SPECTRAL THEORY FOR NEUTRON TRANSPORT
INTRODUCTION
Mustapha Mokhtar-Kharroubi
(In memory of Seiji Ukaï)
Chapter 1
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Aim of the lectures
The aim of these lectures is twofold:
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Aim of the lectures
The aim of these lectures is twofold:
We provide an introduction to spectral theory of
non-self-adjoint operators in Banach spaces
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Aim of the lectures
The aim of these lectures is twofold:
We provide an introduction to spectral theory of
non-self-adjoint operators in Banach spaces
We show how neutron transport …ts into this general theory
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Outline
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Outline
Besides a formal introduction given in Chapter 1
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Outline
Besides a formal introduction given in Chapter 1
Chapter 2: Fundamentals in spectral theory
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Outline
Besides a formal introduction given in Chapter 1
Chapter 2: Fundamentals in spectral theory
Chapter 3: Spectral analysis of weighted shift semigroups
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Outline
Besides a formal introduction given in Chapter 1
Chapter 2: Fundamentals in spectral theory
Chapter 3: Spectral analysis of weighted shift semigroups
Chapter 4: Spectra of perturbed operators with application to
transport theory
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Abstract of Chapter 1
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Abstract of Chapter 1
In this introductory chapter, we outline various models used in
nuclear reactor theory and some important spectral results already
obtained in the …fties and sixties.
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What transport theory is about ?
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What transport theory is about ?
Transport theory provides a statistical description of large populations
of "particles" moving in a host medium.
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For instance
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For instance
The transport of neutrons through the uranium fuel elements of a nuclear
reactor.
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Pulsed neutron experiments:
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Pulsed neutron experiments: injection of a burst of fast neutrons into a
sample of material followed by a measurement of the time decay of the
neutron population in view, e.g. of cross section information.
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The transport of photons through planetary or stellar atmospheres
(radiative transfert)
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The transport of photons through planetary or stellar atmospheres
(radiative transfert) or light transport in tissues in diagnostic medicine
(e.g. in computerized tomography).
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The motion of gas molecules colliding with each another (gas
dynamics).
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The motion of charged particles (ions in plasma physics or electrons in
semiconductor theory...) accelerated by external (e.g. electric) …elds.
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Various kinetic equations (structured population models) in
biology....
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On linearity
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On linearity
In a nuclear reactor, the proportion of neutrons with respect to the
atomes of the host medium, is in…nitesimal (about 10 11 ),
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On linearity
In a nuclear reactor, the proportion of neutrons with respect to the
atomes of the host medium, is in…nitesimal (about 10 11 ),
so the possible collisions between neutrons are negligible in
comparison with the collisions of neutrons with the atomes of the
host material.
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On linearity
In a nuclear reactor, the proportion of neutrons with respect to the
atomes of the host medium, is in…nitesimal (about 10 11 ),
so the possible collisions between neutrons are negligible in
comparison with the collisions of neutrons with the atomes of the
host material.
Thus (in absence of feedback temperature) neutron transport
equations (as well as radiative transfert equations for photons) are
genuinely linear,
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On linearity
In a nuclear reactor, the proportion of neutrons with respect to the
atomes of the host medium, is in…nitesimal (about 10 11 ),
so the possible collisions between neutrons are negligible in
comparison with the collisions of neutrons with the atomes of the
host material.
Thus (in absence of feedback temperature) neutron transport
equations (as well as radiative transfert equations for photons) are
genuinely linear,
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On linearity
In a nuclear reactor, the proportion of neutrons with respect to the
atomes of the host medium, is in…nitesimal (about 10 11 ),
so the possible collisions between neutrons are negligible in
comparison with the collisions of neutrons with the atomes of the
host material.
Thus (in absence of feedback temperature) neutron transport
equations (as well as radiative transfert equations for photons) are
genuinely linear, in contrast, e.g. to Boltzmann equation in rare…ed
gas dynamics.
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Linearized Boltzmann equations are formaly similar to neutron
transport equations.
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Linearized Boltzmann equations are formaly similar to neutron
transport equations. There is however a big di¤erence:
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Linearized Boltzmann equations are formaly similar to neutron
transport equations. There is however a big di¤erence: the scattering
kernel is not nonnegative !!
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Density function
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Density function
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Density function
The population of particles is described by a density function
f (t, x, v )
(the density of particles at time t > 0 at position x and velocity v ).
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Density function
The population of particles is described by a density function
f (t, x, v )
(the density of particles at time t > 0 at position x and velocity v ).
In particular
Z Z
f (t, x, v )dxdv
is the (expected) number of particles at time t > 0.
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Density function
The population of particles is described by a density function
f (t, x, v )
(the density of particles at time t > 0 at position x and velocity v ).
In particular
Z Z
f (t, x, v )dxdv
is the (expected) number of particles at time t > 0.
L1 spaces are natural settings in transport theory !
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MODELS USED IN NUCLEAR REACTOR THEORY
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Inelastic model for neutron transport
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Inelastic model for neutron transport
∂f
∂f
+ v . + σ(x, v )f (t, x, v ) =
∂t
∂x
Z
V
k (x, v , v 0 )f (t, x, v 0 )dv 0
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Inelastic model for neutron transport
∂f
∂f
+ v . + σ(x, v )f (t, x, v ) =
∂t
∂x
with V = v 2 R3 ; c0
(x, v ) 2 Ω
jv j
c1
Z
V
V, Ω
(0
k (x, v , v 0 )f (t, x, v 0 )dv 0
R3
c0 < c1 < ∞ )
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Inelastic model for neutron transport
∂f
∂f
+ v . + σ(x, v )f (t, x, v ) =
∂t
∂x
with V = v 2 R3 ; c0
(x, v ) 2 Ω
jv j
c1
Z
V
V, Ω
(0
f (0, x, v ) = f0 (x, v )
k (x, v , v 0 )f (t, x, v 0 )dv 0
R3
c0 < c1 < ∞) and
(initial condition)
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Inelastic model for neutron transport
∂f
∂f
+ v . + σ(x, v )f (t, x, v ) =
∂t
∂x
with V = v 2 R3 ; c0
(x, v ) 2 Ω
jv j
c1
Z
V
V, Ω
(0
f (0, x, v ) = f0 (x, v )
k (x, v , v 0 )f (t, x, v 0 )dv 0
R3
c0 < c1 < ∞) and
(initial condition)
f (t, x, v )jΓ = 0 (boundary condition)
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Inelastic model for neutron transport
∂f
∂f
+ v . + σ(x, v )f (t, x, v ) =
∂t
∂x
with V = v 2 R3 ; c0
(x, v ) 2 Ω
jv j
c1
Z
V
k (x, v , v 0 )f (t, x, v 0 )dv 0
V, Ω
(0
f (0, x, v ) = f0 (x, v )
R3
c0 < c1 < ∞) and
(initial condition)
f (t, x, v )jΓ = 0 (boundary condition)
where
Γ := f(x, v ) 2 ∂Ω
V ; v .n(x ) < 0g
(n(x ) is the unit exterior normal at x 2 ∂Ω).
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Other boundary conditions
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Other boundary conditions
Periodic boundary conditions (Transport on the torus)
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Other boundary conditions
Periodic boundary conditions (Transport on the torus)
Boundary operator (more suitable for kinetic theory of gases)
f = H (f+ )
relating the outgoing and ingoing ‡uxes
f : = fj Γ
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Other boundary conditions
Periodic boundary conditions (Transport on the torus)
Boundary operator (more suitable for kinetic theory of gases)
f = H (f+ )
relating the outgoing and ingoing ‡uxes
f : = fj Γ
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Other boundary conditions
Periodic boundary conditions (Transport on the torus)
Boundary operator (more suitable for kinetic theory of gases)
f = H (f+ )
relating the outgoing and ingoing ‡uxes
f : = fj Γ
where
Γ := (x, v ) 2 ∂Ω
Γ+ := (x, v ) 2 ∂Ω
R3 ; v .n(x ) < 0
R3 ; v .n(x ) > 0 .
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Multiple scattering
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Multiple scattering
This physical model di¤ers from the previous reactor model by the fact
that
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Multiple scattering
This physical model di¤ers from the previous reactor model by the fact
that
Ω = R3
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Multiple scattering
This physical model di¤ers from the previous reactor model by the fact
that
Ω = R3
but
σ (x, v) and k(x, v, v0 ) are compactly supported in space.
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The presence of delayed neutrons
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The presence of delayed neutrons
Besides the prompt neutrons (appearing instantaneously in a …ssion
process), some neutrons appear after a time delay as a decay product of
radioactive …ssion fragments
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The presence of delayed neutrons
Besides the prompt neutrons (appearing instantaneously in a …ssion
process), some neutrons appear after a time delay as a decay product of
radioactive …ssion fragments
∂f
∂f
+ v . + σ(x, v )f (t, x, v ) =
∂t
∂x
Z
m
R3
k (x, v , v 0 )f (t, x, v 0 )dv 0 + ∑ λi gi
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i =1
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The presence of delayed neutrons
Besides the prompt neutrons (appearing instantaneously in a …ssion
process), some neutrons appear after a time delay as a decay product of
radioactive …ssion fragments
∂f
∂f
+ v . + σ(x, v )f (t, x, v ) =
∂t
∂x
Z
m
R3
k (x, v , v 0 )f (t, x, v 0 )dv 0 + ∑ λi gi
Z
dgi
= λi gi +
ki (x, v , v 0 )f (t, x, v 0 )dv 0
dt
R3
λi > 0 are the radioactive decay constants.
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i =1
(1
i
m)
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Multigroup models
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Multigroup models
(Motivation: numerical calculations)
m
∂fi
∂fi
+ v . + σi (x, v )fi (t, x, v ) = ∑
∂t
∂x
j =1
Z
Vj
ki,j (x, v , v 0 )fj (t, x, v 0 )µj (dv 0 ),
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Multigroup models
(Motivation: numerical calculations)
m
∂fi
∂fi
+ v . + σi (x, v )fi (t, x, v ) = ∑
∂t
∂x
j =1
Z
Vj
ki,j (x, v , v 0 )fj (t, x, v 0 )µj (dv 0 ),
the spheres
Vj := v 2 R3 , jv j = cj , 1
j
m, (cj > 0)
are endowed with surface measures µj and fi (t, x, v ) is the density of
neutrons (at time t > 0 located at x 2 ∂Ω) with velocity v 2 Vi .
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Partly inelastic models (Larsen and Zweifel 1974)
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Partly inelastic models (Larsen and Zweifel 1974)
∂f
∂f
+ v . + σ(x, v )f (t, x, v ) = Ke f + Kin f
∂t
∂x
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Partly inelastic models (Larsen and Zweifel 1974)
∂f
∂f
+ v . + σ(x, v )f (t, x, v ) = Ke f + Kin f
∂t
∂x
where
Kin f =
Z
R3
k (x, v , v 0 )f (x, v 0 )dv 0 (inelastic operator)
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Partly inelastic models (Larsen and Zweifel 1974)
∂f
∂f
+ v . + σ(x, v )f (t, x, v ) = Ke f + Kin f
∂t
∂x
where
Kin f =
Z
R3
k (x, v , v 0 )f (x, v 0 )dv 0 (inelastic operator)
and
Ke f =
Z
S2
k (x, ρ, ω, ω 0 )f (x, ρω 0 )dS (ω 0 ) (elastic operator)
where v = ρω.
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Di¤usive models
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Di¤usive models
∂f
by
4x
∂x
(Laplacian in the position variable x 2 Ω with Neumann or Dirichlet
boundary condition).
Replace v .
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Space homogeneous models
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Space homogeneous models
For instance
∂f
+ σ(v )f (t, v ) =
∂t
Z
R3
k (v , v 0 )f (t, v 0 )dv 0
(Drop the x variable !)
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Well-posedness
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Well-posedness
If we ignore scattering (i.e. k (x, v , v 0 ) = 0), the density of neutral
particles (e.g. neutrons) is governed by
∂f
∂f
+ v . + σ(x, v )f (t, x, v ) = 0
∂t
∂x
(with initial condition f0 ) and is solved explicitly by
f (t, x, v ) = e
Rt
0
σ (x τv ,v )d τ
f0 ( x
tv , v )1ft
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s (x ,v )g
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Well-posedness
If we ignore scattering (i.e. k (x, v , v 0 ) = 0), the density of neutral
particles (e.g. neutrons) is governed by
∂f
∂f
+ v . + σ(x, v )f (t, x, v ) = 0
∂t
∂x
(with initial condition f0 ) and is solved explicitly by
f (t, x, v ) = e
Rt
0
σ (x τv ,v )d τ
f0 ( x
tv , v )1ft
s (x ,v )g
with …rst exit time function
s (x, v ) = inf fs > 0; x
sv 2
/ Ωg
(method of characteristics).
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This de…nes a C0 -semigroup fU (t ); t > 0g on Lp (Ω
U (t ) : g ! e
Rt
0
σ(x τv ,v )d τ
g (x
tv , v )1ft
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R3 )
s (x ,v )g
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This de…nes a C0 -semigroup fU (t ); t > 0g on Lp (Ω
U (t ) : g ! e
Rt
0
σ(x τv ,v )d τ
g (x
tv , v )1ft
R3 )
s (x ,v )g
with generator T given by
D (T ) =
g 2 Lp (Ω
Tg =
v.
R3 ); v .
∂g
∂x
∂g
2 Lp , gjΓ = 0
∂x
σ(x, v )g (x, v ).
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Perturbation theory
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Perturbation theory
If the scattering operator
K :g !
is bounded in Lp (Ω
Z
R3
k (x, v , v 0 )g (x, v 0 )dv 0
R3 ) then
A := T + K
(D (A) = D (T ))
generates a C0 -semigroup fV (t ); t > 0g .
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Two basic eigenvalue problems
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Two basic eigenvalue problems
1
The "time eigenelements" (λ, g )
∂g
v.
∂x
σ(x, v )g (x, v ) +
Z
V
k (x, v , v 0 )g (x, v 0 )dv 0 = λg (x, v )
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Two basic eigenvalue problems
1
The "time eigenelements" (λ, g )
∂g
v.
∂x
σ(x, v )g (x, v ) +
Z
V
k (x, v , v 0 )g (x, v 0 )dv 0 = λg (x, v )
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Two basic eigenvalue problems
1
The "time eigenelements" (λ, g )
∂g
v.
∂x
σ(x, v )g (x, v ) +
Z
V
k (x, v , v 0 )g (x, v 0 )dv 0 = λg (x, v )
and their connection with time asymptotic behaviour (t ! +∞) of
fV (t ); t > 0g .
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Two basic eigenvalue problems
1
The "time eigenelements" (λ, g )
∂g
v.
∂x
2
σ(x, v )g (x, v ) +
Z
V
k (x, v , v 0 )g (x, v 0 )dv 0 = λg (x, v )
and their connection with time asymptotic behaviour (t ! +∞) of
fV (t ); t > 0g .
Criticality eigenvalue problem
Z
∂g
0 =
v.
σ(x, v )g (x, v ) +
ks (x, v , v 0 )g (x, v 0 )dv 0
∂x
V
Z
1
kf (x, v , v 0 )g (x, v 0 )dv 0
+
γ V
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Chapter 1
27 / 45
Two basic eigenvalue problems
1
The "time eigenelements" (λ, g )
∂g
v.
∂x
2
σ(x, v )g (x, v ) +
Z
V
k (x, v , v 0 )g (x, v 0 )dv 0 = λg (x, v )
and their connection with time asymptotic behaviour (t ! +∞) of
fV (t ); t > 0g .
Criticality eigenvalue problem
Z
∂g
0 =
v.
σ(x, v )g (x, v ) +
ks (x, v , v 0 )g (x, v 0 )dv 0
∂x
V
Z
1
kf (x, v , v 0 )g (x, v 0 )dv 0
+
γ V
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CIMPA School Muizemberg July 22-Aug 4
Chapter 1
27 / 45
Two basic eigenvalue problems
1
The "time eigenelements" (λ, g )
∂g
v.
∂x
2
σ(x, v )g (x, v ) +
Z
V
k (x, v , v 0 )g (x, v 0 )dv 0 = λg (x, v )
and their connection with time asymptotic behaviour (t ! +∞) of
fV (t ); t > 0g .
Criticality eigenvalue problem
Z
∂g
0 =
v.
σ(x, v )g (x, v ) +
ks (x, v , v 0 )g (x, v 0 )dv 0
∂x
V
Z
1
kf (x, v , v 0 )g (x, v 0 )dv 0
+
γ V
where ks (x, v , v 0 ) and kf (x, v , v 0 ) are the scattering kernel and the
…ssion kernel, see e.g. J. Mika (1971).
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Chapter 1
27 / 45
A model case (J. Lehner and G. Milton Wing 1955)
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Chapter 1
28 / 45
A model case (J. Lehner and G. Milton Wing 1955)
Af =
∂f
µ
∂x
σf (x, µ) + c
Z +1
1
f (x, µ0 )d µ0
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CIMPA School Muizemberg July 22-Aug 4
Chapter 1
28 / 45
A model case (J. Lehner and G. Milton Wing 1955)
Af =
∂f
µ
∂x
σf (x, µ) + c
(x, µ) 2 [ a, a]
Z +1
1
f (x, µ0 )d µ0
[ 1, +1]
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CIMPA School Muizemberg July 22-Aug 4
Chapter 1
28 / 45
A model case (J. Lehner and G. Milton Wing 1955)
Af =
∂f
µ
∂x
σf (x, µ) + c
(x, µ) 2 [ a, a]
Z +1
1
f (x, µ0 )d µ0
[ 1, +1]
f ( a, µ) = 0 (µ > 0); f (a, µ) = 0 (µ < 0).
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Chapter 1
28 / 45
A model case (J. Lehner and G. Milton Wing 1955)
Af =
∂f
µ
∂x
σf (x, µ) + c
(x, µ) 2 [ a, a]
Z +1
1
f (x, µ0 )d µ0
[ 1, +1]
f ( a, µ) = 0 (µ > 0); f (a, µ) = 0 (µ < 0).
Theorem
σg
fRe λ
σ (A)
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Chapter 1
28 / 45
A model case (J. Lehner and G. Milton Wing 1955)
Af =
∂f
µ
∂x
σf (x, µ) + c
(x, µ) 2 [ a, a]
Z +1
1
f (x, µ0 )d µ0
[ 1, +1]
f ( a, µ) = 0 (µ > 0); f (a, µ) = 0 (µ < 0).
Theorem
σg
fRe λ
σ(A) and σ (A) \ fRe λ > σg consists of a …nite
(nonempty) set of real eigenvalues with …nite algebraic multiplicities.
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CIMPA School Muizemberg July 22-Aug 4
Chapter 1
28 / 45
On Jorgens’paper (1958)
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Chapter 1
29 / 45
On Jorgens’paper (1958)
Theorem
Let Ω be bounded and convex, V = v 2 R3 ; c0
the scattering kernel k (., ., .) be bounded.
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jv j
c1 < ∞ and
Chapter 1
29 / 45
On Jorgens’paper (1958)
Theorem
Let Ω be bounded and convex, V = v 2 R3 ; c0
the scattering kernel k (., ., .) be bounded. If
jv j
c1 < ∞ and
c0 > 0.
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Chapter 1
29 / 45
On Jorgens’paper (1958)
Theorem
Let Ω be bounded and convex, V = v 2 R3 ; c0
the scattering kernel k (., ., .) be bounded. If
jv j
c1 < ∞ and
c0 > 0.
Then V (t ) is compact in L2 (Ω
V ) for t large enough.
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Chapter 1
29 / 45
On Jorgens’paper (1958)
Theorem
Let Ω be bounded and convex, V = v 2 R3 ; c0
the scattering kernel k (., ., .) be bounded. If
jv j
c1 < ∞ and
c0 > 0.
Then V (t ) is compact in L2 (Ω V ) for t large enough. In particular, for
any α 2 R
σ(A) \ fRe λ > αg
consists at most of …nitely many eigenvalues with …nite algebraic
multiplicities fλ1 , ....λm g with spectral projections fP1 , ....Pm g .
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Chapter 1
29 / 45
On Jorgens’paper (1958)
Theorem
Let Ω be bounded and convex, V = v 2 R3 ; c0
the scattering kernel k (., ., .) be bounded. If
jv j
c1 < ∞ and
c0 > 0.
Then V (t ) is compact in L2 (Ω V ) for t large enough. In particular, for
any α 2 R
σ(A) \ fRe λ > αg
consists at most of …nitely many eigenvalues with …nite algebraic
multiplicities fλ1 , ....λm g with spectral projections fP1 , ....Pm g . For some
ε > 0 and Dj := (T λj )Pj
m
V (t ) =
∑ e λ t e tD Pj + O (e βt )
j
j
( β < α ).
j =1
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Chapter 1
29 / 45
On small velocities
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Chapter 1
30 / 45
On small velocities
Theorem
(Albertoni-Montagnini 1966) Let Ω be bounded. We assume that V is
not bounded away from zero.
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Chapter 1
30 / 45
On small velocities
Theorem
(Albertoni-Montagnini 1966) Let Ω be bounded. We assume that V is
not bounded away from zero. Then:
(i) There exists λ > 0 such that
σ(T ) = fλ; Re λ
λ g.
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CIMPA School Muizemberg July 22-Aug 4
Chapter 1
30 / 45
On small velocities
Theorem
(Albertoni-Montagnini 1966) Let Ω be bounded. We assume that V is
not bounded away from zero. Then:
(i) There exists λ > 0 such that
σ(T ) = fλ; Re λ
(ii) fλ; Re λ
λ g
λ g.
σ (T + K ).
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Chapter 1
30 / 45
On small velocities
Theorem
(Albertoni-Montagnini 1966) Let Ω be bounded. We assume that V is
not bounded away from zero. Then:
(i) There exists λ > 0 such that
σ(T ) = fλ; Re λ
(ii) fλ; Re λ
λ g
λ g.
σ (T + K ).
For most physical models
λ = inf σ(x, v ).
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Chapter 1
30 / 45
Compactness results (Montagnini, Demeru, Ukaï,
Borysiewicz, Mika, Vidav)
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Chapter 1
31 / 45
Compactness results (Montagnini, Demeru, Ukaï,
Borysiewicz, Mika, Vidav)
If some power of (λ
T)
1K
is compact (Re λ >
σ(T + K ) \ fλ; Re λ >
λ ) then
λ g
consists at most of isolated eigenvalues with …nite algebraic multiplicities.
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CIMPA School Muizemberg July 22-Aug 4
Chapter 1
31 / 45
Compactness results (Montagnini, Demeru, Ukaï,
Borysiewicz, Mika, Vidav)
If some power of (λ
T)
1K
is compact (Re λ >
σ(T + K ) \ fλ; Re λ >
λ ) then
λ g
consists at most of isolated eigenvalues with …nite algebraic multiplicities.
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CIMPA School Muizemberg July 22-Aug 4
Chapter 1
31 / 45
Compactness results (Montagnini, Demeru, Ukaï,
Borysiewicz, Mika, Vidav)
If some power of (λ
T)
1K
is compact (Re λ >
σ(T + K ) \ fλ; Re λ >
λ ) then
λ g
consists at most of isolated eigenvalues with …nite algebraic multiplicities.
Tool: Analytic Fredholm alternative.
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CIMPA School Muizemberg July 22-Aug 4
Chapter 1
31 / 45
Compactness results (Montagnini, Demeru, Ukaï,
Borysiewicz, Mika, Vidav)
If some power of (λ
T)
1K
is compact (Re λ >
σ(T + K ) \ fλ; Re λ >
λ ) then
λ g
consists at most of isolated eigenvalues with …nite algebraic multiplicities.
Tool: Analytic Fredholm alternative.
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CIMPA School Muizemberg July 22-Aug 4
Chapter 1
31 / 45
Compactness results (Montagnini, Demeru, Ukaï,
Borysiewicz, Mika, Vidav)
If some power of (λ
T)
1K
is compact (Re λ >
σ(T + K ) \ fλ; Re λ >
λ ) then
λ g
consists at most of isolated eigenvalues with …nite algebraic multiplicities.
Tool: Analytic Fredholm alternative.
For most physical models (λ T ) 1 K is compact in L2 (Ω
(Vladimirov’s trick).
University of FrancheComté Besançon France (Institute)
CIMPA School Muizemberg July 22-Aug 4
V)
Chapter 1
31 / 45
Compactness results (Montagnini, Demeru, Ukaï,
Borysiewicz, Mika, Vidav)
If some power of (λ
T)
1K
is compact (Re λ >
σ(T + K ) \ fλ; Re λ >
λ ) then
λ g
consists at most of isolated eigenvalues with …nite algebraic multiplicities.
Tool: Analytic Fredholm alternative.
For most physical models (λ T ) 1 K is compact in L2 (Ω
(Vladimirov’s trick).
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CIMPA School Muizemberg July 22-Aug 4
V)
Chapter 1
31 / 45
Compactness results (Montagnini, Demeru, Ukaï,
Borysiewicz, Mika, Vidav)
If some power of (λ
T)
1K
is compact (Re λ >
σ(T + K ) \ fλ; Re λ >
λ ) then
λ g
consists at most of isolated eigenvalues with …nite algebraic multiplicities.
Tool: Analytic Fredholm alternative.
For most physical models (λ T ) 1 K is compact in L2 (Ω V )
(Vladimirov’s trick).
Compactness of K (λ T ) 1 K in weighted L1 space (Suhadolc
1969).
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CIMPA School Muizemberg July 22-Aug 4
Chapter 1
31 / 45
Compactness results (Montagnini, Demeru, Ukaï,
Borysiewicz, Mika, Vidav)
If some power of (λ
T)
1K
is compact (Re λ >
σ(T + K ) \ fλ; Re λ >
λ ) then
λ g
consists at most of isolated eigenvalues with …nite algebraic multiplicities.
Tool: Analytic Fredholm alternative.
For most physical models (λ T ) 1 K is compact in L2 (Ω V )
(Vladimirov’s trick).
Compactness of K (λ T ) 1 K in weighted L1 space (Suhadolc
1969).
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CIMPA School Muizemberg July 22-Aug 4
Chapter 1
31 / 45
Compactness results (Montagnini, Demeru, Ukaï,
Borysiewicz, Mika, Vidav)
If some power of (λ
T)
1K
is compact (Re λ >
σ(T + K ) \ fλ; Re λ >
λ ) then
λ g
consists at most of isolated eigenvalues with …nite algebraic multiplicities.
Tool: Analytic Fredholm alternative.
For most physical models (λ T ) 1 K is compact in L2 (Ω V )
(Vladimirov’s trick).
Compactness of K (λ T ) 1 K in weighted L1 space (Suhadolc
1969).
If
λ := sup fRe λ; λ 2 σ(T + K )g > λ
University of FrancheComté Besançon France (Institute)
CIMPA School Muizemberg July 22-Aug 4
Chapter 1
31 / 45
Compactness results (Montagnini, Demeru, Ukaï,
Borysiewicz, Mika, Vidav)
If some power of (λ
T)
1K
is compact (Re λ >
σ(T + K ) \ fλ; Re λ >
λ ) then
λ g
consists at most of isolated eigenvalues with …nite algebraic multiplicities.
Tool: Analytic Fredholm alternative.
For most physical models (λ T ) 1 K is compact in L2 (Ω V )
(Vladimirov’s trick).
Compactness of K (λ T ) 1 K in weighted L1 space (Suhadolc
1969).
If
λ := sup fRe λ; λ 2 σ(T + K )g > λ
then λ is the leading eigenvalue associated to a nonnegative
eigenfunction (Peripheral spectral theory via positivity arguments).
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Chapter 1
31 / 45
Absence of eigenvalues
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Chapter 1
32 / 45
Absence of eigenvalues
Theorem
(Albertoni-Montagnini 1966) Under "suitable assumptions",
σ(T + K ) \ fλ; Re λ > λ g = ? if the diameter of Ω is small enough.
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Chapter 1
32 / 45
Absence of eigenvalues
Theorem
(Albertoni-Montagnini 1966) Under "suitable assumptions",
σ(T + K ) \ fλ; Re λ > λ g = ? if the diameter of Ω is small enough.
Theorem
(Ukaï-Hiraoka 1972) If k (v , v 0 ) = k (jv j , jv 0 j) = 0 for jv j > jv 0 j
(superthermal particle transport: no upscattering) then
σ(T + K ) \ fλ; Re λ > λ g = ? 8 Ω.
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Chapter 1
32 / 45
Isotropic models
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Chapter 1
33 / 45
Isotropic models
Theorem
(Albertoni-Montagnini, Ukaï, Mika) If σ(x, v ) = σ(jv j) and
k (x, v , v 0 ) = k (jv j , v 0 ) = k ( v 0 , jv j)
then σ(T + K ) \ fλ; Re λ >
λ g
R.
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Chapter 1
33 / 45
Time asymptotic behaviour (Dunford calculus)
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CIMPA School Muizemberg July 22-Aug 4
Chapter 1
34 / 45
Time asymptotic behaviour (Dunford calculus)
1
V (t )f = lim
γ!+∞ 2i π
Z ρ +i γ
ρ iγ
e λt (λ
A)
1
fd λ (ρ > ω = type).
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CIMPA School Muizemberg July 22-Aug 4
Chapter 1
34 / 45
Time asymptotic behaviour (Dunford calculus)
1
V (t )f = lim
γ!+∞ 2i π
Z ρ +i γ
ρ iγ
e λt (λ
A)
1
fd λ (ρ > ω = type).
and
8ε > 0, σ(T + K ) \ fλ; Re λ >
λ + εg = fλ1 , ....λm g
(with spectral projections fP1 , ....Pm g) is …nite.
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CIMPA School Muizemberg July 22-Aug 4
Chapter 1
34 / 45
Time asymptotic behaviour (Dunford calculus)
1
V (t )f = lim
γ!+∞ 2i π
Z ρ +i γ
ρ iγ
e λt (λ
A)
1
fd λ (ρ > ω = type).
and
8ε > 0, σ(T + K ) \ fλ; Re λ >
λ + εg = fλ1 , ....λm g
(with spectral projections fP1 , ....Pm g) is …nite. If this set is not empty
then shift the path of integration and pick the residues
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CIMPA School Muizemberg July 22-Aug 4
Chapter 1
34 / 45
Time asymptotic behaviour (Dunford calculus)
Z ρ +i γ
1
V (t )f = lim
γ!+∞ 2i π
ρ iγ
e λt (λ
A)
1
fd λ (ρ > ω = type).
and
8ε > 0, σ(T + K ) \ fλ; Re λ >
λ + εg = fλ1 , ....λm g
(with spectral projections fP1 , ....Pm g) is …nite. If this set is not empty
then shift the path of integration and pick the residues
m
V (t )f =
∑ e λ t e tD Pj f
j
j
+ Of (e βt ) ( β <
λ + ε );
j =1
for f 2 D (A2 ); see, e.g. Borysiewicz and Mika (1969).
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Chapter 1
34 / 45
Time asymptotic behaviour (Dunford calculus)
Z ρ +i γ
1
V (t )f = lim
γ!+∞ 2i π
ρ iγ
e λt (λ
A)
1
fd λ (ρ > ω = type).
and
8ε > 0, σ(T + K ) \ fλ; Re λ >
λ + εg = fλ1 , ....λm g
(with spectral projections fP1 , ....Pm g) is …nite. If this set is not empty
then shift the path of integration and pick the residues
m
V (t )f =
∑ e λ t e tD Pj f
j
j
+ Of (e βt ) ( β <
λ + ε );
j =1
for f 2 D (A2 ); see, e.g. Borysiewicz and Mika (1969).
Drawback of the approach: we need smooth initial data.
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Chapter 1
34 / 45
Spectra of perturbed semigroups (Vidav 1970)
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Chapter 1
35 / 45
Spectra of perturbed semigroups (Vidav 1970)
V (t ) = ∑n∞=0 Un (t ) where U0 (t ) = U (t ) is the streaming semigroup and
Un + 1 ( t ) =
Z t
0
U (t
s )KUn (s )ds
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CIMPA School Muizemberg July 22-Aug 4
(n > 0).
Chapter 1
35 / 45
Spectra of perturbed semigroups (Vidav 1970)
V (t ) = ∑n∞=0 Un (t ) where U0 (t ) = U (t ) is the streaming semigroup and
Un + 1 ( t ) =
Z t
0
U (t
s )KUn (s )ds
(n > 0).
Theorem
(Vidav 1970) If some remainder term Rn (t ) := ∑j∞=n Uj (t ) is compact for
large t
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Chapter 1
35 / 45
Spectra of perturbed semigroups (Vidav 1970)
V (t ) = ∑n∞=0 Un (t ) where U0 (t ) = U (t ) is the streaming semigroup and
Un + 1 ( t ) =
Z t
0
U (t
s )KUn (s )ds
(n > 0).
Theorem
(Vidav 1970) If some remainder term Rn (t ) := ∑j∞=n Uj (t ) is compact for
n
o
large t then σ(V (t )) \ µ; jµj > e λ t consists at most of isolated
eigenvalues with …nite multiplicities.
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Chapter 1
35 / 45
Spectra of perturbed semigroups (Vidav 1970)
V (t ) = ∑n∞=0 Un (t ) where U0 (t ) = U (t ) is the streaming semigroup and
Un + 1 ( t ) =
Z t
U (t
0
s )KUn (s )ds
(n > 0).
Theorem
(Vidav 1970) If some remainder term Rn (t ) := ∑j∞=n Uj (t ) is compact for
n
o
large t then σ(V (t )) \ µ; jµj > e λ t consists at most of isolated
eigenvalues with …nite multiplicities. In particular,
8ε > 0, σ(T + K ) \ fλ; Re λ > λ + εg = fλ1 , ....λm g is …nite and
m
V (t ) =
∑ e λ t e tD Pj + O (e βt )
j
j
in operator norm
j =1
where β <
λ + ε.
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CIMPA School Muizemberg July 22-Aug 4
Chapter 1
35 / 45
E¤ective existence of a fundamental mode
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CIMPA School Muizemberg July 22-Aug 4
Chapter 1
36 / 45
E¤ective existence of a fundamental mode
see S. Ukaï and T. Hiraoka (1972) (isotropic case)
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CIMPA School Muizemberg July 22-Aug 4
Chapter 1
36 / 45
Probability generating function of neutron chain …ssions
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Chapter 1
37 / 45
Probability generating function of neutron chain …ssions
Conventional neutron transport theory deals with the expected (or mean
behaviour) of neutrons.
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CIMPA School Muizemberg July 22-Aug 4
Chapter 1
37 / 45
Probability generating function of neutron chain …ssions
Conventional neutron transport theory deals with the expected (or mean
behaviour) of neutrons. In order to describe the ‡uctuations from the
mean value of neutron populations, probabilistic formulations of neutron
chain …ssions were proposed very early, in particular by L. Pàl, G. I. Bell
and others.
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Chapter 1
37 / 45
In a multiplying medium occupying a region Ω a neutron interacting with
a nucleus of the host material may be absorbed
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CIMPA School Muizemberg July 22-Aug 4
Chapter 1
38 / 45
In a multiplying medium occupying a region Ω a neutron interacting with
a nucleus of the host material may be absorbed or scattered in random
directions
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CIMPA School Muizemberg July 22-Aug 4
Chapter 1
38 / 45
In a multiplying medium occupying a region Ω a neutron interacting with
a nucleus of the host material may be absorbed or scattered in random
directions or may produce (instantaneously) by a …ssion process more than
one neutron.
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CIMPA School Muizemberg July 22-Aug 4
Chapter 1
38 / 45
In a multiplying medium occupying a region Ω a neutron interacting with
a nucleus of the host material may be absorbed or scattered in random
directions or may produce (instantaneously) by a …ssion process more than
one neutron. The probability that a neutron located at x 2 Ω, with
velocity v , yields, by a …ssion process, i neutrons (1 i m ) with
velocities
v10 , v20 , ...vi0
is given by
ci (x, v , v10 , v20 , ...vi0 ),
(1
i
University of FrancheComté Besançon France (Institute)
CIMPA School Muizemberg July 22-Aug 4
m ).
Chapter 1
38 / 45
In a multiplying medium occupying a region Ω a neutron interacting with
a nucleus of the host material may be absorbed or scattered in random
directions or may produce (instantaneously) by a …ssion process more than
one neutron. The probability that a neutron located at x 2 Ω, with
velocity v , yields, by a …ssion process, i neutrons (1 i m ) with
velocities
v10 , v20 , ...vi0
is given by
ci (x, v , v10 , v20 , ...vi0 ),
(1
i
m ).
In particular
m
c0 (x, v ) +
∑
k =1
Z
Vk
ck (x, v , v10 , v20 , ...vk0 )dv10 ...dvk0 = 1
where c0 (x, v ) is the probability (for a neutron located at x 2 Ω, with
velocity v ) of being absorbed.
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Chapter 1
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Let
pj (tf , x, v , t )
j = 0, 1, ...
be the probability that a neutron, born at time t at position x 2 Ω with
velocity v , gives rise to j neutrons at time tf > t.
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Chapter 1
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Let
pj (tf , x, v , t )
j = 0, 1, ...
be the probability that a neutron, born at time t at position x 2 Ω with
velocity v , gives rise to j neutrons at time tf > t. Then the functions
pj (tf , x, v , t ) j = 0, 1, ... are governed by in…nitly many coupled
equations.
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Chapter 1
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Let
pj (tf , x, v , t )
j = 0, 1, ...
be the probability that a neutron, born at time t at position x 2 Ω with
velocity v , gives rise to j neutrons at time tf > t. Then the functions
pj (tf , x, v , t ) j = 0, 1, ... are governed by in…nitly many coupled
equations. On the other hand, the probability generating function
∞
G (z, x, v , t, tf ) :=
∑ z j pj (tf , x, v , t )
( t < tf )
j =0
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Chapter 1
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Let
pj (tf , x, v , t )
j = 0, 1, ...
be the probability that a neutron, born at time t at position x 2 Ω with
velocity v , gives rise to j neutrons at time tf > t. Then the functions
pj (tf , x, v , t ) j = 0, 1, ... are governed by in…nitly many coupled
equations. On the other hand, the probability generating function
∞
G (z, x, v , t, tf ) :=
∑ z j pj (tf , x, v , t )
( t < tf )
j =0
is governed by a nonlinear backward equation with …nal condition
G (z, x, v , tf , tf ) = z
and (non-homogeneous) boundary condition
G (z, x, v , t, tf ) = 1 if (x, v ) 2 Γ+ (t < tf ).
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Chapter 1
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Mathematically speaking, it is more expedient to consider
f (z, x, v , t ) := 1
G (z, x, v , tf
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t, tf )
Chapter 1
40 / 45
Mathematically speaking, it is more expedient to consider
f (z, x, v , t ) := 1
G (z, x, v , tf
t, tf )
which is governed by
∂f
∂t
v.
∂f
+ σ(x, v )f (t, x, v )
∂x
m
= σ(x, v )(1
(1
c0 (x, v )
f (t, x, v10 )...(1
∑
Z
k
k =1 V
ck (x, v , v10 , .., vk0 )
f (t, x, vk0 )dv10 ...dvk0 )
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Chapter 1
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Mathematically speaking, it is more expedient to consider
f (z, x, v , t ) := 1
G (z, x, v , tf
t, tf )
which is governed by
∂f
∂t
v.
∂f
+ σ(x, v )f (t, x, v )
∂x
m
= σ(x, v )(1
(1
c0 (x, v )
f (t, x, v10 )...(1
with initial condition f (0, x, v ) = 1
condition f (t, x, v )jΓ+ = 0.
∑
Z
k
k =1 V
ck (x, v , v10 , .., vk0 )
f (t, x, vk0 )dv10 ...dvk0 )
z and homogeneous boundary
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Chapter 1
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Mathematically speaking, it is more expedient to consider
f (z, x, v , t ) := 1
G (z, x, v , tf
t, tf )
which is governed by
∂f
∂t
v.
∂f
+ σ(x, v )f (t, x, v )
∂x
m
= σ(x, v )(1
(1
c0 (x, v )
f (t, x, v10 )...(1
∑
Z
k
k =1 V
ck (x, v , v10 , .., vk0 )
f (t, x, vk0 )dv10 ...dvk0 )
with initial condition f (0, x, v ) = 1 z and homogeneous boundary
condition f (t, x, v )jΓ+ = 0. See G. I. Bell (1965).
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Chapter 1
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Link with expected value theory
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Chapter 1
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Link with expected value theory
Once G (z, x, v , t, tf ) is obtained then pj (tf , x, v , t ) is obtained by
pj (tf , x, v , t ) =
1 dj
G (z, x, v , t, tf )
j ! dz j
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Chapter 1
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Link with expected value theory
Once G (z, x, v , t, tf ) is obtained then pj (tf , x, v , t ) is obtained by
pj (tf , x, v , t ) =
and
1 dj
G (z, x, v , t, tf )
j ! dz j
∞
∑ jpj (tf , x, v , t )
0
is governed by the conventional (expected value) neutron transport
equation.
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Chapter 1
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Nonlinear eigenvalue problems
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Chapter 1
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Nonlinear eigenvalue problems
In the "supercritical case"
f (t, x, v ) ! ϕ(x, v ) as t ! +∞
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Chapter 1
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Nonlinear eigenvalue problems
In the "supercritical case"
f (t, x, v ) ! ϕ(x, v ) as t ! +∞
where ϕ is governed by a nonlinear eigenvalue problem
∂ϕ
+ σ(x, v ) ϕ(x, v )
∂x
v.
m
= σ(x, v )(1
∑
c0 (x, v )
k =1
(1
ϕ(x, v10 )...(1
Z
Vk
ck (x, v , v10 , .., vk0 )
ϕ(x, vk0 )dv10 ...dvk0 )
with
ϕ(x, v )jΓ+ = 0, 0
ϕ(x, v )
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1.
Chapter 1
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Nonlinear eigenvalue problems
In the "supercritical case"
f (t, x, v ) ! ϕ(x, v ) as t ! +∞
where ϕ is governed by a nonlinear eigenvalue problem
∂ϕ
+ σ(x, v ) ϕ(x, v )
∂x
v.
m
= σ(x, v )(1
∑
c0 (x, v )
k =1
(1
ϕ(x, v10 )...(1
Z
Vk
ck (x, v , v10 , .., vk0 )
ϕ(x, vk0 )dv10 ...dvk0 )
with
ϕ(x, v )jΓ+ = 0, 0
ϕ(x, v )
1.
ϕ is the probability of a divergent chain reaction.
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Chapter 1
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The analysis of this nonlinear eigenvalue problem relies completely
on spectral theory of linearized neutron transport operator. See:
A. Pazy and P. Rabinowitz, Arch. Rat. Mech. Anal, 32 (1969)
226-246.
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Chapter 1
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The analysis of this nonlinear eigenvalue problem relies completely
on spectral theory of linearized neutron transport operator. See:
A. Pazy and P. Rabinowitz, Arch. Rat. Mech. Anal, 32 (1969)
226-246.
A. Pazy and P. Rabinowitz, Arch. Rat. Mech. Anal, 51 (1973)
153-164.
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Chapter 1
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The analysis of this nonlinear eigenvalue problem relies completely
on spectral theory of linearized neutron transport operator. See:
A. Pazy and P. Rabinowitz, Arch. Rat. Mech. Anal, 32 (1969)
226-246.
A. Pazy and P. Rabinowitz, Arch. Rat. Mech. Anal, 51 (1973)
153-164.
M. M-K, Proc. Roy. Soc. Edimburg, 121 A (1992) 253-272.
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Chapter 1
43 / 45
The analysis of this nonlinear eigenvalue problem relies completely
on spectral theory of linearized neutron transport operator. See:
A. Pazy and P. Rabinowitz, Arch. Rat. Mech. Anal, 32 (1969)
226-246.
A. Pazy and P. Rabinowitz, Arch. Rat. Mech. Anal, 51 (1973)
153-164.
M. M-K, Proc. Roy. Soc. Edimburg, 121 A (1992) 253-272.
K. Jarmouni and M. M-K, Nonlinear Anal, 31(3-4) (1998) 265-293.
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Chapter 1
43 / 45
The analysis of this nonlinear eigenvalue problem relies completely
on spectral theory of linearized neutron transport operator. See:
A. Pazy and P. Rabinowitz, Arch. Rat. Mech. Anal, 32 (1969)
226-246.
A. Pazy and P. Rabinowitz, Arch. Rat. Mech. Anal, 51 (1973)
153-164.
M. M-K, Proc. Roy. Soc. Edimburg, 121 A (1992) 253-272.
K. Jarmouni and M. M-K, Nonlinear Anal, 31(3-4) (1998) 265-293.
M. M-K and S. Salvarani, Acta. Appli. Math, 113 (2011) 145-165.
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Chapter 1
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Main tools in spectral analysis of neutron transport
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Chapter 1
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Main tools in spectral analysis of neutron transport
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Chapter 1
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Main tools in spectral analysis of neutron transport
Neutron transport semigroups are non-self-adjoint.
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Chapter 1
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Main tools in spectral analysis of neutron transport
Neutron transport semigroups are non-self-adjoint.
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Chapter 1
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Main tools in spectral analysis of neutron transport
Neutron transport semigroups are non-self-adjoint.
The main issue is the understanding of their time asymptotic
behaviour as t ! +∞.
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Chapter 1
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Main tools in spectral analysis of neutron transport
Neutron transport semigroups are non-self-adjoint.
The main issue is the understanding of their time asymptotic
behaviour as t ! +∞.
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Chapter 1
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Main tools in spectral analysis of neutron transport
Neutron transport semigroups are non-self-adjoint.
The main issue is the understanding of their time asymptotic
behaviour as t ! +∞.
Fortunately, we need just a good understanding of "peripheral
spectral theory".
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Chapter 1
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Main tools in spectral analysis of neutron transport
Neutron transport semigroups are non-self-adjoint.
The main issue is the understanding of their time asymptotic
behaviour as t ! +∞.
Fortunately, we need just a good understanding of "peripheral
spectral theory".
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CIMPA School Muizemberg July 22-Aug 4
Chapter 1
44 / 45
Main tools in spectral analysis of neutron transport
Neutron transport semigroups are non-self-adjoint.
The main issue is the understanding of their time asymptotic
behaviour as t ! +∞.
Fortunately, we need just a good understanding of "peripheral
spectral theory".
Compactness arguments
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Chapter 1
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Main tools in spectral analysis of neutron transport
Neutron transport semigroups are non-self-adjoint.
The main issue is the understanding of their time asymptotic
behaviour as t ! +∞.
Fortunately, we need just a good understanding of "peripheral
spectral theory".
Compactness arguments
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CIMPA School Muizemberg July 22-Aug 4
Chapter 1
44 / 45
Main tools in spectral analysis of neutron transport
Neutron transport semigroups are non-self-adjoint.
The main issue is the understanding of their time asymptotic
behaviour as t ! +∞.
Fortunately, we need just a good understanding of "peripheral
spectral theory".
Compactness arguments
Exploitation of positivity
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Chapter 1
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References
Albertoni and Montagnini, J. Math. Anal. Appl, 13 (1966) 19-48.
Bell. Stochastic theory of neutron transport, J. Nucl. Sc. Eng, 21
(1965) 390-401.
Borysiewicz and Mika, J. Math. Anal. Appl, 26 (1969) 461-478.
Demeru and Montagnini, J. Math. Anal. Appl, 12 (1965) 49-57.
Duderstadt and Martin: Transport Theory, John Wiley 1979
Hiraoka and Ukaï, J. Nucl. Sc. Tech, 9 (1972) 36-46.
Jorgens, Com. Pure. Appl. Math, 11 (1958) 219-242.
Larsen and Zweifel, J. Math. Phys, 15 (1974) 1987-1997.
Lehner and Wing, Com. Pure. Appl. Math, 8 (1955) 207-230.
Mika, J. Quant. Spectr. Radiat. Transfert,11 (1971) 879-891.
Ukaï, J. Math. Anal. Appl, 30 (1967) 297-314.
Vidav, J. Math. Anal. Appl, 22 (1968) 144-155.
Vidav, J. Math. Anal. Appl, 30 (1970) 264-279.
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Chapter 1
45 / 45