t. padmanabhan (iucaa, pune, india) · finite gravitating systems – general description – •...
TRANSCRIPT
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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
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Phases of Chandra’s life
[1939] [1942] [1950]
[1961] [1983] [1995]
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)r
Gm2
GRAVITATIONAL N−BODY PROBLEM
)
N particles, mass m, U(r) = −
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)r
Gm2
Non cosmological;no bg. expansion
Cosmological;bg. expansion a(t)
GRAVITATIONAL N−BODY PROBLEM
)
N particles, mass m, U(r) = −
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)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) = −
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)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) = −
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)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) = −
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)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) = −
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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
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BASICS OF STATISTICAL MECHANICS
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BASICS OF STATISTICAL MECHANICS
• Density of states:
Γ(E) =
∫
dpdqθD[E −H(p, q)]; g(E) =dΓ(E)
dE
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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
)
.
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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
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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 )
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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).
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THE TROUBLE WITH GRAVITATING SYSTEMS
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THE TROUBLE WITH GRAVITATING SYSTEMS
• Long range interaction with no shielding. Energy is not
extensive.
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THE TROUBLE WITH GRAVITATING SYSTEMS
• Long range interaction with no shielding. Energy is not
extensive.
• Microcanonical and Canonical descriptions are not
equivalent.
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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.
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FINITE GRAVITATING SYSTEMS
– GENERAL DESCRIPTION –
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FINITE GRAVITATING SYSTEMS
– GENERAL DESCRIPTION –
• Typical systems have two natural energy scales: E1 = (−Gm2/a) and
E2 = (−Gm2/R); a ≪ R.
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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.
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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 .
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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.
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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.
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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.
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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.
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GENERIC BEHAVIOUR
T. Padmanabhan, Physics Reports, 188, 285 (1990).
EnsembleMicrocanonical
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GENERIC BEHAVIOUR
T. Padmanabhan, Physics Reports, 188, 285 (1990).
Ensemble
0U ~ U
Low Temperature Phase
Microcanonical
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GENERIC BEHAVIOUR
T. Padmanabhan, Physics Reports, 188, 285 (1990).
Ensemble
0
2K ~ 3PV
HighTemperaturePhase
U ~ U
Low Temperature Phase
Microcanonical
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GENERIC BEHAVIOUR
T. Padmanabhan, Physics Reports, 188, 285 (1990).
Negative SpecificHeat
0
2K + U ~ 0
2K ~ 3PV
HighTemperaturePhase
U ~ U
Low Temperature Phase
MicrocanonicalEnsemble
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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
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Mean field description of many-body systems
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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
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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).
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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
)
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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]
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Mean field description of self-gravitating systems
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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)
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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.
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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.
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ANTONOV INSTABILITY
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ANTONOV INSTABILITY
1. Systems with (RE/GM2) < −0.335 cannot evolve into isothermal
spheres. Entropy has no extremum for such systems.
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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.
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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.
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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).
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Isothermal sphere
• Basic equation:
∇2φ = 4πGρce−β[φ(x)−φ(0)]; ρc = ρ(0)
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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)] .
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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
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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
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LANE-EMDEN VARIABLES
• Invariant under the transformation y → y + a ; x→ kx with k2 = ea.
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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.
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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
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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.
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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.
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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.
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Chandra (1939) Introduction to the study of stellar structure
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[D.Lynden-Bell, R. Wood, (1968), MNRAS, 138, p.495.]
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Chandra’s comments on Emden’s work
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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
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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}
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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}.
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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
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[T. Padmanabhan, Physics Reports, 188, 285 (1990).]
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COLLISIONAL RELAXATION IN GRAVITATING SYSTEMS
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COLLISIONAL RELAXATION IN GRAVITATING SYSTEMS
• Transverse velocity in an encounter:
δv⊥ ≃(
Gm/b2)
(2b/v) = (2Gm/bv)
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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)
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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.
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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
)
.
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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
)
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Publication date: 1942
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Pages 48 to 73 gives the derivation of TE and TD!
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Pages 317 to 320 contain the derivation by Jeans!
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First appearance of lnN
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CHANDRA’S COMMENT ON THE WORK OF JEANS
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SECOND ASPECT OF DIFFUSION IN VELOCITY SPACE
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SECOND ASPECT OF DIFFUSION IN VELOCITY SPACE
• This can’t be the whole story!
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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.
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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!!
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DIFFUSION IN VELOCITY SPACE:
A UNIFIED APPROACH
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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
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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
)
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TWO FOR THE PRICE OF ONE!
• Current:
Jα =B0
2
∫
d l′{
f∂f ′
∂lβ− f ′ ∂f
∂lβ
}
.
{
δαβ
k−kαkβ
k3
}
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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.
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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!
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AN ILLUSTRATIVE TOY MODEL
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AN ILLUSTRATIVE TOY MODEL
• Note that
Jα(l) ≡ aα(l)f(l) −1
2
∂
∂lβ
{
σ2αβf}
where aα = (∂η/∂lα) , σ2αβ = (∂2ψ/∂lα∂lβ)
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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)
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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
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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)
]
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• The mean velocity decays to zero:
< v >= v0e−αt
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• 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
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• 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.
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HISTORY: OVERLOOKING LANDAU (1936)!
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HISTORY: OVERLOOKING LANDAU (1936)!
Footnote is incorrect; Landau’s expression has both dynamical friction and velocity
dispersion!
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GRAVITATIONAL CLUSTERING IN EXPANDING UNIVERSE
– SOME KEY ISSUES –
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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?
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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?
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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 ?
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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?
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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?
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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)
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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)]
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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);
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“Renormalizability” of gravity
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“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)
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“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)
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Evolution at large scales
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Evolution at large scales
• Ignore the terms Ak and Bk. Then:
δk + 2a
aδk = 4πGρbδk
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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
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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.
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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)
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Bagla, T.P, (1997) MNRAS, 286, 1023
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k3P(k)2a
Lk−1
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k3P(k)2a aP k
4 4
Lk−1
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LINEAR REGIME
QUASI−LINEAR REGIME
Prediction from stable clustering
NON−LINEAR REGIME
Bagla, Engineer, TP (1998), Ap.J.,495, 25
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‘EQUIPARTITION’ OF ENERGY FLOW
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‘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)
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‘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 ;
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‘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 ;
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‘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.
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k3P(k)2a aP k
4 4
Lk−1
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k3P(k)2a
aP k−12
aP k4 4
Lk−1
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aP k−12
aP k−22
k3
P(k)
2a
k−1
aP 2
f(k)
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Injected initial power spectrum
P(k) ~ k −1
Bagla, T.P, (1997) MNRAS, 286, 1023
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Late time power spectra for 3 differentinitial injected spectra
P(k) ~ k −1
Bagla, T.P, (1997) MNRAS, 286, 1023
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Asymptotic universality in nonlinear clustering
(T.P., S. Ray, 2006)
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Summary
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Summary
• Chandra pioneered the use of statistical physics in the
study of gravitating systems.
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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.
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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.
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Acknowledgements
• D. Lynden-Bell (IOA)
• D. Narasimha (TIFR)
• R. Nityananda (NCRA)
• S. Sridhar (RRI)
• K. Subramanian (IUCAA)