T. Padmanabhan (IUCAA, Pune, INDIA) · FINITE GRAVITATING SYSTEMS – GENERAL DESCRIPTION – •...

Statistical Mechanics of Gravitating Systems

...and some curious history of Chandra’s rare misses!

T. Padmanabhan

(IUCAA, Pune, INDIA)

Chandra Centenary Conference

Bangalore, India

8 December 2010

Phases of Chandra’s life

[1939] [1942] [1950]

[1961] [1983] [1995]

GRAVITATIONAL N−BODY PROBLEM

N particles, mass m, U(r) = −

Non cosmological;no bg. expansion

Cosmological;bg. expansion a(t)

CONDENSED MATTER PHYSICS

PHASE TRANSITIONS

POWERTRANSFER

UNIVERSAL POWER SPECTRUM,NON−LINEAR SCALING RELATIONS

FLUIDTURBULENCE

PHASE TRANSITIONS

POWERTRANSFER

FLUIDTURBULENCE

PHASE TRANSITIONS

RENORMALISATION GROUP

POWERTRANSFER

PLAN OF THE TALK

• FINITE SELF-GRAVITATING SYSTEMS OF PARTICLES

– General features

– Mean field description: Isothermal sphere

– Antonov instability

• RELAXATION TIME AND DYNAMICAL FRICTION

– Chandra’s contribution

– Historical background

• GRAVITATING PARTICLES IN EXPANDING BACKGROUND

– General features, Open questions

– Power transfer, Inverse cascade

– Universality

BASICS OF STATISTICAL MECHANICS

• Density of states:

Γ(E) =

dpdqθD[E −H(p, q)]; g(E) =dΓ(E)

Γ(E) =

• Microcanonical description: Entropy S(E) = ln g(E) where

T (E) =

;P = T

Γ(E) =

T (E) =

;P = T

• Example: Ideal gas with H ∝∑

p2i has:

Γ ∼ V NE3N/2 ∼ g(E); T (E) = (2E/3N); P/T = N/V

Γ(E) =

T (E) =

;P = T

p2i has:

• Canonical description:

Z(T ) =

dpdq exp[−βH(p, q)] =

dEg(E) exp[−βE] = e−βF

E = −(∂ lnZ/∂β); P = T (∂ lnZ/∂V )

Γ(E) =

T (E) =

;P = T

p2i has:

• Canonical description:

Z(T ) =

dpdq exp[−βH(p, q)] =

dEg(E) exp[−βE] = e−βF

E = −(∂ lnZ/∂β); P = T (∂ lnZ/∂V )

• If energy is extensive, these descriptions are equivalent for most systems to

O(lnN/N).

THE TROUBLE WITH GRAVITATING SYSTEMS

• Long range interaction with no shielding. Energy is not

extensive.

• Microcanonical and Canonical descriptions are not

equivalent.

extensive.

• Microcanonical and Canonical descriptions are not

equivalent.

• The g(E) and S(E) requires short-distance cutoff for

finiteness.

FINITE GRAVITATING SYSTEMS

– GENERAL DESCRIPTION –

• Typical systems have two natural energy scales: E1 = (−Gm2/a) and

E2 = (−Gm2/R); a ≪ R.

E2 = (−Gm2/R); a ≪ R. Virial theorem gives: 2K + U = 3PV + U0.

• For E ≫ E2, gravity is not strong enough and the system behaves like a gas

confined by the container. High temperature phase with positive specific heat;

2K ≈ 3PV .

• For E1 ≪ E ≪ E2, the system is unaffected by either the box or the short distance

cutoff; dominated entirely by gravity and has negative specific heat. 2K + U ≈ 0.

2K ≈ 3PV .

• Since canonical ensemble cannot lead to negative specific heat, microcanonical

and canonical ensembles differ drastically in this range. Canonical ensemble leads

to a phase transition.

2K ≈ 3PV .

• As we go to E ≃ E1, the hard core nature of the particles begins to be felt and

the gravity is again resisted; low temperature phase with positive specific heat.

U ≈ U0.

2K ≈ 3PV .

• As we go to E ≃ E1, the hard core nature of the particles begins to be felt and

the gravity is again resisted; low temperature phase with positive specific heat.

U ≈ U0.

• Increasing R increases the range over which the specific heat is negative.

GENERIC BEHAVIOUR

T. Padmanabhan, Physics Reports, 188, 285 (1990).

EnsembleMicrocanonical

GENERIC BEHAVIOUR

Ensemble

0U ~ U

Low Temperature Phase

Microcanonical

GENERIC BEHAVIOUR

Ensemble

2K ~ 3PV

HighTemperaturePhase

Microcanonical

GENERIC BEHAVIOUR

Negative SpecificHeat

2K + U ~ 0

2K ~ 3PV

MicrocanonicalEnsemble

GENERIC BEHAVIOUR

Ensemble

Negative SpecificHeat

2K + U ~ 0

2K ~ 3PV

Microcanonical

Canonical EnsemblePhase Transition in the

Mean field description of many-body systems

• A system of N particles interacting through the two-body potential U(x, y). The

entropy S of this system

eS = g(E) =1

d3Nxd3Npδ(E −H) ∝1

E −1

U(xi, xj)

eS = g(E) =1

E −1

U(xi, xj)

• Divide spatial volume V be divided into M cells of equal size. Integration over the

particle coordinates (x1, x2, ..., xN)= sum over the number of particles na in the cell

centered at xa (where a = 1, 2, ...,M).

eS = g(E) =1

E −1

U(xi, xj)

• This gives:

eS ≈

∞∑

N −∑

expS[{na}]

S[{na}] =3N

E −1

naU(xa, xb)nb

eS = g(E) =1

E −1

U(xi, xj)

• This gives:

eS ≈

∞∑

N −∑

expS[{na}]

S[{na}] =3N

E −1

naU(xa, xb)nb

−M∑

• The mean field approximation: retain in the sum only the term for which the

summand reaches the maximum value∑

eS[na] ≈ eS[na,max]

Mean field description of self-gravitating systems

• Take the continuum limit with

na,maxM

V= ρ(xa);

ρ(x) = A exp(−βφ(x)); where φ(x) =

d3yU(x, y)ρ(y)

na,maxM

V= ρ(xa);

d3yU(x, y)ρ(y)

• In the case of gravitational interaction:

ρ(x) = A exp(−βφ(x)); φ(x) = −G

ρ(y)d3y

|x − y|.

gives the configuration of extremal entropy for a gravitating system

in the mean field limit.

na,maxM

V= ρ(xa);

d3yU(x, y)ρ(y)

• In the case of gravitational interaction:

ρ(x) = A exp(−βφ(x)); φ(x) = −G

ρ(y)d3y

|x − y|.

gives the configuration of extremal entropy for a gravitating system

in the mean field limit.

• For gravitational interactions without a short distance cut-off, the

quantity eS is divergent. A short distance cut-off is needed to justify

the entire procedure.

ANTONOV INSTABILITY

1. Systems with (RE/GM2) < −0.335 cannot evolve into isothermal

spheres. Entropy has no extremum for such systems.

ANTONOV INSTABILITY

2. Systems with ((RE/GM2) > −0.335) and (ρ(0) > 709 ρ(R)) can exist

in a meta-stable (saddle point state) isothermal sphere

configuration. The entropy extrema exist but they are not local

maxima.

ANTONOV INSTABILITY

maxima.

3. Systems with ((RE/GM2) > −0.335) and (ρ(0) < 709 ρ(R)) can form

isothermal spheres which are local maximum of entropy.

ANTONOV INSTABILITY

maxima.

3. Systems with ((RE/GM2) > −0.335) and (ρ(0) < 709 ρ(R)) can form

isothermal spheres which are local maximum of entropy.

• Reference:

V.A. Antonov, V.A. : Vest. Leningrad Univ. 7, 135 (1962);

Translation: IAU Symposium 113, 525 (1985).

T.Padmanabhan: Astrophys. Jour. Supp. , 71, 651 (1989).

Isothermal sphere

• Basic equation:

∇2φ = 4πGρce−β[φ(x)−φ(0)]; ρc = ρ(0)

Isothermal sphere

• Basic equation:

∇2φ = 4πGρce−β[φ(x)−φ(0)]; ρc = ρ(0)

• Let

L0 ≡ (4πGρcβ)1/2 , M0 = 4πρcL30, φ0 ≡ β−1 =

with dimensionless variables:

x ≡r

, n ≡ρ

ρc, m =

, y ≡ β [φ− φ (0)] .

Isothermal sphere

• Basic equation:

∇2φ = 4πGρce−β[φ(x)−φ(0)]; ρc = ρ(0)

• Let

L0 ≡ (4πGρcβ)1/2 , M0 = 4πρcL30, φ0 ≡ β−1 =

x ≡r

, n ≡ρ

ρc, m =

, y ≡ β [φ− φ (0)] .

• Then1

dx(x2dy

dx) = e−y; y(0) = y′(0) = 0

Isothermal sphere

• Basic equation:

∇2φ = 4πGρce−β[φ(x)−φ(0)]; ρc = ρ(0)

• Let

L0 ≡ (4πGρcβ)1/2 , M0 = 4πρcL30, φ0 ≡ β−1 =

x ≡r

, n ≡ρ

ρc, m =

, y ≡ β [φ− φ (0)] .

• Then1

dx(x2dy

dx) = e−y; y(0) = y′(0) = 0

• Singular solution:

,m = 2x, y = 2 lnx

LANE-EMDEN VARIABLES

• Invariant under the transformation y → y + a ; x→ kx with k2 = ea.

• Reduce the degree by choosing:

v ≡m

x; u ≡

v ≡m

x; u ≡

du= −

(u− 1)

(u+ v − 3); v(u = 3) = 0; v′(u = 3) = −

v ≡m

x; u ≡

du= −

(u− 1)

(u+ v − 3); v(u = 3) = 0; v′(u = 3) = −

• The singular point is at: us = 1, vs = 2; corresponding to the

solution n = (2/x2),m = 2x.

v ≡m

x; u ≡

du= −

(u− 1)

(u+ v − 3); v(u = 3) = 0; v′(u = 3) = −

• All solutions tend to this asymptotically for large r by spiralling

around the singular point in the u− v plane.

v ≡m

x; u ≡

du= −

(u− 1)

(u+ v − 3); v(u = 3) = 0; v′(u = 3) = −

• All solutions tend to this asymptotically for large r by spiralling

around the singular point in the u− v plane.

• Finite total mass for the system requires a large distance cut-off at

some r = R.

Chandra (1939) Introduction to the study of stellar structure

[D.Lynden-Bell, R. Wood, (1968), MNRAS, 138, p.495.]

Chandra’s comments on Emden’s work

ENERGY OF THE ISOTHERMAL SPHERE

• The potential and kinetic energies are

U = −

drdr = −

∫ x0

m(x0) =GM2

∫ x0

nx2dx; x0 = R/L0

U = −

drdr = −

∫ x0

m(x0) =GM2

∫ x0

nx2dx; x0 = R/L0

• So:

E = K + U =GM2

∫ x0

dx(3nx2 − 2mnx)

∫ x0

dx{2nx3 − 3m} =

{n0x30 −

U = −

drdr = −

∫ x0

m(x0) =GM2

∫ x0

nx2dx; x0 = R/L0

• So:

E = K + U =GM2

∫ x0

dx(3nx2 − 2mnx)

∫ x0

dx{2nx3 − 3m} =

{n0x30 −

• The combination (RE/GM2) is a function of (u, v) alone.

λ =RE

{u0 −3

U = −

drdr = −

∫ x0

m(x0) =GM2

∫ x0

nx2dx; x0 = R/L0

• So:

E = K + U =GM2

∫ x0

dx(3nx2 − 2mnx)

∫ x0

dx{2nx3 − 3m} =

{n0x30 −

• The combination (RE/GM2) is a function of (u, v) alone.

λ =RE

{u0 −3

• An isothermal sphere must lie on the curve

; λ ≡RE

[T. Padmanabhan, Physics Reports, 188, 285 (1990).]

COLLISIONAL RELAXATION IN GRAVITATING SYSTEMS

• Transverse velocity in an encounter:

δv⊥ ≃(

Gm/b2)

(2b/v) = (2Gm/bv)

δv⊥ ≃(

Gm/b2)

(2b/v) = (2Gm/bv)

• ”Hard” collisions: ∆v⊥ ≈ v; so b ≤ bc = Gm/v2. The relevant relaxation timescale:

thard ≃1

(nσv)≃

N (G2m2)≃NR3v3

G2M2≈ N(R/v)

δv⊥ ≃(

Gm/b2)

(2b/v) = (2Gm/bv)

thard ≃1

(nσv)≃

N (G2m2)≃NR3v3

G2M2≈ N(R/v)

• More subtle is the effect of ”soft” collisions which is diffusion in the velocity

space in which the (∆v⊥)2 add up linearly with time.

δv⊥ ≃(

Gm/b2)

(2b/v) = (2Gm/bv)

thard ≃1

(nσv)≃

N (G2m2)≃NR3v3

G2M2≈ N(R/v)

〈(δv⊥)2〉total ≃ ∆t

∫ b2

(2πbdb) (vn)

=2πnG2m2

v∆t ln

δv⊥ ≃(

Gm/b2)

(2b/v) = (2Gm/bv)

thard ≃1

(nσv)≃

N (G2m2)≃NR3v3

G2M2≈ N(R/v)

〈(δv⊥)2〉total ≃ ∆t

∫ b2

(2πbdb) (vn)

=2πnG2m2

v∆t ln

• Take b2 = R= size of the system, b1 = bc. Then

(b2/b1) ≃(

Rv2/Gm)

Rv2/GM)

in virial equilibrium. So

tsoft ≃v3

2πG2m2n lnN≃

Publication date: 1942

Pages 48 to 73 gives the derivation of TE and TD!

Pages 317 to 320 contain the derivation by Jeans!

First appearance of lnN

CHANDRA’S COMMENT ON THE WORK OF JEANS

SECOND ASPECT OF DIFFUSION IN VELOCITY SPACE

• This can’t be the whole story!

• We need a dynamical friction to reach steady state with

Maxwellian distribution of velocities.

• We need a dynamical friction to reach steady state with

Maxwellian distribution of velocities.

• Chandra seems to have realized this soon after the

publication of the book!!

DIFFUSION IN VELOCITY SPACE:

A UNIFIED APPROACH

• Diffusion current as the source term:

dt=∂f

∂t+ v.

∂x−∇φ.

∂v= −

∂Jα

∂pα

A UNIFIED APPROACH

• Diffusion current as the source term:

dt=∂f

∂t+ v.

∂x−∇φ.

∂v= −

∂Jα

∂pα

• Form of the current can be shown to be:

Jα =B0

d l′{

f∂f ′

∂lβ− f ′∂f

∂lβ

δαβ

k−kαkβ

where B0 = 4πG2m5L; and L =b2∫

TWO FOR THE PRICE OF ONE!

• Current:

Jα =B0

d l′{

f∂f ′

∂lβ− f ′ ∂f

∂lβ

δαβ

k−kαkβ

• Current:

Jα =B0

d l′{

f∂f ′

∂lβ− f ′ ∂f

∂lβ

δαβ

k−kαkβ

• The term proportional to f gives dynamical friction.

The term proportional to (∂f/∂lβ) increases the velocity

dispersion.

• Current:

Jα =B0

d l′{

f∂f ′

∂lβ− f ′ ∂f

∂lβ

δαβ

k−kαkβ

• The term proportional to f gives dynamical friction.

The term proportional to (∂f/∂lβ) increases the velocity

dispersion.

• The current Jα vanishes for Maxwell distribution, as it

should!

AN ILLUSTRATIVE TOY MODEL

• Note that

Jα(l) ≡ aα(l)f(l) −1

∂lβ

σ2αβf}

where aα = (∂η/∂lα) , σ2αβ = (∂2ψ/∂lα∂lβ)

• Note that

∂lβ

σ2αβf}

• Aside:

∇2ψ = η; ∇2l η(l) = ∇2

dl′f(l′)

|l − l′|

= −8πf(l)

• Note that

∂lβ

σ2αβf}

• Aside:

∇2ψ = η; ∇2l η(l) = ∇2

dl′f(l′)

|l − l′|

= −8πf(l)

• Treat the coefficients as constants to understand the structure of

the equation:

∂f(v, t)

(αv)f +σ2

≡ −∂J

• Note that

∂lβ

σ2αβf}

• Aside:

∇2ψ = η; ∇2l η(l) = ∇2

dl′f(l′)|l − l′|

= −8πf(l)

• Treat the coefficients as constants to understand the structure of

the equation:

∂f(v, t)

(αv)f +σ2

≡ −∂J

• An initial distribution f(v, 0) = δD(v − v0) evolves to:

f(v, t) =

πσ2(1 − e−2αt)

−α(v − v0e−αt)2

σ2(1 − e−2αt)

• The mean velocity decays to zero:

< v >= v0e−αt

• The distribution tends to Maxwellian limit with velocity dispersion

< v2 > − < v >2=σ2

α(1 − e−2αt) →

< v >= v0e−αt

• The distribution tends to Maxwellian limit with velocity dispersion

< v2 > − < v >2=σ2

α(1 − e−2αt) →

• The interplay between the two effects is obvious.

HISTORY: OVERLOOKING LANDAU (1936)!

Footnote is incorrect; Landau’s expression has both dynamical friction and velocity

dispersion!

GRAVITATIONAL CLUSTERING IN EXPANDING UNIVERSE

– SOME KEY ISSUES –

• If the initial power spectrum is sharply peaked in a narrow band of

wavelengths, how does the evolution transfer the power to other

scales?

• What is the asymptotic nature of evolution for the self gravitating

system in an expanding background?

scales?

• Does the gravitational clustering at late stages wipe out the

memory of initial conditions ?

scales?

• Do the virialized structures formed in an expanding universe due to

gravitational clustering have any invariant properties? Can their

structure be understood from first principles?

scales?

• Do the virialized structures formed in an expanding universe due to

gravitational clustering have any invariant properties? Can their

structure be understood from first principles?

• How can one connect up the local behaviour of gravitating systems

to the evolution of clustering in the universe?

BASIC DEFINITIONS

• Density:

ρ(x, t) =m

δD[x − xi(t)]

• Mean density:

ρb(t) ≡

Vρ(x, t) =

a3V=ρ0

• Density contrast:

1 + δ(x, t) ≡ρ(x, t)

δD[x − xi(t)] =

dqδD[x − xT (t, q)].

• Density contrast in Fourier space:

δk(t) ≡

d3xe−ik·xδ(x, t) =

d3q exp[−ik.xT (t, q)] − (2π)3δD(k)

THE EXACT (BUT USELESS) DESCRIPTION

• Density contrast in Fourier space satisfies:

δk + 2a

aδk = 4πGρbδk + Ak −Bk

Ak = 4πGρb

d3k′

(2π)3δk′δk−k′

k.k′

d3q (k.xT )2 exp [−ik.xT (t, q)]

THE EXACT (BUT USELESS) DESCRIPTION

• Density contrast in Fourier space satisfies:

δk + 2a

aδk = 4πGρbδk + Ak −Bk

Ak = 4πGρb

d3k′

(2π)3δk′δk−k′

k.k′

d3q (k.xT )2 exp [−ik.xT (t, q)]

• Coupled exact equations:

φk + 4a

aφk = −

(2π)3φ 1

2k+pφ 1

2k−p

+ p2 − 2

x + 2a

ax = −

a2∇xφ = −

ikφk exp i(k · x);

“Renormalizability” of gravity

• One can then show that: The term (Ak −Bk) receives contribution

only from particles which are not bound to any of the clusters to

the order O(k2R2). (Peebles, 1980)

“Renormalizability” of gravity

• One can then show that: The term (Ak −Bk) receives contribution

only from particles which are not bound to any of the clusters to

the order O(k2R2). (Peebles, 1980)

• Allows one to ignore contributions from virialised systems and treat

the rest in Zeldovich (-like) approximation. Then one gets:

φk = −2

d3p(2π)3

k+pφL1

2k−p

∫ ′ d3p

(2π)3φ1

2k+pφ1

2k−p

+ p2 − 2

(T.P, 2002)

Evolution at large scales

• Ignore the terms Ak and Bk. Then:

δk + 2a

aδk = 4πGρbδk

δk + 2a

aδk = 4πGρbδk

• For a ∝ t2/3, ρb ∝ a−3, the growing solution is:

δk ∝ a; P (k) = |δk|2 ∝ a2; ξ(a, x) ∝ a2

δk + 2a

aδk = 4πGρbδk

δk ∝ a; P (k) = |δk|2 ∝ a2; ξ(a, x) ∝ a2

• BUT: If δk → 0 for certain range of k at t = t0 (but is

nonzero elsewhere) then (Ak −Bk) ≫ 4πGρbδk and the

growth of perturbations around k will be entirely

determined by nonlinear effects.

δk + 2a

aδk = 4πGρbδk

δk ∝ a; P (k) = |δk|2 ∝ a2; ξ(a, x) ∝ a2

• BUT: If δk → 0 for certain range of k at t = t0 (but is

nonzero elsewhere) then (Ak −Bk) ≫ 4πGρbδk and the

growth of perturbations around k will be entirely

determined by nonlinear effects.

• There is inverse cascade of power in gravitational

clustering! (Zeldovich, 1965)

Bagla, T.P, (1997) MNRAS, 286, 1023

k3P(k)2a

Lk−1

k3P(k)2a aP k

Lk−1

LINEAR REGIME

QUASI−LINEAR REGIME

Prediction from stable clustering

NON−LINEAR REGIME

Bagla, Engineer, TP (1998), Ap.J.,495, 25

‘EQUIPARTITION’ OF ENERGY FLOW

• If the local power law index is n, then one can show that in the

linear regime:

EL(a, x) ≈ ax−(n+2)

linear regime:

EL(a, x) ≈ ax−(n+2)

• In the quasilinear regime:

EQL(a, x) ≈ a2−n

n+4x−2n+5

linear regime:

EL(a, x) ≈ ax−(n+2)

n+4x−2n+5

• In the nonlinear regime:

ENL(a, x) ≈ a1−n

n+5x−2n+4

linear regime:

EL(a, x) ≈ ax−(n+2)

n+4x−2n+5

• In the nonlinear regime:

ENL(a, x) ≈ a1−n

n+5x−2n+4

• NEW FEATURE: The energy flow is form invariant

(“equipartition”) when n = −1 in QL and n = −2 in the NL regimes!

Then E ∝ a in all three regimes.

k3P(k)2a aP k

Lk−1

k3P(k)2a

aP k−12

aP k4 4

Lk−1

aP k−12

aP k−22

Injected initial power spectrum

P(k) ~ k −1

Bagla, T.P, (1997) MNRAS, 286, 1023

Late time power spectra for 3 differentinitial injected spectra

P(k) ~ k −1

Bagla, T.P, (1997) MNRAS, 286, 1023

Asymptotic universality in nonlinear clustering

(T.P., S. Ray, 2006)

Summary

• Chandra pioneered the use of statistical physics in the

study of gravitating systems.

Summary

• His approach to this subject shows the characteristic

rigour employed as a matter of policy rather than out of

necessity.

Summary

• His approach to this subject shows the characteristic

rigour employed as a matter of policy rather than out of

necessity.

• The subject is alive and well and still has several open

questions especially in the context of cosmology.

Acknowledgements

• D. Lynden-Bell (IOA)

• D. Narasimha (TIFR)

• R. Nityananda (NCRA)

• S. Sridhar (RRI)

• K. Subramanian (IUCAA)

T. Padmanabhan (IUCAA, Pune, INDIA) · FINITE GRAVITATING SYSTEMS – GENERAL DESCRIPTION – •...

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