
2
Grav. Potential
2
V  
GmM
R
2
2 K  V
E  K V  K
E  0
Energy decreasing
Negative
specific Heat
K  0
“Temperature” increasing
5
6
Antonov problem
Presence of secular instability and/or equilibrium properties of
stellar system had been considered in a very idealistic situation.
(Antonov 1962)
Adiabatic wall
(perfectly reflecting boundary)
Self-gravitating
N-body system
radius：re
mass：M=N×m
energy：E
re
Isothermal state & its stability
7
Boltzmann3
3
S


d
x
d
v f ( x, v ) ln f ( x, v )
BG
Gibbs entropy

Extremization
0   SBG    M    E
Isothermal distribution
l – reE/GM 2
f ( x , v )  exp(    ) ;
Large
l crit = 0.335
re
  v / 2   (x )
2
D  c / 
No stable isothermale state
 exists at
c
Large re :
re  lcrit GM
 /(  E )
2
e
High density-contrast :
D  Dcrit
D crit = 709
D= c/ e
Gravothermal instability
Gravothermal Catastrophe
CV  0
Heat flow
core = self-gravitating
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Negative specific heat
CV  0
core halo
halo = normal system
CV  0
Tcore > Thalo
Heat flow from
core to halo
Negative specific heat
CV  0
Tcore↑
CV  0
Thalo↑
sufficiently large wall
re > rc
Tcore > Thalo
heat flow does not stop!!
Core-collapse !!
extended halo has large heat capacity
Consequence of instability
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The system does not always approach the thermal equilibrium
(isothermal distribution)
t dyn
Initial
conditions
t relax
Dynamical equilibrium
(virialized)
Thermal
equilibrium
Core collapse
!!
(isothermal state)
To clarify the fate of the system,
Description of
non-equilibrium states
is essential.
transient states
Rest of this talk
An attempt to characterize the evolutionary states of N-body system
both from thermostatistical and dynamical point-of-view
Theoretical investigation
10
Particularly focusing on the setup of Antonov problem,
Thermostatistical approach
analytical
generalized thermostatistical formalism
Physica A 322 (2003) 285
Dynamical approach
numerical
Ｎ体シミュレーション
Phys.Rev.Lett. 90 (2003) 181101; MNRAS (2005)
Kinetic-theory approach
semi-analytical
Fokker-Planck eq. + 一般化された変分原理
A.Tatuya, Okamura & Sakagami (2007), in preparation
A naïve generalization of BG statistics
11
As a possible generalization of thermostatistical treatment,
q-entropy
3
3
Sq   1  d x d v
q 1
  p ( x , v )
q
 p(x, v)

BG limit q→1
Tsallis, J.Stat.Phys.52 (1988) 479
One-particle distribution function
identified with escort distribution
q
f ( x , v)  M
Power-law distribution
{ p ( x , v)}
 d x d v { p( x, v)}
3
3
q
p  p p
f ( x , v )  A  0   
q /( 1  q )
  v / 2  ( x )
2
Stellar polytrope as quasi-equilibrium state
12
This power-law type distribution is well-known in subject
of stellar dynamics, referred to as
“stellar polytrope”
(e.g., Binney & Tremaine 1987)
Polytropic equation of state
P( r )  K n 
11 / n
1
1
Polytrope
n

index
1 q 2
(r)
n→∞
BG limit
Stellar polytrope as quasi-equilibrium state
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n=6
Energy-density contrast
relation for stellar polytrope
stable
unstable
oscillatory behavior appears
when n>5.
For larger densiy D>Dcrit,
unstable state appears at n>5
(gravothermal instability)
Stellar dynamical simulations
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Perfectly reflecting
N-body simulation
Long-term evolution of N-body system
confined in an adiabatic wall
Adiabatic
wall
※ we use GRAPE-6 @ NAOJ
E=M=const.
re
Unit: G=M=re=1
Initial conditions
( n=3, 5, 6, 

Group A
Stellar polytropes

Group B
A family of stellar model with cusped profile
(non-polytropic models)
1
 (r ) 
r
3 
(r  a )
1 
Tremaine et al. AJ 107 (1994) 634
Overview of the N-body results
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Stellar polytropes are not stable in timescale of two-body relaxation.
However, focusing on their transients, we found :
Quasi-equilibrium property
Transient states approximately follow a sequence of stellar
polytropes with gradually changing polytrope index “n”.
Quasi-attractive behavior
Even starting from non-polytropic states, system soon
settles into a sequence of stellar polytropes.
Survey results of group (A)
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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.
Run n3A (1)
Density profile
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Stellar polytrope
(n=3,D=10 4)
One-particle distribution function
1
Fitting to stellar polytropes is quite good until t ~ 30 trh,i.
2
v   ( x)
2
18
Run n3A (2)
unstable
stable
Time evolution of polytrope
index “n” fitted to N-body
simulations
Polytrope index monotonically
grows on relaxation time-scale
t rh, i
 rh

 0 . 138
ln( 0 . 1 N )  GM
N
3




1/ 2
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Survey results of group (B)
Stellar model:
Case (A)
1
 (r ) 
r
3 
(r  a )
1 
model parameters → , a
Case (B)
Fitting failed
Fitting to polytrope is good
Region where stable isothermal
(BG) state can exist.
Quasi-attractive behaviors appear when 1< h & ~
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Cases with a/re=0.5 (1)
η=1
Distribution function
Rapid Core Collapse (×)
η=1.5
Approaching to
Isothermal（△）
η=3
Approaching to polytrope (○)
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Cases with a/re=0.5 (2)
η=1
Density profile
(×) small core → rapid collapse
Fitting failed
η=1.5
almost Isothermal
Larger core approaches to Polytrope
Fitting is GOOD
η=3
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Kinetic-theory approach
For a better understanding of the quasi-equilibrium states,
Fokker-Planck (F-P) model for stellar dynamics
orbit-averaged F-P eq.
  f ( ) 
   ( ) 






 t

  t
;   16  G m ln 
2
2
2
  ln f ( )  ln f ( ) 
 ( )   d  f ( ) f ( ) min[  ( ),  ( )] 









2
16 
3/2
2


 ( ) 
dr
r
2
[



(
r
)]
phase space volume

3
Complicated, but helpful for semi-analytic understanding
F-P eq.に対する一般化された変分原理
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Glansdorff & Prigogine (1971)
Local potential
Inagaki & Lynden-Bell (1990)
 f 0 

 ln f  
  ln f
( f , f 0 )   d 
ln
f




d

d

f
f
min(

,

)



0 0
0
0 

4
  
 t 
 
Variation
w.r.t. f

 f
( f , f0 )
0
F-P equation for f 0
f  f0
Absolute minimum at a solution
f0
  ( f , f 0 )  ( f 0 , f 0 )  0
Application:
Takahashi & Inagaki (1992); Takahashi (1993ab)
2
熱伝導 の場合
 T0
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 T0
2

t
x
2
Local potential
 ( T , T0 ) 
 dx L ( T , T
variation w.r.t. T
  ( T , T0 )
T
1
0
L ( T , T0 ) 
)
1

2  T
T 0 
2
 x
2

 T0
 
T
t

1
1
0
 T 02

   2
t
T
 T0
subsidiary condition
 T0
T  T0
t
2
  2T
 T   T0 

 2   0

2
x x  T 
 x
 T0
2

x
2
absolute minimum
   ( T , T 0 )   ( T 0 , T 0 ) 
2
2   
dx

T
 0
0 

2
 x 
1
 T
1
1
 T0
Local Potential の導出
e
エネルギー保存
t
熱伝導
 
q  l
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q
e ：内部エネルギー密度
e  e0   e
x
T
 lT
x
2
T
1
 e
t
x
 
q
x
q

：熱流
 e0
t
過剰エントロピー生成
1 
 S    dx  T
2 t
2
1
 e
t

 dx  T
l T  l T 0   l T
2
1
l 2  T
  dx T 0  
2
 x
O  T 

O  T 
2
2


1
2
 dx  l T
2
2
1
  q  e0 



t 
 x

 dx
l
1
2  T
T  
2
 x
e0  cV T0
2

e
  0  T
t

  l cV
2

e
  0  T
t

    T
1


 x 
2
1
(T , T0 )  0
1
Stellar Dynamics への適用
K.Takahashi, PASJ 45,233 (1993)
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pre-collapse
post-collapse
C

C
t / t rhi
t / t rhi
C
t / t rhi
r
一般化された変分原理による “n”の発展方程式
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Assuming stellar polytropes with time-varying polytrope index as
quasi-equilibrium state,
trial function
n ( t ) 3 / 2
f ( )  A(t )[ 0 (t )   ]

n
( f , f 0 )
0
( A and  0 are the functions of n)
n, E, M の関数
f  f0

 l    
  ln f
 l     




d


(

)




 







 l 
  e  n  n  e  n  e   e  n 



n (t )  
2




 e n
 l    ln f 

 l    ln f  




d

f
(

)


 


  n 
 



n



 e 
 e 
e n 
 e  n 

Semi-analytic prediction: evolution of “n”
Time evolution of
polytrope index “n” fitted
to N-body simulations
Time-scale of quasi-equilibrium states is successfully reproduced
from semi-analytic approach based on variational method.
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29
Numerical n-dn/dt curves: linear
plot
Units : G  M  re  1 and   1
ハイブリッド法による数値積分
(基研でのバッチジョブ)
Note-.
l=0.1
For n>5 and l<0.335, dn/dt
curves approach a linear curve.
0.15
0.2
0.25
0.3
0.35
0.45
l=1.2
0.5
0.4
30
Numerical n-dn/dt curves: log-log plot
Units : G  M  re  1 and   1
l=0.1
Note-.
0.15
For l>0.335, dn/dt eventually
becomes vanishing at ncrit.
0.2
0.25
0.3
l>0.335
0.35
0.4
l=1.2
0.5
0.45
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n-dn/dt curves ： approximation
ncrit is function of l and
is determined by solving
For the curves with l0.335,
dn
dt
  c*
n  n crit
n crit  5
dl / d e  0
; c *  0 . 16
n ( t )  n crit  { n crit  n ( 0 )} e
 t / *
;  *  ( n crit  5 ) / c *
(n>5)
For the curves with l<0.335,
dn
dt
3
 A1 ( l )  A 2 ( l ) ( n  5 ) ; Ai ( l ) 

a ik ( l  0 . 335 )
k
(i=1,2)
k 1
n ( t )  5  A1 / A2  { n ( 0 )  5  A1 / A2 } e
A2 t
(n>5)
( units: G  M  re  1 and   1 )
32
Analytic estimate of n(t): comparison
Analytic estimate based on
variational method successfully
reproduces the N-body results.
Half-mass
relaxation time
t rh, i
3
N  rh

 0 . 138
ln   GM




1/ 2
33
Summary
（１） 重力多体系
（２） 準定常状態
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
ポリトロープ状態の系列で記述できる
P( r )  K n 
11 / n
(r)
n
1
1 q

1
2
(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 Progress
(1) ポリトロープ
準定常状態： 他の例はあるか
２次元ＨＭＦモデルの解析
(2) ポリトロープ
準定常状態： 長距離相互作用が本質？
Yukawa 型相互作用での解析
(3) ポリトロープによる準定常状態の記述の限界
ポリトロープは core collapse 前しか適用できない？
34
Polytrope による準定常状態の記述の限界
Fokker-Planck eq. によるCore-Collapse の解析
35
self-similar evolution
CV  0
Heat flow
core halo
CV  0
self-similar core collapse が
始まると polytrope で fit できない
H.Cohn Ap.J 242 p.765 (1980)
Self-similar sol. of F-P eq.

36
Heggie and Stevenson,
MN 230 p.223 (1988)

power law envelope
ln 
 r
 2 . 21
n  9 .7
ln r
isothermal core
fitting of self-similar sol. with double polytrope
f ( )  A
37
n
n 


 0    1  c  0    2


Black dots: Numerical Self-Similar sol.
by Heggie and Stevenson
f ( )
Magenta lines: fitting by double polytrope
n 1  9 . 538

n 2  16 . 73
c  3 . 59

r
Summary & Discussions
Stellar Polytrope は，球状星団（collisional self-gravitating
N-body system） の状態を記述するのに適している
collapse 前 single polytrope, index n が大きくなる
collapse 後
central core と envelope
２つの stellar polytrope の重ね合わせで表せる
n1
n2 

f (  )  A  0     c  0   


n 1 , n 2 , c ...
の時間発展はFokker-Planck 方程式に
に対する一般化された変分原理から導出
できるはず．．．
38
39
2次元ＨＭＦモデル
Antoni&Torcini PRE 57(1998) R6233
Antoni, Ruffo&Torcini, PRE 66(2003) 025103R
H 
1
N
(p

2
2
x ,i
 p y ,i )
2
i 1

1
2N
N
  3  cos( x
i
 x j )  cos( y i  y j )  cos( x i  x j ) cos( y i  y j ) 
i, j
平均場と相互作用： 長距離相互作用系
１次元ＨＭＦモデルの準定常状態は精力的に研究されている
２次元に拡張することで，２体散乱によるエネルギー輸送を
含める
U  1 . 95
Magnetization
T-U 曲線 熱平衡 平均場
T
U
N  10 , N  9  10
4
熱平衡
準定常状態
polytrope ?
t/N
Vlasov phase ?
分布関数
分布関数
vx
4
vx
40
U  1 .8
N  9  10
q3
t  2 . 5  10
5
q  1 .2
Magnetization
t  5  10
4
3
t
t  2 . 5  10
q  1 .5
4
41