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

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On Low Temperature Kinetic Theory; Spin Diffusion, Bose Einstein Condensates, Anyons 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 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 temperatures not far from absolute zero and with the focus on models evolving through pair-collisions. 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 gas 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 collision 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 particles 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∗d!. The kernel B is typically of the type b(!)jv − v∗j with −3 < s ≤ 1. The collision term is linearly proportional to the density of each of the two participating molecules. Mass and kinetic energy are each conserved, and the monotone in time entropy 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 motion 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- particle 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 gases 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 anyon 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 Boltzmann equation 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 \ L1-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 1 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 2 H2(C) which is the Wigner transform of the one-atom density operator for the system being modelled. The 3 domain of ρ(t; x; p) is taken as positive time t, p 2 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 2 H2(C) with (Ac;As) 2 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 f[ρ10 ; ρ~1]+Tr(~ρ2ρ20 ) − [~ρ10 ; ρ1]+Tr(ρ2ρ~20 )g − f[ρ10 ρ~2ρ20 ; ρ~1]+ − [~ρ10 ρ2ρ~20 ; ρ1]+g ; whereρ ~ = I − ρ, and [:; :]+ denotes an anti-commutator. The number density 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 momentum density pQndp = 0, and the energy R 2 R density p Qndp = 0, and for Qm the magnetization density Qmdp = 0.
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