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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]
)r
Gm2
GRAVITATIONAL N−BODY PROBLEM
)
N particles, mass m, U(r) = −
)r
Gm2
Non cosmological;no bg. expansion
Cosmological;bg. expansion a(t)
GRAVITATIONAL N−BODY PROBLEM
)
N particles, mass m, U(r) = −
)r
Gm2
Non cosmological;no bg. expansion
Cosmological;bg. expansion a(t)
CONDENSED MATTER PHYSICS
GRAVITATIONAL N−BODY PROBLEM
PHASE TRANSITIONS
)
N particles, mass m, U(r) = −
)r
Gm2
Non cosmological;no bg. expansion
Cosmological;bg. expansion a(t)
CONDENSED MATTER PHYSICS
GRAVITATIONAL N−BODY PROBLEM
PHASE TRANSITIONS
POWERTRANSFER
)
UNIVERSAL POWER SPECTRUM,NON−LINEAR SCALING RELATIONS
N particles, mass m, U(r) = −
)r
Gm2
Non cosmological;no bg. expansion
Cosmological;bg. expansion a(t)
CONDENSED MATTER PHYSICS
GRAVITATIONAL N−BODY PROBLEM
FLUIDTURBULENCE
PHASE TRANSITIONS
POWERTRANSFER
)
UNIVERSAL POWER SPECTRUM,NON−LINEAR SCALING RELATIONS
N particles, mass m, U(r) = −
)r
Gm2
Non cosmological;no bg. expansion
Cosmological;bg. expansion a(t)
CONDENSED MATTER PHYSICS
GRAVITATIONAL N−BODY PROBLEM
FLUIDTURBULENCE
PHASE TRANSITIONS
RENORMALISATION GROUP
POWERTRANSFER
)
UNIVERSAL POWER SPECTRUM,NON−LINEAR SCALING RELATIONS
N particles, mass m, U(r) = −
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
BASICS OF STATISTICAL MECHANICS
• Density of states:
Γ(E) =
∫
dpdqθD[E −H(p, q)]; g(E) =dΓ(E)
dE
BASICS OF STATISTICAL MECHANICS
• Density of states:
Γ(E) =
∫
dpdqθD[E −H(p, q)]; g(E) =dΓ(E)
dE
• Microcanonical description: Entropy S(E) = ln g(E) where
T (E) =
(
∂S
∂E
)−1
;P = T
(
∂S
∂V
)
.
BASICS OF STATISTICAL MECHANICS
• Density of states:
Γ(E) =
∫
dpdqθD[E −H(p, q)]; g(E) =dΓ(E)
dE
• Microcanonical description: Entropy S(E) = ln g(E) where
T (E) =
(
∂S
∂E
)−1
;P = T
(
∂S
∂V
)
.
• Example: Ideal gas with H ∝∑
p2i has:
Γ ∼ V NE3N/2 ∼ g(E); T (E) = (2E/3N); P/T = N/V
BASICS OF STATISTICAL MECHANICS
• Density of states:
Γ(E) =
∫
dpdqθD[E −H(p, q)]; g(E) =dΓ(E)
dE
• Microcanonical description: Entropy S(E) = ln g(E) where
T (E) =
(
∂S
∂E
)−1
;P = T
(
∂S
∂V
)
.
• Example: Ideal gas with H ∝∑
p2i has:
Γ ∼ V NE3N/2 ∼ g(E); T (E) = (2E/3N); P/T = N/V
• Canonical description:
Z(T ) =
∫
dpdq exp[−βH(p, q)] =
∫
dEg(E) exp[−βE] = e−βF
with
E = −(∂ lnZ/∂β); P = T (∂ lnZ/∂V )
BASICS OF STATISTICAL MECHANICS
• Density of states:
Γ(E) =
∫
dpdqθD[E −H(p, q)]; g(E) =dΓ(E)
dE
• Microcanonical description: Entropy S(E) = ln g(E) where
T (E) =
(
∂S
∂E
)−1
;P = T
(
∂S
∂V
)
.
• Example: Ideal gas with H ∝∑
p2i has:
Γ ∼ V NE3N/2 ∼ g(E); T (E) = (2E/3N); P/T = N/V
• Canonical description:
Z(T ) =
∫
dpdq exp[−βH(p, q)] =
∫
dEg(E) exp[−βE] = e−βF
with
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
THE TROUBLE WITH GRAVITATING SYSTEMS
• Long range interaction with no shielding. Energy is not
extensive.
THE TROUBLE WITH GRAVITATING SYSTEMS
• Long range interaction with no shielding. Energy is not
extensive.
• Microcanonical and Canonical descriptions are not
equivalent.
THE TROUBLE WITH GRAVITATING SYSTEMS
• Long range interaction with no shielding. Energy is not
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 –
FINITE GRAVITATING SYSTEMS
– GENERAL DESCRIPTION –
• Typical systems have two natural energy scales: E1 = (−Gm2/a) and
E2 = (−Gm2/R); a ≪ R.
FINITE GRAVITATING SYSTEMS
– GENERAL DESCRIPTION –
• Typical systems have two natural energy scales: E1 = (−Gm2/a) and
E2 = (−Gm2/R); a ≪ R. Virial theorem gives: 2K + U = 3PV + U0.
FINITE GRAVITATING SYSTEMS
– GENERAL DESCRIPTION –
• Typical systems have two natural energy scales: E1 = (−Gm2/a) and
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 .
FINITE GRAVITATING SYSTEMS
– GENERAL DESCRIPTION –
• Typical systems have two natural energy scales: E1 = (−Gm2/a) and
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.
FINITE GRAVITATING SYSTEMS
– GENERAL DESCRIPTION –
• Typical systems have two natural energy scales: E1 = (−Gm2/a) and
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.
• 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.
FINITE GRAVITATING SYSTEMS
– GENERAL DESCRIPTION –
• Typical systems have two natural energy scales: E1 = (−Gm2/a) and
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.
• 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.
• 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.
FINITE GRAVITATING SYSTEMS
– GENERAL DESCRIPTION –
• Typical systems have two natural energy scales: E1 = (−Gm2/a) and
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.
• 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.
• 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
T. Padmanabhan, Physics Reports, 188, 285 (1990).
Ensemble
0U ~ U
Low Temperature Phase
Microcanonical
GENERIC BEHAVIOUR
T. Padmanabhan, Physics Reports, 188, 285 (1990).
Ensemble
0
2K ~ 3PV
HighTemperaturePhase
U ~ U
Low Temperature Phase
Microcanonical
GENERIC BEHAVIOUR
T. Padmanabhan, Physics Reports, 188, 285 (1990).
Negative SpecificHeat
0
2K + U ~ 0
2K ~ 3PV
HighTemperaturePhase
U ~ U
Low Temperature Phase
MicrocanonicalEnsemble
GENERIC BEHAVIOUR
T. Padmanabhan, Physics Reports, 188, 285 (1990).
Ensemble
Negative SpecificHeat
0
2K + U ~ 0
2K ~ 3PV
HighTemperaturePhase
U ~ U
Low Temperature Phase
Microcanonical
Canonical EnsemblePhase Transition in the
Mean field description of many-body systems
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
N !
∫
d3Nxd3Npδ(E −H) ∝1
N !
∫
d3Nx
[
E −1
2
∑
i,j
U(xi, xj)
]3N2
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
N !
∫
d3Nxd3Npδ(E −H) ∝1
N !
∫
d3Nx
[
E −1
2
∑
i,j
U(xi, xj)
]3N2
• 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).
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
N !
∫
d3Nxd3Npδ(E −H) ∝1
N !
∫
d3Nx
[
E −1
2
∑
i,j
U(xi, xj)
]3N2
• 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).
• This gives:
eS ≈
∞∑
n1=1
∞∑
n2=1
....
∞∑
nM=1
δ
(
N −∑
a
na
)
expS[{na}]
S[{na}] =3N
2ln
[
E −1
2
M∑
a,b
naU(xa, xb)nb
]
−
M∑
a=1
na ln
(
naM
eV
)
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
N !
∫
d3Nxd3Npδ(E −H) ∝1
N !
∫
d3Nx
[
E −1
2
∑
i,j
U(xi, xj)
]3N2
• 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).
• This gives:
eS ≈
∞∑
n1=1
∞∑
n2=1
....
∞∑
nM=1
δ
(
N −∑
a
na
)
expS[{na}]
S[{na}] =3N
2ln
[
E −1
2
M∑
a,b
naU(xa, xb)nb
]
−M∑
a=1
na ln
(
naM
eV
)
• The mean field approximation: retain in the sum only the term for which the
summand reaches the maximum value∑
{na}
eS[na] ≈ eS[na,max]
Mean field description of self-gravitating systems
Mean field description of self-gravitating systems
• Take the continuum limit with
na,maxM
V= ρ(xa);
M∑
a=1
→M
V
∫
.
gives
ρ(x) = A exp(−βφ(x)); where φ(x) =
∫
d3yU(x, y)ρ(y)
Mean field description of self-gravitating systems
• Take the continuum limit with
na,maxM
V= ρ(xa);
M∑
a=1
→M
V
∫
.
gives
ρ(x) = A exp(−βφ(x)); where φ(x) =
∫
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.
Mean field description of self-gravitating systems
• Take the continuum limit with
na,maxM
V= ρ(xa);
M∑
a=1
→M
V
∫
.
gives
ρ(x) = A exp(−βφ(x)); where φ(x) =
∫
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
ANTONOV INSTABILITY
1. Systems with (RE/GM2) < −0.335 cannot evolve into isothermal
spheres. Entropy has no extremum for such systems.
ANTONOV INSTABILITY
1. Systems with (RE/GM2) < −0.335 cannot evolve into isothermal
spheres. Entropy has no extremum for such systems.
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
1. Systems with (RE/GM2) < −0.335 cannot evolve into isothermal
spheres. Entropy has no extremum for such systems.
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.
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
1. Systems with (RE/GM2) < −0.335 cannot evolve into isothermal
spheres. Entropy has no extremum for such systems.
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.
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 =
GM0
L0
with dimensionless variables:
x ≡r
L0
, n ≡ρ
ρc, m =
M (r)
M0
, 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 =
GM0
L0
with dimensionless variables:
x ≡r
L0
, n ≡ρ
ρc, m =
M (r)
M0
, y ≡ β [φ− φ (0)] .
• Then1
x2
d
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 =
GM0
L0
with dimensionless variables:
x ≡r
L0
, n ≡ρ
ρc, m =
M (r)
M0
, y ≡ β [φ− φ (0)] .
• Then1
x2
d
dx(x2dy
dx) = e−y; y(0) = y′(0) = 0
• Singular solution:
n =(
2/x2)
,m = 2x, y = 2 lnx
LANE-EMDEN VARIABLES
• Invariant under the transformation y → y + a ; x→ kx with k2 = ea.
LANE-EMDEN VARIABLES
• Invariant under the transformation y → y + a ; x→ kx with k2 = ea.
• Reduce the degree by choosing:
v ≡m
x; u ≡
nx3
m=nx2
v.
LANE-EMDEN VARIABLES
• Invariant under the transformation y → y + a ; x→ kx with k2 = ea.
• Reduce the degree by choosing:
v ≡m
x; u ≡
nx3
m=nx2
v.
u
v
dv
du= −
(u− 1)
(u+ v − 3); v(u = 3) = 0; v′(u = 3) = −
5
3
LANE-EMDEN VARIABLES
• Invariant under the transformation y → y + a ; x→ kx with k2 = ea.
• Reduce the degree by choosing:
v ≡m
x; u ≡
nx3
m=nx2
v.
u
v
dv
du= −
(u− 1)
(u+ v − 3); v(u = 3) = 0; v′(u = 3) = −
5
3
• The singular point is at: us = 1, vs = 2; corresponding to the
solution n = (2/x2),m = 2x.
LANE-EMDEN VARIABLES
• Invariant under the transformation y → y + a ; x→ kx with k2 = ea.
• Reduce the degree by choosing:
v ≡m
x; u ≡
nx3
m=nx2
v.
u
v
dv
du= −
(u− 1)
(u+ v − 3); v(u = 3) = 0; v′(u = 3) = −
5
3
• The singular point is at: us = 1, vs = 2; corresponding to the
solution n = (2/x2),m = 2x.
• All solutions tend to this asymptotically for large r by spiralling
around the singular point in the u− v plane.
LANE-EMDEN VARIABLES
• Invariant under the transformation y → y + a ; x→ kx with k2 = ea.
• Reduce the degree by choosing:
v ≡m
x; u ≡
nx3
m=nx2
v.
u
v
dv
du= −
(u− 1)
(u+ v − 3); v(u = 3) = 0; v′(u = 3) = −
5
3
• The singular point is at: us = 1, vs = 2; corresponding to the
solution n = (2/x2),m = 2x.
• 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 = −
∫ R
0
GM(r)
r
dM
drdr = −
GM20
L0
∫ x0
0
mnxdx
K =3
2
M
β=
3
2
GM20
L0
m(x0) =GM2
0
L0
3
2
∫ x0
0
nx2dx; x0 = R/L0
ENERGY OF THE ISOTHERMAL SPHERE
• The potential and kinetic energies are
U = −
∫ R
0
GM(r)
r
dM
drdr = −
GM20
L0
∫ x0
0
mnxdx
K =3
2
M
β=
3
2
GM20
L0
m(x0) =GM2
0
L0
3
2
∫ x0
0
nx2dx; x0 = R/L0
• So:
E = K + U =GM2
0
2L0
∫ x0
0
dx(3nx2 − 2mnx)
=GM2
0
2L0
∫ x0
0
dxd
dx{2nx3 − 3m} =
GM20
L0
{n0x30 −
3
2m0}
ENERGY OF THE ISOTHERMAL SPHERE
• The potential and kinetic energies are
U = −
∫ R
0
GM(r)
r
dM
drdr = −
GM20
L0
∫ x0
0
mnxdx
K =3
2
M
β=
3
2
GM20
L0
m(x0) =GM2
0
L0
3
2
∫ x0
0
nx2dx; x0 = R/L0
• So:
E = K + U =GM2
0
2L0
∫ x0
0
dx(3nx2 − 2mnx)
=GM2
0
2L0
∫ x0
0
dxd
dx{2nx3 − 3m} =
GM20
L0
{n0x30 −
3
2m0}
• The combination (RE/GM2) is a function of (u, v) alone.
λ =RE
GM2=
1
v0
{u0 −3
2}.
ENERGY OF THE ISOTHERMAL SPHERE
• The potential and kinetic energies are
U = −
∫ R
0
GM(r)
r
dM
drdr = −
GM20
L0
∫ x0
0
mnxdx
K =3
2
M
β=
3
2
GM20
L0
m(x0) =GM2
0
L0
3
2
∫ x0
0
nx2dx; x0 = R/L0
• So:
E = K + U =GM2
0
2L0
∫ x0
0
dx(3nx2 − 2mnx)
=GM2
0
2L0
∫ x0
0
dxd
dx{2nx3 − 3m} =
GM20
L0
{n0x30 −
3
2m0}
• The combination (RE/GM2) is a function of (u, v) alone.
λ =RE
GM2=
1
v0
{u0 −3
2}.
• An isothermal sphere must lie on the curve
v =1
λ
(
u−3
2
)
; λ ≡RE
GM2
[T. Padmanabhan, Physics Reports, 188, 285 (1990).]
COLLISIONAL RELAXATION IN GRAVITATING SYSTEMS
COLLISIONAL RELAXATION IN GRAVITATING SYSTEMS
• Transverse velocity in an encounter:
δv⊥ ≃(
Gm/b2)
(2b/v) = (2Gm/bv)
COLLISIONAL RELAXATION IN GRAVITATING SYSTEMS
• Transverse velocity in an encounter:
δv⊥ ≃(
Gm/b2)
(2b/v) = (2Gm/bv)
• ”Hard” collisions: ∆v⊥ ≈ v; so b ≤ bc = Gm/v2. The relevant relaxation timescale:
thard ≃1
(nσv)≃
R3v3
N (G2m2)≃NR3v3
G2M2≈ N(R/v)
COLLISIONAL RELAXATION IN GRAVITATING SYSTEMS
• Transverse velocity in an encounter:
δv⊥ ≃(
Gm/b2)
(2b/v) = (2Gm/bv)
• ”Hard” collisions: ∆v⊥ ≈ v; so b ≤ bc = Gm/v2. The relevant relaxation timescale:
thard ≃1
(nσv)≃
R3v3
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.
COLLISIONAL RELAXATION IN GRAVITATING SYSTEMS
• Transverse velocity in an encounter:
δv⊥ ≃(
Gm/b2)
(2b/v) = (2Gm/bv)
• ”Hard” collisions: ∆v⊥ ≈ v; so b ≤ bc = Gm/v2. The relevant relaxation timescale:
thard ≃1
(nσv)≃
R3v3
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⊥)2〉total ≃ ∆t
∫ b2
b1
(2πbdb) (vn)
(
G2m2
b2v2
)
=2πnG2m2
v∆t ln
(
b2
b1
)
.
COLLISIONAL RELAXATION IN GRAVITATING SYSTEMS
• Transverse velocity in an encounter:
δv⊥ ≃(
Gm/b2)
(2b/v) = (2Gm/bv)
• ”Hard” collisions: ∆v⊥ ≈ v; so b ≤ bc = Gm/v2. The relevant relaxation timescale:
thard ≃1
(nσv)≃
R3v3
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⊥)2〉total ≃ ∆t
∫ b2
b1
(2πbdb) (vn)
(
G2m2
b2v2
)
=2πnG2m2
v∆t ln
(
b2
b1
)
.
• Take b2 = R= size of the system, b1 = bc. Then
(b2/b1) ≃(
Rv2/Gm)
= N(
Rv2/GM)
≃ N
in virial equilibrium. So
tsoft ≃v3
2πG2m2n lnN≃
(
N
lnN
)(
R
v
)
≃
(
thard
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
SECOND ASPECT OF DIFFUSION IN VELOCITY SPACE
• This can’t be the whole story!
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.
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.
• Chandra seems to have realized this soon after the
publication of the book!!
DIFFUSION IN VELOCITY SPACE:
A UNIFIED APPROACH
DIFFUSION IN VELOCITY SPACE:
A UNIFIED APPROACH
• Diffusion current as the source term:
df
dt=∂f
∂t+ v.
∂f
∂x−∇φ.
∂f
∂v= −
∂Jα
∂pα
DIFFUSION IN VELOCITY SPACE:
A UNIFIED APPROACH
• Diffusion current as the source term:
df
dt=∂f
∂t+ v.
∂f
∂x−∇φ.
∂f
∂v= −
∂Jα
∂pα
• Form of the current can be shown to be:
Jα =B0
2
∫
d l′{
f∂f ′
∂lβ− f ′∂f
∂lβ
}
.
{
δαβ
k−kαkβ
k3
}
where B0 = 4πG2m5L; and L =b2∫
b1
dbb
= ln(
b2b1
)
TWO FOR THE PRICE OF ONE!
• Current:
Jα =B0
2
∫
d l′{
f∂f ′
∂lβ− f ′ ∂f
∂lβ
}
.
{
δαβ
k−kαkβ
k3
}
TWO FOR THE PRICE OF ONE!
• Current:
Jα =B0
2
∫
d l′{
f∂f ′
∂lβ− f ′ ∂f
∂lβ
}
.
{
δαβ
k−kαkβ
k3
}
• The term proportional to f gives dynamical friction.
The term proportional to (∂f/∂lβ) increases the velocity
dispersion.
TWO FOR THE PRICE OF ONE!
• Current:
Jα =B0
2
∫
d l′{
f∂f ′
∂lβ− f ′ ∂f
∂lβ
}
.
{
δαβ
k−kαkβ
k3
}
• 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
AN ILLUSTRATIVE TOY MODEL
• Note that
Jα(l) ≡ aα(l)f(l) −1
2
∂
∂lβ
{
σ2αβf}
where aα = (∂η/∂lα) , σ2αβ = (∂2ψ/∂lα∂lβ)
AN ILLUSTRATIVE TOY MODEL
• Note that
Jα(l) ≡ aα(l)f(l) −1
2
∂
∂lβ
{
σ2αβf}
where aα = (∂η/∂lα) , σ2αβ = (∂2ψ/∂lα∂lβ)
• Aside:
∇2ψ = η; ∇2l η(l) = ∇2
l
{
2
∫
dl′f(l′)
|l − l′|
}
= −8πf(l)
AN ILLUSTRATIVE TOY MODEL
• Note that
Jα(l) ≡ aα(l)f(l) −1
2
∂
∂lβ
{
σ2αβf}
where aα = (∂η/∂lα) , σ2αβ = (∂2ψ/∂lα∂lβ)
• Aside:
∇2ψ = η; ∇2l η(l) = ∇2
l
{
2
∫
dl′f(l′)
|l − l′|
}
= −8πf(l)
• Treat the coefficients as constants to understand the structure of
the equation:
∂f(v, t)
∂t=
∂
∂v
{
(αv)f +σ2
2
∂f
∂v
}
≡ −∂J
∂v
AN ILLUSTRATIVE TOY MODEL
• Note that
Jα(l) ≡ aα(l)f(l) −1
2
∂
∂lβ
{
σ2αβf}
where aα = (∂η/∂lα) , σ2αβ = (∂2ψ/∂lα∂lβ)
• Aside:
∇2ψ = η; ∇2l η(l) = ∇2
l
{
2
∫
dl′f(l′)|l − l′|
}
= −8πf(l)
• Treat the coefficients as constants to understand the structure of
the equation:
∂f(v, t)
∂t=
∂
∂v
{
(αv)f +σ2
2
∂f
∂v
}
≡ −∂J
∂v
• An initial distribution f(v, 0) = δD(v − v0) evolves to:
f(v, t) =
[
α
πσ2(1 − e−2αt)
]1/2
exp
[
−α(v − v0e−αt)2
σ2(1 − e−2αt)
]
• The mean velocity decays to zero:
< v >= v0e−αt
• 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) →
σ2
α
• 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) →
σ2
α
• The interplay between the two effects is obvious.
HISTORY: OVERLOOKING LANDAU (1936)!
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 –
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?
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?
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?
• Does the gravitational clustering at late stages wipe out the
memory of initial conditions ?
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?
• Does the gravitational clustering at late stages wipe out the
memory of initial conditions ?
• 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?
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?
• Does the gravitational clustering at late stages wipe out the
memory of initial conditions ?
• 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
a3(t)
∑
i
δD[x − xi(t)]
• Mean density:
ρb(t) ≡
∫
d3x
Vρ(x, t) =
m
a3(t)
(
N
V
)
=M
a3V=ρ0
a3
• Density contrast:
1 + δ(x, t) ≡ρ(x, t)
ρb=V
N
∑
i
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
with
Ak = 4πGρb
∫
d3k′
(2π)3δk′δk−k′
[
k.k′
k′2
]
Bk =
∫
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
with
Ak = 4πGρb
∫
d3k′
(2π)3δk′δk−k′
[
k.k′
k′2
]
Bk =
∫
d3q (k.xT )2 exp [−ik.xT (t, q)]
• Coupled exact equations:
φk + 4a
aφk = −
1
2a2
∫
d3p
(2π)3φ 1
2k+pφ 1
2k−p
[
(
k
2
)2
+ p2 − 2
(
k.p
k
)2]
+
(
3H20
2
)∫
d3q
a
(
k.x
k
)2
eik.x
x + 2a
ax = −
1
a2∇xφ = −
1
a2
∫
ikφk exp i(k · x);
“Renormalizability” of gravity
“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:
H20
(
ad2
da2+
7
2
d
da
)
φk = −2
3
∫
d3p(2π)3
φL12
k+pφL1
2k−p
[
(
k
2
)2
−
(
k.pk
)2]
−1
2
∫ ′ d3p
(2π)3φ1
2k+pφ1
2k−p
[
(
k
2
)2
+ p2 − 2
(
k.p
k
)2]
(T.P, 2002)
Evolution at large scales
Evolution at large scales
• Ignore the terms Ak and Bk. Then:
δk + 2a
aδk = 4πGρbδk
Evolution at large scales
• Ignore the terms Ak and Bk. Then:
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
Evolution at large scales
• Ignore the terms Ak and Bk. Then:
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
• 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.
Evolution at large scales
• Ignore the terms Ak and Bk. Then:
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
• 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
4 4
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
‘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)
‘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)
• In the quasilinear regime:
EQL(a, x) ≈ a2−n
n+4x−2n+5
n+4 ;
‘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)
• In the quasilinear regime:
EQL(a, x) ≈ a2−n
n+4x−2n+5
n+4 ;
• In the nonlinear regime:
ENL(a, x) ≈ a1−n
n+5x−2n+4
n+5 ;
‘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)
• In the quasilinear regime:
EQL(a, x) ≈ a2−n
n+4x−2n+5
n+4 ;
• In the nonlinear regime:
ENL(a, x) ≈ a1−n
n+5x−2n+4
n+5 ;
• 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
4 4
Lk−1
k3P(k)2a
aP k−12
aP k4 4
Lk−1
aP k−12
aP k−22
k3
P(k)
2a
k−1
aP 2
f(k)
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
Summary
• Chandra pioneered the use of statistical physics in the
study of gravitating systems.
Summary
• Chandra pioneered the use of statistical physics in the
study of gravitating systems.
• His approach to this subject shows the characteristic
rigour employed as a matter of policy rather than out of
necessity.
Summary
• Chandra pioneered the use of statistical physics in the
study of gravitating systems.
• 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)