Introduction
The Vlasov equation
Virial theorem
Summary
The Vlasov equation for cold dark matter and
gravity
Alaric Erschfeld
Ruprecht-Karls-Universität Heidelberg
Master seminar, 25.11.2016
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Introduction
The Vlasov equation
Virial theorem
Summary
Table of Contents
1
Introduction
2
The Vlasov equation
3
Virial theorem
4
Summary
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Introduction
The Vlasov equation
Virial theorem
Summary
Table of Contents
1
Introduction
2
The Vlasov equation
3
Virial theorem
4
Summary
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Introduction
The Vlasov equation
Virial theorem
Summary
Kinetic theory
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Introduction
The Vlasov equation
Virial theorem
Summary
Kinetic theory
Ensembles of particles
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Introduction
The Vlasov equation
Virial theorem
Summary
Kinetic theory
Ensembles of particles
Lagrangian system fully characterised by
~q = (q1 , ..., qd ), p~ = (p1 , ..., pd )
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Introduction
The Vlasov equation
Virial theorem
Summary
Kinetic theory
Ensembles of particles
Lagrangian system fully characterised by
~q = (q1 , ..., qd ), p~ = (p1 , ..., pd )
Statistical mechanics: evolution in phase-space cell
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Introduction
The Vlasov equation
Virial theorem
Summary
Phase-space distribution
Probability for system to be at (~q, p~),
dP (d) (t, ~q, p~) = f (d) (t, ~q, p~)dd qdd p
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Introduction
The Vlasov equation
Virial theorem
Summary
Phase-space distribution
Probability for system to be at (~q, p~),
dP (d) (t, ~q, p~) = f (d) (t, ~q, p~)dd qdd p
Interested in reduced phase-space distribution,
f (k) (t, q1 , ..., qk , p1 , ..., pk ) =
Z
Z
dqk+1 ...dqd
dpk+1 ...dpd f (d) (t, ~q, p~)
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Introduction
The Vlasov equation
Virial theorem
Summary
Liouville’s theorem
Distribution function conserved along its characteristics,
df (d)
∂f (d)
∂f (d)
∂f (d)
=
+ q̇i
+ ṗj
dt
∂t
∂qi
∂pj
=
∂f (d) ∂H ∂f (d) ∂H ∂f (d)
+
−
∂t
∂pi ∂qi
∂qj ∂pj
=0
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Introduction
The Vlasov equation
Virial theorem
Summary
BBGKY hierarchy
Integrate over d − k degrees of freedom
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Introduction
The Vlasov equation
Virial theorem
Summary
BBGKY hierarchy
Integrate over d − k degrees of freedom
f (k) depends on f (k+1)
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Introduction
The Vlasov equation
Virial theorem
Summary
BBGKY hierarchy
Integrate over d − k degrees of freedom
f (k) depends on f (k+1)
Closure condition needed
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Introduction
The Vlasov equation
Virial theorem
Summary
Boltzmann equation
One-particle distribution function,
f (t, ~q, p~)d3 qd3 p = dN
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Introduction
The Vlasov equation
Virial theorem
Summary
Boltzmann equation
One-particle distribution function,
f (t, ~q, p~)d3 qd3 p = dN
Boltzmann equation for Hamiltonian system,
∂f
~ p~ H · ∇
~ q~f − ∇
~ q~H · ∇
~ p~ f = C[f ]
+∇
∂t
∂f
p~ ~
~ p~ f = C[f ]
+
· ∇q~f + F~ · ∇
∂t
m
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Introduction
The Vlasov equation
Virial theorem
Summary
Table of Contents
1
Introduction
2
The Vlasov equation
3
Virial theorem
4
Summary
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Introduction
The Vlasov equation
Virial theorem
Summary
Vlasov equation for plasma
Vlasov’s original idea for a plasma with long-ranged Coulomb
interaction,
~
~ + ~v × B)
F~L = q(E
c
~ ~x f + q(E
~+
∂t f + ~v · ∇
~v
~ ·∇
~ p~ f = 0
× B)
c
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Introduction
The Vlasov equation
Virial theorem
Summary
Vlasov equation for gravity
Using the CBE with gravity for dark matter description,
~ ~x φ
f~g = −∇
~ ~x f − ∇
~ ~x φ · ∇
~ ~v f = 0
∂t f + ~v · ∇
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Introduction
The Vlasov equation
Virial theorem
Summary
Vlasov equation for gravity
Using the CBE with gravity for dark matter description,
~ ~x φ
f~g = −∇
~ ~x f − ∇
~ ~x φ · ∇
~ ~v f = 0
∂t f + ~v · ∇
On an expanding background
∂t f +
vi
∂ x f − a ∂ xi φ ∂ v i f = 0
a i
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Introduction
The Vlasov equation
Virial theorem
Summary
Vlasov transport equation
Weighted integration conservation equations,
Z
h
d3 v ∂t f +
N
iY
vi
v kj
∂ xi f − a ∂ xi φ ∂ v i f
=0
a
a
j=1
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Introduction
The Vlasov equation
Virial theorem
Summary
Vlasov transport equation
Weighted integration conservation equations,
Z
h
d3 v ∂t f +
N
iY
vi
v kj
∂ xi f − a ∂ xi φ ∂ v i f
=0
a
a
j=1
Solve moments for physical observables,
Z
d3 v f
N
Y
1 ...kN
v kj = n mk(N
)
j=1
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Introduction
The Vlasov equation
Virial theorem
Summary
Continuity equation
Zeroth moment of the transport equation,
Z
∂t
1
d vf + ∂xi
a
3
Z
3
d v f v − a (∂xi φ)
Z
Z
i
Z
d3 v ∂vi f = 0
d3 v f = n
d3 v f v i = nhv i i
Z
d3 v ∂vi f = 0
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Introduction
The Vlasov equation
Virial theorem
Summary
Continuity equation
Zeroth moment of the transport equation,
Z
∂t
1
d vf + ∂xi
a
3
Z
3
d v f v − a (∂xi φ)
Z
Z
i
Z
d3 v ∂vi f = 0
d3 v f = n
d3 v f v i = nhv i i
Z
d3 v ∂vi f = 0
yields,
∂t n + ∂xi (nui ) = 0
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Introduction
The Vlasov equation
Virial theorem
Summary
Jeans equation
First moment of the transport equation,
1
∂t
a
Z
1
d vf v + 2 ∂xi
a
j
3
Z
Z
Z
3
i j
d v f v v − (∂xi φ)
Z
d3 v (∂vi f )v j = 0
d3 v f v j = nhv j i
d3 v v i v j f = nhv i v j i = n[σ ij + hv i ihv j i]
Z
d3 v (∂vi f )v j = −nδ ij
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Introduction
The Vlasov equation
Virial theorem
Summary
Jeans equation
First moment of the transport equation,
1
∂t
a
Z
1
d vf v + 2 ∂xi
a
j
3
Z
Z
Z
3
i j
d v f v v − (∂xi φ)
Z
d3 v (∂vi f )v j = 0
d3 v f v j = nhv j i
d3 v v i v j f = nhv i v j i = n[σ ij + hv i ihv j i]
Z
d3 v (∂vi f )v j = −nδ ij
yields,
∂t uj + Huj + ui ∂xi uj = −∂xj φ −
1
∂ i (nσ̃ ij )
n x
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Introduction
The Vlasov equation
Virial theorem
Summary
Moments
Moments of a distribution function,
Z
d3 v f
N
Y
v kj
j=1
a
1 ...kN
= n mk(N
)
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Introduction
The Vlasov equation
Virial theorem
Summary
Moments
Moments of a distribution function,
Z
d3 v f
N
Y
v kj
j=1
a
1 ...kN
= n mk(N
)
Moment generating function,
M (~l) ≡
=
Z
d3 v f e
Z
d3 v f
li v i
a
X
m∈N0
1 li v i N
N! a
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Introduction
The Vlasov equation
Virial theorem
Summary
Moments
Moments of a distribution function,
Z
d3 v f
N
Y
v kj
1 ...kN
= n mk(N
)
a
j=1
Moment generating function,
M (~l) ≡
=
Z
d3 v f e
Z
d3 v f
li v i
a
X
m∈N0
1 li v i N
N! a
Moments,
∂lk1 ...∂lkN M ~l=0
k1 ...kN
= n m(N
)
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Introduction
The Vlasov equation
Virial theorem
Summary
Cumulants
Cumulant generating function,
C(~l) = ln(M (~l))
∂lk1 ...∂lkN C ~l=0
1 ...kN
= ck(N
)
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Introduction
The Vlasov equation
Virial theorem
Summary
Cumulants
Cumulant generating function,
C(~l) = ln(M (~l))
∂lk1 ...∂lkN C ~l=0
1 ...kN
= ck(N
)
Equations of motion,
~ ~)C + ∇
~ ~x C · ∇
~ ~ C + (∇
~ ~x · ∇
~ ~)C = −~l · ∇
~ ~x φ
∂t C + H(~l · ∇
l
l
l
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Introduction
The Vlasov equation
Virial theorem
Summary
Energy equation
~l = 0 and first derivatives yield,
∂t n + ∂xi (nui ) = 0
∂t uj + Huj + ui ∂xi uj = −∂xj φ −
1
∂ i (nσ̃ ij )
n x
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Introduction
The Vlasov equation
Virial theorem
Summary
Energy equation
~l = 0 and first derivatives yield,
∂t n + ∂xi (nui ) = 0
∂t uj + Huj + ui ∂xi uj = −∂xj φ −
1
∂ i (nσ̃ ij )
n x
Applying second derivatives,
1
∂t σ̃ ij +2H σ̃ ij +(uk ∂xk )σ̃ ij + σ̃ jk ∂xk ui + σ̃ ik ∂xk uj = − ∂xk (nΠ̃ijk )
n
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Introduction
The Vlasov equation
Virial theorem
Summary
Orbit crossing
Perfect pressureless fluid approximation,
f (~x, ~v , t) = n(~x, t)δD (v i − aui (~x, t))
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Introduction
The Vlasov equation
Virial theorem
Summary
Orbit crossing
Perfect pressureless fluid approximation,
f (~x, ~v , t) = n(~x, t)δD (v i − aui (~x, t))
yields,
C(~x, t, l) = ln[n(~x, t)] + li ui (~x, t)
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Introduction
The Vlasov equation
Virial theorem
Summary
Orbit crossing
Generation of multiple streams due to orbit crossing. Figure taken from [3]
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Introduction
The Vlasov equation
Virial theorem
Summary
Orbit crossing
Resulting cumulant generating function,
h
C(~x, t, l) = ln
X
~
ns el·~us
i
streams at ~
x
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Introduction
The Vlasov equation
Virial theorem
Summary
Table of Contents
1
Introduction
2
The Vlasov equation
3
Virial theorem
4
Summary
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Introduction
The Vlasov equation
Virial theorem
Summary
Virial theorem
Jeans equation,
1
1
∂t (ρhv j i) + 2 ∂xk (ρhv k v j i) + ρ∂xj φ = 0
a
a
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Introduction
The Vlasov equation
Virial theorem
Summary
Virial theorem
Jeans equation,
1
1
∂t (ρhv j i) + 2 ∂xk (ρhv k v j i) + ρ∂xj φ = 0
a
a
weighted integration,
1
a
Z
1
d x x ∂t (ρhv i) = − 2
a
3
i
j
Z
3
i
k j
Z
d x x ∂xk (ρhv v i)−
d3 x xi ρ∂xj φ
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Introduction
The Vlasov equation
Virial theorem
Summary
Virial theorem
Inertial-tensor,
Z
1
d3 x xi ∂t (ρhv j i) + xj ∂t (ρhv i i)
2
d2 I ij
=
dt2
d3 x xi ∂t (ρhv j i) =
Z
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Introduction
The Vlasov equation
Virial theorem
Summary
Virial theorem
Kinetic energy,
Z
3
i
k j
d x x ∂xk (ρhv v i) =
Z
=−
=−
i
3
k j
d x ∂xk (x ρhv v i) −
Z
Z
d3 x ρhv i v j i
Z
d3 x ρ[hv i ihv j i + σ ij ]
d3 x ρhv k v j i∂xk xi
= −T ij − Πij
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Introduction
The Vlasov equation
Virial theorem
Summary
Virial theorem
Potential energy,
−
Z
d3 x xi ρ∂xj φ = U ij
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Introduction
The Vlasov equation
Virial theorem
Summary
Virial theorem
Tensorial virial theorem,
d2 I˜ij
= T̃ ij + Π̃ij + U ij
dt2
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Introduction
The Vlasov equation
Virial theorem
Summary
Table of Contents
1
Introduction
2
The Vlasov equation
3
Virial theorem
4
Summary
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Introduction
The Vlasov equation
Virial theorem
Summary
Summary
Kinetic theory provides Boltzmann equation
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Introduction
The Vlasov equation
Virial theorem
Summary
Summary
Kinetic theory provides Boltzmann equation
CBE and gravity yield the Vlasov equation for gravity and
dark matter
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Introduction
The Vlasov equation
Virial theorem
Summary
Summary
Kinetic theory provides Boltzmann equation
CBE and gravity yield the Vlasov equation for gravity and
dark matter
VE describes time evolution of the distribution function in
phase-space
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Introduction
The Vlasov equation
Virial theorem
Summary
Summary
Kinetic theory provides Boltzmann equation
CBE and gravity yield the Vlasov equation for gravity and
dark matter
VE describes time evolution of the distribution function in
phase-space
Approximations have to be well motivated since the VE is not
trivially solvable
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Introduction
The Vlasov equation
Virial theorem
Summary
Summary
Kinetic theory provides Boltzmann equation
CBE and gravity yield the Vlasov equation for gravity and
dark matter
VE describes time evolution of the distribution function in
phase-space
Approximations have to be well motivated since the VE is not
trivially solvable
Still many applications to the VE for gravity and dark matter
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Introduction
The Vlasov equation
Virial theorem
Summary
References I
[1] M. Bartelmann, Theoretical Astrophysics An Introduction
(WILEY-VCH, Weinheim, 2013)
[2] A. A. Vlasov, J. Exp. Theor. Phys. 8 (3), 291 (1938)
[3] S. Pueblas, R. Scoccimarro, Phys. Rev. D 80, 043504 (2009)
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