Stability of equilibria for a Hartree equation for random fields
Introduction
The present work concerns the following Hartree equation for a random field: Here is a time-dependent random field over the Euclidean space , and is the underlying probability space. The expectation is with respect to this probability space: . The convolution product is denoted by ⁎ and w is a real-valued pair interaction potential. By this, we mean that we only consider interactions between two particles (and no more), and that this interaction is characterized by w. Equation (1) has been introduced in [12] as an effective dynamics for a large, possibly infinite, number of fermions in a mean field regime.
Indeed, consider the evolution of a finite number of fermions interacting through the potential w. Under some mean-field hypothesis, as the number of particles tend to infinity, the system is approximated to leading order by the following system of N coupled Hartree equations on , for an orthonormal family (to be compliant with the Pauli principle): Let us associate to this orthonormal family the operator , which is the orthogonal projection onto . There exists a large literature about the derivation of this system of equations and about other related approximation results. In particular, it has been shown that, if the wave function of the original fermionic system is close to a Slater determinant, then, in the mean-field limit and under sufficient conditions for w, the associated one-particle density matrix converges to the above operator γ. For the derivation of Equation (2) from many body quantum mechanics we refer to [2], [3], [4], [5], [13], [16]. Note that the so-called exchange term appearing in the Hartree-Fock equation is not present in (2), which is motivated by the fact that it is of lower order in certain regimes, see the aforementioned references.
To deal with infinitely many particles, it is customary to use the density matrices framework, which is an operator formalism. Namely, the family solves Equation (2) if and only if the operator γ defined above solves the corresponding Hartree equation: Above, denotes the commutator, and is the density of particles, that is the diagonal of the integral kernel of the operator γ. An infinite number of particles can then be modeled by a solution of (3) which is not of finite trace (the trace of the operator being, by the derivation of the model, the number of particles). Solutions of (3) with an infinite number of particles were studied previously in [7], [8], [9], [26] for example, and more recently in [10], [11], [22], [23].
In [12], the second author proposed (1) as an alternative equation to (3). It generalizes Equation (2) for a finite number of particles in the following sense. To an orthonormal family , one can associate the random variable , where is any orthonormal family in . The family then solves (2) if and only if the random variable X solves (1). Equation (1) is also in close correspondence with the Hartree equation for density matrices (3). Indeed, for infinite numbers of particles, taking X to be a Gaussian random field with covariance operator γ, we have that X solves (1) if and only if γ solves (3) (provided γ is taken in an appropriate metric space). We refer to [12] for how to relate the solutions of the two Equations (1) and (3) and their corresponding equilibria. One reason behind the study of (1) is that this equation shares more direct resemblances with the commonly studied nonlinear Schrödinger equation.
The existence of solutions of Equation (1) in is investigated in [12]. The local well-posedness is established in the case of localized initial data, as well as in the case of localized perturbations of the equilibria described below. In particular, almost everywhere in the probability space, the random variable solves the corresponding Schrödinger equation in integrated formulation. Though the results are stated and proved in the case of a Dirac potential , their adaptation to the present case of regular interaction potentials is straightforward. Moreover, as for the defocussing nonlinear Schrödinger equation, scattering for large but localized solutions is expected for Equation (1), at least in energy critical and subcritical regimes. This has been showed in dimension 3 in the case of a Dirac potential in [12]. Nontrivial equilibria are thus non-localized, which corresponds to being not of trace-class in the framework of density matrices.
The equilibria at stake in the present paper are the following. For Equation (3), any non-negative Fourier multiplier with symbol is a stationary solution, where denotes the Fourier transform. Note that density operators are non-negative, we write the symbol in the form to be able to give an analogous equilibrium in the random framework. The analogous equilibrium for Equation (1) are given by Wiener integrals for a distribution function (note that this equilibrium has the same law if we replace f by so that we can assume ). Above denotes infinitesimal complex Gaussians characterized by and the scalar m is given by . We refer to [25], Chapter 1, and [1], Part I, for more information on random Gaussian fields. The function is a solution of Equation (1). It is not a steady state but it is an equilibrium, for as a random field its law is invariant by time and space translations (and in particular is not localized).
In the seminal work [22], the authors show the stability of the above equilibria for the Equation (3) for density matrices in dimension 2. Important tools are dispersive estimates for orthonormal systems [14], [15]. This work has been extended to higher dimension in [11]. Note that in higher dimension, some structural hypothesis is made on the interaction potential w, to solve some technical difficulties about a singularity in low frequencies of the equation that we will identify precisely in the sequel. The stability result corresponds to a scattering property in the vicinity of these equilibria: any small and localized perturbation evolves asymptotically into a linear wave which disperses. We mention equally [10], [23] about problems of global well-posedness for the equation on density matrices.
The problem of the stability of the equilibria (4) for Equation (1) shares similarities with the stability of the trivial solution for the Gross-Pitaevskii equation . In both problems the linearized dynamics has distinct dispersive properties at low and high frequencies, making the nonlinear stability problem harder, especially in low dimensions. The proof of scattering for small data for the Gross-Pitaevskii equation was done in [19], [18], [20], [21]. We here use spaces with different regularities at low and high frequencies, inspired by [21].
The result of the present work is the stability of the equilibria (4) for Equation (1), via the proof of scattering of perturbations to linear waves in their vicinity. Our techniques however differ from those used in [11], [22], see the strategy of the proof after Theorem 1.1. We hope that the present proof provides different insights than the ones in the framework of operators, as well as relaxing some of the hypotheses on the potential w.
In what follows, denotes the usual Japanese bracket. Given , denotes the usual ceiling function applied to s, and and are the nonnegative and nonpositive parts of s. We write with an abuse of notation with , if f has spherical symmetry. The space is the set of measurable functions such that almost surely and
Theorem 1.1 Stability of equilibria for Equation (1) Let . Let . Let f be a spherically symmetric function in . Assume the following hypotheses hold true: Assumptions on the equilibrium: , , for , writing h the inverse Fourier transform of , for all with , .
Assumptions on the potential: is even with where is a constant depending on f.
Remark 1.1 Note that since w is even, its Fourier transform is real, and thus its positive and negative parts ( and ) are well-defined.
Relating the framework of random fields to that of density operators, from the above Theorem 1.1 one obtains a scattering result for the operator: with respect to the one associated to the equilibrium : which is the Fourier multiplier by . This convergence holds in Hilbert-Schmidt Sobolev spaces (where below is the standard Hilbert-Schmidt norm):
Corollary 1.2 Under the hypotheses of Theorem 1, setting: there holds and the convergence:
Remark 1.2 The conditions on f are satisfied by thermodynamical equilibria for bosonic or fermionic gases at a positive temperature T: respectively, but it is not the case of the fermionic gases at zero temperature:
Remark 1.3 The smallness assumption on , corresponds to the fact that the equation is not too focusing. The one on enables the equation (1) linearized around to have enough dispersion. Note that these assumptions appear both in [22] and [11].
Remark 1.4 The smallness of the initial datum in cannot be improved as is the critical regularity. The smallness of the initial datum in is related to the smallness of in . Of course, if is orthogonal to the Wiener process W, this term is null. This hypothesis on the initial datum may thus be improved. The optimal space for the initial datum is unclear to us, but taking in low integrability Lebesgue spaces is related to taking the initial datum in [11], [22] in low Schatten spaces.
Remark 1.5 Let us compare briefly with the related works [11], [22]. The framework used in the present article is that of random fields, which provides a new point of view, and strengthen the understanding of equation (1). The comparison between the aforementioned results and ours is not straightforward, the connexion between the frameworks being explained in the introduction. First, the dimension treated differs, [22] providing with a complete proof in dimension 2, [11] with one for all dimensions , and the present paper for (see the remark on lower dimensions below). All works use a linear stability result that is the invertibility of the linear response operator (see Proposition 5.7). Additional dispersive effects are used in [11], [22] by means of new Strichartz estimates in the framework of density matrices [6], [11], [14], [15]. We solely use here standard Strichartz estimates, but find that the response of the equilibrium also enjoys an improved decay at low frequencies due to randomness (Lemmas (5.3) and 5.4). This improved decay does not appear in [11], [22], and in [11] a singularity that appears at low frequencies is handled by making stronger assumptions on their interaction potential. Our analysis eventually yields a scattering result at critical regularity, as in [22], whereas [11] holds for supercritical regularity.
Remark 1.6 The stability is not a consequence of a strong convergence to 0 of the perturbation in almost everywhere in probability, but of the dispersion for the linear dynamics implying in particular local convergence. The solution converges back to the same equilibrium . In particular, it does not trigger modulational instabilities.
Remark 1.7 In [21], which solves scattering for the Gross-Pitaevskii equation in dimension greater than 4, a normal form is needed to close the argument and in particular to deal with the low-frequency singularity. We do not require the use of a normal form transformation because of random cancellations. First, exact computations of the linearized equation around the equilibrium enable us to lose only 1/2 derivatives on in the low frequencies, while in [21] it is required to lose 1 full derivative on the nonlinearity of the Gross-Pitaevskii equation, which contains quadratic and cubic terms. The second issue is that V, even if it contains a linear and a quadratic term in the perturbation , behaves in terms of Lebesgue integration as , which makes the analysis easier. Note that the potential of interaction plays a role to gain derivatives in high frequencies, we do not use it to gain derivatives in low frequencies.
Remark 1.8 The strategy of the present paper does not directly apply to dimensions 2 and 3, as dispersive effects are weaker. The proof fails at Proposition 6.1. In dimension 2, the first two Picard iterations display a singularity at low frequency, and would have to be treated separately. This has been successfully done in [22]. In dimension 3, we believe that only the first iteration would have such singularity. In [11], a related singularity at low frequency is compensated by hypotheses on the interaction potential. We believe that our strategy could be modified along the lines of [22] for .
Remark 1.9 The function f may be referred as momentum distribution function for the equilibrium. In the case of where T is the temperature and μ the chemical potential, it may be referred as the Fermi-Dirac distribution. Even in the defocussing case, some other equilibria than the ones described by a momentum distribution function can have instabilities. Two plane waves, which are orthogonal in probability, propagating in opposite directions, are linearly unstable, which is showed In Section 9. We do not claim nonlinear instability as we did not prove that a non trivial codimensional stability cannot arise. However, we believe that it would follow from the analysis of the linearized around the equilibrium equation since high and low frequencies interact through the cubic term of the nonlinearity. Note that the equilibria of Theorem 1.1 can be seen as a superposition of infinitely many plane waves propagating in different directions, hence this shows the importance of regularity of the underlying function f.
The strategy of the proof is the following. First note that the dynamics of Equation (1) near the above equilibria is somewhat similar to that of the Gross-Pitaevskii equation. We use a more direct fixed point argument which does not involve iterations of the wave operator as in [11], [22], and dispersion properties at low and high frequencies which are inspired from [21].
We start by reducing the proof to finding a correct functional framework for our contraction argument. Namely, instead of solving an equation for the perturbation , we solve a fixed point for the perturbation and the induced potential where , at the same time. The idea behind this is that even if V contains a linear term in Z, it behaves more like a quadratic term in Z (in the sense of the Lebesgue spaces to which they belong) and thus, we can put it in better spaces regarding dispersion.
The fixed point is solved in a classical way by finding the right Banach space Θ for and proving suitable estimates on the linear and nonlinear terms. The Lipschitz-continuity of the quadratic part is treated in a classical fashion, in the sense that it requires that Θ is included in classical Lebesgue, Besov, or Sobolev spaces.
The difficulty comes from the linear term. It can be written see (12), where corresponds to the analogue of the linear term in [11], [22]. The invertibility and continuity of had been dealt with in both papers and their treatment is more or less sufficient for our argument. Note that this term is the linear response of the equilibrium, related to the so-called Lindhard function [17], [24].
But is where singularities in low frequencies occurs. To get the continuity of , V needs to be in a space that compensates this singularity, namely inhomogeneous Besov spaces, with two levels of regularity, one for the low frequencies and one for the high frequencies. But V contains and we cannot close the fixed point argument for Z in a space that compensates singularities in low frequencies. This is where we use bilinear estimates on inhomogeneous Besov spaces coming from the scattering for Gross-Pitaevskii literature, [21].
The paper is organized as follows. In Section 2, we state a few definitions and known results that we use in the rest of the paper. In Section 3, we set up the fixed point argument. In Section 4, estimates related to some direct embeddings are given. The linear terms are studied in Section 5 where explicit formulas, continuity estimates and invertibility conditions are obtained. Estimates for the quadratic terms are proven in Section 6, and estimates for the source terms are showed in Section 7. Theorem 1.1 is then proved in Section 8. The last Section 9 is devoted to an instability result in a defocussing case when the momentum distribution function is not smooth. In the appendix A we prove Corollary 1.2.
Notation 1.3 Fourier transform We define the Fourier transform with the following constants: for , and the inverse Fourier transform by
Notation 1.4 Time-space norms For , we denote by the space For , we denote by the space In the case we also write it or . When , we may write for . For , we denote by the space The proper definition of inhomogeneous Besov spaces is given in Section 2, Definition 2.3.
Notation 1.5 Probability-time-space norms For , we denote by the space For , we denote by the space In the case we also write it or . For , we denote by the space
Notation 1.6 Time-space-probability norms For , we denote by the space In the case we also write it , and note that . In the case , we also write it . For , we denote by the space induced by the norm
Section snippets
Toolbox
In this section, we present existing results in the literature, either classical or more recent ones such as dispersive estimates and matters related to Littlewood-Paley decomposition.
Take η a smooth function with support included in the annulus , and define for , . We assume that on , . For any tempered distribution , we write where is the Fourier multiplier by i.e. .
Notation 2.1 We have and we call this the Littlewood-Paley
Set-up
In this section, we reduce the problem to finding a correct functional setting for our fixed point problem.
To lighten the notation, and f being fixed, we sometimes write Y instead of .
Writing , the perturbation Z satisfies Let an initial perturbation , we have that Z solves the Cauchy problem if and only if the couple perturbation/induced potential solves the Cauchy problem
Embeddings and Strichartz estimates
In this section, we check assumption 1 in Proposition 3.1, and dispersive estimates for the linear flow which induce assumption 4. In the whole section, .
Proposition 4.1 The space is embedded in as in for all ,
Proof We have that is embedded in . It remains to prove that is embedded in . The temporal part of the norm works by Hölder inequality. We are left with proving the embedding in
Linear term
We study in this section the linearized operator L defined in (12). We prove that is continuous, invertible with continuous inverse on . We recall that hence it is sufficient to prove that is continuous, invertible with continuous inverse from to and that is continuous from to . We start with the continuity of .
Proposition 5.1 The operator is continuous from to .
Proposition 5.1 is a corollary of the following estimates.
Proposition 5.2 Let
The remaining quadratic term
We have already dealt with the quadratic terms and in Proposition 4.1, Proposition 4.3 respectively, it remains to prove assumption 5 in Proposition 3.1. In what follows, we assume , it is the only part of the paper that does not work in dimension 3.
Proposition 6.1 There exists such that for all ,
Proof The proof relies on a duality argument. We first prove the estimate. Let . We write with where we use the
A space for the initial datum
One has the following compatibility result between the space for the perturbation at initial time, and the leading order term for the solution and the potential as given in (11). Here, we prove assumption 2 in Proposition 3.1 for dimension higher than 4 but the proof can be adapted to dimension 3.
Lemma 7.1 There exists a universal constant such that for all , one has with
Proof We have that belongs to because of
Proof of Theorem 1.1
We recall that the proof of Theorem 1.1 relies on finding a solution to the fixed point equation (9) for the perturbation Z and the induced potential V. According to (10), the fixed point equation can be written in the form: This is now a standard routine to solve the above equation thanks to the various estimates derived previously. To solve the above equation, one defines as the mapping Let us denote by
Example of instability for a rough momentum distribution function
We study here the linearization of the dynamics near the superposition of two waves which are orthogonal in probability and propagate in opposite directions. This corresponds to an equilibrium of the form (4) with a rough momentum distribution function f. Even in the defocussing case , the form of f is involved to ensure linear stability. Indeed, we will prove linear instability for the present example.
Consider a potential w satisfying, without loss of generality if the equation is
Acknowledgements
C. Collot is supported by the ERC-2014-CoG 646650 SingWave. A-S de Suzzoni is supported by ESSED ANR-18-CE40-0028. Part of this work was done when C. Collot was visiting IHÉS and he thanks the institute.
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