A note on dependent random variables in quantum dynamics

We consider the many-body time evolution of weakly interacting bosons in the mean field regime for initial coherent states. We show that bounded k-particle operators, corresponding to dependent random variables, satisfy both, a law of large numbers and a central limit theorem.


INTRODUCTION AND MAIN RESULTS
We consider N weakly interacting bosons in the mean-field regime described on L 2 s (R 3N ), the symmetric subspace of L 2 (R 3N ), by the Hamilton operator with two-body interaction potential v satisfying for a positive constant C > 0. The mean-field regime is characterized through weak and long-range interactions of the particles. Trapped Bose gases at extremely low temperatures, as prepared in the experiments, are known to relax to the ground state. The ground state ψ gs N of (1.1) exhibits Bose-Einstein condensation [17], i.e. the associated ℓ-particle reduced density for all ℓ ∈ N, where ϕ ∈ L 2 (R 3 ) denotes the condensate wave function, known to be the Hartree minimizer. However, we remark that the factorized state ϕ ⊗N does not approximate the ground state due to correlations of the particles [11].
1.1. Law of large numbers. Turning to the probabilistic picture, the property of Bose-Einstein condensation (1.4) implies a law of large numbers for bounded one-particle operators [3]. To be more precise, for k ∈ N we denote with O (k) a bounded, self-adjoint k-particle operator on L 2 (R 3k ) and with i k the multi-index i k = (i 1 , . . . , i k ) ∈ I  i k as a random variable with probability distribution determined through ψ N by where χ A denotes the characteristic function of the set A ⊂ R.
For one-particle operators, factorized states correspond to i.i.d. random variables as for any subsets A 1 , A 2 ⊂ R and i, j ∈ I (1) (1.8) In particular, for factorized states Chebychef's inequality implies a law of large numbers for the centred averaged sum In contrast to one-particle operators, for k-particle operators with k ≥ 2, factorized states do not correspond to i.i.d. random variables. In fact, for k ≥ 2, we have for all i k = j k for which i k contains at least one element of j k . We conclude that in this case, the random variables are correlated and, thus, dependent. In contrast, whenever i k does not intersect with j k , the random are independent (following from arguments similarly to (1.8)). Consequently, for factorized states, the random variables {O N denote a sequence of m-dependent of random variables with m ∈ R. Still, as the following theorem shows, the centred averaged sum 1 satisfies a law of large numbers.
Theorem 1.1 (Law of Large Numbers). For k ∈ N, let O (k) denote a self-adjoint bounded k-particle operator, ϕ ∈ L 2 (R 3 ) and ψ N ∈ L 2 s R 3N a bosonic wave function satisfying γ for all ℓ ∈ N. Then, for any fixed k ∈ N and δ > 0, the averaged sum O (k) For factorized states, we have γ (ℓ) ϕ ⊗N = |ϕ ϕ| ⊗ℓ , and a law of large numbers follows from Theorem 1.1. In particular, Theorem 1.1 shows that the property of condensation (1.12) implies a law of large numbers for bounded k-particle operators for fixed k ∈ N. Thus, Theorem 1.1 generalizes known results from [3] for bounded one-particle operators to k-particle operators with fixed k ∈ N. We recall that the ground state ψ gs N of (1.1) can not be approximated by a factorized state, nonetheless the condensation property (1.4) ensures that bounded k-particle operators satisfy a law of large numbers for ψ gs N , too. We are interested in the dynamics of initially trapped Bose gases. Removing the trap, the bosons evolve with respect to the Schrödinger equation (1.14) with H N the mean-field Hamiltonian given in (1.1). In the following, we consider coherent initial data, i.e. initial data of the form where Ω denotes the vacuum of the bosonic Fock space F = n≥0 L 2 (R 3 ) ⊗ n s equipped with creation and annihilation operators a * (f ), a(f ) for f ∈ L 2 (R 3 ), W (f ) = e a * (f )−a(f ) denotes the Weyl operator and ϕ ∈ H 1 (R 3 ) the condensate wave function. Coherent states of the form (1.15) exhibit Bose-Einstein condensation in the quantum state ϕ, i.e. they satisfy (1.4). Thus, it follows from Theorem 1.1 that initially, the random variables O (k) i k satisfy a law of large numbers. The property of condensation is preserved along the many-body time evolution [4, Theorem 3.1], i.e. the ℓ-particle reduced density γ (ℓ) where ϕ t ∈ H 1 (R 3 ) denotes the solution to the Hartree equation with initial data ϕ 0 = ϕ ∈ H 1 (R 3 ) (for further references see e.g. [1,2,7,9,10,15,21,22] satisfies a law of large numbers for positive times t > 0, too, i.e. for any δ > 0 While the law of large numbers characterizes the mean of the probability distribution, fluctuations around are governed through the central limit theorem. Before stating our result on a central limit theorem for fluctuations of order and furthermore, for t ∈ R, 0 ≤ s ≤ t and j ∈ {1, . . . , k} the function f (j) with the anti-linear operator Jf = f for any f ∈ L 2 (R 3 ), q t = 1 − |ϕ t ϕ t |, the Hartree Hamiltonian h H defined in (1.17) and the operators be a self-adjoint bounded k-particle operator and ϕ t the solution to the Hartree equation where G t denotes the Gaussian random variable with variance given by We remark that for a factorized state, we can explicitly compute the variance where we introduced the centred k-particle operator The last sum of the r.h.s. of (1.25) vanishes. Furthermore, the first sum vanishes whenever j k does not intersect with i k and we find for the remaining terms (1.20). In particular, we observe that the variance scales as σ 2 N = O(N 2k−1 ) and thus, we expect fluctuations to be O(N k−1/2 ).
We observe that Theorem 1.2 shows that the fluctuations of the many-body dynamics scale similarly to the fluctuations of a factorized state. Moreover, for t = 0 the variance σ 2 0 of the many-body dynamics defined in (1.24) agrees with the covariance matrix M ϕ ⊗N (i, j) in (1.28) of a factorized state.
We remark that for k = 1, i.e. considering bounded one-particle observables, Theorem 1.2 generalizes known results [3,6] to more general one-particle observables. This generalization is due a different strategy of the proof of Theorem 1.3 than in [3,6] . We follow the ideas of [6], however, we use as a first step in Lemma 4.1 directly the norm approximation (4.1) of the many-body time evolution (for more details see Section 4.2).
Recently, for one-particle operators the probability distribution's tails were characterized through large deviation estimates [14,20], showing that t,0 is defined similarly to (1.21), but using the projected kernels K j,s (x, y) = q s K j,s (x, y)q s .
Furthermore, for one-particle operators, a central limit theorem is proven for stronger particles' interactions in the intermediate regime [18], interpolating between the mean-field and the Gross-Pitaevski regime. In the Gross-Pitaevski regime of singular particles' interaction, a central limit theorem is proven for quantum fluctuations in the ground state [19], too. Theorem 1.2 follows from an approximation of the random variable's characteristic function given in the following: In the following, we will now first turn to the proof of Theorem 1.1 in Section 2, then prove Theorem 1.2 from Theorem 1.3 in Section 3 and finally prove Theorem 1.3 in Section 4.

PROOF OF THEOREM 1.1
We generalize ideas from [3] on a law of large numbers for bounded one-particle observables to the case of k-particle operators.
Proof. By Chebycheff's inequality, we have where we used the notation O (k) defined in (1.26). Furthermore, we denote with ♯{i k , j k } the number of elements of i k agreeing with j k . Then, we can write we can express the r.h.s. of (2.1) in terms of j-particle reduced density matrices defined in (1.3) and find Plugging (2.3) into the r.h.s. of (2.1), we find For ℓ = 0, the term of the sum of the r.h.s. of (2.4) is given by (2.5) Since ψ N exhibits Bose-Einstein condensation, it follows by assumption (1.12) and by definition (1.26) of O (k) , we arrive at For ℓ ≥ 1, the terms of the sum of the r.h.s. of (2.4) consists of (2k − ℓ)-particle operators whose expectation values are computed with (2k − ℓ)-particle operators. In particular, we find We conclude with (2.7), (2.8) and (2.4) by Proof. We consider the difference where χ [a,b] denotes the characteristic function of the set [a, b]. We observe that for g ∈ L 1 (R) with Fourier transform g ∈ L 1 R, (1 + s 2k ) ds , we have on the one hand and on the other hand (3.3) and, in particular by Theorem 1.3 Thus, in order to find an estimate for (3.1), we shall find an approximation from above f +,ε and from below f −,ε of the characteristic function χ [a,b] which satisfy f −,ε , f +,ε ∈ L 1 (R 3 ) and f −,ε , f +,ε ∈ L 1 (R, (1 + s 2k )ds). For this, let η ∈ C ∞ 0 (R) with η ≥ 0, η(s) = 0 for all |s| ≥ 1 and´ds η(s) = 1. Furthermore, for ε > 0, let η ε (s) = ε −1 η(s/ε). Then, for any ε > 0, we define Moreover, the Fourier transform is given by Thus it follows from (3.1), (3.6) and with (3.4), (3.7) we arrive at Similarly, using f +,ε we have Now, we optimize with respect to ε > 0 and arrive at (1.23).

4.1.
Fluctuations around the Hartree dynamics. In the following, we consider the bosonic N -particle wave function ψ N,t as an element of the bosonic Fock space F = n≥0 L 2 (R 3 ) ⊗ n s with creation and annihilation operators a * (f ), a(f ) for f ∈ L 2 (R 3 ). Theorem 1.3 characterizes the fluctuations around the Hartree dynamics which are well described by the approximation of the many-body time evolution [4,Theorem 4.1] resp. [6,Proposition 3.3] where the limiting dynamics U ∞ (t; 0) is given by with generator Here, dΓ(A) =´dxdy A(x; y)a * x a y denotes the second quantization of an operator A on L 2 (R 3 ), h H (t) the Hartree Hamiltonian defined in (1.17), and K j,t denote the operators defined in (1.22). For further references, see also [8,13,12,16]. The generator L ∞ (t) is quadratic in creation and annihilation operators, and thus [3, Theorem 2.2] (see also [5,18] gives rise to a Bogoliubov dynamics, i.e. there exists bounded operators U (t; 0), V (t; 0) on L 2 (R 3 ) such that for f, g ∈ L 2 (R 3 ) and the operator A(f, g) = a(f ) + a * (g), we have where Jf = f for any f ∈ L 2 (R 3 ). In particular, for the operator For the first step, Lemma 4.1, we use similarly to the strategy in [18,19], directly the norm approximation (4.1). This allows to consider more general k-(resp. one-) particle operators than in [3,6] where the difference of the limiting fluctuation dynamics U ∞ (t; 0) defined in (4.2) to the full many-body dynamics was estimated in (4.10) by Duhamel's formula and a Gronwall estimate. The remaining steps use similar ideas as in [3,6].
Then, there exists C > 0 such that Proof. We have (4.10) The operator O (k) is a self-adjoint operator, thus e iN −k+1/2 O k N op ≤ 1 and we find with (4.8) and (4.1) Under the same assumptions as in Theorem 1.2, let φ(f ) be defined as in (4.5) and Then, there exists C > 0 such that Proof. We observe that on the bosonic Fock space, we have where we used the second quantization In order to compute the operator we use the Weyl operators' shifting properties on creation and annihilation operators, i.e.