ON A BOLTZMANN EQUATION FOR HALDANE STATISTICS

. The study of quantum quasi-particles at low temperatures includ-ing their statistics, is a frontier area in modern physics. In a seminal paper Haldane [10] proposed a deﬁnition based on a generalization of the Pauli exclusion principle for fractional quantum statistics. The present paper is a study of quantum quasi-particles obeying Haldane statistics in a fully non-linear kinetic Boltzmann equation model with large initial data on a torus. Strong L 1 solutions are obtained for the Cauchy problem. The main results concern existence, uniqueness and stabililty. Depending on the space dimension and the collision kernel, the results obtained are local or global in time.


1.
Haldane statistics and the Boltzmann equation. In a previous paper [2], we studied the Cauchy problem for a space-dependent anyon Boltzmann equation [5], The collision operator Q α in [2] depends on a parameter α ∈]0, 1[, and is given by and the filling factor F α Let us recall the definition of anyon. Consider the wave function ψ(R, θ, r, ϕ) for two identical particles with center of mass coordinates (R, θ) and relative coordinates 324 LEIF ARKERYD AND ANNE NOURI (r, ϕ). Exchanging them, ϕ → ϕ + π, gives a phase factor e 2πi for bosons and e πi for fermions. In three or more dimensions those are all possibilities. Leinaas and Myrheim proved in 1977 [11], that in one and two dimensions any phase factor is possible in the particle exchange. This became an important topic after the first experimental confirmations in the early 1980-ies, and Frank Wilczek in analogy with the terms bos(e)-ons and fermi-ons coined the name any-ons for the new quasiparticles with any phase.
By moving from spin to a definition in terms of a generalized Pauli exclusion principle, Haldane [10] extended this to a fractional exclusion statistics valid for any dimension. The conventional Bose-Einstein and Fermi-Dirac statistics are commonly associated with integer spin bosonic elementary particles resp. half integer spin fermionic elementary particles, whereas the Haldane fractional statistics is connected with quasi-particles corresponding to elementary excitations in many-body interacting quantum systems.
The collision operator Q is given by Strong solutions to the space-homogeneous case were obtained in [1] for any dimension bigger than one in velocity. Strong solutions to the space-inhomogeneous case were obtained in [2] in a periodic slab for two-dimensional velocities. There the proof depends on the two-dimensional velocities setting. In the present paper we prove local in time well-posedness of the Cauchy problem for k = 1 and collision kernels similar to those used in [2], and for k ∈ {1, 2, 3} global in time well-posedness under the supplementary assumption of very soft potential at infinity [15]. The solutions conserve mass, momentum and energy.
2. The main results. With cosθ = n · v−v * |v−v * | , the kernel B(|v − v * |, n) will from now on be written B(|v − v * |, θ) and be assumed measurable with for some B 0 > 0. It is also assumed for some γ, γ > 0, that The initial datum f 0 (x, v), periodic in x, is a measurable function with values in [0, 1 α ], (2.4) and such that for some positive constants c 0 andc 0 , for any subset X of R 3 of positive measure, (2.9) Strong solutions to the Cauchy problem with initial value f 0 associated to the quantum Boltzmann equation (1.1) are considered in the following sense. and The main results of the present paper are given in the following theorems.
) on the initial L 1 -datum. It conserves mass, momentum and energy.
Remarks. Theorem 2.1 is restricted to the slab case, since its proof below uses an estimate for the Bony functional only valid in one space dimension. Theorems 2.1 and 2.2 also hold with the same proofs in the fermion case where α = 1, in particular giving strong solutions to the Fermi-Dirac equation.
Theorems 2.1 and 2.2 also hold with a limit procedure when α → 0 in the boson case where α = 0, in particular giving strong solutions to the Boltzmann Nordheim equation [14]. It is the object of a separate paper [4] (see also [9], [13] and [7]) Theorems 2.1 and 2.2 also hold for v ∈ R n , n ≥ 3. The proofs in [2] strongly rely on the property that for any unit vector n with direct orthogonal unit vector n ⊥ , either n 1 or n ⊥1 is bigger that 1 √ 2 , where n 1 (resp. n ⊥1 ) is the component of n (resp. n ⊥ ) along the x-axis. This allows to control the mass density of the solution from its Bony functional. This is no more the case in the three-dimensional velocity setting of the present paper. It is why our results are local in time under the same assumptions on the collision kernel B as in [2]. They are global in time under the supplementary assumption of a very soft potential at infinity.
The paper is organized as follows. Approximations are introduced in Section 3 for k ∈ {1, 2, 3} together with for k = 1, a control of their Bony functional. Their mass density is uniformly controlled under the assumptions of Theorem 2.1 (resp. Theorem 2.2) in Section 4 (resp. Section 5). The well-posedness of the Cauchy problem is proven in Section 6. Conservation of mass, momentum and energy is proven in Section 7.

Preliminaries on solution approximations and the Bony functional.
Let k ∈ {1, 2, 3}. For any j ∈ N * , denote by ψ j , the cut-off function with ψ j (r) = 0 if r > j 2 and ψ j (r) = 1 if r ≤ j 2 , and set Denote by Q j (resp. Q + j , and Q − j to be used later), the operator and its loss part For j ∈ N * , let a mollifier ϕ j be defined by 3) The following lemma concerns a corresponding approximation of (1.1)-(1.2) for k ∈ {1, 2, 3}.
There is η j > 0 such that f j takes its values in [0, 1 α − η j ]. The solution conserves mass, momentum and energy.
Proof of Lemma 3.1. Let T > 0 be given. We shall first prove by contraction that for T 1 > 0 and small enough, there is a unique solution to (3.4). Let the map C be defined on periodic in x functions in The previous linear partial differential equation has a unique periodic solution ).
For f with values in [0, 1 α ], g takes its values in [0, 1 α ]. Indeed, denoting bȳ it holds that for T 1 > 0 small enough only depending on j, since the derivative of the map F j is bounded by (3jα α−1 + 1)j 1−α on [0, 1 α ]. Let f j be its fixed point, i.e. the solution of (3.4) on [0, T 1 ]. The argument can be repeated and the solution continued up to t = T . By Duhamel's form for f j (resp. 1 − αf j ), If there were another nonnegative local solutionf j to (3.4), defined on [0, T ] for some T ∈]0, T ], then by the exponential form it would strictly stay below 1 implying that the difference would be identically zero on [0, T ]. Thus f j is the unique solution on [0, T ] to (3.4), and has its range contained In Lemma 3.2 the tails for large velocities of the mass are controlled with respect to the mass density.
where c T only depends on T and |v| 2 f 0 (x, v)dxdv.
Proof of Lemma 3.2. Denote f j by f for simplicity. By the non-negativity of f , Here in the last integral, either |v | or |v * | is the largest and larger than λ √ 2 . The two cases are symmetric, and we discuss the case |v | ≥ |v * |. After a translation in x, the integrand of the r.h.s of the former inequality is estimated from above by The change of variables (v, v * , n) → (v , v * , −n) and the integration over give the bound The lemma follows.
Lemma 3.3. For any n ∈ S 2 , denote by n 1 the component of n along the x-axis. It holds that Consequently, using the conservation of mass and energy of f , It results from and the conservations of the mass, momentum and energy of f that And so, by (3.8), Here, c is a constant depending only on f 0 (x, v)dxdv and where c is a constant only depending on f 0 (x, v)dxdv and |v| 2 f 0 (x, v)dxdv.

ON A BOLTZMANN EQUATION FOR HALDANE STATISTICS 331
After a change of variables the left hand side can be written by an exchange of the variables v and v * . Moreover, exchanging first the variables v and v * , for any β > 0. It follows that Proof of Lemma 4.1. Denote f j by f for simplicity. By (3.6), For any (v, v * ) ∈ R 3 × R 3 , let N be the set of n ∈ S 2 with max{n 1 , n ⊥1 } < , where n ⊥ is the unit vector in the direction v − v * (orthogonal to n) in the plane defined by v − v * and n, and n 1 is the component of n along the x-axis. Let N c be the complement of N in S 2 . Denote by (3.7) also holds with n 1 replaced by n ⊥1 . Integrating (4.2) with respect to (x, v) and using (2.2) and Lemma 3.3 leads to Moreover, And so, (4.1) holds with Denote by Integrating (4.5) with respect to (x, v), using Lemma 3.3, the 1 α (resp. α α−1 ) bound from above of f (resp. F j (y),y ∈ [0, 1 α ]), gives for any x 0 ∈ [0, 1], λ > 0 and Λ > 0 that for an appropriate choice of (Λ, λ). Moreover, Taking =c 1+t   The lemma follows.

Proof of Lemma 4.3. Denote by E(x) the integer part of
where, for > 0, δ > 0 and λ that will be fixed later, In A 1 , A 2 and A 3 , bound the factor sup τ ∈[0,t] f (τ, x + s(v 1 − v * 1 ), v * ) by its supremum over x ∈ [0, 1], and make the change of variables

LEIF ARKERYD AND ANNE NOURI
Choose λ = −1 , δ = 5 and = 1 16 min{ 1 (4.13) which is sufficient for the polynomial in (4.12) to have two nonnegative roots and take a negative value at 2c 0 . Recalling that M j (0) = c 0 and M j is continuous by the continuity in time and space of f j , it follows that

Control of the mass density under the assumptions of Theorem 2.2.
Let k ∈ {1, 2, 3}. Under the supplementary assumption (2.11), we prove a uniform control with respect to j of the mass density M j (t) defined in (3.5). It relies on the two following lemmas.
Proof of Lemma 5.1 Denote f j by f for simplicity. By the non-negativity of f , it holds Using the 1 α bound for f (τ, x + τ (v − v * ), v * ), and (2.11) leads to sup by the mass conservation.  6. Well-posedness of the Cauchy problem. Let T 0 be supremum of the times up to which it has been proved that the mass densities of the approximations are uniformly bounded. Recall that T 0 may be finite (resp. is infinite) under the assumptions of Theorem 2.1 (resp. 2.2). We prove in this section that for any T ∈ [0, T 0 [ there is a unique solution to the Cauchy problem (1.1)-(1.2). This section is divided into three steps. In the first step, we study initial layers for the approximations. In the second step, the existence of a solution f to (1.1) on [0, T ] for T ∈]0, T 0 [ is shown. Finally the third step proves the uniqueness and stability result stated in Theorems 2.1 and 2.2.
First step: initial layers.
It follows from (6.1) that ν j (f ) andν j (f ) are bounded from above uniformly with respect to j. Denote by c 2 (T ) a bound from above of (ν j (f ) ) j∈N . Let us prove that (ν j (f ) ) is bounded from below for large j on [0, T ] × [0, 1] k × {v; |v| < V } for any V > 0. By definition, Using Duhamel's form for the solution, (6.1) and (2.8), one gets that and denote it by U . Let n be the vector with polar coordinates (θ, ϕ) with respect to v * − v. Choose a coordinates system with the first (resp. second, resp. third) axis in the direction of v * − v (resp. orthogonal to v * − v in the plane defined by v * − v and n, resp. orthogonal to the two first axes). The map U maps the volume since up to second order terms with respect to . And so the Jacobian of U equals cos 4 θ. Using these changes of variables and (6.1), it holds that Consequently, the measure of the set is bounded by 2c1(T ) (γ ) 4 , uniformly with respect to (x, v, θ, ϕ) with | cos θ| > γ , t ∈ [0, T ], and j ∈ N * . Take j T so large that 4 3 πj 3 T is at least twice this uniform bound. Notice that here j T only depends on T , f 0 (x, v)dxdv and |v| 2 f 0 (x, v)dxdv.
Using a median property for the restriction of 3 , which is a bounded measurable Lebesgue function, there are two disjoint sets Ω 1 and Ω 2 of equal volume, such that Hence, by (2.3), for j ≥ j T and a.a. (t, Applying (2.8) to Ω 1 , this is a positive bound from below of are uniformly bounded from above with respect to j by that is continuous and decreasing to zero at x = 1 α . Hence there is Consequently, for j ≥ j T and |v| < V , This gives a maximum time which excludes such a possibility. It follows that The previous estimates leading to the definition of t m are independent of j ≥ j T .
Second step: existence of a solution f to (1.1). Let T ∈ [0, T 0 [ where T 0 , defined at the beginning of this section, may be finite under the hypothesis of Theorem 2.1 and is infinite under those of Theorem 2.2. We shall prove the convergence in L 1 ([0, T ] × [0, 1] k × R 3 )) of the sequence (f j ) to a solution f of (1.1) by proving that it is a Cauchy sequence. Let us first prove that it is a Cauchy sequence in L 1 ([0, T 0 ] × [0, 1] k × R 3 )) for some T 0 ∈]0, T [, i.e. for any β > 0, there exists a ≥ max{1, j T } such that where The sequence (f j ) will be proven to be a Cauchy sequence in in an analogous way. By the uniform boundedness of energy of (g j ), there is V > 0 such that The function g j satisfies the equation x, v)|dxdv, by Lemmas 4.3 and 5.2, (6.10) and By Lemmas 4.1, 4.3 and 5.1, 5.2, this integral restricted to the set where 1 − αf a (t, x, v)) ≤ 2 a , hence where a α , is bounded by c a α for some constant c > 0. For the remaining domain of integration where 1 − αf a (t, x, v)) ≥ 2 a , it holds And so, Split the (x, v)-domain of integration of the latest integral into It holds that |g j (t, x, v)|dxdv, by (6.6), |g j (t, x, v)|dxdv.
Third step: uniqueness of the solution to (1.1) and stability results. The previous line of arguments can be followed to obtain that the solution is unique. Namely, assuming the existence of two possibly local solutions f 1 and f 2 to (1.1) with the same initial datum and bounded energy, Lemma 6.1 holds for both solutions. The difference g = f 1 − f 2 satisfies The first line in the r.h.s. of the former equation gives rise to c |g (t, x, v)|dxdv in the bound from above of d dt |g (t, x, v)|dxdv, whereas the two last lines in the r.h.s of the former equation give rise to the bound c(1 + t α−1 ) |g (t, x, v)|dxdv. Consequently, This implies that |g (t, x, v)|dxdv is identically zero, since it is zero initially. The proof of stability is similar.
7. Conservations of mass, momentum and energy. The conservation of mass and momentum of f follow from the boundedness of the total energy. The energy is non-increasing by the construction of f . Energy conservation will follow if the energy is non-decreasing. This requires the preliminary control of the mass density over large velocities, performed in the following lemma.
For v , v * outside of the angular cutoff (2.2), let n be the unit vector in the direction v − v and n ⊥ its orthogonal unit vector in the direction v − v * . Split C into C = 0≤i≤2 C i , where f # (s, x + s(v 1 − v * 1 ), v * )dv * dnds dv,