Interacting Brownian motions and the Gross-Pitaevskii formula
aa r X i v : . [ m a t h . P R ] S e p Interacting Brownian motions and theGross-Pitaevskii formula
Stefan Adams and Wolfgang K¨onigOctober 29, 2018
Abstract
We review probabilistic approaches to the Gross-Pitaevskii theory de-scribing interacting dilute systems of particles. The main achievementare large deviations principles for the mean occupation measure of a largesystem of interacting Brownian motions in a trapping potential. The cor-responding rate functions are given as variational problems whose solutionprovide effective descriptions of the infinite system.
The phenomenon known as Bose-Einstein condensation (hereafter abbreviatedBEC) was predicted by Einstein (1925) on the basis of ideas of the Indian physi-cist Bose (1924) concerning statistical description of the quanta of light: In asystem of particles obeying Bose statistics and whose total number is conserved,there should be a temperature below which a finite fraction of all the particles“condense” into the same one-particle state. Einstein’s original prediction wasfor a non-interacting gas of particles. The predicted phase transition is asso-ciated with the condensation of atoms in the state of lowest energy and is theconsequence of quantum statistical effects.For a long time these predictions were considered as a curiosity of non-interacting gases and had no practical impact. After the observation of super-fluidity in liquid He below the λ temperature (2.17 K) was made, London (1938)suggested that, despite the strong interatomic interactions, BEC indeed occursin this system and is responsible for the superfluidity properties. This sugges-tion has stood the test of time and is the basis of our modern understanding ofthe properties of the superfluid phase.The first self-consistent theory of super-fluids was developed by Landau(1941) in terms of the spectrum of elementary excitations of the fluid. In 1947Bogoliubov developed the first microscopic theory of interacting Bose gases,based on the concept of Bose-Einstein condensation. This initiated severaltheoretical studies; a recent account on the state of the art can be found inAdams, S. and Bru, J.-B. (2004a,b) and on its contribution to superfluidity the-ory in Adams, S. and Bru, J.-B. (2004c). After Landau, L.D. and Lifshitz, E.M.11951) had appeared, Penrose (1951) and Onsager, L. and Penrose, O. (1956)introduced the concept of the non-diagonal long-range order and discussed its re-lationship with BEC. An important development in the field took place with theprediction of quantised vortices by Onsager (1949) and Feynman (1955). Theexperimental studies on dilute atomic gases were developed much later, startingfrom the 1970s, benefiting from the new techniques developed in atomic physicsbased on magnetic and optical trapping, and advanced cooling mechanisms.In 1995, the first experimental realisations of BEC were achieved in a systemthat is as different as possible from He, namely, in dilute atomic alkali gasestrapped by magnetic fields. These realisations are due to Anderson, M.H. et al.(1995), Bradley, C.C. et al. (1995), Davis, K.B. et al. (1995), after appropri-ate cooling methods had been developed. For this remarkable achievement,the Nobel prize in physics 2001 was awarded to E.A. Cornell, W. Ketterleand C.E. Wieman. Over the last few years these systems have been the sub-ject of an explosion of research, both experimental and theoretical. A com-prehensive account on Bose-Einstein condensation is the recent monographPitaevskii, L. and Stringari, S. (2003).Perhaps the most fascinating aspect of BEC is best illustrated by the coverof
Science magazine of December 22, 1995, in which the Bose condensate isdeclared as the “molecule of the year”. The Bose condensate is pictured as aplatoon of soldiers marching in lookstep: every atom in the condensate mustbehave in exactly the same way. One of the most striking consequences is thateffects, which are so small that they are practically invisible at the level of asingle atom, are spectacularly amplified.Motivated by the experimental success, in a series of papers Lieb, E.H. et al.(2000a), Lieb, E.H. et al. (2000b), Lieb, E.H. et al. (2001), Lieb, E.H. and Seiringer, R.(2002) obtained a mathematical foundation of Bose-Einstein condensation atzero temperature. The mathematical formulation of the N -particle Boson sys-tem is in terms of an N -particle Hamilton operator, H N , whose ground statesdescribe the Bosons under the influence of a trap potential and a pair potential,see Section 2. Lieb et al. rigorously proved that the ground state energy perparticle of H N (after proper rescaling of the pair potential) converges towardsthe energy of the well-known Gross-Pitaevskii functional. The ground state isapproximated by the N -fold product of the Gross-Pitaevskii minimiser mulit-plied by a correlated term involving the solution of the associated scatteringequation. Moreover, they also showed the convergence of the reduced densitymatrix, which implies the Bose-Einstein condensation. As had been generallypredicted, the scattering length of the pair interaction potential plays a key rolein this description.These rigorous results are only for zero temperature, whereas the experi-ments show BEC at very low, but positive temperature. The mathematicalunderstanding of BEC at positive temperature is rather incomplete yet. Itsanalysis represents an important challenging and ambitious research area inthe field of many-particle systems. Thermodynamic equilibrium states are de-scribed by traces of e − β H N , where β ∈ (0 , ∞ ) is the inverse temperature and H N is the N -particle Hamilton operator. Via the Feynman-Kac formula (see e.g.2eynman (1953) and Ginibre (1970)), these traces are expressed as exponentialexpectations of N interacting Brownian motions with time horizon [0 , β ]. Thisopens up the possibility to use probabilistic approaches for the study of thesetraces, in particular stochastic analysis and the theory of large deviations.In this review we present our probabilistic approaches to dilute systemsof interacting many-particle systems at positive temperature using the Gross-Pitaevskii approximation. Using the the theory of large deviations, we char-acterise the large- N and the large- β behaviour of various exponential expec-tations of N interacting Brownian motions with time horizon [0 , β ] in termsof variants of the Gross-Pitaevskii variational formula. In particular we intro-duce and analyse a new model, which we call the Hartree model, whose groundstates are the ground product states of the Hamilton operator H N . Their large- N behaviour is characterised in terms of the Gross-Pitaevskii formula, withthe scattering length replaced by the integral of the pair interaction potential.This nice assertion is complemented by an analogous result for positive tem-perature. Our programme started with Adams, S. et al. (2006a,b), which wesummarise here. Further aspects are considered in Adams, S. and Dorlas, T.(2007a), Adams, S. and K¨onig, W. (2007), Adams, S. and Dorlas, T. (2007b)and Adams (2007a). Under current development are Adams (2007c), Adams(2007d), Adams, S. et al. (2007) and Adams, S. et al. (2007) in which non-dilutesystems are studied.We give a brief introduction to the physics of dilute quantum gases andtheir mathematical treatment at zero temperature in Section 2. In particular weintroduce the Gross-Pitaevskii formula and the scattering length and describethe results by Lieb et al. and our results of the ground product state. Ourprobabilistic models are introduced in Section 3. Section 4 is devoted to ourlarge deviations results and the variational analysis. We introduce the modelling of the Gross-Pitaevskii theory which will be thestarting point for our probabilistic models in Section 3. Let us comment brieflyon some issues of the 1995 experiments as these are the motivation for therenewed interest in the Gross-Pitaevskii theory and its analytical proof by Liebet al.The experimental systems are collections of individual neutral alkali-gasatoms (e.g., Li, Li and Rb,
Cs,
Yb, Rb ,and Li )), with total number N ranging from a few hundreds up to ∼ ,confined by magnetic and/or optical means to a relatively small region of space.Their densities range from ∼ cm − to ∼ × cm − , and their tempera-tures are typically in the range of a few tenths of nK up to ∼ µ K.In a typical system, we are faced with several length scales. One of themis the two-body interaction energy ~ /mα , where m is the reduced mass ofthe two particles, ~ is Heisenberg’s constant and α is the scattering length(see Section 2.1 below), expressing the strength of the interatomic interaction.3 second one is the mean interparticle spacing r int , and a third one is theoscillator frequency a osc of the confining trap potential. Note that the first scaledoes not depend on the trap geometry, whereas the oscillator frequency a osc ,the mean interparticle spacing, the transition temperature T c and the mean-field energy U (to be specified later) do depend on the shape of the confiningpotential. Introduce the “healing length” ξ = (2 mnU ~ ) − / and the de Brogliewavelength λ DB . Note that a osc is the zero-point spread of the ground-statewave function of a free particle in the trap. The relations between these scalesare as follows. α ≪ r int ∼ λ DB ≤ ξ ≪ a osc . Typical values are α ∼
50 ˚A, r int ∼ ξ ∼ a osc ∼ µ . If onecompares these numbers with those of liquid helium, one sees that the dilutegas condition α ≪ r int , which is characteristic for the BEC of alkali gases, isvery far from satisfied for liquid helium. As a consequence, liquid helium isa much more strongly interacting system than BEC gases, by many orders ofmagnitude.We now turn to a mathematical modelling and introduce the potentialsand the scattering length in Section 2.1 and the Gross-Pitaevskii theory inSection 2.2. Our two fundamental ingredients are a trap potential, W , and a pair-interactionpotential, v . We restrict ourselves to dimensions d ∈ { , } . Our assumptionson W are the following. W : R d → [0 , ∞ ] is measurable and locally integrable on { W < ∞} withlim R →∞ inf | x | >R W ( x ) = ∞ . (1)In order to avoid trivialities, we assume that { W < ∞} is either equal to R d oris a bounded connected open set containing the origin.Our assumptions on v are the following. By B r ( x ) we denote the open ballwith radius r around x ∈ R d . v : [0 , ∞ ) → R ∪ { + ∞} is measurable and bounded from below, a := sup { r ≥ v ( r ) = ∞} ∈ [0 , ∞ ) , v | [ η, ∞ ) is bounded ∀ η > a. (2)Note that we also admit v ( a ) = + ∞ . We are mainly interested in the casewhere v has a singularity, i.e., either a >
0, or a = 0 and lim r ↓ v ( r ) = ∞ .Examples include also super-stable potentials and potentials of Lennard-Jonestype (Ruelle (1969)). According to integrability properties near the origin, wedistinguish two different classes as follows. We call the interaction potential v a soft-core potential if a = 0 and R B (0) v ( | x | ) d x < + ∞ . Otherwise (i.e., if a >
0, or if a = 0 and R B (0) v ( | x | ) d x = + ∞ ), we call the interaction potentiala hard-core potential. 4e shall need the following dN -dimensional versions of the trap and theinteraction potential: W ( x ) = N X i =1 W ( x i ) and v ( x ) = X ≤ i 3. Let u : [0 , ∞ ) → [0 , ∞ ) be a solution of thescattering equation, u ′′ = 12 uv on (0 , ∞ ) , u (0) = 0 . (3)Then the scattering length α ( v ) ∈ [0 , ∞ ], of v is defined as α ( v ) = lim r →∞ h r − u ( r ) u ′ ( r ) i . (4)If v (0) > 0, then α ( v ) > 0, and if R ∞ a +1 v ( r ) r d − d r < ∞ , then α ( v ) < ∞ . In thepure hard-core case, i.e., v = ∞ [0 ,a ) , we have α ( v ) = a . It is easily seen fromthe definition that the scattering length of the rescaled potential ξ − v ( · ξ − ) isequal to ξα ( v ), for any ξ > u in (3); positive multiples of u are also solutions, but the factor drops out in (4). We like to normalise u byrequiring that lim R →∞ u ′ ( R ) = 1. It is easily seen that (where ω d denotes thearea of the unit sphere in R d ), Z R d v ( | x | ) u ( | x | ) | x | d − d x = ω d Z ∞ v ( r ) u ( r ) r d r = 2 ω d Z ∞ u ′′ ( r ) r d r = 2 ω d lim R →∞ (cid:16) u ′ ( r ) r (cid:12)(cid:12)(cid:12) R − Z R u ′ ( r ) d r (cid:17) = 2 ω d lim R →∞ (cid:0) u ′ ( R ) R − u ( R ) (cid:1) = 2 ω d α ( v ) . (5)As a consequence, in dimension d = 3, we have α ( v ) < e α ( v ). Indeed, u is a nonnegative convex function whose slope is always below one because oflim R →∞ u ′ ( R ) = 1. By u (0) = 0, we have that u ( r ) < r = r d − for any r > πα ( v ) = 2 ω d α ( v ) < R R d v ( | x | ) d x =8 π e α ( v ).In d = 2, the definition of the scattering length is slightly different. We treatfirst the case that supp( v ) ⊂ [0 , R ∗ ] for some R ∗ > R > R ∗ , the solution u : [0 , R ] → [0 , ∞ ) of the scattering equation u ′′ = 12 uv on [0 , R ] , u ( R ) = 1 , u (0) = 0 . u ( r ) = log rα ( v ) / log Rα ( v ) for R ∗ < r < R for some α ( v ) ≥ 0, which is bydefinition the scattering length of v in the case that supp( v ) ⊂ [0 , R ∗ ]. Notethat α ( v ) does not depend on R . Hence,log α ( v ) = log r − u ( r ) log R − u ( R ) , R ∗ < r < R. For general v (i.e., not necessarily having finite support), v is approximated bycompactly supported potentials, and the scattering length of v is put equal tothe limit of the scattering lengths of the approximations.The dilute gas condition ensures that the scattering length is a satisfactorymeasure of the interaction strength. This approximation neglects any higherenergy scattering processes. We finally discuss briefly the effects of the atom-atom scattering on the properties of the many-body alkali-gas system. Thefundamental result is that under some conditions the true interaction potential v of two atoms of reduced mass m may be replaced by a delta function ofstrength 2 π ~ α/m . The effective interaction is v eff ( x ) = 4 πα ~ m δ ( x ) , x ∈ R d . This motivates to scale the potential in such a way that it approximates thedelta function in the large N -limit. This will be done in the so-called Gross-Pitaevskii scaling in Subsection 2.2, which is a particular approximation of thedelta function. The simplest possible approximation for the wave function of a many-bodysystem is a (correctly symmetrised) product of single-particle wave functions,i.e., the Hartree-Fock ansatz. In the case of a BEC system at temperature T = 0, this approximation usually leads to the Gross-Pitaevskii approximation.Basically the Gross-Pitaevskii approximation suggest to replace the evolution(time-dependent or stationary) of the many-body wave functions, governed bya system of Schr¨odinger equations, by a one-particle non-linear Schr¨odingerequation (see Gross (1961), Pitaevskii (1961)):i ∂ t Ψ( x, t ) = (cid:16) − ∇ + W + 4 πα | Ψ( x, t ) | (cid:17) Ψ( x, t ) , x ∈ R d , t ∈ R + . In the stationary case the Gross-Pitaevskii theory gives an approximation for thequantum mechanical ground state for many particles (i.e., in the limit N → ∞ )as a variational problem for a single particle in an effective potential. Hence wefirst summarise some ground state properties for finitely many particles.The ground-state energy per particle of the N -particle Hamilton operator H N = − ∆ + W + v on L ( R d ) , 6s given by χ N = 1 N inf h ∈ H ( R d ): k h k =1 n k∇ h k + h W , h i + h v , h i o , (6)Here H ( R d ) = { f ∈ L ( R d ) : ∇ f ∈ L ( R d ) } is the usual Sobolev space, and ∇ is the distributional gradient. It is standard to proof that there is a unique,continuously differentiable, minimiser h ∗ ∈ H ( R d ) on the right hand side of(6), and that it satisfies the variational equation∆ h ∗ = W h ∗ + v h ∗ − N χ N h ∗ . Now we turn to the above mentioned product ansatz. Introduce the groundproduct state energy of H N , that is, χ ( ⊗ ) N = 1 N inf h ,...,h N ∈ H ( R d ): k h i k =1 ∀ i (cid:10) h ⊗ · · · ⊗ h N , H N h ⊗ · · · ⊗ h N (cid:11) . (7)The replacement of the ground state energy, χ N , by the ground product state en-ergy, χ ( ⊗ ) N , is known as the Hartree-Fock approach (see Dickhoff, W.H. and Van Neck, D.(2005)). Sometimes, the formula in (7) is called the Hartree formula. Obviously, χ ( ⊗ ) N ≥ χ N . We can also write χ ( ⊗ ) N = 1 N inf h ,...,hN ∈ H R d ): k hi k ∀ i n N X i =1 n k∇ h i k + h W, h i i o + X ≤ i 1) inf v ) d/ for any i ∈ { , . . . , N } , where C d > dependson the dimension d only. iii) Let v be soft-core, assume that d ∈ { , } , and let ( h , . . . , h N ) be anyminimiser. Assume that v | (0 ,η ) ≥ for some η > . In d = 3 , furthermoreassume that Z B (0) (cid:12)(cid:12) v ( | y | ) (cid:12)(cid:12) δ d y < ∞ , for some δ > . Then every h i is positive everywhere in R d and continuously differentiable,and all first partial derivatives are α -H¨older continuous for any α < .(iv) Let v be hard-core, assume that d ∈ { , } , and let ( h , . . . , h N ) be anyminimiser. Then every h i is continuously differentiable in the interior ofits support, and all first partial derivatives are α -H¨older continuous forany α < . Remark 2.2 (i) Unlike for the ground states of H N in (6) , there is no con-vexity argument available for the formula in (7) . This is due to the factthat a convex combination of tensor-products of functions is not tensor-product in general, and hence the domain of the infimum in (7) is not aconvex subset of H ( R dN ) . However, for h , . . . , h N fixed, the minimisa-tion over h enjoys the analogous convexity properties on H ( R d ) as theminimisation in (6) .(ii) If v is hard-core, it is easy to see that the distances between the supportsof h , . . . , h N have to be no smaller than a (see (2) ) in order to make thevalue of h h ⊗· · ·⊗ h N , H N h ⊗· · ·⊗ h N i finite. The potential P j = i V h j isequal to ∞ in the a -neighbourhood of the union of the supports of h j with j = i , and h i is equal to zero there (we regard · ∞ as ). In particular,minimisers of (7) are not of the form ( h, . . . , h ) . In the soft-core case,this statement is not obvious at all. A partial result on this question in d = 3 will be a by-product of Section 2.2 below. ✸ We study now our main variational formulas, χ N and χ ( ⊗ ) N , and their minimisersin the limit for diverging number N of particles. In particular, we point outsome significant differences between χ N and its product state version χ ( ⊗ ) N inthe soft-core and the hard-core case, respectively.First we report on recent results by Lieb, Seiringer and Yngvason on thelarge- N behaviour of χ N . Let the pair functional v be as in (2) and assumeadditionally that v ≥ v (0) > v by the rescaling v N ( · ) = ξ − N v ( · ξ − N ), for some appropriate ξ N tending to zero sufficiently fast. This will provide the dilute gas conditionneeded. Hence, the reach of the repulsion is of order ξ N , and its strength oforder ξ − N . Furthermore, the scattering length of v , α ( v ), is rescaled such that α ( v N ) = α ( v ) β N . If β N ↓ α ( v N ) ≪ N − /d . This means that the interparticledistance is much bigger than the range of the interaction potential strength.8ore precisely, the decay of β N will be chosen in such a way that the pair-interaction has the same order as the kinetic term.The mathematical description of the large- N behaviour of χ N in this scal-ing, and hence the theoretical foundation of the above mentioned physicalexperiments, has been successfully accomplished in a recent series of papersLieb, E.H. et al. (2000a), Lieb, E.H. and Yngvason, J. (2001), Lieb, E.H. et al.(2001), Lieb, E.H. and Seiringer, R. (2002). It turned out that the well-knownGross-Pitaevskii formula adequately describes the limit of the ground states andits energy. This variational formula was first introduced in Gross (1961) andGross (1963) and independently in Pitaevskii (1961) for the study of superfluidHelium. After its importance for the description of Bose-Einstein condensationof dilute gases in magnetic traps was realised in 1995, the interest in this for-mula considerably increased; see Dalfovo, F. et al. (1999) for a summary and themonograph Pitaevskii, L. and Stringari, S. (2003) for a comprehensive accounton Bose-Einstein condensation.The Gross-Pitaevskii formula has a parameter α > χ (GP) α = inf φ ∈ H ( R d ): k φ k =1 (cid:8) k∇ φ k + h W, φ i + 4 πα k φ k (cid:9) . It is known Lieb, E.H. et al. (2000a) that χ (GP) α possesses a unique minimiser φ (GP) α , which is positive and continuously differentiable with H¨older continuousderivatives of order one.Since v (0) > 0, its scattering length α ( v ) is positive. The condition Z ∞ a +1 v ( r ) r d − d r < ∞ implies that α ( v ) < ∞ . Furthermore, note that the rescaled potential ξ − v ( · ξ − )has scattering length ξα ( v ) for any ξ > Theorem 2.3 (Large- N asymptotic of χ N in d ∈ { , } ) [Lieb, E.H. et al.(2000a), Lieb, E.H. and Yngvason, J. (2001), Lieb, E.H. et al. (2001)] . Assumethat d ∈ { , } , that v ≥ with v (0) > , and R ∞ a +1 v ( r ) r d − d r < ∞ . Replace v by v N ( · ) = ξ − N v ( · ξ − N ) with ξ N = 1 /N in d = 3 and ξ N = α ( v ) − e − N/α ( v ) N k φ (GP) α ( v ) k − in d = 2 . Let h N ∈ H ( R dN ) be the unique minimiser on the right hand side of (6) , and define φ N ∈ H ( R d ) as the normalised first marginal of h N , i.e., φ N ( x ) = Z R d ( N − h N ( x, x , . . . , x N ) d x · · · d x N , x ∈ R d . Then we have lim N →∞ χ N = χ (GP) α ( v ) and φ N → (cid:0) φ (GP) α ( v ) ) in weak L ( R d ) -sense. In particular, the proofs show that the ground state, h N , approaches, forlarge N , the function( x , . . . , x N ) N Y i =1 (cid:16) φ (GP) α ( v ) ( x i ) k φ (GP) α ( v ) k ∞ f (cid:0) min {| x i − x j | : j < i } (cid:1)(cid:17) , f ( r ) = u ( r ) /r and u is the solution of the scattering equation (3). Inorder to obtain the Gross-Pitaevskii formula as the limit of χ N also in d = 2, therescaling of v in Theorem 2.3 has to be chosen in such a way that the repulsionstrength is the inverse square of the repulsion reach and such that this reachdecays exponentially, which is rather unphysical.There is an analogue of Theorem 2.3 for the Hartree model in the soft-corecase, see Adams, S. et al. (2006a). It turns out that the ground product stateenergy χ ( ⊗ ) N also converges towards the Gross-Pitaevskii formula. However, in d = 2, it turns out that the potential v has to be rescaled differently. Further-more, in d ∈ { , } , the scattering length α ( v ) is replaced by the number e α ( v ) := 18 π Z R d v ( | y | ) d y. Theorem 2.4 (Large- N asymptotic of χ ( ⊗ ) N , soft-core case) Let d ∈ { , } .Assume that v is a soft-core pair potential with v ≥ and v (0) > and e α ( v ) < ∞ . In dimension d = 3 , additionally assume that (iii) of Lemma 2.1holds. Replace v by v N ( · ) = N d − v ( · N ) and let ( h ( N ) , . . . , h ( N ) N ) be any min-imiser for the ground product state energy. Define φ N = N P Ni =1 ( h ( N ) i ) . Thenwe have lim N →∞ χ ( ⊗ ) N = χ (GP) e α ( v ) and φ N → (cid:0) φ (GP) e α ( v ) (cid:1) , where the convergence of φ N is in the weak L ( R d ) -sense and weakly for theprobability measures φ n ( x ) d x towards the measure ( φ (GP) e α ( v ) ) ( x ) d x . Note that, in d = 3, the interaction potential is rescaled in the same way inTheorems 2.3 and 2.4. However, the two relevant parameters depend on differ-ent properties of the potential (the scattering length, respectively the integral)and have different values, since α ( v ) < e α ( v ) (see Section 2.1). In particular,for N large enough, the ground state of χ N is not a product state. This im-plies the strictness of the inequality for the two ground state energies, for v replaced by v N ( · ) = N v ( · N ). The phenomenon that (unrestricted) groundstates are linked with the scattering length has been theoretically predictedfor more general N -body problems (see Fetter, A.L. and Walecka, J.D. (1971,Ch. 14), Popov (1983)). Indeed, Landau combined a diagrammatic method (aBorn approximation of the scattering length) with Bogoliubov’s approximationsto almost reconstruct the scattering length from the L -norm of v ◦ | · | in the(non-dilute) ground state. However, the relation between the L -norm and theproduct ground states was not rigorously known before.In d = 2, a more substantial difference between the large- N behaviours of χ N and χ ( ⊗ ) N is apparent. Not only the asymptotic relation between the reachand the strength of the repulsion is different, but also the order of this rescalingin dependence on N . We can offer no intuitive explanation for this.Interestingly, in the hard-core case, χ ( ⊗ ) N shows a rather different large- N behaviour, which we want to roughly indicate in a special case. Assume that W and v are purely hard-core potentials, for definiteness we take W = ∞ B (0) c v = ∞ [0 ,a ] . We replace v by v N ( · ) = v ( · /ξ N ) for some ξ N ↓ χ ( ⊗ ) N is equal to N times the minimum over the sum ofthe principal Dirichlet eigenvalues of − ∆ in N subsets of the unit ball havingdistance ≥ aβ N to each other, where the minimum is taken over the N sets. Itis clear that the volumes of these N sets should be of order N , independentlyof the choice of ξ N . Then their eigenvalues are at least of order N /d . Hence,one arrives at the statement lim inf N →∞ N − /d χ ( ⊗ ) N > 0, i.e., χ ( ⊗ ) N tends to ∞ at least like N /d . Much thermodynamic information about the Boson system is contained in thetraces of the Boltzmann factor e − β H N for β > 0, like the free energy, or thepressure. Since the 1960ies, interacting Brownian motions are generally usedfor probabilistic representations for these traces. The parameter β , which isinterpreted as the inverse temperature of the system, is then the length of thetime interval of the Brownian motions.However, the traces do not contain much information about the ground state.Since the pioneering work of Donsker and Varadhan in the early 1970ies it isbasically known that the ground states are intimately linked with the Brownianoccupation measures. This link is rigorously established via the theory of largedeviations for diverging time, which corresponds to vanishing temperature.We introduce two different models of interacting Brownian motions. Thesemodels are given in terms of transformed measures for paths of length β in termsof certain Hamiltonians. Let a family of N independent Brownian motions,( B (1) t ) t ≥ , . . . , ( B ( N ) t ) t ≥ , in R d with generator − ∆ be given. The Hamiltoniansof both models possess a trap part and a pair-interaction part. The trap partis for both models the same, namely H N,β = N X i =1 Z β W ( B ( i ) s ) d s. (8)The Hamiltonian of our first model consists of two parts: the trap part given in(8), and a pair-interaction part, G N,β = X ≤ i Fix N ∈ N . i) lim β →∞ N β log E (cid:0) exp( − H N,β − G N,β ) (cid:1) = − χ N , where χ N is the ground-state energy per particle of the N -particle operator H N given in (6) .(ii) As β → ∞ , the distribution of µ β on M ( R dN ) under b P N,β satisfies aprinciple of large deviation with speed β and rate function I N given by I N ( µ ) = I N ( µ ) + h W , µ i + h v , µ i − N χ N for µ ∈ M ( R dN ) . (iii) The distribution of µ β under b P N,β converges weakly towards the measure h ∗ ( x ) d x , where h ∗ is the unique minimiser in (6) . Remark 4.2 It is well-known Ginibre (1970) that the bottom of the spectrumof H N is related to the large- β behaviour of the trace of e − β H N , more precisely, χ N = − lim β →∞ N β log Tr (cid:0) e − β H N (cid:1) . ✸ Theorem 4.3 (Hartree model at late times) Assume that W and v arecontinuous in { W < ∞} resp. in { v < ∞} . Furthermore, assume in the soft-core case that there exists an ε > and a decreasing function e v : (0 , ε ) → R with v ≤ e v on (0 , ε ) , which satisfies R B ε (0) G (0 , y ) e v ( | y | ) d y < ∞ , where G denotes theGreen’s function of the free Brownian motion on R d . Fix N ∈ N .(i) lim β →∞ N β log E (cid:0) exp( − H N,β − K N,β ) (cid:1) = − χ ( ⊗ ) N . (ii) As β → ∞ , the distribution of the tuple ( µ (1) β , . . . , µ ( N ) β ) of Brownian oc-cupation measures on M ( R d ) N under b P ( ⊗ ) N,β satisfies a large deviationprinciple with speed β and rate function I ( ⊗ ) N ( µ , . . . , µ N ) = N X i =1 I ( µ i ) + h W , µ ⊗ i + h v , µ ⊗ i − N χ ( ⊗ ) N , with µ , . . . , µ N ∈ M ( R d ) where I is defined in (10) , and µ ⊗ = µ ⊗· · · ⊗ µ N is the product measure.(iii) The distribution of ( µ (1) β , . . . , µ ( N ) β ) under b P ( ⊗ ) N,β is attracted by the set ofminimisers for ground product state energy χ ( ⊗ ) N . .2 Large systems at Positive Temperature We now formulate our results on the behaviour of the Hartree model in the limitas N → ∞ , with β > v by v N ( · ) = N d − v ( · N ); we write K ( N ) N,β for the Hamiltonian introducedin (9).First we introduce an important functional, which will play the role of aprobabilistic energy functional. Define J β : M ( R d ) → [0 , ∞ ] as the Legendre-Fenchel transform of the map C b ( R d ) ∋ f β log E [e R β f ( B s ) d s ] on the set C b ( R d ) of continuous bounded functions on R d , where ( B s ) s ≥ is one of theabove Brownian motions. That is, J β ( µ ) = sup f ∈C b ( R d ) n h µ, f i − β log E (cid:0) e R β f ( B s ) d s (cid:1)o , µ ∈ M ( R d ) . Here M ( R d ) denotes the set of probability measures on R d . Note that J β depends on the initial distribution of the Brownian motion. One can show that J β is not identical to + ∞ . Clearly, J β is a lower semi continuous and convexfunctional on M ( R d ), which we endow with the topology of weak convergenceinduced by test integrals against continuous bounded functions. However, J β is not a quadratic form coming from any linear operator. We wrote h µ, f i = R R d f ( x ) µ (d x ) and use also the notation h f, g i = R R d f ( x ) g ( x ) d x for integrablefunctions f, g . If µ possesses a Lebesgue density φ for some L -normalised φ ∈ L , then we also write J β ( φ ) instead of J β ( µ ). It turns out that J β ( µ ) = ∞ if µ fails to have a Lebesgue density, see Adams, S. et al. (2006b).In the language of the theory of large deviations, J β is the rate function thatgoverns a large deviations principle. The object that satisfies this principle isthe mean of the N normalised occupation measures, µ N,β = 1 N N X i =1 µ ( i ) β , N ∈ N . Roughly speaking, this principle says that, as N → ∞ , P ( µ N,β ≈ µ ) ≈ e − NJ β ( µ ) , µ ∈ M ( R d ) . The principle follows from Cram´er’s theorem, together with the exponentialtightness of the sequence ( µ N,β ) N ∈ N .To apply this principle, we have to express our Hamiltonians H N,β and K N,β as functionals of µ N,β . For the first this is an easy task and can be done for anyfixed N : H N,β = N β Z R d W ( x ) 1 N N X i =1 µ ( i ) β (d x ) = N β (cid:10) W, µ N,β (cid:11) . Now we rewrite the second Hamiltonian, which will need Brownian intersec-tion local times and an approximation for large N . Let us first introduce the15ntersection local times, see Geman, D. et al. (1984). For the following, we haveto restrict to the case d ∈ { , } .Fix 1 ≤ i < j ≤ N and consider the process B ( i ) − B ( j ) , the so-called confluentBrownian motion of B ( i ) and − B ( j ) . This two-parameter process possesses alocal time process, i.e., there is a random process ( L ( i,j ) β ( x )) x ∈ R d such that, forany bounded and measurable function f : R d → R , Z R d f ( x ) L ( i,j ) β ( x ) d x = 1 β Z β d s Z β d t f (cid:0) B ( i ) s − B ( j ) t (cid:1) = Z R d Z R d µ ( i,j ) β (d x ) µ ( i,j ) β (d y ) f ( x − y ) . Hence, we may rewrite K ( N ) N,β as follows: K ( N ) N,β = βN d − X ≤ i 7→ k d µ d x k , this heuristic explanation is finished by E (cid:16) e − H N,β − K ( N ) N,β e N h f,µ N,β i (cid:17) ≈ E (cid:16) exp n − N β h(cid:10) W − f, µ N,β (cid:11) − πα ( v ) (cid:13)(cid:13)(cid:13) d µ N,β d x (cid:13)(cid:13)(cid:13) io(cid:17) ≈ e − Nβχ ( ⊗ ) α ( v ) ( f ) , χ ( ⊗ ) α ( β ) = inf φ ∈ L ( R d ): k φ k =1 n J β ( φ ) + h W, φ i + 4 πα || φ || o . (11)Here we substituted φ ( x ) d x = µ (d x ), we may restrict the infimum overprobability measures to the set of their Lebesgue densities φ .Let us now give the precise formulation of our results. Theorem 4.4 (Many-particle limit for the Hartree model) Assume that d ∈ { , } and let W and v satisfy Assumptions (W) and (v), respectively. In-troduce α ( v ) := Z R d v ( | y | ) d y < ∞ . Fix β > . Then, as N → ∞ , the mean µ N,β = N P Ni =1 µ ( i ) β of the normalisedoccupation measures satisfies a large deviation principle on M ( R d ) under themeasure with density e − H N,β − K ( N ) N,β with speed N β and rate function I ( ⊗ ) β ( µ ) = ( J β ( φ ) + h W, φ i + α ( v ) || φ || if φ = d µ d x exists, ∞ otherwise.The level sets { µ ∈ M ( R d ) : I ( ⊗ ) β ( µ ) ≤ c } , c ∈ R , are compact. Lemma 4.5 (Analysis of χ ( ⊗ ) α ( β )) Fix β > and α > .(i) There exists a unique L -normalised minimiser φ ∗ ∈ L ( R d ) ∩ L ( R d ) ofthe right hand side of (11) .(ii) For any neighbourhood N ⊂ L ( R d ) ∩ L ( R d ) of φ ∗ , inf φ ∈ L ( R d ): k φ k =1 ,φ/ ∈N n J β ( φ ) + h W, φ i + 4 πα || φ || o > χ ( ⊗ ) α ( β ) . Here ‘neighbourhood’ refers to any of the three following topologies: weaklyin L , weakly in L , and weakly in the sense of probability measures, if φ is identified with the measure φ ( x ) d x . 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