A kinetic equation for spin polarized Fermi systems

This paper a kinetic Boltzmann equation having a general type of collision kernel and modelling spin-dependent Fermi gases at low temperatures modelled by a kinetic equation of Boltzmann type. The distribution functions have values in the space of positive hermitean 2x2 complex matrices. Global existence of bounded weak solutions is proved in L1 to the initial value problem in a periodic box.

The experimental study of spin polarized Fermi gases at low temperatures and their kinetic modelling is well established in physics, an early mathematical physics text in the area being [S]. The first experiments concerned very dilute solutions of 3 He in superfluid 4 He with -in comparison with classical Boltzmann gases -interesting new properites such as spin waves (see [NTLCL]). The experimentalists later turned to other set-ups, in particular laser-trapped low temperature gases (see [JR]). Before turning to the mathematical modelling of such phenomena, we recall some basics about the Pauli spin matrices Denoting by [σ i , σ j ] the commutator σ i σ j − σ j σ i , the Pauli matrices satisfy the Pauli spin vector, (1.1) is equivalent to σ × σ = 2iσ. Let M 2 (C) denote the space of 2 × 2 complex matrices with H 2 (C) the subspace of hermitean matrices. H 2 (C) is linearly isomorphic to R 4 using the decomposition ρ = A c I + A s · σ and identifying ρ ∈ H 2 (C) with (A c , A s ) ∈ R 4 . For A, B ∈ M 2 (C) the contracted product of A and B is A : B = A ijBij . The contracted product of two Pauli matrices σ i and σ j gives σ i : σ j = 2δ ij , where δ ij is Kronecker's delta. More generally for two vectors v = (v 1 , v 2 , v 3 ) and w = (w 1 , w 2 , w 3 ), it holds (v · σ) : (w · σ) = 2v · w. With I the identity matrix, (v · σ)(w · σ) = v · wI + i(v × w) · σ, which implies [v·σ, w·σ] = 2i(v×w)·σ. For ρ † the conjugate transpose of ρ, it holds ρ : ρ † = Trρρ † =: ρ 2 .
A dilute spin polarized gas with spin 1 2 , can be modelled kinetically by a distribution function ρ(t, x, p) with values in H 2 (C), which is the Wigner transform of the one-atom density operator for the system. The domain of ρ(t, x, p) is positive time t, p ∈ R 3 , and for simplicity in this paper x ∈ T 3 , i.e. periodic 3-dimensional position space with period one. The number density of particles is given by f := Tr(ρ(t, x, p)), and the magnetization of particles by the vectorσ(t, x, p) := T r(σρ(t, x, p)). It follows that ρ = 1 2 (f I +σσ). In the fermion case it is assumed that 0 ≤ f ≤ 2, min((2 − f ) 2 , f 2 ) ≥σ ·σ. That condition is equivalent to the hermitean matrices ρ and I − ρ having non-negative eigenvalues, which will be denoted ρ ≥ 0, I − ρ ≥ 0. A common model in mathematical physics considers the evolution Here Q n is the number density part and Q m the magnetization part of the collision operator. The energy matrix is split into V is the inter-particle potential,B an external magnetic field, and γ is the gyromagnetic ratio. Open mathematical problems of interest for physics concern existence, regularity, validation, the relaxation times for spin-diffusion, and time asymptotic behaviour in general.
In this paper p 1 , p 2 and p ′ 1 , p ′ 2 denote post-collisional and pre-collisional moments in an elastic collision, where ω ∈ S 2 and p 1 , p 2 , p ′ 1 , p ′ 2 ∈ R 3 . The distribution function ρ with values in H 2 (C) is assumed to evolve according to the kinetic equation with the collision term of [JM], (1.4) Hereρ = I − ρ, and [., .] + denotes an anti-commutator. This collision term with kernel B = 1, also appears as the dissipative collision term in the Hubbard-Boltzmann model, which describes the evolution of interacting spin-1 2 fermions on a lattice (see [LMS]). In the present paper the collision kernel B(z, ω) is assumed to satisfy . Obviously the two terms in the integrand of (1.4) with a trace factor, are hermitean. For the remaining two terms that property follows by a change of variables ω → −ω. Separating the I-and ρ-part ofρ, we notice that the ensuing terms in (1.4) without any I-factor, formally cancel each others. The main result of the paper is the following global existence theorem for the initial value problem of (1.3).
. The number density f conserves mass and first moments, and has the kinetic energy bounded by its initial value.
The extension of this result to the more general and physically important equation (1.2) remains open. Linearized versions of (1.3) are also discussed in [JM], but again we are not aware of any related mathematical studies. However, phenomenologically modelled, simplified linear Boltzmann equations with spin, introduced in spintronics for semiconductor hetero-structures, such as the equation have been analyzed mathematically. Here E is an electric field, and Q is the collision operator for collisions without spin-reversal in the linear BGK approximation with M denoting a normalized Maxwellian. The spin-orbit coupling generates an effective field Ω making the spins precess. The corresponding spin-orbit interaction term Q SO (ρ) is given by is a spin-flip collision operator, in relaxation time approximation given by Then R := 1 2 (F I + Σσ) ≥ 0, and (I − R) ≥ 0. Consider the equation Lemma 2.1 The equation (2.1) for the truncated kernel B j with initial value ρ 0 , has locally in time a unique hermitean L ∞ -solution ρ = 1 2 (f I +σσ). Proof Set Obviously ρ is a solution of the the initial value problem for (2.1) with collision operator Q j , if and only if it is a fixed point of T j . So it is enough to prove that the operator T j is contracting in L ∞ when 0 ≤ t ≤ t 0 for some t 0 > 0 and small enough. Now Assume that ρ 1# and ρ 2# are continuous in t with respect to the norm After some computations and using the bounds on F 1 , F 2 , Σ 1 , Σ 2 , one obtains Here C j denotes a generic constant. But it is easy to see from its definition, that the mapping ρ → R is Lipschitz continuous in the . ∞ -norm, and so Hence T j is contracting on [0, t 0 ] for t 0 > 0 and sufficiently small, with t 0 independent of ρ 0 . Moreover, T j preserves the hermitean property, hence the solution ρ is hermitean.
It follows that equation (1.3) for Q = Q j holds locally in time if ρ = R, which is equivalent to ρ ≥ 0 and I − ρ ≥ 0. The collision term (1.4) coincides after a change of variables with the collision term C diss of [LMS], where a splitting into gain and loss term is introduced. The gain term is hermitean, if ρ is hermitean. It holds With the loss term becomes The loss term is hermitean together with D j . The collision operator Q j can now be written and (2.1) becomes We next consider the special initial data ρ 0 such that for some η j > 0 and for all |p| < j, uniformly in x Using the bounds on the norm of T j , it follows that on a (short and j-dependent) positive timeinterval [0, t j ) which can be taken maximal in [0, t 0 ], the solution satisfies (2.4) with a decreasing The inverse U −1 , which exists initially by continuity since U (0) = I, satisfies the This gives The integral term is positive since G j (ρ) is positive, and the term U −1 (t j )ρ(0)U †−1 (t j ) satisfies (2.4) for some η > 0. It follows that t j = t 0 and (2.4) holds at t 0 for some η > 0. An analogous reasoning holds for I − ρ. Global existence with positivity of ρ and I − ρ follow by iterating the arguments. Approximating by the above type of uniformly positive initial values an arbitrary initial value ρ 0 having ρ 0 ≥ 0 and I − ρ 0 ≥ 0, the existence result of Theorem 1.1 for Q = Q j follows by continuity.
To prove the conservation properties for the number density f j , consider the equation for f j resulting from (1.3) with B = B j (and dropping the j from f j ,σ j ) where the number density part Q n of the collision operator is Conservation of mass, first moments, and kinetic energy follow by the usual change of variables argument.
End of proof of Theorem 1.1. It remains to prove that the initial value problem for (1.3) has a solution ρ also in the limit j → ∞, i.e. for B with the full domain R 3 . Extracting subsequences, let (f,σ) denote the weak L 1 -limit of (f j ,σ j ). Mass and first moments of f are conserved, and its kinetic energy is bounded by the initial value. The weak limit ρ defined by (f,σ), satisfies (1.3). That can now be proved by the type of weak compactness arguments for (ρ j ) j∈N that were introduced for the scalar Fermi-Dirac case of the Boltzmann equation by [PLL]. In fact, that proof holds step by step, when applied not just to the non-cancelling terms of for the number density f j (i.e. those terms with two or three factors), but to each (two-fold and triple) combination of number density f j and of spin-componentsσ j k , k = 1, 2, 3 that appears. Averaging lemma arguments and weak convergence steps can here be applied separately to each of the four scalar component equations of (1.3), by using that the number density f j satisfies 0 ≤ f j ≤ 2, and the spinsσ j = (σ j 1 ,σ j 2 ,σ j 3 ) satisfy |σ j k | ≤ min(f j , 2 − f j ). The weak limit (extracting subsequences) in the collision term, of each such product of f j s andσ j k s, proves in this way to equal the corresponding product of elements of the limits f andσ k s.
In all such proofs, only an entropy dissipation argument in [PLL] to control various properties/estimates for Q + (f ) = dp 2 dωB( , has to be replaced in the present case by e.g. direct control. The following example shows how that may be done. The L 1 -convergence of a sequence Q + j (f ) is in [(134) of PLL] carried out in terms of an already known convergence of the corresponding sequence Q − j (f ). That may be replaced by the following direct proof. Let K be a fixed compact set in R 3 p . For p 1 ∈ K and j large, max(|p ′ 1 |, |p ′ 2 |) > |p 2 | 2 ) when p 2 1 + p 2 2 > j. Here the B j 's are the earlier truncated versions of B. In the integral Q + (f )(p 1 ) = dp 2 dωB( , make a change of variable from p 2 to the one of p ′ 1 and p ′ 2 giving max(|p ′ 1 |, |p ′ 2 |), and estimate the other factors in f and 1 − f by one. With ω = (θ, ϕ), in these changes of variable the corresponding cos θ or sin θ is bounded away from zero. With A j (p 1 , p 2 ) = B j (p 1 , p 2 , ω)dω and using the bounds on mass and energy, this gives Replacing in our present case, one or more f 's byσ k 's with |σ k | ≤ f , does not change the argument, nor does the introduction of further truncations of B.
In the [PLL] proof, the entropy dissipation argument also appears once before the case discussed in (2.6), namely for a weak L 1 -convergence of Q ± (f j ) (extracting subsequences if necessary). There Q − (f j ) is bounded from above by the usual Boltzmann collision frequency which converges with j, weakly in L 1 on compacts. Then for Q + (f j ), instead of comparing Q + with Q − as in [PLL], consider in our case directly integrals K dp 1 ψ(p 1 )Q + (f j ) for test functions ψ ∈ L ∞ with support in a compact K ⊂ R 3 p , and argue similarly to (2.5) to prove a weak L 1 -convergence of Q + (f j ) with respect to K.
Similarly working through all the many cases of (2.7-8), completes the proof of Theorem 1.1.