Homogen. Boltzmann- Nordheim Non Isotropic Cauchy Theory for the Boltzmann Nordheim Equations Equation for Bosons. Bose Einstein Condensa- tion Known Amit Einav, University of Cambridge 1 Results Local Cauchy Theory for Boltzmann- Nordheim Equation Nonlocal Nonlinear Partial Differential Equations and Applications Strategy of Anacapri, Italy the Proof Global Existence 17th of September, 2015 for the Boltzmann- Nordheim Equation Final Remarks 1 Joint Work with Marc Briant The Author was supported by EPSRC grant EP/L002302/1
Table of Contents Homogen. 1 The Spatially Homogeneous Boltzmann-Nordheim equation for Bosons Boltzmann- Nordheim Equations 2 Bose Einstein Condensation Bose Einstein Condensa- tion 3 Known Results Known Results Local 4 A Local in Time Cauchy Theory for the Boltzmann-Nordheim Equation Cauchy Theory for Boltzmann- Nordheim Equation 5 Strategy of the Proof Strategy of the Proof Global 6 Global Existence for the Boltzmann-Nordheim Equation Existence for the Boltzmann- Nordheim 7 Final Remarks Equation Final Remarks
The Spatially Homogeneous Boltzmann Equation One of the most influential equation in the kinetic theory of gases is the Homogen. so-called Boltzmann equation, describing the time evolution of the Boltzmann- Nordheim probability density of a particle in a classical dilute gas. Equations In its spatially homogeneous form it reads as Bose Einstein Condensa- � t > 0 , v ∈ R d ∂ t f ( v ) = Q B ( f )( v ) tion Known f | t =0 = f 0 , Results Local with Cauchy Theory for Boltzmann- � f ( v ′ ) f ( v ′ Nordheim � � Q B ( f )( v ) = R d × S d − 1 B ( v , v ∗ , σ ) ∗ ) − f ( v ) f ( v ∗ ) dv ∗ d σ, Equation Strategy of the Proof where d σ is the uniform probability measure on the sphere, B is the Global collision kernel, containing all the physical information about the Existence for the interactions between the particles, and Boltzmann- Nordheim Equation v ′ = v + v ∗ + | v − v ∗ | σ ∗ = v + v ∗ − | v − v ∗ | σ v ′ , . Final 2 2 2 2 Remarks
The Collision Kernel Most normal physical situations correspond to the case where B ( v , v ∗ , σ ) = Φ ( | v − v ∗ | ) b (cos θ ) , Homogen. Boltzmann- Nordheim � � Equations v − v ∗ and Φ( z ) = C Φ z γ . with cos θ = | v − v ∗ | , σ Bose Einstein The power γ represents the ’hardness’ of the potential. In what will Condensa- follow we will assume that γ ∈ [0 , 1], a regime containing the interesting tion cases of the Maxwell Molecules ( γ = 0) and Hard Spheres ( γ = 1). Known Results In general, the angular part of the collision kernel, b , satisfies Local Cauchy Theory for b (cos θ ) sin d − 2 ( θ ) θ → 0 + b 0 θ − (1+ ν ) , ∼ Boltzmann- Nordheim Equation for some ν ∈ ( −∞ , 2). Removing the singularity, i.e. requiring that Strategy of the Proof � π Global b (cos θ ) sin d − 2 ( θ ) d θ < ∞ l b = Existence for the 0 Boltzmann- Nordheim corresponds to the so-called Grad’s angular cut off condition, which we Equation will also assume in what follows. In fact, we will require that Final Remarks b ∞ = � b � L ∞ < ∞ .
The Spatially Homogeneous Boltzmann-Nordheim Equation The Boltzmann equation arises in classical mechanics, and bears no consideration to possible quantum statistical effects. Homogen. This was corrected in 1928 by Nordheim, who suggested a new equation Boltzmann- Nordheim that takes into account the fact that in the quantum statistics the Equations probability to end up in a state may also depend on the number of Bose particles already occupying that state. His modification to the original Einstein Condensa- equation reads as tion Known � t > 0 , v ∈ R d ∂ t f ( v ) = Q BN ( f )( v ) Results (1) Local f | t =0 = f 0 , Cauchy Theory for Boltzmann- where Nordheim Equation � � f ( v ′ ) f ( v ′ Q BN ( f )( v ) = R d × S d − 1 B ( v , v ∗ , σ ) ∗ ) (1 + α f ( v )) (1 + α f ( v ∗ )) Strategy of the Proof Global Existence � for the 1 + α f ( v ′ ) 1 + α f ( v ′ � � � � − f ( v ) f ( v ∗ ) ∗ ) dvdv ∗ d σ, Boltzmann- Nordheim Equation with α = 1 for bosons (the probability to occupy the same state as Final Remarks another particle is increased) and α = − 1 for fermions (the probability to occupy the same state as another particle is decreased).
Conserved Quantities and the Entropy Homogen. Much like the classic Boltzmann equation, the Boltzmann-Nordheim Boltzmann- Nordheim equation for both bosons and fermions conserves mass, momentum and Equations energy: Bose Einstein Condensa- 1 1 M 0 tion � � f ( t , v ) dv = f 0 ( v ) dv = . v v u Known Results R d | v | 2 R d | v | 2 M 2 Local Cauchy The Boltzmann-Nordheim equation also admits an entropy functional. In Theory for Boltzmann- the bosonic gas case it is given by Nordheim Equation � Strategy of S ( f ) = R d ((1 + f ( v )) log(1 + f ( v )) − f ( v ) log f ( v )) dv . the Proof Global Existence Under the Boltzmann-Nordheim flow the entropy increases with time, yet for the Boltzmann- as a difference of two terms it gives no control over possible blow-ups. Nordheim Equation Final Remarks
The Bose Einstein Condensation Homogen. Boltzmann- It has been shown that for a given mass M 0 , momentum u , and energy Nordheim Equations M 2 there exists a unique maximiser to the entropy S which is of the form Bose 1 Einstein F BE ( v ) = m 0 δ ( v − u ) + , Condensa- β 2 ( ( v − u ) 2 − µ ) − 1 tion e Known Results with Local m 0 ≥ 0 and m 0 or µ must be zero. Cauchy β ∈ (0 , ∞ ] is the inverse of the equilibrium temperature. Theory for Boltzmann- µ ∈ ( −∞ , 0] is the chemical potential. Nordheim Equation This suggests that any solution to the Boltzmann-Nordheim equation Strategy of should converge to an appropriate F BE as time goes to infinity. The the Proof phenomena of the appearance of a delta function at the average Global Existence momentum u corresponds to the physical aggregation of all the particles for the Boltzmann- at the same velocity. This is known as the Bose-Einstein Condensation. Nordheim Equation Final Remarks
The Bose Einstein Condensation Cont. Homogen. Boltzmann- Nordheim Equations The Condensation phenomena can be expressed in a simple, physical, Bose Einstein condition depending on the mass and energy of the gas. Condensa- tion In the case when d = 3 one has that m 0 = 0 if and only if Known Results � 3 � 4 π ζ (3 / 2) Local 5 3 5 M 0 ≤ M 2 , Cauchy 3 3 Theory for ( ζ (5 / 2)) 5 Boltzmann- Nordheim Equation where ζ is the Riemann Zeta function. Strategy of This can be recast in terms of a critical temperature condition. the Proof Global Existence for the Boltzmann- Nordheim Equation Final Remarks
Known Results Homogen. Boltzmann- Up till now, most results for the Boltzmann-Nordheim equation for Nordheim Equations bosons have been obtained in the isotropic setting. Bose X. Lu: Global in time Cauchy theory for isotropic initial data with Einstein Condensa- 1 + | v | 2 � bounded mass and energy in L 1 � . Lu also developed the theory tion for distributional solution and has shown long time convergence towards Known Results equilibrium. Local Cauchy Escobedo and Vel´ azques: Local in time Cauchy Theory in Theory for L ∞ � 1 + | v | 6+0 � . Additionally, Escobedo and Vel´ azques gave conditions Boltzmann- Nordheim under which, in the isotropic setting, a blow up (implying possible Equation condensation) in finite time must occur. Strategy of the Proof The main goal of the presented work is to develop a robust Local in time Global Existence Cauchy Theory for the Boltzmann-Nordheim equation in a general for the framework . Boltzmann- Nordheim Equation Final Remarks
Local in Time Cauchy Theory for the Boltzmann-Nordheim Equation - Notations Homogen. Boltzmann- Nordheim Equations In what follows we will use the following notations: Bose Einstein Condensa- � � L p f ∈ L p s , v = � (1 + | v | s ) f ( v ) � L p tion s , v = v ( R ) | � f � L p v ( R ) < ∞ Known Results � R d | v | α f ( v ) dv M α = Local Cauchy Theory for Boltzmann- b ∞ = � b � L ∞ Nordheim Equation � Strategy of l b = S d − 1 b (cos θ ) d σ the Proof Global Existence for the Boltzmann- Nordheim Equation Final Remarks
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