1. 京都大学大学院人間・環境学研究科 阪上雅昭
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
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)
Collaboration with
熱海合宿08/04/01

2. Self-gravitating Stellar System
A System with N Particles (stars) (N>>1)
Particles interact with Newtonian gravity each other
Typical example of a system under long-range force
確認のため、一応、自己重力多体系の定義を述べておく
Key word
Negative Specific Heat
Long-term (thermodynamic) instability
Antonov 1962
Lynden-Bell & Wood 1968
Gravothermal instability

3. Two typical examples of self-gravitating stellar systems
Globular cluster
collisional system
(discussed in this lecture)
relaxation time << age of universe
Elliptical galaxy
collisionless system
(NOT discussed )
relaxation time >> age of universe

4. Time scales of Self-gravitating System
free-fall time
Motion driven by mean gravitational potential
Dynamical time scale
Mean mass density
two-body relaxation time
Time scale for loss of memory by two-body collision
time scale for approaching to thermal equilibrium
relaxation

5. Intuitive explanation of Negative Specific Heat
Circular Motion under Gravity
Eq. of Motion
Kinetic Energy
Grav. Potential
mathematica
Virial Theorem
Total Energy
Negative
specific Heat
Energy decreasing
“Temperature” increasing

6. Self-gravitating N-body system
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 re
Self-gravitating N-body system
mass：M=N×m
energy：E
radius：re
(perfectly reflecting boundary)
In essence, the gravothermal catastrophe occurs when we consider the relaxation process in the inhomogeneous media. To see this, we hereafter consider the very idealistic situation, the so-called Antonov problem. This is the setup of the Antonov problem. From this, Antonov discovered that the thermal equilibrium cannot be always stable. Later, Lynden-Bell & Wood considered this issue based on the turning-point analysis. In this setup, we treat the many-body particle system confined in an sphere of radius re. The wall of the sphere is perfectly reflecting boundary, which keeps the energy conserved.

7. Isothermal state & its stability
Boltzmann-Gibbs entropy
Extremization
Isothermal distribution
D= rc/ re
l= – reE/GM 2
Large re
= 0.335
= 709
No stable isothermal state exists at
Large re :
High density-contrast :
According to the standard analysis of thermostatistics, the equilibrium property of the system confined in an adiabatic wall is determined by the maximum entropy principle. If one traditionally uses the Boltzmann-Gibbs entropy defined by this, where the f is the one-particle distribution function, the extremization under keeping the mass and the energy fixed leads to the exponential form of the distribution function, referred to as the isothermal distribution. It is important to note that while the velocity distribution is uniform Maxwellian, the spatial distribution of particles inevitably becomes inhomogeneous due to the long-range attractivity. The non-uniform structure of density distribution must be obtained self-consistently by solving the Poisson equation. After that, the equilibrium properties of this system is characterized by the energy-density contrast diagram shown here. The vertical axis means the dimensionless “minus” energy and the horizontal axis means the density contrast, the central density divided by the edge density. The red line is the equilibrium sequence of isothermal distribution. This figure clearly shows that there exists the energy bound at some critical-energy point and beyond the critical density contrast, the energy-density contrast relation becomes multi-valued, indicating the system becomes unstable. On the other hand, below the critical density contrast and/or for a smaller wall radius, the equilibrium sequence keeps to exist and is single-valued, implying the system is stable.
Although a more rigorous discussion concerning the stability/instability is necessary by analyzing the second variation of entropy, it turns out that the turning point shown here is precisely the marginal stability point and from this figure, the critical values can be read as follows.
Gravothermal instability

8. Gravothermal Catastrophe
Negative specific heat
Heat flow
core = self-gravitating
core
halo
Negative specific heat
halo = normal system
DTcore↑
Heat flow from
core to halo
Tcore > Thalo
DThalo↑
sufficiently large wall
extended halo has large heat capacity
re > rc
DTcore > DThalo
heat flow does not stop!!
Core-collapse !!

9. Consequence of instability
The system does not always approach the thermal equilibrium
(isothermal distribution)
Dynamical equilibrium
(virialized)
Thermal equilibrium
(isothermal state)
Core collapse !!
Initial conditions
To clarify the fate of the system,
Description of
is essential.
non-equilibrium states
transient states
The consequence of this result is as follows. In usual sense, one may think that the system starting with an appropriate initial condition first reach the dynamical equilibrium on dynamical timescales. Then, the effect of two-body encounter becomes significant and the system finally approach the stable thermal equilibrium. However, the actual end-point is quite different. During the relaxation processes, the thermodynamic instability appears and the particles at the center becomes highly concentrated, finally undergoing the core-collapse. This indicates that the self-gravitating system is, in nature, non-equilibrium system and the description of non-equilibrium states away from the thermal equilibrium is needed to be investigated. This is the main motivation of this work. Hence, the rest of this talk especially focuses on the attempt to characterize the transient states before entering the (self-similar) core-collapse phase.
An attempt to characterize the evolutionary states of N-body system
both from thermostatistical and dynamical point-of-view
Rest of this talk

10. Theoretical investigation
Particularly focusing on the setup of Antonov problem,
analytical
Thermostatistical approach
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) ; MNRAS (2005)
The characterization of the evolutionary sequences consists of three approach. First is the thermostatistical approach based on a generalized thermostatistical treatment. Second is the dynamical approach based on the N-body simulation in order to check the thermostatistical prediction. Third is the kinetic-theory approach to understand the results of N-body simulation semi-analytically, in which we present the new results. I will discuss these treatments in turn.
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

11. A naïve generalization of BG statistics
As a possible generalization of thermostatistical treatment,
BG limit q→1
q-entropy
Tsallis, J.Stat.Phys.52 (1988) 479
One-particle distribution function identified with escort distribution
Let us first consider the thermostatistical approach. To characterize the non-equilibrium transient states, it is clear that the standard Boltzmann-Gibss formalism is inadequate. If possible, it would be better to incorporate the formalism into some additional parameter to characterize the non-equilibrium behaviors. One such naïve generalization is the non-extensive formalism proposed by C. Tsallis, in which he introduced the q-deformed entropy. This entropy coincides with BG entropy when the only additional parameter q approaches unity. By using the q-entropy formalism, the one-particle distribution function is given by this. And owing to the entropy principle, extremum state of the q-entropy is derived. The resultant distribution becomes power-law function and the index of power is controlled by the q-parameter.
Power-law distribution

12. Stellar polytrope as quasi-equilibrium state
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
Polytrope index
n→∞
BG limit
This power-law form of one-particle distribution function is well-known in the subject of stellar dynamics, referred to as the stellar polytropes. The reason why we call this distribution polytropes comes from the fact that when calculating the pressure and the local density, one obtains the polytropic equation of state given by this equation. The polytrope index “n” is related to the q-parameter of the non-extensive entropy, given by this expression. Here is the energy-density contrast diagram for the stellar polytropic distribution confined in an adiabatic wall. The trajectories with different color indicates the stellar polytrope with different polytrope index or q-value. As one can see, the trajectory covers a wide area of energy-density contrast diagram including the equilibrium trajectory of BG thermal state. A closer look at each trajectory shows that the oscillatory behavior appears when n>5. A more detailed analysis based on the eigen-value problem of second variation of entropy implies that the stellar polytropic distribution becomes thermodynamically unstable beyond the boundary denoted by dotted curve.
Anyway, the stellar polytropic states as one-parameter family of stellar model provide a variety of stellar structure. Potentially, these can characterize the evolutionary sequence of stellar system. At this point, however, it remains unclear whether such one-parameter model really characterizes the actual non-equilibrium states or not.

13. Stellar polytrope as quasi-equilibrium state
n=6
Energy-density contrast
relation for stellar polytrope
stable
unstable
For larger densiy D>Dcrit,
unstable state appears at n>5
(gravothermal instability)
oscillatory behavior appears when n>5.
This power-law form of one-particle distribution function is well-known in the subject of stellar dynamics, referred to as the stellar polytropes. The reason why we call this distribution polytropes comes from the fact that when calculating the pressure and the local density, one obtains the polytropic equation of state given by this equation. The polytrope index “n” is related to the q-parameter of the non-extensive entropy, given by this expression. Here is the energy-density contrast diagram for the stellar polytropic distribution confined in an adiabatic wall. The trajectories with different color indicates the stellar polytrope with different polytrope index or q-value. As one can see, the trajectory covers a wide area of energy-density contrast diagram including the equilibrium trajectory of BG thermal state. A closer look at each trajectory shows that the oscillatory behavior appears when n>5. A more detailed analysis based on the eigen-value problem of second variation of entropy implies that the stellar polytropic distribution becomes thermodynamically unstable beyond the boundary denoted by dotted curve.
Anyway, the stellar polytropic states as one-parameter family of stellar model provide a variety of stellar structure. Potentially, these can characterize the evolutionary sequence of stellar system. At this point, however, it remains unclear whether such one-parameter model really characterizes the actual non-equilibrium states or not.

14. Stellar dynamical simulations
N-body simulation
Long-term evolution of N-body system
confined in an adiabatic wall
E=M=const.
Unit: G=M=re=1 re
Adiabatic
Perfectly reflecting
wall
※ we use GRAPE-6 @ NAOJ
Initial conditions
Group A
Stellar polytropes
( n=3, 5, 6, ∞ )
In order to check the non-equilibrium status of stellar distribution quantitatively, we performed the long-term stellar dynamical simulations in the setup of Antonov problem. To do this, we use a special purpose hardware, GRAPE-6, which accelerates the force calculation even with a large number of particles. In our systematic survey, we examined the various cases of initial distribution, but these are summarized as two kinds of initial distribution. Shortly speaking, one is the power-law distribution characterized by stellar polytropes. The other is the stellar models with singular density profile, which have the non-power-law behavior of the distribution function.
Group B
A family of stellar model with cusped profile
(non-polytropic models)
Tremaine et al. AJ 107 (1994) 634

15. Overview of the N-body results
However, focusing on their transients, we found :
Stellar polytropes are not stable in timescale of two-body relaxation.
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.
We will mainly show the quasi-equilibrium behavior.

16. Survey 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.
Let us first present the survey results of group (A), which shows the quasi-equilibrium properties. We summarize the results in energy-density contrast diagram. In this plot, the thick arrow shows the evolutionary tracks of each simulation run. Since the total energy is conserved in our system, the evolutionary tracks just move horizontally. It is interesting to note that the evolutionary track keeps the direction increasing the polytrope index “n”. Further, once exceeding the critical value, central density rapidly increases, finally leading to a core collapse. We now show the representative results taken from this run.

17. Run n3A (1) Density profile One-particle distribution function
Stellar polytrope
(n=3,D=10 ) 4
Density profile
One-particle distribution function
Fitting to stellar polytropes is quite good until t ~ 30 trh,i.
(Snapshot)
Keeping to pay an attention on the pre-collapse stage, we plot the snapshot of density profiles and distribution functions.
This is the superposition of the snapshots of the simulation run. Left shows the density profiles and right shows the distribution function as function of specific energy. Since this plot is not easy to see, we shift each snapshot to two-digits below. Further, superpose the initial density profile and the initial one-particle distribution. Clearly, the system gradually evolves and deviates from the initial state. We then try to fit the transient states to the stellar polytropes by changing the polytrope index. The fitting results are shown here. As one sees, the fitting to the stellar polytropes is remarkably good.

18. Polytrope index monotonically grows on relaxation time-scale
Run n3A (2)
unstable
stable
Time evolution of polytrope index “n” fitted to N-body simulations
Polytrope index monotonically grows on relaxation time-scale
This is the fitting results of polytrope index as function of time. The three curves represent the different simulation runs with different particle numbers. As one can see, the polytrope index monotonically increases until the curve exceeds the marginal stability line. The fitting results from three different runs almost coincides with each other, indicating that time re-scaling by two-body relaxation is successful.

19. Survey results of group (B)
Stellar model:
model parameters → h, a
Case (A)
Case (B)
Fitting to polytrope is good
Fitting failed
Next, let us present the survey results of group (B), which shows the quasi-attractive behaviors. Here we plot the energy-scale radius diagram. Each curve represent an equilibrium sequence of stellar model with cusped density profile for a fixed slope index “\eta”. The symbols mean the fitting results to the stellar polytropes. Roughly speaking, if the system is not so far away from the region where stable isothermal equilibrium exists, the quasi-attractive behavior was found and the transient state can be reasonably fitted to the stellar polytropic distribution.
Region where stable isothermal (BG) state can exist.
Quasi-attractive behaviors appear when 1< h & l ~ 0.335

20. Distribution function
Cases with a/re=0.5 (1)
Distribution function
η=1
Rapid Core Collapse (×)
Approaching to Isothermal（△）
η=1.5
η=3
Approaching to polytrope (○)

21. (×) small core → rapid collapse
Cases with a/re=0.5 (2)
Density profile
(×) small core → rapid collapse
η=1
Fitting failed
η=1.5
almost Isothermal
Larger core approaches to Polytrope
Fitting is GOOD
η=3

22. 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.
phase space volume
Complicated, but helpful for semi-analytic understanding

23. F-P eq.に対する一般化された変分原理 Local potential F-P equation for
Glansdorff & Prigogine (1971)
Inagaki & Lynden-Bell (1990)
Local potential
Variation w.r.t. f
F-P equation for
Absolute minimum at a solution
Application:
Takahashi & Inagaki (1992); Takahashi (1993ab)

24. 熱伝導 の場合 Local potential variation w.r.t. subsidiary condition
absolute minimum

25. Local Potential の導出
：内部エネルギー密度
エネルギー保存
：熱流
熱伝導
過剰エントロピー生成

26. Stellar Dynamics への適用 pre-collapse post-collapse
K.Takahashi, PASJ 45,233 (1993)
pre-collapse
post-collapse

27. 一般化された変分原理による “n”の発展方程式
Assuming stellar polytropes with time-varying polytrope index as quasi-equilibrium state,
trial function
の関数

28. 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.

29. Numerical n-dn/dt curves: linear plot
Units :
and
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-.
For n>5 and l<0.335, dn/dt curves approach a linear curve.
l=0.1
0.15
0.2
0.25
0.3
0.35
0.45
0.4
l=1.2
0.5

30. Numerical n-dn/dt curves: log-log plot
Units :
and
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
0.45
0.5
l=1.2

31. n-dn/dt curves ： approximation
For the curves with l>0.335,
is function of l and
is determined by solving
(n>5)
For the curves with l<0.335,
(i=1,2)
(n>5)
( units:
and )

32. Analytic estimate of n(t): comparison
Half-mass relaxation time
Analytic estimate based on variational method successfully reproduces the N-body results.

33. Summary 長距離力（引力） 比熱が負 small system （１） 重力多体系 非平衡進化 準定常状態が存在 （２） 準定常状態
（１）　重力多体系
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
(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

34. 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
(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？

35. Polytrope による準定常状態の記述の限界
self-similar evolution
Fokker-Planck eq. によるCore-Collapse の解析
Heat flow
core
halo
self-similar core collapse が
始まると polytrope で fit できない
H.Cohn Ap.J 242 p.765 (1980)

36. Self-similar sol. of F-P eq.
Heggie and Stevenson,
MN 230 p.223 (1988)
power law envelope
isothermal core

37. fitting of self-similar sol. with double polytrope
Black dots: Numerical Self-Similar sol.
by Heggie and Stevenson
Magenta lines: fitting by double polytrope

38. 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
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．．．

39. 2次元ＨＭＦモデル 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
Antoni&Torcini PRE 57(1998) R6233
Antoni, Ruffo&Torcini, PRE 66(2003) R
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

40. T-U 曲線 熱平衡 平均場 Magnetization T U t / N Vlasov phase ? 分布関数 分布関数 熱平衡
準定常状態
polytrope ? U
t / N
Vlasov phase ?
分布関数
分布関数

41. Magnetization t