On the Weak Coupling Limit of Quantum Many-body Dynamics and the Quantum Boltzmann Equation

The rigorous derivation of the Uehling-Uhlenbeck equation from more fundamental quantum many-particle systems is a challenging open problem in mathematics. In this paper, we exam the weak coupling limit of quantum N-particle dynamics. We assume the integral of the microscopic interaction is zero and we assume W^{4,1} per-particle regularity on the coressponding BBGKY sequence so that we can rigorously commute limits and integrals. We prove that, if the BBGKY sequence does converge in some weak sense, then this weak-coupling limit must satisfy the infinite quantum Maxwell-Boltzmann hierarchy instead of the expected infinite Uehling-Uhlenbeck hierarchy, regardless of the statistics the particles obey. Our result indicates that, in order to derive the Uehling-Uhlenbeck equation, one must work with per-particle regularity bound below W^{4,1}.


Introduction
The rigorous derivation of the celebrated Uehling-Uhlenbeck equation from more fundamental quantum many-particle systems is a challenging open problem in mathematics. This problem has received a lot of attentions in recent years. In particular, Erdös, Salmhofer and Yau have given, in [4], a formal derivation of the spatially homogeneous Uehling-Uhlenbeck equation as the thermodynamic limit from the Fock space model. Around the same time, in [1,2,3], Benedetto, Castella, Esposito, and Pulvirenti initiated a different study of the problem with the "classical N -particle Bogoliubov-Born-Green-Kirkwood-Yvon (BBGKY) hierarchy" approach. Here, "classical BBGKY hierarchy" means the usual BBGKY hierarchy in R 3N +1 . Moreover, Benedetto, Castella, Esposito, and Pulvirenti consider the N → ∞ limit of N particles in R 3 instead of the thermodynamic limit. In this paper, we follow the classical BBGKY hierarchy approach in [1,2,3]. Let t ∈ R, x k = (x 1 , ..., x k ) , v k = (v 1 , ..., v k ) ∈ R 3k , ε = N − 1 3 , and φ be an even pair interaction. We consider the following quantum BBGKY hierarchy We would like to immediately remark that, we have not assumed anything about the statistics the particles obey, though it seems that writing down hierarchy (1.1) like [ in [3, (2.14)] is N −k √ ε . Since N = ε −3 , the difference −k √ ε B ε j,k+1 must tend to zero as long as N √ ε B ε j,k+1 = ε −3 √ ε B ε j,k+1 tends to a definite limit as ε → 0 for every fixed k. Hence, we have not assumed anything about the statistics the particles obey.
We shall not go into the details about the rise of hierarchy (1.1). We refer the interested readers to [2] and [3]. The ε → 0 limit of hierarchy (1.1) is called the weak-coupling limit of quantum many-body dynamics. As mentioned in [3], this is characterized by the fact that the potential interaction is weak in the sense that it is of order √ ε and the density of particles is 1. Therefore the number of collisions per unit time is ε −1 . Since the quantum mechanical cross-section in the Born approximation (justified because the potential is small) is quadratic in the potential interaction, the accumulated effect is of the order number of collisions × [potential interaction] 2 = 1/ε × ε = 1. We are concerned with the central question of identifying the weak coupling limit ε → 0 (or equivalently N → ∞) for such a quantum BBGKY hierarchy (1.1), even at a formal level. The expected ε → 0 limit of hierarchy (1.1) is the infinite Uehling-Uhlenbeck hierarchy which is defined by where the two particle term Q 1,j,k+1 is given by and the three particle term Q 2,j,k+2 is given by and θ = ±1 for bosons and fermions respectively. The expected mean-field equation for the infinite Uehling-Uhlenbeck hierarchy (1.2) is exactly the Uehling-Uhlenbeck equation [8]. It arises as the special solution to hierarchy (1.2), provided that f satisfies However, to our surprise, we find that, as long as f (k+1) N is of W 4,1 per-particle regularity, the infinite Uehling-Uhlenbeck hierarchy (1.2) is not the ε → 0 limit of the BBGKY hierarchy (1.1) regardless of the statistics the particles obey. We find that, regardless of the statistics the particles obey, the ε → 0 limit of the BBGKY hierarchy (1.1) is the infinite quantum Maxwell-Boltzmann hierarchy coming from [3], defined as (1.4) in this paper. In fact, if one formally commutes integrals and lim ε→0 in approriate places, one finds that, regardless of the statistics the particles obey, the formal ε → 0 limit of the BBGKY hierarchy (1.1) must be the infinite quantum Maxwell-Boltzmann hierarchy (1.4) instead of the infinite Uehling-Uhlenbeck hierarchy (1.2).
To this end, we define the quantum Maxwell-Boltzmann hierarchy to be The C j,k+1 in (1.4) is certainly the Boltzmann collision operator. We use A ε and B ε to denote the inhomogeneous terms in (1.1) because neither of the ε → 0 limit of A ε nor B ε along gives the collision operator C. The collision operator C j,k+1 in (1.4) arises as the ε → 0 limit of a suitable composition of A ε and B ε .
Notice that, the mean-field equation of hierarchy (1.4) is not the Uehling-Uhlenbeck equation (1.3). If one assumes Maxwell-Boltzmann statistics as well asφ(0) = 0, a term by term convergence from (1.1) to (1.4) was rigorously established in [3]. The mean-field equation in this case is the quantum Boltzmann equation: where the collision operator Q is given by is a solution to hierarchy (1.4) provided that f solves (1.6).
In our main theorem, we assume W 4,1 per-particle regularity on the BBGKY sequence f (k) N N k=1 so that we can rigorously commute limits and integrals in suitable places. We are then able to prove that, if the BBGKY sequence f does converge in some weak sense, then the limit sequence must satisfy the infinite quantum Maxwell-Boltzmann hierarchy (1.4) instead of the infinite Uehling-Uhlenbeck hierarchy (1.2), regardless of the statistics the particles obey. We work in the space W 4,1 k which is W 4,1 (R 3k × R 3k ) equiped with the weak topology. We work with the norm Our main theorem is the following.
Theorem 1 (Main Theorem). Assume the interaction potential φ is an even Schwarz class function and satisfies the vanishing condition:φ vanishes at the origin to at least 11th order. Suppose a subsequence converges weakly to some Γ = f (k) ∞ k=1 in the following sense: There is a C > 0 such that Then Γ = f (k) ∞ k=1 satisfies the infinite quantum Boltzmann hierarchy (1.4), regardless of the form of the initial datum f We remark that Theorem 1 certainly does not imply that the Uehling-Uhlenbeck equation (1.3) is not derivable as a mean-field limit. Our result is merely an indication that, in order to derive the Uehling-Uhlenbeck equation, one must work with per-particle regularity bound below W 4,1 . It is certainly an interesting question to lower the regularity requirement of Theorem 1. But we are not able to do so currently.
Before delving into the proof of Theorem 1, we would like to discuss the assumptions of Theorem 1. First of all, not only we are not specifying the statistics f to take a special form, e.g. tensor product form or quasi free form, to make Theorem 1 to hold. Compared with the work by King [6] and Landford [7] 1 on deriving the classical Boltzmann equation from models with hard spheres collision and singular potentials, the interparticle interaction φ we are considering here, is smooth, and hence the regularity assumption in Theorem 1 is not impossible. The proof of Theorem 1 suggests that the assumption φ = 0 orφ(0) = 0 might actually be a necessary condition such that the quantum BBGKY hierarchy (1.1) has a N → ∞ limit. See §2.4 for a discussion. For completeness, we include, in the appendix, a discussion about the cubic term of the Uehling-Uhlenbeck equation (1.3) whenφ(0) = 0.
1.1. Acknowledgement. The first author would like to thank P. Germain, E. Lieb, B. Schlein, C. Sulem, and J. Yngvason for discussions related to this work.

Proof of the Main Theorem
For notational simplicity, it suffices to prove the main theorem for k = 1 and with the assumption that the whole sequence Γ N = f (k) N N k=1 N has only one limit point. Our goal is to prove the absence of cubic Uehling-Uhlenbeck terms in the limit. Let S (k) (t) be the solution operator to the equation We will prove that every limit point To this end, we use the BBGKY hierarchy (1.1) .Write hierarchy (1.1) in integral form, we have Iterate hierarchy (2.2) once and get to f (1)

3)
1 See also [5]. where On the one hand, iterating hierarchy (2.2) once gives the terms which are quadratic in φ and hence are the central part of the quantum Boltzmann hierarchy (1.4) and the Uehling-Uhlenbeck hierarchy (1.2). On the other hand, we remark that one will not obtain the infinite Uehling-Uhlenbeck hierarchy (1.2) corresponding to the Uehling-Uhlenbeck equation (1.3) even one iterates (2.2) more than once and then considers its limit as ε → 0. The easiest way to see this is to notice that the new terms will not be quadratic in φ.
If one believes the mean-field limit where f satisfies some mean-field equation, then in the ε → 0 limit, IV in (2.3) will generate a nonlinearity which is quadratic in f and φ in the mean-field equation, and V in (2.3) will produce a term which is cubic in f and quadratic in φ. With the above discussion in mind, alert reader can immediately tell that the main part of the proof of Theorem 1 is proving that the Boltzmann collision operator C j,k+1 defined in (1.5) arises as the ε → 0 limit of IV, and the ε → 0 limit of V is zero and thus there is no Uehling-Uhlenbeck term in the limit.
in the sense stated in the main theorem (Theorem 1), we know by definition that Moreover, it has been shown in [1,3] that the terms II and III tend to zero as ε → 0. We are left to prove the emergence of the quardratic collision kernel C j,k from IV and the possible cubic term V is in fact zero as ε → 0.
2.1. Emergence of the Quardratic Collision Kernel. IV is the most important term since it contributes (1.5) in the limit. Recall IV We write and hence obtain the quardratic collision kernel which is the rightmost term in (2.1). Notice that is simply another test function for all t 1 and t 2 . Hence, to establish (2.4), it suffices to prove the following proposition.

Proposition 1. Under the assumptions in Theorem 1, we have
Proof. We prove the propopsed limit with a direct computation. We start by writing out C ε 1,2 f The h 1 and h 2 integrals are highly oscillatory. We change variables to move the h ′ s away from f N : first the x part, Then the v part To evaluate the above integral, we substitute like [1, (2.15)] and have Taking the ε → 0 limit (justified in §2.3), we arrive at Using the fact that Put in spherical coordinates for the dh 2 integration: we let h 2 = rω, where r ∈ R + and ω ∈ S 2 , to get for short at the moment. Notice that in the middle of (2.7), we have σ1,σ2=±1 Do the substitution, ω new = −ω old in terms C and D, we then find that C =Ā and D =B. So (2.8) is actually if we denote the Heaviside function by H.
Putting the above computation of (2.8) into (2.7), we have Insert a Heaviside function H(r) to do the dr integral, which is exactly Whence we conclude the proof of Proposition 1.

2.2.
The Cubic Term is Zero. Here, we investigate the limit of We write N .
If the ε → 0 limit of Q ε 1,3 f N is nonzero, it will correspond to a cubic nonlinearity in the mean-field equation. On the one hand, as we remarked earlier in the paper, for the Uehling-Uhlenbeck equation (1.3) to rise as the mean-field equation, lim ε→0 Q ε 1,3 f N must not be zero. On the other hand, lim ε→0 Q ε 1,3 f (3) N has to be zero for Theorem 1 to hold. Hence, we compute lim ε→0 Q ε,1 1,3 f N and lim ε→0 Q ε,2 1,3 f N in complete detail.

Treatment of Q ε,1
1,3 f N . We prove that the limit N (t 2 )dx 1 dv 1 = 0 by direct computation. Since the proposed limit is zero, we drop the prefactor (−i) (2π) 3 in B ε so that we do not need to keep track of it. We write Different from the quadratic term treated in §2.1, S (2) has no effect on (x 3 , v 3 ), so it becomes Then We move all h's away from f N . First, we substitute the x-part with Then the v-substitution: Redo the change of variable: we then have Write out the phase, Rearrange the phase, Now we need to perform one more change of variable to take care of [ih 2 · (x 2 − x 3 )] /ε. We do and arrive at . Taking the ε → 0 limit inside, which is justified in §2.3, we have Do the dx 2 dξ 1 dh 2 integration, = σ1,σ2=±1 Sinceφ(0) = 0, the above is zero and hence N dx 1 dv 1 = 0.
Notice that ifφ(0) = 0, the ds 1 integral yields an infinity. We formally see that it is necessary for φ to have zero integration in order to have the quantum Boltzmann hierarchy (1.4) and hence the quantum Boltzmann equation (1.6). We will go back to (2.10) in §2. 4 to discuss more about this. It is natural to wonder if Q ε,2 1,3 f N will carry a negative sign and hence cancel out Q ε,1 1,3 f N . Such a guess is not true. The term N dx 1 dv 1 actually equals to (2.11) with no sign difference. (See (2.13).) In below we treat Q ε,2 1,3 f N .

Treatment of
Again, we change variables to move all h's away from f N . We use the new x-variables: Then the new velocity variables: we then have . Write out the phase, Another change of variable takes us to which is zero under the same reasoning as in the treatment of Q ε,1 1,3 f N . At this point, we have proven that the possible cubic term is zero in the ε → 0 limit. Therefore, we have proven that relation (2.1) holds for f (k) and hence established Theorem 1. The rest of this section is to prove that we can take the limits inside the integrals under the assumptions of Theorem 1.
In fact, rewrite which makes we can then transform A ε (s 1 , ξ 1 ) into where B(εs 1 , y 1 , y 2 , εx 3 , w 1 , N (t 2 − εs 1 , Clearly, for bounded ξ 1 , x 3 and s 1 ,such a integral is finite. We only need to control large ξ 1 , x 3 and s 1 to pass to the limit. Upon using standard smooth cutoff functions, we only need to concentrate the most singular region of we may further assume that in such a region, All the other cases are simpler and can be controlled similarly. We first integrate by part in y 1 1 , w 1 1 repeatedly, (since εs 1 is bounded), to obtain we then take integration by part in h 1 2 four times as above to get the worse term, in terms of vanishing order ofφ(εξ 1 − h 2 )φ(h 2 ) in B as [φ(h 2 )φ(εξ 1 − h 2 )] × B j (εs 1 , y 1 , y 2 , εx 3 , w 1 , w 2 , εξ 1 , h 2 , v 3 ).

2.4.
It is Necessary to Have φ = 0. Recall (2.10), J(x 1 , v 1 )Q ε, 1 1,3 f To see that it is necessary to have φ = 0 in order to have a ε → 0 limit for the BBGKY hierarchy (1.1) and hence a possible derivation for the quantum Boltzmann hierarchy (1.4) and the quantum Boltzmann equation (1.6), we analysis the size of (2.17). To avoid some technical issues, let us assume that J(...)f Hereĝ means the Fourier transform in x 3 . We then find that, for every t 2 , x 2 , v 3 , we effectively have a δ(h 2 ), so that the dh 2 integral is restricted to have size |h 2 | ε. Now, say t 2 1, we know |s 1 h 2 | 1 and hence e −ih2·(s1(v1+σ2 h 2 2 )−s1v2) ∼ 1 which then makes the ds 1 integral to blow up as ε → 0 ifφ(0) = 0.