Derivation of effective evolution equations from many body quantum dynamics
aa r X i v : . [ m a t h - ph ] O c t Derivation of effective evolution equationsfrom many body quantum dynamics
Benjamin SchleinDPMMS, University of Cambridge,Cambridge, CB3 0WB, UKE-mail: [email protected] 8, 2018
Abstract
We review some recent results concerning the derivation of effective evolution equations fromfirst principle quantum dynamics. In particular, we discuss the derivation of the Hartree equationfor mean field systems and the derivation of the Gross-Pitaevskii equation for the time evolutionof Bose-Einstein condensates.
We consider a quantum mechanical system of N spinless bosons in three dimensions. The systemis described on the Hilbert space H N = L s ( R N ), consisting of all functions in L ( R N ) whichare symmetric with respect to arbitrary permutations of the N particles. The time evolution of aquantum mechanical system of N bosons is governed by the N -particle Schr¨odinger equation i∂ t ψ N,t = H N ψ N,t (1.1)for the wave function ψ N,t ∈ H N of the system. Here H N is a self-adjoint operator on H N , known asthe Hamilton operator. We will consider Hamilton operators with two-body interactions describedby a potential V , having the form H N = N X j =1 − ∆ x j + N X i A mean-field system is described by an N -body Hamilton operator of the form H mf N = N X j =1 − ∆ j + 1 N N X i Under appropriate assumptions on the interaction potential V , let ψ N = ϕ ⊗ N forsome ϕ ∈ H ( R ) with k ϕ k = 1 , and ψ N,t = e − iH N t ψ N (with H N defined in (2.2)). Then, for everyfixed k ≥ and t ∈ R , we have γ ( k ) N,t → | ϕ t ih ϕ t | ⊗ k as N → ∞ (2.4) in the trace-norm topology. Here ϕ t is the solution to the nonlinear Hartree equation i∂ t ϕ t = − ∆ ϕ t + ( V ∗ | ϕ t | ) ϕ t (2.5) with initial data ϕ t =0 = ϕ . The first proof of this theorem was obtained by Spohn in [17] under the assumption of a boundedpotential. The approach introduced by Spohn is based on the study of the time evolution of thereduced density matrices, which is governed by the BBGKY Hierarchy (for k = 1 , . . . , N ) i∂ t γ ( k ) N,t = k X j =1 h − ∆ x j , γ ( k ) N,t i + 1 N k X i 0. If we fix k ≥ N → ∞ , the BBGKY Hierarchy formally converges to theinfinite hierarchy i∂ t γ ( k ) ∞ ,t = k X j =1 h − ∆ x j , γ ( k ) ∞ ,t i + k X j =1 Tr k +1 h V ( x j − x k +1 ) , γ ( k +1) ∞ ,t i . (2.7)It is then worth noticing that this infinite hierarchy has a factorized solution. In fact, the family γ ( k ) ∞ ,t = | ϕ t ih ϕ t | ⊗ k solves (2.7) if and only if ϕ t solves the nonlinear Hartree equation (2.5). Thissimple observation implies that in order to obtain a rigorous proof of Theorem 2.1, it is enough tocomplete the following three steps. • Prove the compactness of the sequence of densities γ ( k ) N,t with respect to an appropriate weaktopology. • Prove the convergence to the infinite hierarchy. In other words, show that every limit point ofthe sequence γ ( k ) N,t solves (2.7). • Prove the uniqueness of the solution of the infinite hierarchy.These three steps immediately imply that γ ( k ) N,t → | ϕ t ih ϕ t | ⊗ k as N → ∞ (the convergence is first ina weak sense, but since the limit is a rank one projection, this automatically implies convergence inthe trace norm topology).Implementing this three step strategy becomes more difficult for potentials with singularities. In[8], Erd˝os and Yau extended Spohn’s result to the case of a Coulomb potential V ( x ) = ± / | x | . Todeal with the singularity of V , they proved that arbitrary limit points γ ( k ) ∞ ,t of the sequence of reduceddensities γ ( k ) N,t satisfy strong a-priori bounds of the formTr (1 − ∆ x ) . . . (1 − ∆ x k ) γ ( k ) ∞ ,t ≤ C k (2.8)for some finite constant C > k ∈ N , t ∈ R . Hence, it is enough to prove the uniquenessof the solution of (2.7) in the class of densities satisfying (2.8), where the Coulomb singularity canbe controlled by the operator inequality | x | − ≤ C (1 − ∆). A similar technique was used in [1], ajoint work with A. Elgart, to handle bosons with a relativistic dispersion law interacting through amean field Coulomb potential (the relativistic case is more complicated, because one can only proveweaker a-priori estimates).A disadvantage of the strategy exposed above is the lack of effective bounds on the rate ofconvergence in (2.4). Following a different approach, proposed initially by Hepp in the slightlydifferent context of the classical limit of quantum mechanics and later extended by Ginibre andVelo to a larger class of potentials (see [11, 9]), we recently managed in [16], a joint work with I.Rodnianski, to obtain effective bounds on the difference between the full Schr¨odinger evolution andthe Hartree approximation. For potentials with at most a Coulomb singularity, we show that thereexist constants C, K > (cid:12)(cid:12)(cid:12) γ (1) N,t − | ϕ t ih ϕ t | (cid:12)(cid:12)(cid:12) ≤ Ce Kt √ N (2.9)for all t ∈ R (similar bounds holds for higher order marginals as well). Note that the bound on ther.h.s. of (2.9) is not expected to be optimal in its N dependence (in fact, for bounded potential, itwas proven in [3], a joint work with L. Erd˝os, that the l.h.s. is at most of the order 1 /N for every3xed t ). The proof of (2.9) is based on a Fock space representation of the many boson system, andon the study of the time evolution of coherent states.Recently, further progress has been ahieved in the analysis of the dynamics of mean-field systems.In [12], Knowles and Pickl obtain effective estimates on the rate of convergence to the Hartreeequation, for potential with strong singularity (their proof is based on a method developed by Picklin [15]). In [10], Grillakis, Machedon, and Margetis found second order correction to the mean-fieldevolution of coherent states. Another class of systems for which an effective evolution equation can be derived consists of diluteBose gases in the so called Gross-Pitaevskii scaling limit. The Hamiltonian of a trapped dilute Bosegas is given by H trap N = N X j =1 (cid:0) − ∆ x j + V ext ( x j ) (cid:1) + N X i 1, as | x | → ∞ . Note that, if a is the scattering length of V , thescattering length of the rescaled potential V N is exactly given by a = a /N . This follows by simplescaling because, if f is the solution of (3.12), then f N ( x ) = f ( N x ) solves the rescaled problem (cid:18) − ∆ + 12 V N (cid:19) f N = 0 . (3.13)In [14], Lieb, Seiringer, and Yngvason proved that, if E N denotes the ground state energy of H trap N , lim N →∞ E N N = min ϕ ∈ L ( R ): k ϕ k =1 E GP ( ϕ )where E GP ( ϕ ) is the so called Gross-Pitaevskii energy functional given by E GP ( ϕ ) = Z d x (cid:0) |∇ ϕ ( x ) | + | V ext ( x ) | + 4 πa | ϕ ( x ) | (cid:1) . (3.14)In [13], Lieb and Seiringer also proved that the ground state ψ trap N of the Hamiltonian H trap N exhibitscomplete Bose-Einstein condensation. More precisely, they proved that, if γ (1) N denotes the one-particle marginal associated with ψ trap N ,lim N →∞ γ (1) N = | φ GP ih φ GP | φ GP denotes the minimizer of (3.14) (the limit is in the trace norm topology). The interpreta-tion of this result is straightforward; in the ground state of H trap N all particles, up to a fraction whichvanishes in the limit N → ∞ , are in the same one-particle state with orbital φ GP .The question now is what happens to the condensate when the trapping potential is turned off.The next theorem, proven in a series of joint works with L. Erd¨os and H.-T. Yau (see [4, 5, 6, 7]) showsthat complete condensation is preserved by the Schr¨odinger dynamics and that the Gross-Pitaevskiitheory correctly predicts the time evolution of the condensate wave function. Theorem 3.1. Suppose V ≥ is bounded and decays sufficiently fast and let H N = N X j =1 − ∆ x j + N X i 2. Inserting this ansatz in (3.17), and using that ( N − V N ( x ) ≃ N V ( N x ) → b δ ( x ) as N → ∞ , we obtain the self-consistent equation i∂ t ϕ t = − ∆ ϕ t + b | ϕ t | ϕ t for ϕ t . This equation has the same form as (3.16), but a different constant in front of the nonlinearity.We get the wrong constant because, in this naive argument, we neglected the correlations charac-terizing the two-particle density γ (2) N,t . It turns out that the solution of the Schr¨odinger equationdevelops a short scale correlation structure which varies exactly on the same length scale of order N − characterizing the interaction potential. If we assume for a moment that the correlations can bedescribed by the solution of the zero-energy scattering equation f N (see 3.13), we may expect thatfor large but finite N the kernel of the one- and the two-particle marginals can be approximated by γ (1) N,t ( x ; x ′ ) ≃ ϕ t ( x ) ϕ t ( x ′ ) γ (2) N,t ( x , x ; x ′ , x ′ ) ≃ f N ( x − x ) f N ( x ′ − x ′ ) ϕ t ( x ) ϕ t ( x ) ϕ t ( x ′ ) ϕ t ( x ′ ) . (3.18)Inserting this new, more precise, ansatz in (3.17), we obtain another self-consistent equation for ϕ t ,which, by (3.11), is exactly the Gross-Pitaevskii equation (3.16).From this heuristic discussion it is already clear that in order to obtain a rigorous proof ofTheorem 3.1, one of the main steps consists in showing that the solution of the N -particle Schr¨odingerequation develops a short scale correlations structure, and that this structure can be described, invery good approximation, by the solution of the zero energy scattering equation f N . In order toreach this goal, we make use of strong a-priori bounds of the form Z d x (cid:12)(cid:12)(cid:12)(cid:12) ∇ x i ∇ x j ψ N,t ( x ) f N ( x i − x j ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ C (3.19)valid for all i = j , uniformly in t and in N , and for every initial wave function ψ N such that h ψ N , H N ψ N i ≤ CN (the bounds (3.19) were derived in [5] under the assumption of sufficientlyweak interaction potential; the case of strong potential was treated in [6], using different a-prioribounds). The estimate (3.19), which is proven using the conservation of the expectation of theHamiltonian squared, identifies the short scale correlation structure of ψ N,t . This can be used toprove the convergence of solutions of the BBGKY hierarchy towards an infinite hierarchy of equationsimilar to (2.7), but with V replaced by 8 πa .To complete the proof of Theorem 3.1 (following the strategy outlined in Section 2), one still needsto prove the uniqueness of the solution of the infinite hierarchy; technically, this is actually the mostdifficult part of the proof. On the one hand, we have to prove that the limit point of the densitiessatisfy a-priori estimates of the form (2.8); the problem here is much more involved compared withthe case of a mean-field Coulomb interaction because of the presence of the singular correlationstructure. On the other hand, when we prove the uniqueness of the solution of the infinite hierarchyin the class of densities satisfying the a-priori estimates, we have to face the problem that, in threedimensions, the delta-interaction is not bounded by the Laplacian (in contrast to | x | − ≤ C (1 − ∆),the inequality δ ( x ) ≤ C (1 − ∆) is not true in three dimensions). Details of the proof of the a-prioribounds can be found in [5, Section 5]. The proof of the uniqueness (in the class of densities satisfyingthe a-priori bounds) can be found in [4]. In the last section, we stressed the fact that the emergence of the scattering length in the Gross-Pitaevskii equation (3.16) is a consequence of the short scale correlation structure characterizing the6olution ψ N,t of the N -particle Schr¨odinger equation. If one assumes that h ψ N , H N ψ N i ≤ CN , thepresence of the correlation structure in the evolved wave function ψ N,t follows from a-priori boundsof the form (3.19). It turns out, however, that Theorem 3.1 can also be applied to completelyfactorized initial data of the form ψ N = ϕ ⊗ N . Therefore, also for initial data with absolutely nocorrelations among the particles, the evolution of the condensate wave function is described by theGross-Pitaevskii equation (3.16) with coupling constant proportional to a . This observation suggeststhat the time evolution ψ N,t of the completely factorized data ψ N develops the short scale correlationstructure within very small time intervals, whose length vanishes in the limit N → ∞ . Note that thea-priori bound (3.19) cannot be used to prove that ψ N,t contains the correct short scale structurebecause h ϕ ⊗ N , H N ϕ ⊗ N i ≃ N ≫ N . In order to prove Theorem 3.1 for factorized initial data ψ N we need therefore an approximation argument to replace ψ N by the N -particle wave function e ψ N,ε ≃ χ ( H N ≤ N ε − ) ψ N having the correct correlation structure. We perform then the wholeanalysis on the evolution of e ψ N,ε and only at the end, after taking N → ∞ , we let ε → h (1 , N = − ∆ − ∆ + N V ( N ( x − x ))and the factorized two-particle wave function ψ ( x , x ) = ϕ ( x ) ϕ ( x ) for a ϕ ∈ L ( R ) smoothand decaying fast enough. In order to monitor the formation of correlations in a window of length N − ≤ ℓ ≪ 1, we introduce the quantity F N ( t ) = Z d x d x θ ( | x − x | ≤ ℓ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∇ x − x e − it h (1 , N ψ ( x , x ) f N ( x − x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . At time t = 0, ψ is factorized and it is simple to show that N F N (0) ≃ 1. The following propositiongives an upper bound on F N ( t ) for t > Proposition 4.1. Suppose that ≤ t ≤ N − and N − ≤ ℓ ≪ . Then N F N ( t ) ≤ (log N t ) N t ( N ℓ ) . In particular, Proposition 4.1 implies that, if ℓ ≃ N − , F N ( t ) ≪ F N (0) for all N − ≪ t ≤ N − ;this is clear evidence for the formation of the correlation structure on length scale of order ℓ ≃ N − within times of order N − (note that, for technical reasons, the result is restricted to times t ≤ N − ;hence, strictly speaking, Proposition 4.1 only shows that correlations form during the time interval N − ≪ t ≤ N − ).To prove Proposition 4.1 it is useful to switch to a center of mass coordinate η = ( x + x ) / x = x − x . In these coordinates, the Hamiltonian takes the form h (1 , N = − ∆ η − x + V N ( x ) =: − ∆ η h N with h N = − x + V N ( x ). If we moreover rescale the relative coordinate letting X = N x , we find h N = N ( − X + V ( X )) =: N h . T = N t , we find N F N ( t ) = Z d η d X θ ( | X | ≤ N ℓ ) (cid:12)(cid:12)(cid:12)(cid:12) ∇ X ( e − iT h ψ N )( η, X ) f ( X ) (cid:12)(cid:12)(cid:12)(cid:12) (4.20)where ψ N ( η, X ) = ϕ ( η + X/N ) ϕ ( η − X/N ). In (4.20), the N -dependence is hidden in the initial data ψ N which has the property of being essentially constant in X , as long as | X | ≪ N . The formationof correlations for T ≫ X , under the dynamics generated by h , to the function f ( X ). Recall herethat f is such that h f = 0; one can prove that, for large | X | , it has the form f ( X ) ≃ − a / | X | .Defining ω ( X ) = 1 − f ( X ), we have, very formally, e − iT h e − iT h (1 − ω ) + e − iT h ω = f + e − iT h ω . The second term disperses away for large T and therefore, if we only look inside a window of orderone ( ℓ ≃ N − ), e − iT h ≃ f . This explains why the quantity under investigation becomes small for T ≫ t ≫ N − ). The main obstacle to make this formal argument rigorousis the lack of decay of ω . Since ω ≃ | X | − at infinity, the standard dispersion estimates for theevolution of ω cannot be applied. Instead, we show new dispersion estimates of the form k e − i ∆ t ϕ k ∞ ≤ Ct − s (cid:16) k ϕ k s + k∇ ϕ k ss +3 + k∇ ϕ k s s +3 (cid:17) (4.21)for all s ≥ / L q -norms of e − i ∆ t ϕ for sufficiently large q ). Choosing s ≥ 3, the bound (4.21) can be applied to the function ω . Using Yajima bounds onthe wave operator (see [18]), estimates of the form (4.21) can then be proven to hold also for thedynamics generated by h .Proposition 4.1 proves that the formation of correlation is a two-particle phenomenon. In orderto understand the formation of correlation in the many particle setting, we would need to control theeffect of the three-body interactions on the correlation structure. This is a very difficult task and,in general, not much can be rigorously established about this problem. Since, however, the manybody system is very dilute, three body interactions are very rare. Therefore we can find an intervalof time during which two body collisions are already effective (and lead therefore to the formationof the correlation structure) while three body collisions are still negligible. Theorem 4.2. Let G N ( t ) = Z d x θ ( | x − x | ≤ ℓ ) (cid:12)(cid:12)(cid:12)(cid:12) ψ N,t ( x ) f N ( x − x ) − ψ N ( x ) (cid:12)(cid:12)(cid:12)(cid:12) for ψ N = ϕ ⊗ N with ϕ ∈ L ( R ) smooth and decaying fast enough, and ψ N,t = e − iH N t ψ N , with H N defined in (3.15). Then, for all ≤ t ≤ N − , G N ( t ) ≤ C G N (0) (cid:18) (log N ) N ( N t ) N ℓ + ( N ℓ ) N t (cid:0) log N t (cid:1) (cid:19) . In particular, the theorem implies that, when considering windows of size ℓ ≃ N − , G N ( t ) ≪ G N (0) if 1 ≪ N t ≪ N . This inequality indicates that, under the many body dynamics, a completely factorized initial datadevelops the short scale correlation structure within times of order N − and keeps it at least up totimes of order N − . (of course, we expect the correlation structure to remain intact up to times oforder one, but proving this claim would require a better understanding of the effect of the many-bodyinteractions). 8 eferences [1] Elgart, A.; Schlein, B.: Mean field dynamics of boson stars. Commun. Pure Appl. Math. (2007), no. 4, 500–545.[2] Erd˝os, L.; Michelangeli, A.; Schlein, B.: Dynamical formation of correlations in a Bose-Einsteincondensate. Comm. Math. Phys. (2009), no. 3, 1171-1210.[3] Erd˝os, L.; Schlein, B.: Quantum dynamics with mean field interactions: a new approach. J. Stat.Phys. (2009), no. 5, 859-870.[4] Erd˝os, L.; Schlein, B.; Yau, H.-T.: Derivation of the cubic nonlinear Schr¨odinger equation fromquantum dynamics of many-body systems. Invent. Math. (2007), 515-614.[5] Erd˝os, L.; Schlein, B.; Yau, H.-T.: Derivation of the Gross-Pitaevskii equation for the dynamicsof Bose-Einstein condensate. Preprint arXiv:math-ph/0606017. To appear in Ann. Math. [6] Erd˝os, L.; Schlein, B.; Yau, H.-T.: Rigorous derivation of the Gross-Pitaevskii equation. Phys.Rev Lett. (2007), no. 4, 040404.[7] Erd˝os, L.; Schlein, B.; Yau, H.-T.: Rigorous derivation of the Gross-Pitaevskii equation with alarge interaction potential. Preprint arXiv:0802.3877. To appear in J. Amer. Math. Soc. [8] Erd˝os, L.; Yau, H.-T.: Derivation of the nonlinear Schr¨odinger equation from a many bodyCoulomb system. Adv. Theor. Math. Phys. (2001), no. 6, 1169–1205.[9] Ginibre, J.; Velo, G.: The classical field limit of scattering theory for nonrelativistic many-bosonsystems. I.-II. Comm. Math. Phys. (1979), no. 1, 37–76, and (1979), no. 1, 45–68.[10] Grillakis, M.; Machedon, M.; Margetis, D.: Second-order corrections to mean field evolution forweakly interacting bosons. I. Preprint arXiv: 0904.0158.[11] Hepp, K.: The classical limit for quantum mechanical correlation functions. Comm. Math. Phys. (1974), 265–277.[12] Knowles, A.; Pickl, P.: Mean-field dynamics: singular potentials and rate of convergence.Preprint arXiv:0907.4313.[13] Lieb, E.H.; Seiringer, R.: Proof of Bose-Einstein condensation for dilute trapped gases. Phys.Rev. Lett. (2002), 170409-1-4.[14] Lieb, E.H.; Seiringer, R.; Yngvason, J.: Bosons in a trap: a rigorous derivation of the Gross-Pitaevskii energy functional. Phys. Rev A (2000), 043602.[15] Pickl, P.: A simple derivation of mean field limits for quantum systems. PreprintarXiv:0907.4464.[16] Rodnianski, I.; Schlein, B.: Quantum fluctuations and rate of convergence towards mean fielddynamics. Comm. Math. Phys. (2009), no. 1, 31–61.[17] Spohn, H.: Kinetic equations from Hamiltonian dynamics. Rev. Mod. Phys. (1980), no. 3,569–615.[18] Yajima, K.: The W k,p -continuity of wave operators for Schr ¨odinger operators. J. Math. Soc.Japan47