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PREPRINT 2011:27

On Low Kinetic ; Spin , Bose Einstein Condensates,

LEIF ARKERYD

Department of Mathematical Sciences Division of Mathematics CHALMERS UNIVERSITY OF TECHNOLOGY UNIVERSITY OF GOTHENBURG Gothenburg Sweden 2011

Preprint 2011:27

On Low Temperature Kinetic Theory; Spin Diffusion, Bose Einstein Condensates, Anyons

Leif Arkeryd

Department of Mathematical Sciences Division of Mathematics Chalmers University of Technology and University of Gothenburg SE-412 96 Gothenburg, Sweden Gothenburg, December 2011

Preprint 2011:27 ISSN 1652-9715

Matematiska vetenskaper Göteborg 2011 ON LOW TEMPERATURE KINETIC THEORY; SPIN DIFFUSION, BOSE EINSTEIN CONDENSATES, ANYONS

Leif Arkeryd∗

Abstract

The lecture discusses various aspects of kinetic theory in the quantum regime for not far from absolute zero and with the focus on models evolving through pair-. No pre- vious knowledge in the area is required.

0 Introduction.

Let us start by recalling some basic facts about the classical Boltzmann case. Consider a parcel of evolving as dx dv = v, = F. dt dt That implies the evolution of the gas density f; ∂f ∂f dx ∂f dv ∂f ∂f ∂f D f(x(t), v(t), t) = + + = + v + F , t ∂t ∂x dt ∂v dt ∂t ∂x ∂v which gives the classical BE when driven by a term Q(f), ∂f ∂f ∂f (D f(x(t), v(t), t) =) + v + F = Q(f). t ∂t ∂x ∂v

0 0 For two having pre-collisional velocities v, v∗, denote the velocities after collision by v , v∗, 0 0 0 0 and write f∗ = f(v∗), f = f(v ), f∗ = f(v∗) . In this notation the classical Boltzmann collision R 0 0 s operator becomes Q(f) = B(f f∗ − ff∗)dv∗dω. The kernel B is typically of the type b(ω)|v − v∗| with −3 < s ≤ 1. The collision term is linearly proportional to the density of each of the two participating . and are each conserved, and the monotone in time R dvdxf log f(t), prevents strong concentrations to form. In the classical case all collisions respecting the conservation laws are permitted,

∗Department of Mathematics, Chalmers University, S-41296 Gothenburg, Sweden.

1 The quantum case is different. Close to absolute zero, permitted energy levels are usually dis- crete, implying fewer collisions. There may exist a condensate, where an excitation may interact only with those modes of its zero-point that will not give away energy. In the quantum regime, also the thermal de Broglie wave length may become much larger than the typical inter- distance, a situation totally absent from classical kinetic theory. The scattering is wave-like and two-body quantum statistics gives a marked imprint on the individual collision process, e.g. the Pauli principle for fermions.

The rich phenomenology of low temperature corresponds to a similarly rich mathematical structure for its models, including the kinetic ones. The quantum kinetic collisions and equations are much more varied and fine-tuned than in the classical case. This talk will consider typical quan- tum kinetic models evolving through pair collisions, in order to illustrate the differences between the classical and quantum cases;

1) The Nordheim Uehling Uhlenbeck model, 2) Spin in the fermionic case, 3) Low temperature bosons and a condensate, 4) A kinetic equation,

The physicists obtain these models from wave mechanics in the Heisenberg setting and from quan- tum field theory. In some cases the models have been validated in a formal mathematical sense (cf [BCEP], [S1]). Based on the examples 1-4, the final section will elaborate on the

5) Differences between quantum and classical Boltzmann theory.

1 The Nordheim Uehling Uhlenbeck model.

Around 1930 the first quantum kinetic equation for boson and fermion gases was proposed in- dependently by Nordheim and by Uehling-Uhlenbeck ([N], [UU]). It is of particular interest in a neighbourhood of and below the transition temperature Tc, where a condensate first appears. The NUU equation is semiclassical; the transport left hand side of this evolution equation is classical, only the collisional right hand side is directly influenced by the quantum effects. Actually the term quantum usually refers to a semiclassical equation.

The collision operator for the NUU equation is Z 0 0 0 0 QNUU (f)(p) = Bδ(p + p∗ − (p + p∗))δ(E(p) + E(p∗) − (E(p ) + E(p∗))) R3×R3×R3 (1.1)  0 0 0 0  0 0 f f∗(1 + f)(1 + f∗) − ff∗(1 + f )(1 + f∗) dp∗dp dp∗, with  = 1 for bosons,  = −1 for fermions, and  = 0 gives the classical case. Here the quartic terms vanish. For  = ±1 the entropy is Z  1  f log f − (1 + f) log(1 + f) dp. 

2 Also this entropy is monotone but, contrary to the classical case, gives no control of concentrations. f In equilibrium Q is zero, and multiplication with log 1+f and integration gives Z Z f 0 0 0 0 0 = Q(f) log dv = Bδ(p + p∗ − (p + p∗))δ(E(p) + E(p∗) − (E(p ) + E(p∗))) 1 + f R3×R3×R3 0 0 0 0  f f∗ f f∗  (1 + f)(1 + f∗)(1 + f )(1 + f∗) 0 0 − 1 + f 1 + f∗ 1 + f 1 + f∗ 0 0  f f∗ f f∗  log log − log 0 log 0 . 1 + f 1 + f∗ 1 + f 1 + f∗ Since the two last factors in the integral have the same sign, that implies 0 0 f f∗ f f∗ 0 0 ≡ . 1 + f 1 + f∗ 1 + f 1 + f∗ f M As in the classical case, we conclude that 1+f = M, a Maxwellian, hence f = 1−M , which for  = ±1 is the Planckian equilibrium function. For fermions ( = −1) the concentrations are bounded by one, because of the factor (1 − f) which preserves positivity together with f. This is stronger than the entropy controlled concentrations from the classical case. There are general existence results for the fermion case in an L1 ∩ L∞-setting due to J. Dolbeault [D] and P.L. Lions [PLL]. The boson case is more intricate. So far there are only partial existence results - for the space-homogeneous, isotropic setting ([Lu1], [Lu2], [EMV]), and for some space-dependent settings close to equilibrium ([R]).

There are also results about existence, uniqueness and asymptotic behaviour ([EM]) for the re- lated Boltzmann-Compton equation ∂f Z ∞ k2 = (f 0(1 + f)B(k0, k; θ) − f(1 + f 0)B(k, k0; θ))dk0 (1.2) ∂t 0 with f photon density, k energy, θ temperature, and with the detailed balance law ek/θB(k0, k; θ) = ek0/θB(k, k0; θ). This equation models a space-homogeneous isotropic photon gas in interaction with a low temperature electron gas in equilibrium, having a Maxwellian distribution of velocities.

2 a) Spin in the fermionic case.

Spin polarized neutral gases at low temperatures have been experimentally studied and kinetically modelled in physics for some time, an early mathematical physics text being [S]. The first experi- ments concerned very dilute solutions of 3He in superfluid 4He with - in comparison with classical Boltzmann gases - interesting new properties such as spin waves. The experimentalists later turned to laser-trapped low-temperature gases. We recall some properties of the Pauli spin matrices  0 1   0 i   1 0  σ = , σ = , σ = . 1 1 0 2 −i 0 3 0 −1

The Pauli spin matrix vector is denoted by σ = (σ1, σ2, σ3). With [σi, σj] denoting the commutator σiσj − σjσi, the Pauli matrices satisfy

[σ1, σ2] = 2iσ3, [σ2, σ3] = 2iσ1, [σ3, σ1] = 2iσ2, and [σi, σi] = 0 for i = 1, 2, 3,

3 which is equivalent to σ × σ = 2iσ. Let M2(C) denote the space of 2 × 2 complex matrices, and H2(C) the subspace of hermitean matrices. 1 A dilute spin polarized gas with spin 2 , will be modelled by a distribution function matrix ρ ∈ H2(C) which is the Wigner transform of the one- density operator for the system being modelled. The 3 domain of ρ(t, x, p) is taken as positive time t, p ∈ R , and for simplicity here with x-space periodic in 4 3d with period one. H2(C) is linearly isomorphic to R , if we use the decomposition ρ = AcI +As ·σ 4 and identify ρ ∈ H2(C) with (Ac,As) ∈ R .We shall focus on the following simplified model, ∂ Dρ := ρ + p · 5 ρ = Q(ρ) (2.1) ∂t x with (in the Born approximation), the collision integral given by Z 0 0 0 0 2 2 02 02 Q(ρ) = dp2dp1dp2δ(p1 + p2 − p1 − p2)δ(p1 + p2 − p1 − p2 ) h i {[ρ10 , ρ˜1]+Tr(˜ρ2ρ20 ) − [˜ρ10 , ρ1]+Tr(ρ2ρ˜20 )} − {[ρ10 ρ˜2ρ20 , ρ˜1]+ − [˜ρ10 ρ2ρ˜20 , ρ1]+} , whereρ ˜ = I − ρ, and [., .]+ denotes an anti-commutator. The of particles of any spin component, is given by f := T r(ρ(t, x, p)), and the magnetization of particles is given by the 1 vectorσ ¯(t, x, p) := T r(σρ(t, x, p)), which implies ρ = 2 (fI +σ ¯ · σ). The resulting equations for f andσ ¯ are

Df = Qn(f, σ¯), (2.2)

Dσ¯ = Qm(f, σ¯), (2.3) where 1 Z Q (f, σ¯) = dp dp0 dp0 δ(p + p − p0 − p0 )δ(p2 + p2 − p02 − p02) n 2 2 1 2 1 2 1 2 1 2 1 2  1 1 1 1 [f 0 − (f f 0 +σ ¯ · σ¯ 0 )][f 0 − (f f 0 +σ ¯ · σ¯ 0 )] − [f − (f f 0 +σ ¯ · σ¯ 0 )][f − (f f 0 +σ ¯ · σ¯ 0 )] 1 2 1 1 1 1 2 2 2 2 2 2 1 2 1 1 1 1 2 2 2 2 2 2 1 1 1 1  −([¯σ 0 − (f 0 σ¯ + f σ¯ 0 )] · [¯σ 0 − (f 0 σ¯ + f σ¯ 0 )] − [¯σ − (f 0 σ¯ + f σ¯ 0 )][¯σ − (f 0 σ¯ + f σ¯ 0 )]) , 1 2 1 1 1 1 2 2 2 2 2 2 1 2 1 1 1 1 2 2 2 2 2 2 1 Z Q (f, σ¯) = dp dp0 dp0 δ(p + p − p0 − p0 )δ(p2 + p2 − p02 − p02) m 2 2 1 2 1 2 1 2 1 2 1 2  1 1 1 1 [f − (f f 0 +σ ¯ · σ¯ 0 )][¯σ − (f 0 σ¯ + f σ¯ 0 )] − [f 0 − (f f 0 +σ ¯ · σ¯ 0 )][¯σ 0 − (f 0 σ¯ + f σ¯ 0 )] 2 2 2 2 2 2 1 2 1 1 1 1 2 2 2 2 2 2 1 2 1 1 1 1 1 1 1 1  −([f 0 − (f f 0 +σ ¯ · σ¯ 0 )][¯σ 0 − (f 0 σ¯ + f σ¯ 0 )] − [f − (f f 0 +σ ¯ · σ¯ 0 )][¯σ − (f 0 σ¯ + f σ¯ 0 )]) . 1 2 1 1 1 1 2 2 2 2 2 2 1 2 1 1 1 1 2 2 2 2 2 2

The collision terms Qn and Qm do not change: R R for Qn the number density Qndp = 0, the linear density pQndp = 0, and the energy R 2 R density p Qndp = 0, and for Qm the magnetization density Qmdp = 0.

2 b) Existence results.

Consider the initial value problem for the equations (2.3-4). The initial values (f0, σ¯0) are assumed 2 to satisfy 0 ≤ f0 ≤ 1, f0 ≥ σ¯0 · σ¯0. First consider the case with a truncation Bn in the domain of

4 2 2 integration for Q, where Bn is the characteristic function of the set p1 + p2 ≤ n. Set F (t, x, p) = f(t, x, p) for 0 ≤ f ≤ 1, = 0 for f < 0, = 1 for f > 1, F σ¯ Σ(t, x, p) =σ ¯(t, x, p) when F 2 ≥ σ¯ · σ,¯ else Σ(t, x, p) = √ (t, x, p). σ¯ · σ¯

Lemma 2.1 [A2] The initial value problem for the truncated problem Df = Qn(F, Σ),Dσ¯ = ∞ ∞ Qm(F, Σ) has a unique local solution in L , when (f0, σ¯0) ∈ L and 0 ≤ f0 ≤ 1.

The system Df = Qn(F, Σ), Dσ = Qm(F, Σ) with the same initial values under the truncation Bn can be solved by a contraction argument under the truncation Bn.

We shall remove the truncation and, using a variant of the approach by [D] and [PLL], study the simplified collision integral, when there is spin in the σ3 direction only; Z 0 0 0 0 2 2 02 02 Q(ρ) = dp2dp1dp2δ(p1 + p2 − p1 − p2)δ(p1 + p2 − p1 − p2 )[{[ρ10 , ρ˜1]+Tr(˜ρ2ρ20 )

−[˜ρ10 , ρ1]+Tr(ρ2ρ˜20 )} + {[ρ10 ρ˜2ρ20 , ρ˜1]+ − [˜ρ10 ρ2ρ˜20 , ρ1]+}], withρ ˜ = I − ρ.

Recalling that the spin is restricted to the σ3 direction, by adding and subtracting the two collision operators 1 Z Q ± Q = dp dp0 dp0 δ(p + p − p0 − p0 )δ(p2 + p2 − p02 − p02) n m 2 2 1 2 1 2 1 2 1 2 1 2  1 1 [f 0 − (f f 0 +σ ¯ · σ¯ 0 )][(f 0 ± σ¯ 0 )(1 − (f ± σ¯ )] 2 2 2 2 2 2 1 1 2 1 1 1 1 −[f − (f f 0 +σ ¯ · σ¯ 0 )][(f ± σ¯ )(1 − (f 0 ± σ¯ 0 )] 2 2 2 2 2 2 1 1 2 1 1 1 1 ∓[¯σ 0 − (f 0 σ¯ + f σ¯ 0 )][(f 0 ± σ¯ 0 )(1 − (f ± σ¯ )] 2 2 2 2 2 2 1 1 2 1 1 1 1  ±[¯σ − (f 0 σ¯ + f σ¯ 0 )][(f ± σ¯ )(1 − (f 0 ± σ¯ 0 ))] 2 2 2 2 2 2 1 1 2 1 1 1 Z = dp dp0 dp0 δ(p + p − p0 − p0 )δ(p2 + p2 − p02 − p02) 2 2 1 2 1 2 1 2 1 2 1 2  1 1 (f 0 ± σ¯ 0 )(1 − (f ± σ¯ ))(f 0 ∓ σ¯ 0 )(1 − (f ∓ σ¯ )) 1 1 2 1 1 2 2 2 2 2 1 1  −(f ± σ¯ )(1 − (f 0 ± σ¯ 0 ))(f ∓ σ¯ )(1 − (f 0 ∓ σ¯ 0 )) . 1 1 2 1 1 2 2 2 2 2 This gives

−C max(f1 ± σ1, 0) ≤ −C(F1 ± Σ1) ≤ Qn(F, Σ) ± Qm(F, Σ). And so

# ∂t(max(Tnf ± Tmσ, 0) ) exp(Ct) ≥ 0.

Here Tnf, Tmf are the right hand sides of the equations in mild form. In the same way 1 ∂ (1 − min(T f ± T σ, 0)#) exp(Ct) ≥ 0. t 2 n m Hence the sums and differences are under bounded control. It follows that f = F, σ¯ = Σ, and then by the method of [PLL] that under the assumptions on 0 ≤ f0 ≤ 1 and −f0 ≤ σ¯0 ≤ f0, there is a

5 3 solution also in the limit p ∈ R to (2.3-4) with 0 ≤ f ≤ 1 and −f ≤ σ¯ ≤ f. We have thus proved

1 Theorem 2.2 [A2] ρ = 2 (fI +σσ ¯ ) solves the original problem (2.1).

A next step would be to extend this proof to the case of varying spin directions, where spin diffu- sion phenomena occur. Other problems of considerable physical interest are (cf [JM]) the relaxation times for spin-diffusion, and the influence of more involved transport terms in (2.3-4) such as the physicists’ version of the problem, namely

∂f X h∂hp ∂σ¯p ∂hp ∂σ¯p i + 5  · 5 f − 5  5 ·f + · − · = Q ∂t p p r p r p p p ∂p ∂r ∂r ∂p n i=xyz i i i i ∂σ¯p X h∂p ∂σ¯p ∂p ∂σ¯p ∂fp ∂hp ∂fp ∂hp i + − + − − 2(h × σ¯ ) = Q . ∂t ∂p ∂r ∂r ∂p ∂r ∂p ∂p ∂r p p m i i i i i i i i i Here 2 Z p 0 1 0  = + dp {V (0) − V (|p − p |)}f 0 (r, t) p 2m 2 p Z 1 0 0 h = − (γB − dp V (p − p )¯σ 0 (r, t), p 2 p V is the inter-particle potential, B an external magnetic field, and γ is the gyromagnetic ratio.

Spintronics, widely studied in physics, tries to control the spin of electrons and use it as a new vector of information. For semiconductor hetero-structures the control can be achieved by magnetic fields, and modelled by a linear spinor Boltzmann equation ∂ ρ + v · 5 ρ + E · 5 ρ = Q(ρ) + Q (ρ) + Q (ρ). (2.4) ∂t x v SO SF Here E is an electric field and Q is the collision operator for collisions without spin-reversal, in the linear BGK approximation Z α(v, v0)(M(v)ρ(v0) − M(v0)ρ(v)dv0, R3 M denoting a normalized Maxwellian. The spin-orbit coupling generates an effective field Ω making i the spins precess. The corresponding spin-orbit interaction term QSO(ρ) is given by 2 [Ω · σ, ρ]−, [, ]− denoting a commutator. Finally QSF (ρ) is a spin-flip collision operator, in relaxation time approximation given by

trρI2 − 2ρ QSF (ρ) = , τsf with τsf > 0 the spin relaxation time. Mathematical properties such as existence, uniqueness, asymptotic behaviour, have been studied in particular by the French group around Ben Abdallah, his coworkers and students (see [BH] and references therein).

6 3 a) Low temperature bosons and a condensate.

For a Bose fluid and for sufficiently low temperatures, only interactions between thermally excited (quasi-)particles and a condensate are of practical importance. We shall now consider one such situ- ation, involving of the same type in both components and allowing transfer of atoms between the components, so that in particular no mass conservation should be expected for an individual component. The model (for validation questions cf [BCEP] and [S1]) is based on the Beliaev-Popov approximation of the Bogoliubov Green-function description which ignores off-diagonal correlations, and the Thomas-Fermi approximation which neglects a quantum term.

Our simplified two-component model consists of a kinetic equation for the distribution function of a gas of bosonic (quasi-)particles interacting with a Bose condensate, which in turn is described by a Gross-Pitaevskii equation (cf [PS]). In the local rest frame, the kinetic equation for the excitations is ∂f 1 1 + 5 (E + v p) · 5 f = Q(f, n ). (3.1) ∂t  p p s x 2 c

Here vs is the superfluid velocity, Ep the Bogoliubov excitation energy, and  a small parameter. The collision term becomes Z 2 Q(f, nc)(p) = nc |A| δ(p1 − p2 − p3)δ(E1 − E2 − E3)[δ(p − p1) (3.2)

−δ(p − p2) − δ(p − p3)]((1 + f1)f2f3 − f1(1 + f2)(1 + f3))dp1dp2dp3. We notice that the domain of integration is only 2d and not 5d as for classical Boltzmann. The transition kernel |A|2 can be explicitly computed by the Bogoliubov approximation scheme.

A := (u3 − v3)(u1u2 + v1v2) + (u2 − v2)(u1u3 + v1v3) − (u1 − v1)(u2v3 + v2u3). Here the Bose coherence factors u and v are

2 ˜p + Ep 2 2 up = , up − vp = 1, 2Ep

p2 with ˜p = 2m + gnc, nc the non-equilibrium density of the atoms in the condensate, m the atomic mass, and g a scattering length defined later.

The collision operator Q(f, nc) can be formally obtained (cf [ST], [EMV], [No]) from the Nordheim- Uehling-Uhlenbeck collision operator Z ˜ 0 0 0 0 QNUU (f)(p) = Bδ(p + p∗ − (p + p∗))δ(E(p) + E(p∗) − (E(p ) + E(p∗))) R3×R3×R3  0 0 0 0  0 0 f f∗(1 + f)(1 + f∗) − ff∗(1 + f )(1 + f∗) dp∗dp dp∗.

Namely, assume that a condensate appears below the Bose-Einstein condensation temperature Tc. 1 That splits the quantum gas distribution function into a condensate part ncδp=0 and an L -density part f(t, x, p), and we obtain ˜ ˜ 2 3 QNUU (f + ncδp=0) = QNUU (f) + Q(f, nc) + bnc + cnc + dncδp=0, where a simple computation shows that b = c = 0. At the low temperatures we have in mind today, the physicists consider the number of excited (quasi-)particles to be small, and neglect the Q˜NUU

7 collision term relative to the collision operator Q.

The usual Gross-Pitaevskii (GP) equation for the wave function ψ (the order parameter) associated with a Bose condensate is ∂ψ h2 ih = − ∆ ψ + (U + g|ψ|2)ψ, ∂t 2m x ext i.e. a Schr¨odingerequation complemented by a non-linear term accounting for two-body interac- tions. Uext is an external potential. For simplicity, in this preliminary study we do not include the strongly space-inhomogeneous trapping potential, an essential ingredient in modern experiments on laser-trapped Bose gases. Modulo a numerical factor, g is the s-scattering length of the two-body interaction potential. In the present context the GP equation is further generalized by letting the condensate move in a self-consistent (Hartree-Fock) mean field 2gn˜ = 2g R f(p)dp produced by the thermally excited atoms, together with a dissipative coupling term associated with the collisions. This gives ∂ψ 1 1 Z i = − ∆ ψ + (U + g|ψ|2 − i C(f, n )(p)dp)ψ, (3.3) ∂t 2 x ext 2m2 c with C(f, n ) = 1 Q(f, n ) and U in our case equal to a constant plus a term of order . It is often c nc c ext √ iθ useful to split the equation for ψ = nce into phase and amplitude variables (polar representation or the Madelung transform) and remove some physically negligible term, leading to ∂n Z 2 c + 5 · (n v ) = − Q(f, n )dp ∂t x c s c ∂θ mv2 = −µ − s , ∂t c 22 with µc a local condensate .

3 b) A space-homogeneous, isotropic case.

Consider the Cauchy problem for the two component model (3.1), (3.3) in a space-homogeneous isotropic situation and in the superfluid rest frame (condensate velocity vs = 5xθ = 0), i.e. the equations ∂f = Q(f, n ), (3.4) ∂t c dn Z c = − Q(f, n )dp, (3.5) dt c with initial values

f(p, 0) = fi(| p |), nc(0) = nci. (3.6)

Here f(p, t) is the density of the quasi-particles, nc(t) the mass of the condensate, and the collision operator Q is given by (3.2).

There are two physically important regimes. One is the low temperature situation where the

8 temperature is smaller than 0.4T , with all |p | << p , i.e. where physically all quasi-particle mo- c i 0 √ menta are much smaller than the characteristic momentum p0 = 2mgnc for the crossover between the linear and quadratic parts of the Bogoliubov excitation energy of the quasi-particles;

r 4 2 2 p gnc 2 p p E(p) := 2 + p ≈ c|p|(1 + ) = c|p|(1 + 2 ). (3.7) 4m m 8gmnc 4p0 q Here c := gnc the speed of Bogoliubov sound. Setting m = √1 gives p = c. In applications with m 2 0 |p| << p0, the right hand side of (3.4) is usually taken as the value of E(p). The Bose coherence factors can then be taken as r s r s gnc 1 E(p) gnc 1 E(p) 2 2 up = + , vp = − , up − vp = 1, 2E(p) 2 2gnc 2E(p) 2 2gnc which gives s s s p 0 0 r 0 0 1 E(p∗)E(p )E(p∗) gnc E(p∗) E(p ) E(p∗) A = 5 3 + ( 0 + 0 − 0 0 ). 2 2 (gnc) 2 2 E(p∗)E(p ) E(p∗)E(p∗) E(p )E(p∗) 0 0 And so recalling that E(p∗) = E(p ) + E(p∗), we obtain

p 0 0 1 E(p∗)E(p )E(p∗) A = 5 3 . 2 2 (gnc) 2 With this A, the collision operator becomes Z E(p1)E(p2)E(p3) Q(f, nc)(p) = χ 3 2 δ(p1 − p2 − p3)δ(E1 − E2 − E3)[δ(p − p1) g nc −δ(p − p2) − δ(p − p3)]((1 + f1)f2f3 − f1(1 + f2)(1 + f3))dp1dp2dp3, (3.8) where χ denotes the truncation for |pi| ≤ λ, 1 ≤ i ≤ 3. The choice of the positive constant λ will be discussed below.

The opposite limit of intermediate temperatures around 0.7Tc, and where all momenta |pi| >> p0, has the dominant excitation of (Hartree-Fock) single particle type. Expanding the square root p2 definition of E in (3.7), we may approximate Ep by 2m + gnc leading to a collision operator of the type Z Q(f, nc) = knc χδ(p1 − p2 − p3)δ(E1 − E2 − E3)[δ(p − p1) R3×R3×R3 −δ(p − p2) − δ(p − p3)]((1 + f1)f2f3 − f1(1 + f2)(1 + f3))dp1dp2dp3 (3.9) for the ’partial local equilibrium regime’. Here χ is the characteristic function of the set of (p, p1, p2, p3) with |p|, |p1|, |p2|, |p3| ≥ α for a given positive constant α.

In the general case, the collision operator is Z 2 Q(f, nc)(p) = nc |A| δ(p1 − p2 − p3)δ(E1 − E2 − E3)[δ(p − p1)

−δ(p − p2) − δ(p − p3)]((1 + f1)f2f3 − f1(1 + f2)(1 + f3))dp1dp2dp3, (3.10)

9 with the excitation energy E defined by r p2 gn E(p) = |p| + c . 4m2 m The kernel |A|2 is bounded by a multiple of  |p |  |p |  |p |  |A¯|2 := √ 1 ∧ 1 √ 2 ∧ 1 √ 3 ∧ 1 , nc nc nc in the physically interesting cases where asymptotically all |pi| << p0, all |pi| >> p0, or one |pi| << p0 and the others >> p0. These three cases are relevant for low respectively intermediate temperatures compared to Tc, and (the third case) for the collision of low energy phonons with high energy excitations (atoms). The asymptotic situation of two |pi| << p0 and one pi >> p0 (with unbounded A) is excluded by the energy condition. Using |A¯|2 as the kernel in the collision operator, we have with Nouri proved the following result.

1 Theorem 3.1 [AN1] Let nci > 0 and fi(p) = fi(|p|) ∈ L be given with fi nonnegative and 2+γ 1 fi(p)|p| ∈ L for some γ > 0. For the collision operator (3.10) with the transition probabil- ¯ 2 1 1 1 ity kernel |A| , there exists a nonnegative solution (f, nc) ∈ C ([0, ∞); L+) × C ([0, ∞)) to the initial value problem (3.1-3) in the space-homogeneous, isotropic case. The condensate density nc is locally bounded away from zero for t > 0. The excitation density f has energy locally bounded in R R 2+γ time. Total mass M0 = nci + fi(p)dp is conserved, and the moment |p| fdp is locally bounded in time. In the intermediate temperature case a total energy quantity is conserved.

The proof is via approximations controlled by a priori estimates and fixed point techniques. An important still open problem is the question of time-asymptotics.

Last spring A. Heintz developed numerical methods for the above evolution problem in connection with an exam project. The recent paper [S2] considers the spatially homogeneous and isotropic kinetic regime of weakly interacting bosons with s-wave scattering. It has a focus on post-nucleation self-similar solutions. Another recent paper, [EPV], studies linearized space homogeneous kinetic problems in settings related to the ones discussed here, and with a focus on large time behaviour.

3 c) A space-dependent, close to equilibrium case.

In equilibrium, the right hand side of the kinetic equation (3.1), i.e. the collision term, vanishes. log f Multiplying the equation with 1+log f and integrating in p, we obtain

0 0 f∗ f f∗ = 0 0 . (3.11) (1 + f∗) (1 + f ) (1 + f∗) M As usual - and ultimately because of a Cauchy equation - equation (3.11) implies that f = (1−M) 2 with M a Maxwellian. We assume that the Maxwellian has zero bulk velocity, M = exp(−β p +δ). √ 2m With p0 = 2mgnc, we restrict to the ’intermediate temperature region’ |p| >> p0 in the super- fluid rest frame. Denoting the mass density of the superfluid by n0, and using the approximation p2 β 1 2m + gn0 for the energy E of (3.7), equation (3.11) holds for δ = − 2m gn0. Hence for m = 2 , the

10 2 particular Maxwellian M0 = exp(−β(p + gn0)) gives an equilibrium for the collision operator Q. 1 Take for simplicity g = β = 1, which fixes the equilibrium limit as (P0, n0) with P0 = 2 , (exp(p +n0)−1) the Planckian corresponding to M0. Require that the ingoing boundary values for the kinetic part 3 are equal to this limit. Take the initial values (f0, n0) as  -small deviations from this equilibrium which retain the total mass M0.

This space-dependent, close to equilibrium evolutionary problem poses interesting questions about local existence and the long time behaviour, including whether the solution converges to equilib- rium when time tends to infinity. In contrast to the NUU case of [R], here the splitting of the linearized collision operator into collision frequency plus a compact operator, requires a (physically acceptable) cut-off for very large velocities. Also the Milne problem is different from the classical case, e.g. in not having a constant mass flow, and again requires a truncation for large p’s.

4 a) A kinetic anyon equation.

The quantum Boltzmann Haldane equation [BH] is a kinetic equation of Boltzmann type for confined quasi-particles with Haldane statistics [H]. This exclusion statistics interpolates between the Fermi and Bose quantum behaviours. In quantum , the number of quantum states of N occupying G states is given by (G + N − 1)! G! and N!(G − 1)! N!(G − N)! in the boson resp. fermion cases. The interpolated number of quantum states for the fractional exclusion of Haldane and Wu is (G + (N − 1)(1 − α))! 0 < α < 1. (4.1) N!(G − αN − (1 − α))! When applied to the fractional quantum Hall effect, the Haldane statistics coincides with the two- dimensional anyon definition (in terms of the braiding of particle trajectories. Haldane statistics may also be realized for neutral fermionic atoms at ultra-low temperature in three dimensions.

Elastic pair collisions in a Boltzmann type collision operator, are expected to preserve mass, lin- ear momentum, and energy. That holds for the collision operator Q of Haldane statistics, first introduced in [BBM], Z 0 0 0 0 Q(f) = B(v − v∗, ω) × [f f∗F (f)F (f∗) − ff∗F (f )F (f∗)]dv∗dω. (4.2) IRd×Sd−1 Here dω corresponds to the Lebesgue probability measure on the sphere. d d−1 The collision kernel B in the variables (z, ω) ∈ IR × S is positive, locally integrable, and only depends on |z| and |(z, ω)|. The filling factor F is given by F (f) = (1 − αf)α(1 + (1 − α)f)1−α, 1 0 < α < 1. The factor (1 − αf) keeps the value of f between 0 and α . F is convex with maximum 1 1 1−2α 1−2α 1 value one at f = 0 for α ≥ 2 , and maximum value ( α − 1) > 1 at f = α(1−α) for α < 2 . With this filling factor, the collision operator vanishes identically for the Haldane equilibrium dis- tribution functions as obtained in [W] under (4.1), but for no other functions. For the limiting cases, representing boson statistics (α = 0) and fermion statistics (α = 1) the quartic terms in the

11 collision integral are canceled, and ideas from Lions’ compactness result for the classical gain term are applicable. This allows approaches related to the weak L1 analysis for the quadratic Boltzmann equation. But for 0 < α < 1 there is no cancelations in the collision term. Moreover, the Lipschitz continuity of the collision term in the Fermi-Dirac case, is now replaced by a weaker H¨oldercontinuity. For the space homogeneous case, I have developed an approach based on strong L1-compactness.

Consider the initial value problem for the Boltzmann equation with Haldane statistics in the space- homogeneous case with velocities in IRd, d ≥ 2, df = Q(f). (4.3) dt 1 Because of the filling factor F , the range for the initial value f0 should belong to [0, α ], which is then formally preserved by the equation. The general BH equation (0 < α < 1) retains important prop- erties from the Fermi-Dirac case, but it has so far not been validated from basic quantum theory. In particular the choice of kernel for the collision operator has to be studied in other ways. The choice β (z,ω) by the physicists has been to use classical kernels such as those for hard , B(z, ω) = |z| b( |z| ) with 0 < β ≤ 1 and Grad angular cut-off, which I will now discuss. That also agrees with the better understood limiting cases α = 0, 1. We assume that 0 < b ≤ c|z|β| sin θ cos θ|d−1, with an initial s ∞ moment (1 + |v| )f0 in L for s ≥ d − 1 + β.

Theorem 4.1 [A3] Consider the space-homogeneous equation (4.3) for hard kernels with

0 < B(z, ω) ≤ C|z|β| sin θ cos θ|d−1, (4.4)

1 where 0 < β ≤ 1, d > 2, and 0 < β < 1, d = 2, and let the initial value 0 < f0 ∈ L have 1 s finite energy. If 0 < f0 ≤ α and esssup(1 + |v| )f0 < ∞ for s = d − 1 + β, then the initial value problem for (4.3) has a solution in the space of functions continuous from t ≥ 0 into L1 ∩ L∞, s0 which conserves mass and energy, and for t0 > 0 given, has esssupv,t≤t0 |v| f(t, v) bounded,where 0 2β(d+1)+2 s = min(s, d ). Idea of proof. This initial value problem is first considered for a family of approximations with 1 bounded support for the kernel B, when 0 < f0 ≤ esssupf0 < α . Starting from approximations with Lipschitz continuous filling factor, the corresponding solutions are shown to stay away uniformly 1 from α , the upper bound for the range. Uniform Lipschitz continuity follows for the approximating operators and leads to well posedness for the limiting problem. Uniform L∞ moment bounds for the approximate solutions hold, using an approach from the classical Boltzmann case [A1]. Based on those preliminary results the global existence result for hard forces of Theorem 4.1, can be proved by strong compactness arguments. Mass and first moments are conserved and energy is bounded by its initial value. That bound on energy in turn implies energy conservation using the arguments for energy conservation from [MW] or [Lu2].

Remarks. i) The proof of the theorem implies the following stability. Given a sequence of initial values (f0n)n∈N with 1 sup esssupv f0n(v) < , n α

12 1 1 and converging in L to f0, then there is a subsequence of the solutions converging in L to a solution with initial value f0. ii) The important question of long time behaviour is open. iii) The space-dependent case will require new ideas.

5 Differences between quantum and classical Boltzmann theory.

The examples discussed, were a selection to illustrate how the quantum kinetic situation in various ways markedly differ from classical kinetic theory. There are also other types of low temperature kinetic evolutions of collision type. One example is collisions involving five quasi-particles, two col- liding ones giving rise to three or conversely, cf [K]. Other examples are Grassmann algebra valued gas densities, i.e. completely anti-commuting densities, sometimes useful to model the Pauli exclu- sion effect, cf [PR], and pair collision terms taking account of the finite duration of a collision, cf [ES]. i) Qualitatively different collision operators. There are many more qualitatively different collision operators in the quantum than in the classical regime, like (1.1), (1.2), (2.2), (2.5), (3.2), (4.2), and the examples mentioned from [K], [PR], and ES]. The quantum influence sometimes appears as entirely new phenomena such as the collision- driven spin waves in the fermionic spin matrix Boltzmann equation mentioned earlier. The thermal de Broglie wave length may become much larger than the typical inter-particle distance, an aspect totally absent from the classical theory. The quantum influence leads to different filling factors, which as for the NUU equation can stabilize as well as destabilize the collision effects. ii) Fewer collisions. There are fewer collisions close to absolute zero, where permitted energy levels usually are discrete. In the example of boson excitations plus a condensate, an excitation may interact only with the modes of the zero-point motion that do not give away energy to it. Also, in that example the domain of integration is 2d instead of the classical 5d. iii) Typical energy range. A quantum situation is usually not scale invariant as in the classical case, but has a typical energy range (cf the example in Section 3b). A particular type of quasi-particles/excitations exists in a particular energy interval, and extending the energy outside this interval, may introduce collision types not observed in the experiments, and physically irrelevant for the modelling at hand. In this way bounded velocity domains may be both physically correct and mathematically convenient in certain quantum situations, like the two component, space-dependent boson example. Similarly, solutions existing only for a bounded interval of time, e.g. depending on relaxation effects, may be the most adequate ones in a particular quantum case. iv) Parameter range and kernel family. The collision terms and energy expressions discussed, were defined from a physical context. Chang- ing a parameter mildly may completely change the mathematical aspects of the problem, as in the anyon example around 0 and 1. Another example where very small changes in a parameter com- pletely changes the physical situation, is the the extremely narrow temperature-pressure domain for the very rich A-phase in 3He. The question of parameter range and kernel family, is in general

13 more delicate for quantum than for classical kinetic theory. v) Questions from classical kinetic theory. For each type of quantum kinetic collision term, the multitude of questions from classical kinetic theory may be posed and studied. New boundary-condition dependent phenomena may appear, an example being a low-temperature gas of excitations in a condensate between rotating cylinders having a vacuum friction type of radiation as in the Zeldovich-Starobinsky effect, cf [V]. vi) New ideas. The set-ups are sometimes variants of the classical case that require new ideas, an examples being the Milne problem for the space-dependent boson system.

Bibliography

[A1] L. Arkeryd, L∞-estimates for the space-homogeneous Boltzmann equation, Jour. Stat. Phys., 31 (1983) 347-361.

[A2] L. Arkeryd, A kinetic equation for spin polarized Fermi systems, in preparation.

A3] L. Arkeryd, A quantum Boltzmann equation for Haldane statistics and hard forces; the space-homogeneous initial value problem, Comm. Math. Phys. 298, 573-583 (2010).

[AN1] L. Arkeryd, A. Nouri, Bose condensates in interaction with excitations - a kinetic model, Comm. Math. Phys., to appear 2012.

[AN2] L. Arkeryd, A. Nouri, Bose condensates in interaction with excitations - a space- dependent, small data, kinetic model, in preparation.

[BBM] R. K. Bhaduri, R. S. Bhalero, M. V. Murthy, Haldane exclusion statistics and the Boltzmann equation, J. Stat. Phys. 82, 1659-1668 (1996).

[BCEP] D. Benedetto, F. Castella, R. Esposito, M. Pulvirenti, Some considerations on the derivation of the nonlinear quantum Boltzmann equation II, J. Stat. Phys. 124, 951-966 (2006).

[BH] N. Ben Abdallah, R. El Hajj, On a hierarchy of macroscopic models for semiconductor spintronics, preprint (2009) IMT, Universit´ede Toulouse.

14 [D] J. Dolbeault, Kinetic models and quantum effects: a modified Boltzmann equation for Fermi-Dirac particles, Arch. Rat. Mech. Anal. 127, 101-131 (1994).

[EM] M. Escobedo, S. Mischler, On a quantum Boltzmann equation for a gas of photons, J. Math. Pures Appl. 80, 471-515 (2001).

[EMV] M. Escobedo, S. Mischler, M. Valle, Homogeneous Boltzmann equation in quantum relativistic kinetic theory, Electronic J. Diff. Eqns., Monograph 04 (2003).

[EPV] M. Escobedo, F. Pezzotti, M. Valle, Analytical approach to relaxation dynamics of condensed Bose gases, arXiv: 1008.0714v1.

[ES] S. F. Edwards, D. Sherrington, A new method of expansion in the quantum many-body problem, Proc. Phys. Soc. 90, 3-22 (1967).

[H] F. D. Haldane, Fractional statistics in arbitrary dimensions: a generalization of the Pauli principle, Phys. Rev. Lett. 67, 937-940 (1991).

[JM] J. W. Jeon, W. J. Mullin, Kinetic equations for dilute, spin-polarized quantum systems, J. Phys. France 49, 1691-1706 (1988).

[K] I. M. Khalatnikov, Theory of superfluidity (in Russian), Nauka, Moskva 1971.

[PLL] P. L. Lions, Compactness in Boltzmann’s equation via Fourier integral operators and applications I, III, Jour. Math. Kyoto Univ. 34, 391-427, 539-584 (1994).

[Lu1] X. Lu, A modified Boltzmann equation for Bose-Einstein particles: isotropic solutions and long time behaviour, Jour. Stat. Phys. 98, 1335-1394 (2000).

[Lu2] X. Lu, On isotropic distributional solutions to the Boltzmann equation for Bose-Einstein particles, Jour. Stat. Phys. 116, 1597-1649 (2004).

[MW] S. Mischler, B. Wennberg, On the spacially homogeneous Boltzmann equation, Ann. Inst. Henri Poincar´e16, 467-501 (1999).

[N] L. W. Nordheim, On the kinetic methods in the new statistics and its applications in the electron theory of conductivity, Proc. Roy. Soc. London Ser. A 119, 689-698 (1928).

[No] A. Nouri, Bose-Einstein condensates at very low temperatures. A mathematical result in the isotropic case, Bull. Inst. Math. Acad. Sin. 2, 649-666 (2007).

[PR] S. J. Putterman, P. H. Roberts, Random waves in a classical nonlinear Grassmann field I, II, Physica A 131, 35-63 (1985).

15 [PS] L. Pitaevski, S. Stringari, Bose-Einstein condensation, Oxford 2003.

[R] G. Royat, Etude de l’´equationd’Uehling-Uhlenbeck: existence de solutions proches de Planckiennes et ´etudenum´erique,Th`ese,Marseille 2010.

[S] V. P. Silin, Introduction to the kinetic theory of gases (in Russian), Nauka Moscow, 1971.

[S1] H. Spohn, Kinetic equations for quantum many-particle systems, Modern encyclopedia of mathematical physics, Springer 2007.

[S2] H. Spohn, Kinetics of the Bose-Einstein condensation, Physica D 239, 627-634 (2010).

[UU] E. A. Uehling, G. E. Uhlenbeck, in Einstein-Bose and Fermi-Dirac gases, Phys. Rev. 43, 552-561 (1933).

[V] G. E. Volovik, The universe in a droplet, Oxford Science Publications, 2003.

[W] Y. S. Wu, Statistical distribution for generalized of fractional-statistics particles, Phys. Rev. Lett. 73, 922-925 (1994).

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