SPECTRAL THEORY FOR NEUTRON TRANSPORT

Transcription

1 SPECTRAL THEORY FOR NEUTRON TRANSPORT INTRODUCTION Mustapha Mokhtar-Kharroubi (In memory of Seiji Ukaï) Chapter 1 niversity of FrancheComté Besançon France (Institute) CIMPA School Muizemberg July 22-Aug 4 Chapter 1 1 / 45

2 Aim of the lectures The aim of these lectures is twofold: niversity of FrancheComté Besançon France (Institute) CIMPA School Muizemberg July 22-Aug 4 Chapter 1 2 / 45

3 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 niversity of FrancheComté Besançon France (Institute) CIMPA School Muizemberg July 22-Aug 4 Chapter 1 2 / 45

4 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 niversity of FrancheComté Besançon France (Institute) CIMPA School Muizemberg July 22-Aug 4 Chapter 1 2 / 45

5 Outline niversity of FrancheComté Besançon France (Institute) CIMPA School Muizemberg July 22-Aug 4 Chapter 1 3 / 45

6 Outline Besides a formal introduction given in Chapter 1 niversity of FrancheComté Besançon France (Institute) CIMPA School Muizemberg July 22-Aug 4 Chapter 1 3 / 45

7 Outline Besides a formal introduction given in Chapter 1 Chapter 2: Fundamentals in spectral theory niversity of FrancheComté Besançon France (Institute) CIMPA School Muizemberg July 22-Aug 4 Chapter 1 3 / 45

8 Outline Besides a formal introduction given in Chapter 1 Chapter 2: Fundamentals in spectral theory Chapter 3: Spectral analysis of weighted shift semigroups niversity of FrancheComté Besançon France (Institute) CIMPA School Muizemberg July 22-Aug 4 Chapter 1 3 / 45

9 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 niversity of FrancheComté Besançon France (Institute) CIMPA School Muizemberg July 22-Aug 4 Chapter 1 3 / 45

10 Abstract of Chapter 1 niversity of FrancheComté Besançon France (Institute) CIMPA School Muizemberg July 22-Aug 4 Chapter 1 4 / 45

11 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. niversity of FrancheComté Besançon France (Institute) CIMPA School Muizemberg July 22-Aug 4 Chapter 1 4 / 45

12 What transport theory is about? niversity of FrancheComté Besançon France (Institute) CIMPA School Muizemberg July 22-Aug 4 Chapter 1 5 / 45

13 What transport theory is about? Transport theory provides a statistical description of large populations of "particles" moving in a host medium. niversity of FrancheComté Besançon France (Institute) CIMPA School Muizemberg July 22-Aug 4 Chapter 1 5 / 45

14 For instance niversity of FrancheComté Besançon France (Institute) CIMPA School Muizemberg July 22-Aug 4 Chapter 1 6 / 45

15 For instance The transport of neutrons through the uranium fuel elements of a nuclear reactor. niversity of FrancheComté Besançon France (Institute) CIMPA School Muizemberg July 22-Aug 4 Chapter 1 6 / 45

16 Pulsed neutron experiments: University of FrancheComté Besançon France (Institute) CIMPA School Muizemberg July 22-Aug 4 Chapter 1 7 / 45

17 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. University of FrancheComté Besançon France (Institute) CIMPA School Muizemberg July 22-Aug 4 Chapter 1 7 / 45

18 The transport of photons through planetary or stellar atmospheres (radiative transfert) niversity of FrancheComté Besançon France (Institute) CIMPA School Muizemberg July 22-Aug 4 Chapter 1 8 / 45

19 The transport of photons through planetary or stellar atmospheres (radiative transfert) or light transport in tissues in diagnostic medicine (e.g. in computerized tomography). niversity of FrancheComté Besançon France (Institute) CIMPA School Muizemberg July 22-Aug 4 Chapter 1 8 / 45

20 The motion of gas molecules colliding with each another (gas dynamics). University of FrancheComté Besançon France (Institute) CIMPA School Muizemberg July 22-Aug 4 Chapter 1 9 / 45

21 The motion of charged particles (ions in plasma physics or electrons in semiconductor theory...) accelerated by external (e.g. electric) elds. University of FrancheComté Besançon France (Institute) CIMPA School Muizemberg July 22-Aug 4 Chapter 1 10 / 45

22 Various kinetic equations (structured population models) in biology... University of FrancheComté Besançon France (Institute) CIMPA School Muizemberg July 22-Aug 4 Chapter 1 11 / 45

23 On linearity niversity of FrancheComté Besançon France (Institute) CIMPA School Muizemberg July 22-Aug 4 Chapter 1 12 / 45

24 On linearity In a nuclear reactor, the proportion of neutrons with respect to the atomes of the host medium, is in nitesimal (about ), niversity of FrancheComté Besançon France (Institute) CIMPA School Muizemberg July 22-Aug 4 Chapter 1 12 / 45

25 On linearity In a nuclear reactor, the proportion of neutrons with respect to the atomes of the host medium, is in nitesimal (about ), so the possible collisions between neutrons are negligible in comparison with the collisions of neutrons with the atomes of the host material. niversity of FrancheComté Besançon France (Institute) CIMPA School Muizemberg July 22-Aug 4 Chapter 1 12 / 45

26 On linearity In a nuclear reactor, the proportion of neutrons with respect to the atomes of the host medium, is in nitesimal (about ), 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, niversity of FrancheComté Besançon France (Institute) CIMPA School Muizemberg July 22-Aug 4 Chapter 1 12 / 45

27 On linearity In a nuclear reactor, the proportion of neutrons with respect to the atomes of the host medium, is in nitesimal (about ), 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, niversity of FrancheComté Besançon France (Institute) CIMPA School Muizemberg July 22-Aug 4 Chapter 1 12 / 45

28 On linearity In a nuclear reactor, the proportion of neutrons with respect to the atomes of the host medium, is in nitesimal (about ), 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. niversity of FrancheComté Besançon France (Institute) CIMPA School Muizemberg July 22-Aug 4 Chapter 1 12 / 45

29 Linearized Boltzmann equations are formaly similar to neutron transport equations. University of FrancheComté Besançon France (Institute) CIMPA School Muizemberg July 22-Aug 4 Chapter 1 13 / 45

30 Linearized Boltzmann equations are formaly similar to neutron transport equations. There is however a big di erence: University of FrancheComté Besançon France (Institute) CIMPA School Muizemberg July 22-Aug 4 Chapter 1 13 / 45

31 Linearized Boltzmann equations are formaly similar to neutron transport equations. There is however a big di erence: the scattering kernel is not nonnegative!! University of FrancheComté Besançon France (Institute) CIMPA School Muizemberg July 22-Aug 4 Chapter 1 13 / 45

32 Density function niversity of FrancheComté Besançon France (Institute) CIMPA School Muizemberg July 22-Aug 4 Chapter 1 14 / 45

33 Density function niversity of FrancheComté Besançon France (Institute) CIMPA School Muizemberg July 22-Aug 4 Chapter 1 14 / 45

34 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). niversity of FrancheComté Besançon France (Institute) CIMPA School Muizemberg July 22-Aug 4 Chapter 1 14 / 45

35 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. niversity of FrancheComté Besançon France (Institute) CIMPA School Muizemberg July 22-Aug 4 Chapter 1 14 / 45

36 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. L 1 spaces are natural settings in transport theory! niversity of FrancheComté Besançon France (Institute) CIMPA School Muizemberg July 22-Aug 4 Chapter 1 14 / 45

37 MODELS USED IN NUCLEAR REACTOR THEORY University of FrancheComté Besançon France (Institute) CIMPA School Muizemberg July 22-Aug 4 Chapter 1 15 / 45

38 Inelastic model for neutron transport niversity of FrancheComté Besançon France (Institute) CIMPA School Muizemberg July 22-Aug 4 Chapter 1 16 / 45

39 Inelastic model for neutron transport f t f + v. x ZV + σ(x, v)f (t, x, v) = k(x, v, v 0 )f (t, x, v 0 )dv 0 niversity of FrancheComté Besançon France (Institute) CIMPA School Muizemberg July 22-Aug 4 Chapter 1 16 / 45

40 Inelastic model for neutron transport f t f + v. x ZV + σ(x, v)f (t, x, v) = k(x, v, v 0 )f (t, x, v 0 )dv 0 (x, v) 2 Ω V, Ω R 3 with V = v 2 R 3 ; c 0 jvj c 1 (0 c 0 < c 1 < ) niversity of FrancheComté Besançon France (Institute) CIMPA School Muizemberg July 22-Aug 4 Chapter 1 16 / 45

41 Inelastic model for neutron transport f t f + v. x ZV + σ(x, v)f (t, x, v) = k(x, v, v 0 )f (t, x, v 0 )dv 0 (x, v) 2 Ω V, Ω R 3 with V = v 2 R 3 ; c 0 jvj c 1 (0 c 0 < c 1 < ) and f (0, x, v) = f 0 (x, v) (initial condition) niversity of FrancheComté Besançon France (Institute) CIMPA School Muizemberg July 22-Aug 4 Chapter 1 16 / 45

42 Inelastic model for neutron transport f t f + v. x ZV + σ(x, v)f (t, x, v) = k(x, v, v 0 )f (t, x, v 0 )dv 0 (x, v) 2 Ω V, Ω R 3 with V = v 2 R 3 ; c 0 jvj c 1 (0 c 0 < c 1 < ) and f (0, x, v) = f 0 (x, v) (initial condition) f (t, x, v) jγ = 0 (boundary condition) University of FrancheComté Besançon France (Institute) CIMPA School Muizemberg July 22-Aug 4 Chapter 1 16 / 45

43 Inelastic model for neutron transport f t f + v. x ZV + σ(x, v)f (t, x, v) = k(x, v, v 0 )f (t, x, v 0 )dv 0 (x, v) 2 Ω V, Ω R 3 with V = v 2 R 3 ; c 0 jvj c 1 (0 c 0 < c 1 < ) and f (0, x, v) = f 0 (x, v) (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). University of FrancheComté Besançon France (Institute) CIMPA School Muizemberg July 22-Aug 4 Chapter 1 16 / 45

44 Other boundary conditions niversity of FrancheComté Besançon France (Institute) CIMPA School Muizemberg July 22-Aug 4 Chapter 1 17 / 45

45 Other boundary conditions Periodic boundary conditions (Transport on the torus) niversity of FrancheComté Besançon France (Institute) CIMPA School Muizemberg July 22-Aug 4 Chapter 1 17 / 45

46 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 := f jγ niversity of FrancheComté Besançon France (Institute) CIMPA School Muizemberg July 22-Aug 4 Chapter 1 17 / 45

47 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 := f jγ niversity of FrancheComté Besançon France (Institute) CIMPA School Muizemberg July 22-Aug 4 Chapter 1 17 / 45

48 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 := f jγ where Γ := (x, v) 2 Ω R 3 ; v.n(x) < 0 Γ + := (x, v) 2 Ω R 3 ; v.n(x) > 0. niversity of FrancheComté Besançon France (Institute) CIMPA School Muizemberg July 22-Aug 4 Chapter 1 17 / 45

49 Multiple scattering niversity of FrancheComté Besançon France (Institute) CIMPA School Muizemberg July 22-Aug 4 Chapter 1 18 / 45

50 Multiple scattering This physical model di ers from the previous reactor model by the fact that niversity of FrancheComté Besançon France (Institute) CIMPA School Muizemberg July 22-Aug 4 Chapter 1 18 / 45

51 Multiple scattering This physical model di ers from the previous reactor model by the fact that Ω = R 3 niversity of FrancheComté Besançon France (Institute) CIMPA School Muizemberg July 22-Aug 4 Chapter 1 18 / 45

52 Multiple scattering This physical model di ers from the previous reactor model by the fact that Ω = R 3 but σ(x, v) and k(x, v, v 0 ) are compactly supported in space. niversity of FrancheComté Besançon France (Institute) CIMPA School Muizemberg July 22-Aug 4 Chapter 1 18 / 45

53 The presence of delayed neutrons niversity of FrancheComté Besançon France (Institute) CIMPA School Muizemberg July 22-Aug 4 Chapter 1 19 / 45

54 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 niversity of FrancheComté Besançon France (Institute) CIMPA School Muizemberg July 22-Aug 4 Chapter 1 19 / 45

55 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 t + v. f x + σ(x, v)f (t, x, v) = Z R 3 k(x, v, v 0 )f (t, x, v 0 )dv 0 + m i=1 λ i g i niversity of FrancheComté Besançon France (Institute) CIMPA School Muizemberg July 22-Aug 4 Chapter 1 19 / 45

56 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 t + v. f x + σ(x, v)f (t, x, v) = Z R 3 k(x, v, v 0 )f (t, x, v 0 )dv 0 + Z dg i dt = λ i g i + k i (x, v, v 0 )f (t, x, v 0 )dv 0 (1 i m) R 3 λ i > 0 are the radioactive decay constants. m i=1 λ i g i niversity of FrancheComté Besançon France (Institute) CIMPA School Muizemberg July 22-Aug 4 Chapter 1 19 / 45

57 Multigroup models niversity of FrancheComté Besançon France (Institute) CIMPA School Muizemberg July 22-Aug 4 Chapter 1 20 / 45

58 Multigroup models (Motivation: numerical calculations) m Z f i t + v. f i x + σ i (x, v)f i (t, x, v) = k i,j (x, v, v 0 )f j (t, x, v 0 )µ j (dv 0 ), j=1 V j niversity of FrancheComté Besançon France (Institute) CIMPA School Muizemberg July 22-Aug 4 Chapter 1 20 / 45

59 Multigroup models (Motivation: numerical calculations) m Z f i t + v. f i x + σ i (x, v)f i (t, x, v) = k i,j (x, v, v 0 )f j (t, x, v 0 )µ j (dv 0 ), j=1 V j the spheres V j := v 2 R 3, jvj = c j, 1 j m, (c j > 0) are endowed with surface measures µ j and f i (t, x, v) is the density of neutrons (at time t > 0 located at x 2 Ω) with velocity v 2 V i. niversity of FrancheComté Besançon France (Institute) CIMPA School Muizemberg July 22-Aug 4 Chapter 1 20 / 45

60 Partly inelastic models (Larsen and Zweifel 1974) niversity of FrancheComté Besançon France (Institute) CIMPA School Muizemberg July 22-Aug 4 Chapter 1 21 / 45

61 Partly inelastic models (Larsen and Zweifel 1974) f t + v. f x + σ(x, v)f (t, x, v) = K ef + K in f niversity of FrancheComté Besançon France (Institute) CIMPA School Muizemberg July 22-Aug 4 Chapter 1 21 / 45

62 Partly inelastic models (Larsen and Zweifel 1974) where f t + v. f x + σ(x, v)f (t, x, v) = K ef + K in f Z K in f = k(x, v, v 0 )f (x, v 0 )dv 0 R 3 (inelastic operator) niversity of FrancheComté Besançon France (Institute) CIMPA School Muizemberg July 22-Aug 4 Chapter 1 21 / 45

63 Partly inelastic models (Larsen and Zweifel 1974) where and f t + v. f x + σ(x, v)f (t, x, v) = K ef + K in f Z K in f = k(x, v, v 0 )f (x, v 0 )dv 0 R 3 (inelastic operator) Z K e f = k(x, ρ, ω, ω 0 )f (x, ρω 0 )ds(ω 0 ) (elastic operator) S 2 where v = ρω. niversity of FrancheComté Besançon France (Institute) CIMPA School Muizemberg July 22-Aug 4 Chapter 1 21 / 45

64 Di usive models niversity of FrancheComté Besançon France (Institute) CIMPA School Muizemberg July 22-Aug 4 Chapter 1 22 / 45

65 Di usive models Replace v. f x by 4 x (Laplacian in the position variable x 2 Ω with Neumann or Dirichlet boundary condition). University of FrancheComté Besançon France (Institute) CIMPA School Muizemberg July 22-Aug 4 Chapter 1 22 / 45

66 Space homogeneous models niversity of FrancheComté Besançon France (Institute) CIMPA School Muizemberg July 22-Aug 4 Chapter 1 23 / 45

67 Space homogeneous models For instance (Drop the x variable!) Z f t + σ(v)f (t, v) = R 3 k(v, v 0 )f (t, v 0 )dv 0 niversity of FrancheComté Besançon France (Institute) CIMPA School Muizemberg July 22-Aug 4 Chapter 1 23 / 45

68 Well-posedness niversity of FrancheComté Besançon France (Institute) CIMPA School Muizemberg July 22-Aug 4 Chapter 1 24 / 45

69 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 t f + v. + σ(x, v)f (t, x, v) = 0 x (with initial condition f 0 ) and is solved explicitly by f (t, x, v) = e R t 0 σ(x τv,v )d τ f 0 (x tv, v)1 fts(x,v )g niversity of FrancheComté Besançon France (Institute) CIMPA School Muizemberg July 22-Aug 4 Chapter 1 24 / 45

70 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 t f + v. + σ(x, v)f (t, x, v) = 0 x (with initial condition f 0 ) and is solved explicitly by f (t, x, v) = e R t 0 σ(x τv,v )d τ f 0 (x tv, v)1 fts(x,v )g with rst exit time function s(x, v) = inf fs > 0; x sv /2 Ωg (method of characteristics). niversity of FrancheComté Besançon France (Institute) CIMPA School Muizemberg July 22-Aug 4 Chapter 1 24 / 45

71 This de nes a C 0 -semigroup fu(t); t > 0g on L p (Ω R 3 ) U(t) : g! e R t 0 σ(x τv,v )d τ g(x tv, v)1 fts(x,v )g niversity of FrancheComté Besançon France (Institute) CIMPA School Muizemberg July 22-Aug 4 Chapter 1 25 / 45

72 This de nes a C 0 -semigroup fu(t); t > 0g on L p (Ω R 3 ) U(t) : g! e R t 0 σ(x τv,v )d τ g(x tv, v)1 fts(x,v )g with generator T given by D(T ) = g 2 L p (Ω R 3 ); v. g x 2 Lp, g jγ = 0 Tg = v. g x σ(x, v)g(x, v). niversity of FrancheComté Besançon France (Institute) CIMPA School Muizemberg July 22-Aug 4 Chapter 1 25 / 45

73 Perturbation theory niversity of FrancheComté Besançon France (Institute) CIMPA School Muizemberg July 22-Aug 4 Chapter 1 26 / 45

74 Perturbation theory If the scattering operator Z K : g! k(x, v, v 0 )g(x, v 0 )dv 0 R 3 is bounded in L p (Ω R 3 ) then A := T + K (D(A) = D(T )) generates a C 0 -semigroup fv (t); t > 0g. niversity of FrancheComté Besançon France (Institute) CIMPA School Muizemberg July 22-Aug 4 Chapter 1 26 / 45

75 Two basic eigenvalue problems University of FrancheComté Besançon France (Institute) CIMPA School Muizemberg July 22-Aug 4 Chapter 1 27 / 45

76 Two basic eigenvalue problems 1 The "time eigenelements" (λ, g) v. g x Z σ(x, v)g(x, v) + k(x, v, v 0 )g(x, v 0 )dv 0 = λg(x, v) V niversity of FrancheComté Besançon France (Institute) CIMPA School Muizemberg July 22-Aug 4 Chapter 1 27 / 45

77 Two basic eigenvalue problems 1 The "time eigenelements" (λ, g) v. g x Z σ(x, v)g(x, v) + k(x, v, v 0 )g(x, v 0 )dv 0 = λg(x, v) V niversity of FrancheComté Besançon France (Institute) CIMPA School Muizemberg July 22-Aug 4 Chapter 1 27 / 45

78 Two basic eigenvalue problems 1 The "time eigenelements" (λ, g) v. g x Z σ(x, v)g(x, v) + k(x, v, v 0 )g(x, v 0 )dv 0 = λg(x, v) V and their connection with time asymptotic behaviour (t! + ) of fv (t); t > 0g. niversity of FrancheComté Besançon France (Institute) CIMPA School Muizemberg July 22-Aug 4 Chapter 1 27 / 45

79 Two basic eigenvalue problems 1 The "time eigenelements" (λ, g) v. g x Z σ(x, v)g(x, v) + k(x, v, v 0 )g(x, v 0 )dv 0 = λg(x, v) V and their connection with time asymptotic behaviour (t! + ) of fv (t); t > 0g. 2 Criticality eigenvalue problem 0 = v. g Z σ(x, v)g(x, v) + k s (x, v, v 0 )g(x, v 0 )dv 0 x V + 1 Z k f (x, v, v 0 )g(x, v 0 )dv 0 γ V University of FrancheComté Besançon France (Institute) CIMPA School Muizemberg July 22-Aug 4 Chapter 1 27 / 45

80 Two basic eigenvalue problems 1 The "time eigenelements" (λ, g) v. g x Z σ(x, v)g(x, v) + k(x, v, v 0 )g(x, v 0 )dv 0 = λg(x, v) V and their connection with time asymptotic behaviour (t! + ) of fv (t); t > 0g. 2 Criticality eigenvalue problem 0 = v. g Z σ(x, v)g(x, v) + k s (x, v, v 0 )g(x, v 0 )dv 0 x V + 1 Z k f (x, v, v 0 )g(x, v 0 )dv 0 γ V University of FrancheComté Besançon France (Institute) CIMPA School Muizemberg July 22-Aug 4 Chapter 1 27 / 45

81 Two basic eigenvalue problems 1 The "time eigenelements" (λ, g) v. g x Z σ(x, v)g(x, v) + k(x, v, v 0 )g(x, v 0 )dv 0 = λg(x, v) V and their connection with time asymptotic behaviour (t! + ) of fv (t); t > 0g. 2 Criticality eigenvalue problem 0 = v. g Z σ(x, v)g(x, v) + k s (x, v, v 0 )g(x, v 0 )dv 0 x V + 1 Z k f (x, v, v 0 )g(x, v 0 )dv 0 γ V where k s (x, v, v 0 ) and k f (x, v, v 0 ) are the scattering kernel and the ssion kernel, see e.g. J. Mika (1971). University of FrancheComté Besançon France (Institute) CIMPA School Muizemberg July 22-Aug 4 Chapter 1 27 / 45

82 A model case (J. Lehner and G. Milton Wing 1955) niversity of FrancheComté Besançon France (Institute) CIMPA School Muizemberg July 22-Aug 4 Chapter 1 28 / 45

83 A model case (J. Lehner and G. Milton Wing 1955) Af = µ f x σf (x, µ) + c Z +1 1 f (x, µ 0 )dµ 0 niversity of FrancheComté Besançon France (Institute) CIMPA School Muizemberg July 22-Aug 4 Chapter 1 28 / 45

84 A model case (J. Lehner and G. Milton Wing 1955) Af = µ f x σf (x, µ) + c Z +1 1 f (x, µ 0 )dµ 0 (x, µ) 2 [ a, a] [ 1, +1] niversity of FrancheComté Besançon France (Institute) CIMPA School Muizemberg July 22-Aug 4 Chapter 1 28 / 45

85 A model case (J. Lehner and G. Milton Wing 1955) Af = µ f x σf (x, µ) + c Z +1 1 f (x, µ 0 )dµ 0 (x, µ) 2 [ a, a] [ 1, +1] f ( a, µ) = 0 (µ > 0); f (a, µ) = 0 (µ < 0). niversity of FrancheComté Besançon France (Institute) CIMPA School Muizemberg July 22-Aug 4 Chapter 1 28 / 45

86 A model case (J. Lehner and G. Milton Wing 1955) Af = µ f x σf (x, µ) + c Z +1 1 f (x, µ 0 )dµ 0 (x, µ) 2 [ a, a] [ 1, +1] f ( a, µ) = 0 (µ > 0); f (a, µ) = 0 (µ < 0). Theorem fre λ σg σ(a) niversity of FrancheComté Besançon France (Institute) CIMPA School Muizemberg July 22-Aug 4 Chapter 1 28 / 45

87 A model case (J. Lehner and G. Milton Wing 1955) Af = µ f x σf (x, µ) + c Z +1 1 f (x, µ 0 )dµ 0 (x, µ) 2 [ a, a] [ 1, +1] f ( a, µ) = 0 (µ > 0); f (a, µ) = 0 (µ < 0). Theorem fre λ σg σ(a) and σ(a) \ fre λ > σg consists of a nite (nonempty) set of real eigenvalues with nite algebraic multiplicities. niversity of FrancheComté Besançon France (Institute) CIMPA School Muizemberg July 22-Aug 4 Chapter 1 28 / 45

88 On Jorgens paper (1958) niversity of FrancheComté Besançon France (Institute) CIMPA School Muizemberg July 22-Aug 4 Chapter 1 29 / 45

89 On Jorgens paper (1958) Theorem Let Ω be bounded and convex, V = v 2 R 3 ; c 0 jvj c 1 < the scattering kernel k(.,.,.) be bounded. and niversity of FrancheComté Besançon France (Institute) CIMPA School Muizemberg July 22-Aug 4 Chapter 1 29 / 45

90 On Jorgens paper (1958) Theorem Let Ω be bounded and convex, V = v 2 R 3 ; c 0 jvj c 1 < the scattering kernel k(.,.,.) be bounded. If and c 0 > 0. niversity of FrancheComté Besançon France (Institute) CIMPA School Muizemberg July 22-Aug 4 Chapter 1 29 / 45

91 On Jorgens paper (1958) Theorem Let Ω be bounded and convex, V = v 2 R 3 ; c 0 jvj c 1 < the scattering kernel k(.,.,.) be bounded. If and c 0 > 0. Then V (t) is compact in L 2 (Ω V ) for t large enough. niversity of FrancheComté Besançon France (Institute) CIMPA School Muizemberg July 22-Aug 4 Chapter 1 29 / 45

92 On Jorgens paper (1958) Theorem Let Ω be bounded and convex, V = v 2 R 3 ; c 0 jvj c 1 < the scattering kernel k(.,.,.) be bounded. If and c 0 > 0. Then V (t) is compact in L 2 (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 fλ 1,...λ m g with spectral projections fp 1,...P m g. niversity of FrancheComté Besançon France (Institute) CIMPA School Muizemberg July 22-Aug 4 Chapter 1 29 / 45

93 On Jorgens paper (1958) Theorem Let Ω be bounded and convex, V = v 2 R 3 ; c 0 jvj c 1 < the scattering kernel k(.,.,.) be bounded. If and c 0 > 0. Then V (t) is compact in L 2 (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 fλ 1,...λ m g with spectral projections fp 1,...P m g. For some ε > 0 and D j := (T λ j )P j V (t) = m e λ j t e td j P j + O(e βt ) j=1 (β < α). University of FrancheComté Besançon France (Institute) CIMPA School Muizemberg July 22-Aug 4 Chapter 1 29 / 45

94 On small velocities University of FrancheComté Besançon France (Institute) CIMPA School Muizemberg July 22-Aug 4 Chapter 1 30 / 45

95 On small velocities Theorem (Albertoni-Montagnini 1966) Let Ω be bounded. We assume that V is not bounded away from zero. University of FrancheComté Besançon France (Institute) CIMPA School Muizemberg July 22-Aug 4 Chapter 1 30 / 45

96 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. University of FrancheComté Besançon France (Institute) CIMPA School Muizemberg July 22-Aug 4 Chapter 1 30 / 45

97 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. (ii) fλ; Re λ λ g σ(t + K ). University of FrancheComté Besançon France (Institute) CIMPA School Muizemberg July 22-Aug 4 Chapter 1 30 / 45

98 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. (ii) fλ; Re λ λ g σ(t + K ). For most physical models λ = inf σ(x, v). University of FrancheComté Besançon France (Institute) CIMPA School Muizemberg July 22-Aug 4 Chapter 1 30 / 45

99 Compactness results (Montagnini, Demeru, Ukaï, Borysiewicz, Mika, Vidav) niversity of FrancheComté Besançon France (Institute) CIMPA School Muizemberg July 22-Aug 4 Chapter 1 31 / 45

100 Compactness results (Montagnini, Demeru, Ukaï, Borysiewicz, Mika, Vidav) If some power of (λ T ) 1 K is compact (Re λ > λ ) then σ(t + K ) \ fλ; Re λ > λ g consists at most of isolated eigenvalues with nite algebraic multiplicities. niversity of FrancheComté Besançon France (Institute) CIMPA School Muizemberg July 22-Aug 4 Chapter 1 31 / 45

101 Compactness results (Montagnini, Demeru, Ukaï, Borysiewicz, Mika, Vidav) If some power of (λ T ) 1 K is compact (Re λ > λ ) then σ(t + K ) \ fλ; Re λ > λ g consists at most of isolated eigenvalues with nite algebraic multiplicities. niversity of FrancheComté Besançon France (Institute) CIMPA School Muizemberg July 22-Aug 4 Chapter 1 31 / 45

102 Compactness results (Montagnini, Demeru, Ukaï, Borysiewicz, Mika, Vidav) If some power of (λ T ) 1 K is compact (Re λ > λ ) then σ(t + K ) \ fλ; Re λ > λ g consists at most of isolated eigenvalues with nite algebraic multiplicities. Tool: Analytic Fredholm alternative. niversity of FrancheComté Besançon France (Institute) CIMPA School Muizemberg July 22-Aug 4 Chapter 1 31 / 45

103 Compactness results (Montagnini, Demeru, Ukaï, Borysiewicz, Mika, Vidav) If some power of (λ T ) 1 K is compact (Re λ > λ ) then σ(t + K ) \ fλ; Re λ > λ g consists at most of isolated eigenvalues with nite algebraic multiplicities. Tool: Analytic Fredholm alternative. niversity of FrancheComté Besançon France (Institute) CIMPA School Muizemberg July 22-Aug 4 Chapter 1 31 / 45

104 Compactness results (Montagnini, Demeru, Ukaï, Borysiewicz, Mika, Vidav) If some power of (λ T ) 1 K is compact (Re λ > λ ) then σ(t + K ) \ fλ; Re λ > 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 L 2 (Ω V ) (Vladimirov s trick). niversity of FrancheComté Besançon France (Institute) CIMPA School Muizemberg July 22-Aug 4 Chapter 1 31 / 45

105 Compactness results (Montagnini, Demeru, Ukaï, Borysiewicz, Mika, Vidav) If some power of (λ T ) 1 K is compact (Re λ > λ ) then σ(t + K ) \ fλ; Re λ > 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 L 2 (Ω V ) (Vladimirov s trick). niversity of FrancheComté Besançon France (Institute) CIMPA School Muizemberg July 22-Aug 4 Chapter 1 31 / 45

106 Compactness results (Montagnini, Demeru, Ukaï, Borysiewicz, Mika, Vidav) If some power of (λ T ) 1 K is compact (Re λ > λ ) then σ(t + K ) \ fλ; Re λ > 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 L 2 (Ω V ) (Vladimirov s trick). Compactness of K (λ 1969). T ) 1 K in weighted L 1 space (Suhadolc niversity of FrancheComté Besançon France (Institute) CIMPA School Muizemberg July 22-Aug 4 Chapter 1 31 / 45

107 Compactness results (Montagnini, Demeru, Ukaï, Borysiewicz, Mika, Vidav) If some power of (λ T ) 1 K is compact (Re λ > λ ) then σ(t + K ) \ fλ; Re λ > 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 L 2 (Ω V ) (Vladimirov s trick). Compactness of K (λ 1969). T ) 1 K in weighted L 1 space (Suhadolc niversity of FrancheComté Besançon France (Institute) CIMPA School Muizemberg July 22-Aug 4 Chapter 1 31 / 45

108 Compactness results (Montagnini, Demeru, Ukaï, Borysiewicz, Mika, Vidav) If some power of (λ T ) 1 K is compact (Re λ > λ ) then σ(t + K ) \ fλ; Re λ > 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 L 2 (Ω V ) (Vladimirov s trick). Compactness of K (λ 1969). If T ) 1 K in weighted L 1 space (Suhadolc λ := sup fre λ; λ 2 σ(t + K )g > λ niversity of FrancheComté Besançon France (Institute) CIMPA School Muizemberg July 22-Aug 4 Chapter 1 31 / 45

109 Compactness results (Montagnini, Demeru, Ukaï, Borysiewicz, Mika, Vidav) If some power of (λ T ) 1 K is compact (Re λ > λ ) then σ(t + K ) \ fλ; Re λ > 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 L 2 (Ω V ) (Vladimirov s trick). Compactness of K (λ 1969). If T ) 1 K in weighted L 1 space (Suhadolc λ := sup fre λ; λ 2 σ(t + K )g > then λ is the leading eigenvalue associated to a nonnegative eigenfunction (Peripheral spectral theory via positivity arguments). University of FrancheComté Besançon France (Institute) CIMPA School Muizemberg July 22-Aug 4 Chapter 1 31 / 45 λ

110 Absence of eigenvalues niversity of FrancheComté Besançon France (Institute) CIMPA School Muizemberg July 22-Aug 4 Chapter 1 32 / 45

111 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. niversity of FrancheComté Besançon France (Institute) CIMPA School Muizemberg July 22-Aug 4 Chapter 1 32 / 45

112 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(jvj, jv 0 j) = 0 for jvj > jv 0 j (superthermal particle transport: no upscattering) then σ(t + K ) \ fλ; Re λ > λ g =? 8 Ω. niversity of FrancheComté Besançon France (Institute) CIMPA School Muizemberg July 22-Aug 4 Chapter 1 32 / 45

113 Isotropic models niversity of FrancheComté Besançon France (Institute) CIMPA School Muizemberg July 22-Aug 4 Chapter 1 33 / 45

114 Isotropic models Theorem (Albertoni-Montagnini, Ukaï, Mika) If σ(x, v) = σ(jvj) and k(x, v, v 0 ) = k(jvj, v 0 ) = k( v 0, jvj) then σ(t + K ) \ fλ; Re λ > λ g R. niversity of FrancheComté Besançon France (Institute) CIMPA School Muizemberg July 22-Aug 4 Chapter 1 33 / 45

115 Time asymptotic behaviour (Dunford calculus) University of FrancheComté Besançon France (Institute) CIMPA School Muizemberg July 22-Aug 4 Chapter 1 34 / 45

116 Time asymptotic behaviour (Dunford calculus) V (t)f = lim γ!+ Z 1 ρ+i γ 2iπ ρ i γ e λt (λ A) 1 fdλ (ρ > ω = type). University of FrancheComté Besançon France (Institute) CIMPA School Muizemberg July 22-Aug 4 Chapter 1 34 / 45

117 Time asymptotic behaviour (Dunford calculus) and V (t)f = lim γ!+ Z 1 ρ+i γ 2iπ ρ i γ e λt (λ 8ε > 0, σ(t + K ) \ fλ; Re λ > A) 1 fdλ (ρ > ω = type). λ + εg = fλ 1,...λ m g (with spectral projections fp 1,...P m g) is nite. University of FrancheComté Besançon France (Institute) CIMPA School Muizemberg July 22-Aug 4 Chapter 1 34 / 45

118 Time asymptotic behaviour (Dunford calculus) and V (t)f = lim γ!+ Z 1 ρ+i γ 2iπ ρ i γ e λt (λ 8ε > 0, σ(t + K ) \ fλ; Re λ > A) 1 fdλ (ρ > ω = type). λ + εg = fλ 1,...λ m g (with spectral projections fp 1,...P m g) is nite. If this set is not empty then shift the path of integration and pick the residues niversity of FrancheComté Besançon France (Institute) CIMPA School Muizemberg July 22-Aug 4 Chapter 1 34 / 45

119 Time asymptotic behaviour (Dunford calculus) and V (t)f = lim γ!+ Z 1 ρ+i γ 2iπ ρ i γ e λt (λ 8ε > 0, σ(t + K ) \ fλ; Re λ > A) 1 fdλ (ρ > ω = type). λ + εg = fλ 1,...λ m g (with spectral projections fp 1,...P m g) is nite. If this set is not empty then shift the path of integration and pick the residues V (t)f = m e λ j t e td j P j f + O f (e βt ) (β < λ + ε); j=1 for f 2 D(A 2 ); see, e.g. Borysiewicz and Mika (1969). niversity of FrancheComté Besançon France (Institute) CIMPA School Muizemberg July 22-Aug 4 Chapter 1 34 / 45

120 Time asymptotic behaviour (Dunford calculus) and V (t)f = lim γ!+ Z 1 ρ+i γ 2iπ ρ i γ e λt (λ 8ε > 0, σ(t + K ) \ fλ; Re λ > A) 1 fdλ (ρ > ω = type). λ + εg = fλ 1,...λ m g (with spectral projections fp 1,...P m g) is nite. If this set is not empty then shift the path of integration and pick the residues V (t)f = m e λ j t e td j P j f + O f (e βt ) (β < λ + ε); j=1 for f 2 D(A 2 ); see, e.g. Borysiewicz and Mika (1969). Drawback of the approach: we need smooth initial data. niversity of FrancheComté Besançon France (Institute) CIMPA School Muizemberg July 22-Aug 4 Chapter 1 34 / 45

121 Spectra of perturbed semigroups (Vidav 1970) niversity of FrancheComté Besançon France (Institute) CIMPA School Muizemberg July 22-Aug 4 Chapter 1 35 / 45

122 Spectra of perturbed semigroups (Vidav 1970) V (t) = n=0 U n (t) where U 0 (t) = U(t) is the streaming semigroup and U n+1 (t) = Z t 0 U(t s)ku n (s)ds (n > 0). niversity of FrancheComté Besançon France (Institute) CIMPA School Muizemberg July 22-Aug 4 Chapter 1 35 / 45

123 Spectra of perturbed semigroups (Vidav 1970) V (t) = n=0 U n (t) where U 0 (t) = U(t) is the streaming semigroup and U n+1 (t) = Z t 0 U(t s)ku n (s)ds (n > 0). Theorem (Vidav 1970) If some remainder term R n (t) := j=n U j (t) is compact for large t niversity of FrancheComté Besançon France (Institute) CIMPA School Muizemberg July 22-Aug 4 Chapter 1 35 / 45

124 Spectra of perturbed semigroups (Vidav 1970) V (t) = n=0 U n (t) where U 0 (t) = U(t) is the streaming semigroup and U n+1 (t) = Z t 0 U(t s)ku n (s)ds (n > 0). Theorem (Vidav 1970) If some remainder term R n (t) := j=n U j (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. niversity of FrancheComté Besançon France (Institute) CIMPA School Muizemberg July 22-Aug 4 Chapter 1 35 / 45

125 Spectra of perturbed semigroups (Vidav 1970) V (t) = n=0 U n (t) where U 0 (t) = U(t) is the streaming semigroup and U n+1 (t) = Z t 0 U(t s)ku n (s)ds (n > 0). Theorem (Vidav 1970) If some remainder term R n (t) := j=n U j (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 = fλ 1,...λ m g is nite and V (t) = m e λ j t e td j P j + O(e βt ) j=1 in operator norm where β < λ + ε. niversity of FrancheComté Besançon France (Institute) CIMPA School Muizemberg July 22-Aug 4 Chapter 1 35 / 45

126 E ective existence of a fundamental mode niversity of FrancheComté Besançon France (Institute) CIMPA School Muizemberg July 22-Aug 4 Chapter 1 36 / 45

127 E ective existence of a fundamental mode see S. Ukaï and T. Hiraoka (1972) (isotropic case) niversity of FrancheComté Besançon France (Institute) CIMPA School Muizemberg July 22-Aug 4 Chapter 1 36 / 45

128 Probability generating function of neutron chain ssions niversity of FrancheComté Besançon France (Institute) CIMPA School Muizemberg July 22-Aug 4 Chapter 1 37 / 45

129 Probability generating function of neutron chain ssions Conventional neutron transport theory deals with the expected (or mean behaviour) of neutrons. niversity of FrancheComté Besançon France (Institute) CIMPA School Muizemberg July 22-Aug 4 Chapter 1 37 / 45

130 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. niversity of FrancheComté Besançon France (Institute) CIMPA School Muizemberg July 22-Aug 4 Chapter 1 37 / 45

131 In a multiplying medium occupying a region Ω a neutron interacting with a nucleus of the host material may be absorbed niversity of FrancheComté Besançon France (Institute) CIMPA School Muizemberg July 22-Aug 4 Chapter 1 38 / 45

132 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 niversity of FrancheComté Besançon France (Institute) CIMPA School Muizemberg July 22-Aug 4 Chapter 1 38 / 45

133 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. niversity of FrancheComté Besançon France (Institute) CIMPA School Muizemberg July 22-Aug 4 Chapter 1 38 / 45

134 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 v 0 1, v 0 2,...v 0 i is given by c i (x, v, v 0 1, v 0 2,...v 0 i ), (1 i m). niversity of FrancheComté Besançon France (Institute) CIMPA School Muizemberg July 22-Aug 4 Chapter 1 38 / 45

135 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 v 0 1, v 0 2,...v 0 i is given by In particular c i (x, v, v 0 1, v 0 2,...v 0 i ), (1 i m). m Z c 0 (x, v) + c k (x, v, v1, 0 v2, 0...v 0 k=1 V k k )dv1...dv 0 k 0 = 1 where c 0 (x, v) is the probability (for a neutron located at x 2 Ω, with velocity v) of being absorbed. niversity of FrancheComté Besançon France (Institute) CIMPA School Muizemberg July 22-Aug 4 Chapter 1 38 / 45

136 Let p j (t f, 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 t f > t. niversity of FrancheComté Besançon France (Institute) CIMPA School Muizemberg July 22-Aug 4 Chapter 1 39 / 45

137 Let p j (t f, 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 t f > t. Then the functions p j (t f, x, v, t) j = 0, 1,... are governed by in nitly many coupled equations. niversity of FrancheComté Besançon France (Institute) CIMPA School Muizemberg July 22-Aug 4 Chapter 1 39 / 45

138 Let p j (t f, 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 t f > t. Then the functions p j (t f, 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, t f ) := z j p j (t f, x, v, t) (t < t f ) j=0 niversity of FrancheComté Besançon France (Institute) CIMPA School Muizemberg July 22-Aug 4 Chapter 1 39 / 45

139 Let p j (t f, 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 t f > t. Then the functions p j (t f, 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, t f ) := z j p j (t f, x, v, t) (t < t f ) j=0 is governed by a nonlinear backward equation with nal condition G (z, x, v, t f, t f ) = z and (non-homogeneous) boundary condition G (z, x, v, t, t f ) = 1 if (x, v) 2 Γ + (t < t f ). niversity of FrancheComté Besançon France (Institute) CIMPA School Muizemberg July 22-Aug 4 Chapter 1 39 / 45

140 Mathematically speaking, it is more expedient to consider f (z, x, v, t) := 1 G (z, x, v, t f t, t f ) niversity of FrancheComté Besançon France (Institute) CIMPA School Muizemberg July 22-Aug 4 Chapter 1 40 / 45

141 Mathematically speaking, it is more expedient to consider which is governed by f (z, x, v, t) := 1 G (z, x, v, t f t, t f ) f t + σ(x, v)f (t, x, v) m Z = σ(x, v)(1 c 0 (x, v) c k (x, v, v1, 0.., v 0 k=1 V k k ) (1 f (t, x, v1)...(1 0 f (t, x, vk 0 )dv1...dv 0 k 0 ) v. f x niversity of FrancheComté Besançon France (Institute) CIMPA School Muizemberg July 22-Aug 4 Chapter 1 40 / 45

142 Mathematically speaking, it is more expedient to consider which is governed by f (z, x, v, t) := 1 G (z, x, v, t f t, t f ) f t + σ(x, v)f (t, x, v) m Z = σ(x, v)(1 c 0 (x, v) c k (x, v, v1, 0.., v 0 k=1 V k k ) (1 f (t, x, v1)...(1 0 f (t, x, vk 0 )dv1...dv 0 k 0 ) v. f x with initial condition f (0, x, v) = 1 condition f (t, x, v) jγ+ = 0. z and homogeneous boundary niversity of FrancheComté Besançon France (Institute) CIMPA School Muizemberg July 22-Aug 4 Chapter 1 40 / 45

143 Mathematically speaking, it is more expedient to consider which is governed by f (z, x, v, t) := 1 G (z, x, v, t f t, t f ) f t + σ(x, v)f (t, x, v) m Z = σ(x, v)(1 c 0 (x, v) c k (x, v, v1, 0.., v 0 k=1 V k k ) (1 f (t, x, v1)...(1 0 f (t, x, vk 0 )dv1...dv 0 k 0 ) v. f x with initial condition f (0, x, v) = 1 z and homogeneous boundary condition f (t, x, v) jγ+ = 0. See G. I. Bell (1965). niversity of FrancheComté Besançon France (Institute) CIMPA School Muizemberg July 22-Aug 4 Chapter 1 40 / 45

144 Link with expected value theory University of FrancheComté Besançon France (Institute) CIMPA School Muizemberg July 22-Aug 4 Chapter 1 41 / 45

145 Link with expected value theory Once G (z, x, v, t, t f ) is obtained then p j (t f, x, v, t) is obtained by p j (t f, x, v, t) = 1 j! d j dz j G (z, x, v, t, t f ) niversity of FrancheComté Besançon France (Institute) CIMPA School Muizemberg July 22-Aug 4 Chapter 1 41 / 45

146 Link with expected value theory Once G (z, x, v, t, t f ) is obtained then p j (t f, x, v, t) is obtained by and p j (t f, x, v, t) = 1 j! 0 d j dz j G (z, x, v, t, t f ) jp j (t f, x, v, t) is governed by the conventional (expected value) neutron transport equation. niversity of FrancheComté Besançon France (Institute) CIMPA School Muizemberg July 22-Aug 4 Chapter 1 41 / 45

147 Nonlinear eigenvalue problems niversity of FrancheComté Besançon France (Institute) CIMPA School Muizemberg July 22-Aug 4 Chapter 1 42 / 45

148 Nonlinear eigenvalue problems In the "supercritical case" f (t, x, v)! ϕ(x, v) as t! + niversity of FrancheComté Besançon France (Institute) CIMPA School Muizemberg July 22-Aug 4 Chapter 1 42 / 45

149 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 v. ϕ + σ(x, v)ϕ(x, v) x m Z = σ(x, v)(1 c 0 (x, v) c k (x, v, v1, 0.., v 0 k=1 V k k ) (1 ϕ(x, v1)...(1 0 ϕ(x, vk 0 )dv1...dv 0 k 0 ) with ϕ(x, v) jγ+ = 0, 0 ϕ(x, v) 1. niversity of FrancheComté Besançon France (Institute) CIMPA School Muizemberg July 22-Aug 4 Chapter 1 42 / 45

150 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 with v. ϕ + σ(x, v)ϕ(x, v) x m Z = σ(x, v)(1 c 0 (x, v) c k (x, v, v1, 0.., v 0 k=1 V k k ) (1 ϕ(x, v1)...(1 0 ϕ(x, vk 0 )dv1...dv 0 k 0 ) ϕ(x, v) jγ+ = 0, 0 ϕ(x, v) 1. ϕ is the probability of a divergent chain reaction. niversity of FrancheComté Besançon France (Institute) CIMPA School Muizemberg July 22-Aug 4 Chapter 1 42 / 45

151 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) niversity of FrancheComté Besançon France (Institute) CIMPA School Muizemberg July 22-Aug 4 Chapter 1 43 / 45

152 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) A. Pazy and P. Rabinowitz, Arch. Rat. Mech. Anal, 51 (1973) niversity of FrancheComté Besançon France (Institute) CIMPA School Muizemberg July 22-Aug 4 Chapter 1 43 / 45

153 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) A. Pazy and P. Rabinowitz, Arch. Rat. Mech. Anal, 51 (1973) M. M-K, Proc. Roy. Soc. Edimburg, 121 A (1992) niversity of FrancheComté Besançon France (Institute) CIMPA School Muizemberg July 22-Aug 4 Chapter 1 43 / 45

154 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) A. Pazy and P. Rabinowitz, Arch. Rat. Mech. Anal, 51 (1973) M. M-K, Proc. Roy. Soc. Edimburg, 121 A (1992) K. Jarmouni and M. M-K, Nonlinear Anal, 31(3-4) (1998) niversity of FrancheComté Besançon France (Institute) CIMPA School Muizemberg July 22-Aug 4 Chapter 1 43 / 45

155 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) A. Pazy and P. Rabinowitz, Arch. Rat. Mech. Anal, 51 (1973) M. M-K, Proc. Roy. Soc. Edimburg, 121 A (1992) K. Jarmouni and M. M-K, Nonlinear Anal, 31(3-4) (1998) M. M-K and S. Salvarani, Acta. Appli. Math, 113 (2011) niversity of FrancheComté Besançon France (Institute) CIMPA School Muizemberg July 22-Aug 4 Chapter 1 43 / 45

156 Main tools in spectral analysis of neutron transport niversity of FrancheComté Besançon France (Institute) CIMPA School Muizemberg July 22-Aug 4 Chapter 1 44 / 45

157 Main tools in spectral analysis of neutron transport niversity of FrancheComté Besançon France (Institute) CIMPA School Muizemberg July 22-Aug 4 Chapter 1 44 / 45

158 Main tools in spectral analysis of neutron transport Neutron transport semigroups are non-self-adjoint. niversity of FrancheComté Besançon France (Institute) CIMPA School Muizemberg July 22-Aug 4 Chapter 1 44 / 45

159 Main tools in spectral analysis of neutron transport Neutron transport semigroups are non-self-adjoint. niversity of FrancheComté Besançon France (Institute) CIMPA School Muizemberg July 22-Aug 4 Chapter 1 44 / 45

160 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! +. niversity of FrancheComté Besançon France (Institute) CIMPA School Muizemberg July 22-Aug 4 Chapter 1 44 / 45

161 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! +. niversity of FrancheComté Besançon France (Institute) CIMPA School Muizemberg July 22-Aug 4 Chapter 1 44 / 45

162 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". niversity of FrancheComté Besançon France (Institute) CIMPA School Muizemberg July 22-Aug 4 Chapter 1 44 / 45

163 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". niversity of FrancheComté Besançon France (Institute) CIMPA School Muizemberg July 22-Aug 4 Chapter 1 44 / 45