Analytical approach to relaxation dynamics of condensed Bose gases
aa r X i v : . [ c ond - m a t . qu a n t - g a s ] J a n Analytical approach to relaxation dynamics of condensed Bosegases
Miguel Escobedo, Federica Pezzotti, and Manuel Valle Departamento de Matem´aticas, Universidad del Pa´ıs Vasco, Apartado 644, E-48080 Bilbao, Spain Departamento de F´ısica Te´orica, Universidad delPa´ıs Vasco, Apartado 644, E-48080 Bilbao, Spain (Dated: August 20, 2018)The temporal evolution of a perturbation of the equilibrium distribution of acondensed Bose gas is investigated using the kinetic equation which describes collisionbetween condensate and noncondensate atoms. The dynamics is studied in the lowmomentum limit where an analytical treatment is feasible. Explicit results are givenfor the behavior at large times in different temperature regimes.
I. INTRODUCTION
A distinctive feature of the quantum kinetic theory of the condensed Bose gas arises fromthe correlations between the superfluid component and the normal fluid part correspondingto the excitations. This causes the occurrence of number-changing processes determiningthe relaxation to equilibrium when excitations collide frequently. As a consequence, in thehydrodynamic regime, a collision integral C describing 1 ↔ ω ( k ) for the energy of quasiparticles and on the matrixelement M of the effective Hamiltonian describing the interaction between them.In a series of papers before the experimental realization of BEC, Kirkpatrick and Dorf-man [1, 2] have derived a kinetic equation in a uniform Bose gas which includes theseprocesses, and they have computed explicit values for the transport coefficients of the two-fluid hydrodynamics. More recently, Zaremba, Nikuni and Griffin [3] have extended thetreatment to a trapped Bose gas by including Hartree-Fock corrections to the energy of theexcitations, and they have derived coupled kinetic equations for the distribution functionsof the normal and superfluid components. Many implications of their approach concerningtransport coefficients and relaxation times have been thoroughly reported in Ref. [4], mainlywhen k B T ≫ gn c , where g = 4 πam − is the interaction coupling constant in terms of the s -wave scattering length a , and n c is the condensate density. In this regime the dispersionlaw approaches the free particle law ω ( k ) ≈ k / (2 m ). In the opposite low temperature limit k B T ≪ gn , the relevant part of the excitation spectrum is ω ( k ) ≈ ck , where c = p gnm − isthe sound speed near zero temperature and n is the particle density. The kinetic equationin this regime has been derived in Ref. [5], but its features have been less considered. Due tothe absence of studies on the eigenvalue problem posed by the linearized kinetic equation,little is known about the time behavior of the solutions of kinetic equations in both regimes.This paper is devoted to deriving some results on the free evolution of initial perturbationsof thermal equilibrium as described by the three-excitation collision term near the criticaland near zero temperature. By making appropriate approximations of the kinetic equationwe shall see that the problem becomes analytically tractable in some limits, and hence theasymptotic behavior of the solutions for large time may be evaluated. In the first regime, k B T ≫ gn c , we consider the small momentum limit where the equilibrium distribution func-tion obeys the condition n ( k ) ≫
1. By approximating the full linearized kinetic equation inthis limit we obtain an integro-differential equation which, after integration by parts, agreeswith linearized evolution equation of wave turbulence [6, 7] with the appropriate indices.This result is not a priori obvious when one starts from the full kinetic equation, becauseof the strong singularity of the integrand. At low temperature, k B T ≪ gn , we consider twoopposite regimes n ( k ) ≫ n ( k ) ≪
1. By performing a low momentum approxima-tion, ck ≪ k B T , we will show that the wave turbulence framework still emerges but in aform not consistent with energy conservation, so an additional improvement is needed inorder to restore this conservation law. In the opposite regime near zero temperature, where ck ≫ k B T , we shall consider an approximation based on the dominance of Beliaev dampingprocesses.A notable feature of the linearized collision terms in the wave limit is the homogeneousdependence on the momentum. Such a property furnishes a systematic way to compare therelative orders of different collision integrals at low momentum. This is achieved by simplyconsidering the degrees of homogeneity, since their values depend on some indices relatedto the dispersion law, the scattering amplitude and the number of bodies in the collision.Thus one can see the dominance of C over the binary collision term C .The plan of the paper is the following. In Sec.II we discuss the general form of the lin-earized kinetic equation as written in terms of irreducible components of the perturbation.Sec.III is first devoted to deriving the low momentum approximation of the evolution equa-tion below the critical temperature. Then, once we show that the wave turbulence pictureis recovered, we briefly review the general procedure for solving this kind of equation, weapply these techniques to the present context and we determine the asymptotic behavior ofthe perturbation for large time. Anisotropic perturbations are also discussed. In Sec. IVwe derive the evolution equation in the thermal regime, with particular attention to therequirement of energy conservation, and we deal with the issue of the dominance of C over C . In Sec. V we consider the case of very low temperature and derive some asymptoticresults specific for this regime. Section VI contains some concluding remarks. II. LINEARIZED COLLISION INTEGRAL IN THE FREDHOLM FORM
The form of the three-excitation collision integral C [ n ] = Z [ R ( k , k , k ) − R ( k , k , k ) − R ( k , k , k )] d k d k , (1)depends essentially on the dispersion law ω ( k ) and the matrix element M of the three-excitation interaction. These quantities determine R by means of R ( k , k , k ) = |M ( k , k , k ) | [ δ ( ω ( k ) − ω ( k ) − ω ( k )) δ ( k − k − k )] × [ n n (1 + n ) − (1 + n )(1 + n ) n ] . (2)In this work we will consider two extreme regimes of the Bogoliubov dispersion law ω ( k ) = " gnm k + (cid:18) k m (cid:19) / , (3)near the critical temperature and near zero temperature. In the first limit k B T ≫ gn c thedispersion law and the squared amplitude become [3] ω ( k ) ≈ k m + gn c , (4) |M ( k , k , k ) | = g n c π = 8 n c a m , (5)where n c is the superfluid density. Since in this regime the average value of the energy ofa quasiparticle is of order k B T the term gn c is subleading in Eq. (4). In the opposite limit k B T ≪ gn we have [5] ω ( k ) ≈ ck + αk , (6) |M ( k , k , k ) | = 9 gkk k π m c = 9 ckk k π mn . (7)The amplitudes M can be derived heuristically by golden rule arguments from the effectiveHamiltonians in these regimes [8]. Note that no distinction is made between n and n c in the scattering amplitude valid at T →
0, because we neglect the small low temperaturedepletion. Since we are interested in the low momentum region, we shall retain the first termof the dispersion law. Here we will consider the homogeneous regime where the distributionfunction is independent of the position.The linearization of the kinetic equation ∂ t n = C [ n ] proceeds by the insertion of n ( k , t ) = n ( ω ( k )) + n ( ω ( k ))[1 + n ( ω ( k ))] χ ( k , t ) , (8)where n ( ω ( k )) is the Bose distribution function without chemical potential with the energygiven by the first term of Eqs. (4) or (6); the dimensionless function χ ( k , t ) parametrizesthe departure from equilibrium for the noncondensate distribution function. The linearizedkinetic equation adopts the form n (1 + n ) ∂χ∂t = L [ χ ]( k , t ) = Z L ( k , k ′ ) χ ( k ′ , t ) d k ′ , (9)which must be completed with the linearized equation for the fluctuation of the condensatedensity δn c , dδn c dt = − δ Γ = − Z L [ χ ]( k ) d k (2 π ) . (10)The kernel L is symmetric in k and k ′ and, due to the Bose functions, falls off sufficientlyas k, k ′ → ∞ . We may then introduce the following scalar product( χ , χ ) = Z n (1 + n ) χ ( k ) χ ( k ) d k , (11)and pose the eigenvalue problem Z L ( k , k ′ ) χ j ( k ′ ) d k ′ = − λ j n (1 + n ) χ j ( k ) . (12)The solution of the linearized equation would be formally written as the following series χ ( k , t ) = X j c j χ j ( k ) e − λ j t , (13)where the coefficients c j would be determined by the initial condition. In this work wedo not intend to follow this procedure since it is uncertain that there exists a spectrumof discrete eigenvalues for this problem. This doubt is based on the observation that thelifetime of a long-lived well-developed sound mode of energy ω = ck varies continuously withthe momentum, τ ( k ) ∝ k − δ , where δ = 1 at T = 0 [9] or δ = 5 at T = 0 [10]. So, thisproperty does not seem to be consistent with the discrete character of the spectrum. Instead,we will try to find an expression for the action of the operator L [ χ ] at low momentum, andthen solve the evolution equation found in this approximation.Formally, the operator L is decomposed in two types of contributions L [ χ ] = −N ( k ) χ ( k , t ) + Z U ( k , k ′ ) χ ( k ′ , t ) d k ′ , (14)where the function U is given by U ( k , k ′ ) = (cid:2) |M ( k , k ′ , k − k ′ ) | δ ( ω ( k ) − ω ( k ′ ) − ω ( k − k ′ )) × n ( ω )[1 + n ( ω ′ )][1 + n ( ω − ω ′ )] + ( k ↔ k ′ )] − |M ( k + k ′ , k , k ′ ) | δ ( ω ( k ) + ω ( k ′ ) − ω ( k + k ′ )) × [1 + n ( ω )][1 + n ( ω ′ )] n ( ω + ω ′ ) , (15)and the explicit form of the coefficient N ( k ) will be not required.The conservation laws of energy and momentum imply the presence of zero modes of L proportional to the energy and the momentum, χ ( k , t ) ∝ ω ( k ) , h ( t ) · k . In particular, L [ βω ( k )] = −N ( k ) βω ( k ) + Z U ( k , k ′ ) βω ( k ′ ) d k ′ = 0 . (16)Thus, the linearized collision operator can be written as L [ χ ] = Z U ( k , k ′ ) (cid:20) χ ( k ′ , t ) − βω ( k ′ ) χ ( k , t ) βω ( k ) (cid:21) d k ′ , (17)which suggests to introduce a dimensionless variable A ( k , t ) defined as A ( k , t ) = χ ( k , t ) βω ( k ) . (18)In terms of A , the linearized kinetic equation has the form n (1 + n ) βω ( k ) ∂ t A ( k, t ) = Z U ( k , k ′ ) βω ( k ′ ) [ A ( k ′ , t ) − A ( k , t )] d k ′ . (19)The rotational invariance of the kernel U ( k , k ′ ) may be exploited by expressing the pertur-bation as a superposition of angular momentum eigenstates A ( k , t ) = X l,m A lm ( k, t ) Y lm (ˆ k ) . (20)Using the addition theorem for spherical harmonics one arrives at the equation n (1 + n ) βω ( k ) ∂ t A lm ( k, t ) = K l [ A lm ]( k, t ) (21)where the operator K l adopts the Fredholm form K l [ A ]( k, t ) = 4 π l + 1 Z ∞ U l ( k, k ′ ) βω ( k ′ ) [ A ( k ′ , t ) − A ( k, t )] k ′ dk ′ − π A ( k, t ) Z ∞ (cid:18) U ( k, k ′ ) − U l ( k, k ′ )2 l + 1 (cid:19) βω ( k ′ ) k ′ dk ′ . (22)Here U l ( k, k ′ ) is the l -coefficient in the series of Legendre polynomials of U : U ( k , k ′ ) = ∞ X l =0 U l ( k, k ′ ) P l (cos θ kk ′ ) . (23)Next we derive two approximations to K l [ A ]( k, t ) at low momentum valid near the criticaltemperature and low temperature. They must formally satisfy energy conservation which isachieved if ddt Z ∞ n (1 + n ) βω ( k ) A ( k, t ) ω ( k ) k dk = Z ∞ K [ A ]( k, t ) ω ( k ) k dk = 0 . (24)Clearly, in view of Eq. (22), the symmetry of the kernel U ( k, k ′ ) is sufficient in order toaccomplish this requirement, so such a property must not be lost in the approximated kernel. III. DYNAMICS NEAR THE CRITICAL TEMPERATUREA. The linearized equation at low momentum
In this section we consider the regime k B T ≫ gn c near the critical temperature. Herethe dispersion is well approximated by ω = k / (2 m ), and the squared amplitude by |M| = Β m k ¢ Β n c U @ k , k ' D k ' FIG. 1: The kernel U ( k, k ′ ) k ′ for k p β/m = 1 / k ′ = k is not integrable. n c a m − . The low momentum approximation to U ( k, k ′ ) is obtained by the rescaling k → ǫk, k ′ → ǫk ′ and the expansion around ǫ = 0. This gives U ( k, k ′ ) ∼ m a n c β (cid:18) θ ( k − k ′ ) k k ′ ( k − k ′ ) + θ ( k ′ − k ) k k ′ ( k ′ − k ) (cid:19) , k → , k ′ → . (25)The kernel U l ( k, k ′ ) has a non-integrable singularity proportional to | k − k ′ | − as k ′ → k (seeFig. 1). In order to manage carefully this singularity, it is convenient to integrate by partsin the exact expression of K l [ A ] given by Eq. (22). It helps to write A ( k ′ , t ) − A ( k, t ) = Z k ′ k dq ∂ q A ( q, t ) , (26)and to perform the change of the order of integration. This yields K l [ A ]( k, t ) = Z k dq J 0. With respect tothe second term in Eq. (27) it is far from obvious that the limit q → J >l ( k, q ) mustbe incorporated since q runs from k to ∞ . The key idea emerges by noting first that theconsidered scattering amplitude involves the assumption that all three momenta are of thesame order. It is then expected to be adequate to take the limit q → k → J >l ( k, q ). Another more formal justification arises from the fact that J >l ( k, q ) diverges asln( q − k ) as q → k . Therefore, by assuming that ∂ q A ( q, t ) falls off sufficiently to insureintegrability, the main contribution to the second term in Eq. (27) as k → q is also close to zero. The leading behavior of the integral in the last termof Eq. (27) as k → l + 1) U and U l canceleach other.The evaluation of the integrals in that limit can be carried out analytically, and theyassume the scaling form J 1) 1 x ln (cid:18) − x (cid:19) , (38) H ( x ) = θ (1 − x ) 1 x ln (cid:18) x − x (cid:19) + θ ( x − (cid:20) x − x ln (cid:18) x + 1 x − (cid:19)(cid:21) , (39) H ( x ) = − θ (1 − x ) 12 x ln (cid:2) (1 + x )(1 − x ) (cid:3) + θ ( x − (cid:20) x − x ln (cid:18) x + x ( x − (cid:19)(cid:21) , (40) I = 0 , (41) I = 2 − ln 16 , (42) I = 32 − 12 ln 256 , (43)Hereafter in this section we will drop the lines over the reduced momentum variables.The most efficient way to manage the convolution integral of the kinetic equation is touse the Mellin transform with respect to momentum and the Laplace transform with respectto time. If F ( s, λ ) denotes the image of A ( k, t ), F ( s, λ ) = Z ∞ dt e − λt Z ∞ A ( k, t ) k s − dk, (44)the evolution equation becomes λ F ( s, λ ) = W H l ( s ) [ − ( s − F ( s − , λ )] − I l F ( s − , λ ) + Ψ( s ) , (45)or alternatively λ F ( s + 1 , λ ) = W V l ( s ) F ( s, λ ) + Ψ( s + 1) , (46)where W V l ( s ) ≡ − sW H l ( s + 1) − I l . (47)Here W H l ( s ) and Ψ( s ) are the Mellin transforms of H l ( x ) and the initial condition A ( k, W V l ( s ) denotes the Mellin image of the kernel V l ( x ) in Eq. (35). We will assumethat A ( k, 0) has compact support to assure that Ψ( s ) is an entire function. Let us firstconcentrate in the l = 0 case. From Eq. (38) we see that the Mellin function is given by W V ( s ) = − γ E − ψ (cid:16) s (cid:17) − π cot (cid:16) πs (cid:17) , Re s ∈ ( − , , (48)where γ E is the Euler constant and ψ ( z ) = Γ ′ ( z ) / Γ( z ) is the digamma function. Note thereflection property W V (1 + s ) = W V (1 − s ), in accordance with the symmetry of U ( k, k ′ ).The fundamental strip where W V ( s ) is analytic is Re s ∈ ( − , s = 0 , 2. It is important for what follows to introduce the winding number of theMellin function W V l ( s ) as the increment divided by 2 π of the argument of W V l ( s ) as s movesfrom σ − i ∞ to σ + i ∞ along the line Re s = σ . In the l = 0 case, one can check numericallythat the winding number κ ( σ ) of the Mellin function vanishes when σ ∈ (0 , F ( s, λ ). (In this paragraph we willassume an arbitrary non-negative value for h .) This can be accomplished by following themethod of Ref. [12], briefly reviewed in the sequel. These authors have shown that whenan interval of zero winding exists within the fundamental strip, ( σ − , σ + ) ⊂ ( a, b ), then theequation (46) has a unique solution. It can be constructed using a particular solution B ( s )of the homogeneous equation B ( s + h ) = − W ( s ) B ( s ) , Re s ∈ ( a, b ) , (49)whose requisite properties have been described in Ref. [12]. The most important featuresof B ( s ) are the meromorphic property in the strip Re s ∈ ( a, b + h ), and its analyticity andabsence of zeros in the strip Re s ∈ ( σ − , σ + + h ). The location of the poles and zeros of B ( s ) outside that interval is dictated by the set of zeros of the Mellin function within thefundamental strip. They determine the asymptotic properties of the solution. In particular,0it turns out that the pole of B ( s ) with maximal real part determines the asymptotics of A ( k, t ) as k → 0. Now, by expressing the Mellin image of the perturbation as F ( s, λ ) = B ( s ) f ( s, λ ) , (50)the function f satisfies the inhomogeneous difference equation with constant coefficients λf ( s, λ ) + f ( s − h, λ ) = Ψ( s ) B ( s ) , Re s ∈ ( σ − , σ + + h ) (51)which has an inverse Mellin image given by( λ + k − h ) a ( k, λ ) = Q ( k )= 12 πi Z σ + i ∞ σ − i ∞ Ψ( s ) B ( s ) k − s ds, σ ∈ ( σ − , σ + + h ) . (52)The end step is to write the convolution corresponding to Eq. (50) and to invert the Laplacetransform. This yields the solution A ( k, t ) = Z ∞ Z (cid:18) kq (cid:19) exp (cid:0) − q − h t (cid:1) Q ( q ) dqq , (53)in terms of the inverse Mellin transform of the base function Z ( k ) = 12 πi Z σ + i ∞ σ − i ∞ B ( s ) k − s ds, Re s ∈ ( σ − , σ + + h ) . (54)Notice that when h > t → ∞ requiresthe determination of the leading behavior of Q ( k ) as k → ∞ . According to Eq. (52), thiscomes from the zero of B ( s ) with minimal real part.Next we turn to the l = 0 case. Due to the presence of the zero of W V ( s ) at s = 0, thefunction B ( s ) has a sequence of simple poles at s = 0 , − , − , . . . , with the behavior B ( s ) ∼ − B (1) W ′V (0) 1 s , s → , (55)then Z ( k ) ∼ − B (1) W ′V (0) , k → , (56)where B (1) is an arbitrary constant. This leads to a behavior independent of k at lowmomentum A ( k, t ) ∼ − B (1) W ′V (0) Z ∞ exp (cid:0) − t/q (cid:1) Q ( q ) dqq , k → . (57)Note that B (1) must cancel another similar factor arising from Q ( q ). On the other hand,the zero of B ( s ) with minimal real part is at s = 3, and it comes from the zero of W V ( s ) at s = 2. Using B ( s + 2) = W V ( s + 1) W V ( s ) B ( s ) when s → B ( s ) ∼ B (1) W V (1) W ′V (2)( s − , s → . (58)1This zero determines the high momentum limit of Q ( k ) in the form Q ( k ) ∼ − Ψ( s = 3) B (1) W V (1) W ′V (2) 1 k , k → ∞ , (59)so, the long time behavior of the solution is given by A ( k, t ) ∼ − Ψ( s = 3) B (1) W V (1) W ′V (2) Z ∞ Z (cid:18) kq (cid:19) exp (cid:0) − t/q (cid:1) dqq , t → ∞ , (60)which possess the self-similarity property A ( k, tv ) = 1 t A (cid:18) kt , v (cid:19) . (61)Finally, it is easy to derive in closed form the asymptotics in the more restricted regime k → , t → ∞ either from Eq. (57) combined with (59), or from Eq. (60) combined with (56).The result is A ( k, t ) ∼ s = 3) W ′V (0) W V (1) W ′V (2) 1 t , k → , t → ∞ , (62)where W ′V (0) W V (1) W ′V (2) = π 144 ( π − ln 16) . (63)The source term δ Γ only receives contribution from the l = 0 component, δ Γ = ddt Z n (1 + n ) χ ( k , t ) d k (2 π ) = 14 π / ddt Z ∞ n (1 + n ) βω ( k ) A ( k, t ) k dk, (64)or using the above low momentum approximation δ Γ ∼ π / ddt Z ∞ A ( k, t ) βω ( k ) k dk = m / π / β / τ ddt F ( s = 1 , t ) , (65)which contains the partial Mellin transform of A ( k, t ) for s = 1. From the convolution inEq. (53) one finds F ( s, t ) = B ( s ) Z ∞ exp (cid:0) − t/q (cid:1) Q ( q ) q s − dq. (66)The substitution of the large momentum behavior of Q ( k ) given in Eq. (59) yields theasymptotics F ( s, t ) ∼ −B ( s ) Ψ(3)Γ(3 − s ) t s − B (1) W V (1) W ′V (2) , t → ∞ , Re s < . (67)Combining this with Eq. (65) then gives the long-time behavior of the source term δ Γ ∼ m / Ψ(3) π / β / τ W V (1) W ′V (2) 1 t , t → ∞ . (68)2Since the energy of the perturbation is proportional to F ( s = 3 , t ), we see that F ( s = 3 , t ) = Ψ(3) , t → ∞ , (69)in accordance with energy conservation.What about the non-isotropic perturbations? The Mellin functions corresponding to the l = 1 , W V ( s ) = − γ E + 2 s − − ψ (cid:18) s (cid:19) − ψ (cid:18) − s (cid:19) , Re s ∈ ( − , , (70) W V ( s ) = − γ E + ln 16 + 6 s ( s − − π cot (cid:16) πs (cid:17) + π (cid:0) πs (cid:1) − ψ (cid:16) s (cid:17) , Re s ∈ ( − , . (71)Now W V ( s ) has a double zero at s = 1, and the winding number κ does not vanish on the en-tire strip of analyticity; the corresponding values are 1 , ( − 1) if Re s ∈ (1 , s ∈ ( − , l = 2, the winding number does not assume the value zero either, but now κ = 1 on theentire interval (-2,4), and the Mellin function does not vanish in the strip of analyticity. Inthese situations of absence of an interval of zero winding, Balk and Zakharov [12] have shownthat the initial value problem for A ( k, 0) is not well posed, signaling the non-uniqueness ofthe solution when κ > l = 2 case), or a strong instability ofexponential growth if κ ≷ l = 1 case). In terms of the Mellin function, we have W V , (0) > 0, which at least is a sufficient condition [12] for the instability of the Rayleigh-Jeans spectrum n ∝ k − under anisotropic perturbations. Thus, it seems praiseworthy toassign this property to the Bose-Einstein equilibrium solution of the kinetic equation witha collision term given by Eqs. (4) and (5).Finally, it may be instructive to compare these findings with those of Ref. [13]. Theapplication of the methods of wave turbulence theory to the study of the binary collisionterm C , approximated when n ≫ 1, showed the Mellin function W ( s ) to be symmetricwith respect to s = 3 / 2, reflecting the symmetry of the corresponding kernel U . At thesame time, it was shown that W ( s = 0) > W ( s = 1) = 0. As a consequence, suchapproximated form of C destroyed energy conservation but was consistent with particlenumber conservation. By simply shifting the minimum of W ( s ) one can get W ( s = 0) = 0and W ( s = 1) = 0. So that another approximated form of C , which conserves the energybut destroys the conservation of the particle number, appears as possible. However, giventhe reflection property of W ( s ) and the location of its zeros, it does not seem possible toget a low momentum approximation maintaining both conservation laws. IV. EVOLUTION OF PERTURBATIONS IN THE THERMAL REGIME βck ≪ In this section we consider the regime ω ( k ) ≪ k B T ≪ gn , where the energy of quasi-particles is ω = ck , with c = p gnm − . Their low-energy interactions are well described by3 Β c k ' - - - - Π m n c Β U @ k , k ' D FIG. 2: The kernel U ( k, k ′ ) in the low temperature regime when βck ≪ 1. The continuous linecorresponds to the exact kernel for βck = 1 / 4. The dashed line is the kernel obtained with U ( k, k ′ )approximated by Eq. (72) for βck = 1 / the squared amplitude |M| given by Eq. (7). By ignoring the next positive contribution(8 m c ) − k in the dispersion law all U l become equal to (2 l + 1) U , due to the collinearity ofwave vectors in the collision integral. Therefore in this approximation, the kinetic equationis the same for all values of l . The low momentum limit of this kernel is U ( k, k ′ ) ∼ − π mnβ c (cid:18) θ ( k − k ′ ) k + θ ( k ′ − k ) k ′ (cid:19) , βck → , βck ′ → . (72)A comparison between the exact and the approximated kernels is shown in Fig. 2. Our firsttask is to find an evolution equation consistent with energy conservation. When Eq. (72) issubstituted into Eq. (22), the latter turns into K [ A ]( k, t ) = − k πmnβ c Z ∞ (cid:18) k ′ k θ ( k − k ′ ) + k ′ k θ ( k ′ − k ) (cid:19) A ( k ′ , t ) dk ′ k ′ + 9 k πmnβ c (cid:18) Λ k − (cid:19) A ( k, t ) , (73)where we have introduced an ultraviolet cutoff Λ ≫ k in the integral which gives thecoefficient of A ( k, t ). We left for the moment this term undetermined. The well-definedconvolution term in the above equation determines the (provisional) class of admissiblesolutions of the approximated evolution equation. Such a class would be formed by functions A ( k, t ) which grow as k → k − , and fall off faster than k − as k → ∞ .In accordance with that, the integral for the Mellin transform of the gain term − Z ∞ (cid:2) x − θ ( x − 1) + x − θ (1 − x ) (cid:3) x s − dx = 1( s − s − , (74)is convergent for 3 < Re s < 4. Note however that the analytic continuation of this integralto s = 0 is the value of the gain term formally evaluated for A ( k ′ , t ) = 1. Based on4energy conservation, we may insist on considering the functions A = constant within theadmissible class of solutions, so the loss term must cancel out this value 1 / 12. Therefore,the replacement Λ / (3 k ) − / → − / 12 produces such a cancellation.The approximated evolution equation obtained in this way reads ∂ t A ( k, t ) = − k πmnβ c Z ∞ (cid:18) k ′ k θ ( k − k ′ ) + k ′ k θ ( k ′ − k ) (cid:19) A ( k ′ , t ) dk ′ k ′ − k πmnβ c A ( k, t ) , (75)which again has the form of a linearized kinetic equation of wave turbulence, where thedegree of homogeneity is now h = − 4. In fact, we will find that the above equation stillintroduces a conflict with energy conservation. This defect is repaired by adding a sourceterm independent of k to the right-hand side of Eq. (75). It has the formΦ( t ) = 98 πmnβ c Z ∞ A ( k ′ , t ) k ′ dk ′ , (76)and produces a partial cancellation of the integral term of Eq. (75) with the result ∂ t A ( k, t ) = − k πmnβ c A ( k, t )+ 9 k πmnβ c Z ∞ k (cid:18) k ′ k − k ′ k (cid:19) A ( k ′ , t ) dk ′ k ′ . (77)This implies that the class of solutions of the improved equation must be changed. Such aclass is formed indeed by functions which fall off faster than k − as k → ∞ . In order tounderstand the role of the source term and how it arises, it is convenient first to discuss thesolution of the wrong evolution based on Eq. (75).By introducing the function V ( x ) = − δ ( x − − x − θ ( x − − x − θ (1 − x ) , (78)we define the Mellin function corresponding to Eq. (75) as W V ( s ) = Z ∞ V ( x ) x s − dx = − s ( s − s − s − , Re s ∈ (3 , , (79)which has zero winding in the entire strip of analyticity, and W V ( s + 7 / 2) = W V (7 / − s ).The corresponding difference equation for the Laplace-Mellin image of A becomes λ F ( s − , λ ) = vW V ( s ) F ( s, λ ) + Ψ( s − , Re s ∈ (3 , , (80)where v = 3 / (32 πmnβ c ). The method used to solve Eq. (80) is the same as before. Weseek an appropriate solution of B ( s − 4) = − W V ( s ) B ( s ) for Re s ∈ (3 , F ( s, λ ) = B ( s ) Z ∞ Q ( k ) vk + λ k s − dk, (81) Q ( k ) = 12 πi Z Re s + i ∞ Re s − i ∞ Ψ( s ) B ( s ) k − s ds. (82)5In view of Eq. (80) and such a factorization, we need a solution B ( s ) which has neither zerosnor poles when Re s ∈ ( − , B ( s ) = Γ (cid:0) − s (cid:1) Γ (cid:0) s − (cid:1) Γ (cid:0) − s (cid:1) Γ (cid:0) s (cid:1) = 3 − ss sin (cid:0) πs (cid:1) sin (cid:0) π ( s + 1) (cid:1) . (83)The other solutions of the homogeneous difference equation in terms of products and frac-tions of Γ-functions, such asΓ (cid:0) − s (cid:1) Γ (cid:0) s − (cid:1) Γ (cid:0) − s (cid:1) Γ (cid:0) s − (cid:1) = 3 − ss sin (cid:0) π (1 + s ) (cid:1) sin (cid:0) π s (cid:1) , (84)Γ (cid:0) − s (cid:1) Γ (cid:0) − s (cid:1) Γ (cid:0) − s (cid:1) Γ (cid:0) − s (cid:1) = Γ (cid:0) s − (cid:1) Γ (cid:0) s − (cid:1) Γ (cid:0) s − (cid:1) Γ (cid:0) s (cid:1) = s − s , (85)are not appropriate for the present problem since they have a pole at s = 0 within the stripRe s ∈ ( − , F ( s, λ ), one finds the solution as theconvolution A ( k, t ) = Z ∞ Z (cid:18) kq (cid:19) exp (cid:0) − vq t (cid:1) Q ( q ) dqq , (86)where Z ( k ) and Q ( k ) are given by the integrations in Eqs. (54) and (52) with σ ∈ ( − , t → ∞ , we must evaluate the lowmomentum behavior of Q ( k ). This is dictated by the pole of B ( s ) − at s = − Q ( k ) ∼ − / Ψ( − π k , k → . (87)If F ( s, t ) denotes the Mellin image of the solution A ( k, t ), the above result permits thecalculation of the leading behavior of the quotient F ( s, t ) / B ( s ) as t → ∞ . It reads F ( s, t ) B ( s ) = Z ∞ exp( − vk t ) Q ( k ) k s − dk ∼ − / Ψ( − π Γ (cid:16) s (cid:17) ( vt ) − − s/ , t → ∞ , Re s > − . (88)We can see now the violation of energy conservation. In the approximation where n ≫ dEdt = 14 π / β ddt Z ∞ A ( k, t ) k dk = 14 π / β ddt F ( s = 3 , t ) = 0 , (89)or F ( s = 3 , t ) = F ( s = 3 , t = 0) = Ψ(3), since Ψ(3) corresponds to the energy of the initialperturbation. But, in accordance with the above asymptotics, the quantity F ( s = 3 , t )assumes a value proportional to Ψ( − B (3) t − / for large time, so dE/dt = 0. Another6indication that Eq. (75) is conflicting comes from the fact that, in order to check energyconservation, the integration of both sides after multiplication by k leads to ddt Z ∞ A ( k, t ) k dk = v Z ∞ A ( q, t ) q dq Z ∞ V ( x ) x dx, (90)which does not converge for the class of functions considered up till now.To obtain an improved evolution equation, let us consider a factorized solution F ( s, t )for a different base function respecting energy conservation. If we assume that Q ( k ) ∼ Dk α as k → 0, then the limit behavior of F ( s, t ) as t → ∞ is F ( s, t ) = B ( s ) Z ∞ exp( − vk t ) Q ( k ) k s − dk ∼ D B ( s )Γ (cid:18) s + α (cid:19) ( vt ) − ( s + α ) / , Re( s + α ) > . (91)Since energy conservation requires that F ( s = 3 , t = ∞ ) = Ψ(3), we see that α = − 3, and B ( s ) must possess a simple zero at s = 3 in order to cancel the pole of the gamma function.Furthermore, to accomplish the low momentum behavior of Q ( k ), the prescription for Re s in Q ( k ) = 12 πi Z Re s + i ∞ Re s − i ∞ Ψ( s ) B ( s ) k − s ds, (92)and the analog of (54) must be Re s > 3. Clearly, if the base function assumes the value B ( s ) = s − s , Re s > , (93)which was discarded before, the requirement F ( s = 3 , t ) = Ψ(3) is accomplished for largetime.Some features of the solution generated by this base function are the following. Theinverse images of Ψ( s ) / B ( s ) and B ( s ) are simply Q ( k ) = A ( k, 0) + 3 k Z ∞ k A ( q, q dq, (94) Z ( k ) = δ ( k − − θ (1 − k ) , (95)and the behavior of Q ( k ) as k → Q ( k ) ∼ k − , k → . (96)Hence the corresponding solution is written as A ( k, t ) = Z ∞ Z (cid:18) kq (cid:19) exp (cid:0) − vq t (cid:1) Q ( q ) dqq , = exp( − vk t ) Q ( k ) − Z ∞ k exp( − vq t ) Q ( q ) dqq , (97)7which for large t assumes the value A ( k, t ) ∼ exp (cid:0) − vk t (cid:1) Q ( k ) , t → ∞ . (98)Similarly, the behavior or the source term as t → ∞ becomes δ Γ = 14 π / βc Z ∞ ∂ t A ( k, t ) k dk = v π / βc (cid:20) − Z ∞ exp( − vk t ) k Q ( k ) dk + 3 Z ∞ dqq exp( − vq t ) q Q ( q ) Z q k dk (cid:21) = v π / βc Z ∞ exp( − vk t ) k Q ( k ) dk, (99)so the insertion of Eq. (96) produces δ Γ ∼ v / Γ(7 / π / βc t / , t → ∞ . (100)We can check directly energy conservation for any time, Z ∞ ∂ t A ( k, t ) k dk = − v Z ∞ exp( − vk t ) k Q ( k ) dk +3 v Z ∞ dqq exp( − vq t ) q Q ( q ) Z q k dk = 0 . (101)It remains to derive the resulting evolution equation satisfied by A ( k, t ) given in Eq. (97).From the homogeneous difference equation evaluated for B ( s ) in Eq. (85) it follows that W V ( s ) B ( s ) = −B ( s − 4) = − s − , Re s ∈ (3 , . (102)Applying the inverse Mellin transform we obtain Z ∞ V (cid:18) kq (cid:19) Z ( q ) dqq = − δ ( k − − k θ ( k − − k − Z ( k ) − k . (103)The origin of the extra term proportional to k − is the pole of B ( s ) at s = 0. With thisresult at hand, the integral term of Eq. (75) evaluated for A ( k, t ) in Eq. (97) becomes Z ∞ V (cid:18) kq (cid:19) A ( q, t ) dqq = 1 vk ∂ t A ( k, t ) − k Z ∞ exp (cid:0) − vq t (cid:1) Q ( q ) q dq = 1 vk ∂ t A ( k, t ) − k F ( s = 4 , t ) B ( s = 4)= 1 vk ∂ t A ( k, t ) − k Z ∞ A ( q, t ) q dq. (104)8The last term can be understood as a self-consistent source which restores energy conserva-tion. In consequence, if one uses the partial cancellation produced by this term, the correctedevolution equation adopts the form ∂ t A ( k, t ) = vk Z ∞ V (cid:18) kq (cid:19) A ( q, t ) dqq , (105)where V ( x ) = − δ ( x − − x − θ (1 − x ) + 12 x − θ (1 − x ) , (106)which gives rise to the same Mellin function as before W V ( s ) = − s ( s − s − s − , Re s > , (107)but with a different strip of analyticity. The relevant interval of zero winding is now Re s ∈ (7 , ∞ ) and the base function B ( s ) has neither zero nor poles in the corresponding stripRe s ∈ (3 , ∞ ). We expect that the evolution equation obtained in this way captures theessential features of the low momentum regime at low temperature. Dominance of C over C . To conclude this section, it is interesting to observe thatthe collision term that we have considered in this and the previous section is the prevailingsummand in the sum C + C in the low momentum regime, n ( k ) ≫ 1. This is based on therules of power counting for the degrees of homogeneity h of the linearized collision integralwhich have been given in Ref. [12]. These authors have shown that when the dispersion lawhas the form ω ( k ) ∝ k δ , and the scattering amplitudes of C and C scale according to M ( ε k , ε k , ε k ) = ε η M ( k , k , k ) , (108) M ( ε k , ε k , ε k , ε k ) = ε κ M ( k , k , k , k ) , (109)the index h is given by h [ C ] = δ − η − d + ν, (110) h [ C ] = δ − κ − d + 2 ν, (111)where d is the spatial dimension, and ν is the exponent in a solution n ( k ) ∝ k − ν of C [ n ] = 0.Near the critical temperature, the scattering amplitude involving four excitations comesfrom the term g ( ψ † ψ ) in the effective Hamiltonian, where ψ † ( ψ ) creates (destroys) ex-citations, while terms as g √ n c ψ † ψψ give rise to the scattering amplitude involving threeexcitations. In both cases κ = η = 0 since M , ∝ g, g √ n c . The insertion into Eq. (111) of δ = 2 , κ = 0 , d = 3 , ν = 2 produces h [ C ] = 0, while h [ C ] = 1. Thus, we can neglect thecontribution of C in the linearized description at low momentum.At very low temperature, when the excitations are phonons, the effective descriptionis made in terms of a Hamiltonian H [ n ( x ) , θ ( x )] depending on the particle density n andthe Goldstone mode or phonon field θ conjugate to n . Now, since each θ -field carries the9 Β c k '0.050.100.1564 Π m n c Β U @ k , k ' D k' FIG. 3: U ( k, k ′ ) k ′ in the quantum regime, βck ≫ 1. The continuous line corresponds to theexact kernel for βck = 10. The dashed line is the kernel obtained with U ( k, k ′ ) approximated byEq. (112) for βck = 10. oscillator factor 1 / p ω ( k ), the conjugate field n goes with p ω ( k ). Thus, a term in theHamiltonian of order n N produces a scattering amplitude M N proportional to Q Ni √ k i ,where k i denotes the momentum of the quasiparticle i in the process. This means that η = 3 / 2, according to Eq. (7), and κ = 4 / δ = 1 , ν = 1, we find h [ C ] = − h [ C ] = − 7, which shows that C prevails over C . V. EVOLUTION OF PERTURBATIONS IN THE QUANTUM REGIME βck ≫ In this section we consider the kinetic equation and its solutions when the term U ( k, k ′ )of the collision integral (22) is approximated in the quantum regime k B T ≪ ck ≪ gn . Thisproceeds by replacing n (1 + n ) → e − βck on the right side of Eq. (21), while U ( k, k ′ ) issymmetrically approximated by its leading behavior as βck → ∞ , (see Fig. 3) U ( k, k ′ ) ∼ π mn (cid:0) e − βck ( k − k ′ ) θ ( k − k ′ )+ e − βck ′ ( k − k ′ ) θ ( k ′ − k ) (cid:17) , βck → ∞ . (112)The relative size of the second term is O ( e − βc ( k ′ − k ) ). The corresponding subleading contri-bution to the collision term may be evaluated by using the Watson’s lemma [14]916 πmn Z ∞ k e − βck ′ ( k − k ′ ) k ′ ( A ( k ′ , t ) − A ( k, t )) dk ′ ∼ e − βck k πmnβ c ∂ k A ( k, t ) , β → ∞ . (113)Although the approximation that arises by ignoring this contribution is necessarily non-energy conserving, it may be interesting to explore its consequences, since we would expect0this drawback to have a negligible effect as T → 0. Therefore we consider the evolutionequation ∂ t A ( k, t ) = 916 πmn k Z k ( k − k ′ ) k ′ ( A ( k ′ , t ) − A ( k, t )) dk ′ = 9 k πmn Z k (cid:18) k ′ k (cid:19) (cid:18) kk ′ − (cid:19) A ( k ′ , t ) dk ′ k ′ − k πmn A ( k, t ) , (114)where, for convenience, we have written the gain term as a convolution. Unsurprisingly wesee that the loss term reduces to − Γ B ( k ) A ( k, t ) where Γ B ( k ) = 3 k / (320 πmn ) is preciselythe decay width of Beliaev damping at T = 0 [10].Although in this regime, n ∼ e − βck ≪ 1, we are not in the vicinity of a power lawspectrum as those appearing in the wave turbulence framework, the above linear opera-tor is homogeneous of degree h = − 5. Therefore, we can analyze the evolution equationwith methods similar to the ones we have described before. We define the Mellin functioncorresponding to Eq. (114) W ( s ) = − Z ∞ ( x − x x s − dx = − s ( s − s + 74)( s − s − s − , Re s < . (115)The interval of zero winding within the strip of analyticity is ( −∞ , F ( s, t ) of A ( k, t ) and the Mellin image Ψ( s ) of the initial condition A ( k, λ F ( s − , λ ) = v B W ( s ) F ( s, λ ) + Ψ( s − , Re s < , (116)where v B = 3 / (320 πmn ). Now we seek an appropriate solution of the difference equation B ( s − 5) = − W ( s ) B ( s ) in the strip of zero winding, Re s ∈ ( −∞ , F ( s, λ ) in the factorized form (50). To establish this particular solution we requirethat B ( s ) has neither zeros nor poles for Re s ∈ ( −∞ , s = 0. It must produce only a simplepole of B ( s ) at s = 0 contained in the strip Re ∈ ( −∞ , B ( s ) = Γ (cid:0) − s (cid:1) Γ (cid:16) − i √ − s (cid:17) Γ (cid:16) + i √ − s (cid:17) Γ (cid:0) − s (cid:1) Γ (cid:0) − s (cid:1) Γ (cid:0) − s (cid:1) , (117)and the inverse images of B and 1 / B are expressed as Mellin-Barnes integrals along a verticalpath with Re s < 0. The inverse of B ( s ) may be computed by making the splitting ( B ( s ) − 1) + 1 which produces a Meijer G function and a delta function: Z ( k ) = 5 G , , " k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , − i √ , − − i √ , , − + δ ( k − 1) = Z ( k ) + δ ( k − . (118)1It remains to find the second factor Q ( k ) of the convolution which gives the solution of theinitial value problem A ( k, t ) = Z ∞ Z (cid:18) kq (cid:19) exp (cid:0) − v B q t (cid:1) Q ( q ) dqq . (119)In particular, if the initial perturbation is A ( k, 0) = C δ ( k − k ), the inversion of Ψ( s ) / B ( s )yields Q ( k ) in the form Q ( k ) = 5 Ck G , , " k k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , , − , − i √ , − − i √ + C δ ( k − k ) = Q ( k ) + C δ ( k − k ) . (120)Since all singularities of B ( s ) and 1 / B ( s ) are located in the half-plane Re s ≥ 0, theMeijer function G , , ( z ) vanishes if 0 ≤ z < 1. Some explicit limiting forms are G , , " z (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , , − , − i √ , − − i √ ∼ − M ( z − , z → + , (121) G , , " z (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , − i √ , − − i √ , , − ∼ p − √ 55 cosh( √ π/ , z → ∞ , (122)where M ≈ . 48. Consequently, the solution may be rewritten as A ( k, t ) = Z k Z (cid:18) kq (cid:19) exp (cid:0) − v B q t (cid:1) Q ( q ) dqq + exp (cid:0) − v B k t (cid:1) Q ( k )= Ck Z (cid:18) kk (cid:19) exp (cid:0) − v B k t (cid:1) θ ( k − k ) + exp (cid:0) − v B k t (cid:1) Q ( k )+ Z k Z (cid:18) kq (cid:19) exp (cid:0) − v B q t (cid:1) Q ( q ) dqq . (123)It is interesting to find the leading behavior of the last integral as t → ∞ . To apply theWatson’s lemma, we use Eq. (121) and thus we obtain Z k Z (cid:18) kq (cid:19) exp (cid:0) − v B q t (cid:1) Q ( q ) dqq ∼ − CMk Z (cid:18) kk (cid:19) θ ( k − k ) × Z ∞ k exp (cid:0) − v B q t (cid:1) (cid:18) q k − (cid:19) dqq ∼ − CMk Z (cid:18) kk (cid:19) exp ( − v B k t ) v k t × θ ( k − k ) , t → ∞ . (124)We see that this integral corresponds to a subleading term of O ((Γ B ( k ) t ) − ) in the expressionfor A ( k, t ). Noting that when k > k the second term in Eq. (123) proportional to Q ( k )is subleading with respect to the first one, we obtain the final result for the long timeasymptotics of the solution A ( k, t ) ∼ Ck Z (cid:18) kk (cid:19) exp (cid:0) − v B k t (cid:1) θ ( k − k )+ C exp (cid:0) − v B k t (cid:1) δ ( k − k ) , t → ∞ , (125)2or A ( k, t ) ∼ C p − √ k cosh( √ π/ 10) exp (cid:0) − v B k t (cid:1) , t → ∞ , k ≫ k . (126) VI. CONCLUDING REMARKS In this paper we have studied the temporal evolution of a perturbation of the equilibriumdistribution of noncondensate atoms in Bose gases as described by a kinetic equation in-cluding only collisions between condensate and noncondensate atoms. Due to the difficultyto address this problem without any simplification, we have considered some approxima-tions to the kinetic equation in different temperature regimes. The evolution equations thatarise by approximating the full kinetic equation when the energy of an excitation is smallcompared to the thermal energy turn out to have definite homogeneity properties, and maybe regarded as kinetic equations for waves in the limit of large occupation numbers. Thisoccurs near the critical temperature when the dispersion law is quadratic, and near zerotemperature for Bogoliubov excitations of very low energy. Notably, by using the explicitexpression that determines the degree of homogeneity, one may easily show the dominanceof C over the binary collision term C at low momentum. This provides a justification toour approach (i. e. dealing only with C ). In the opposite quantum regime where the oc-cupation number of noncondensate atoms decreases exponentially with the momentum, werecover unexpectedly a kinetic equation of the same type of the wave-turbulence equationsin the previous situations.To obtain the solution of the initial value problem in all cases, we have applied themethod developed by Balk and Zakharov [7, 12] to analyze the behavior of weak-turbulentmedia near Kolmogorov spectra. This is a scarcely used technique due to the difficulties inthe analytical calculation of the Mellin function and the base function, mainly for binaryprocesses 2 ↔ ↔ Acknowledgments The work of M.E. is supported by the Spanish MICINN under Grant MTM2008-03541 andby the Basque Government under Grant No. IT-305-07. The work of F.P. has been partiallysupported by Project CBDif-Fr ANR-08-BLAN-0333-01. The work of M.V. is partiallysupported by the Spanish Ministry of Science and Technology under Grant FPA2009-10612,the Spanish Consolider-Ingenio 2010 Programme CPAN (CSD2007-00042) and the BasqueGovernment under Grant No. IT559-10. Appendix: Low momentum analysis of the linearized Boltzmann equation In this appendix we give some details about the derivation of the limiting form of the col-lision term for small momentum. From Eq. (15) in the regime near the critical temperature,and setting u = cos θ kk ′ , the following expression for U ( k , k ′ ) follows easily116 n c a m − U ( k , k ′ ) = [ n ( ω ( k ))[1 + n ( ω ( k ′ ))][1 + n ( ω ( k ) − ω ( k ′ ))] × mθ ( k − k ′ ) kk ′ δ (cid:18) u − k ′ k (cid:19) + ( k ↔ k ′ ) (cid:21) − n ( ω ( k ) + ω ( k ′ ))[1 + n ( ω ( k ))][1 + n ( ω ( k ′ ))] × mkk ′ δ ( u ) , (A.1)where n is the Bose distribution function with zero chemical potential n ( ω ) = 1 e βω − . (A.2)The coefficients U l ( k, k ′ ) are easily evaluated using the standard formula U l ( k, k ′ ) = 2 l + 12 Z − U ( k , k ′ ) P l ( u ) du. (A.3)To extract the low momentum behavior of J < ( k, q ) and J > ( k, q ) we first perform exactlythe integrations of Eqs. (28) and (29). These may be expressed in closed form in termsof polylogarithmic functions Li n ( x ), where x is an exponential of the some combination ofreduced energies. As an example, the expressions for J <,> ( k, q ) near the critical temperature4read J < ( k, q ) = − πn c a βm e βω ( k ) ( e βω ( k ) − k × (cid:20) β q − βmq ln 1 − e − β ( ω ( k ) − ω ( q )) − e − β ( ω ( k )+ ω ( q )) − m Li (cid:0) e − β ( ω ( k ) − ω ( q )) (cid:1) − m Li (cid:0) e − β ( ω ( k )+ ω ( q )) (cid:1) +16 m Li (cid:0) e − βω ( k ) (cid:1)(cid:3) θ ( k − q ) , (A.4) J > ( k, q ) = 8 πn c a βm e βω ( k ) ( e βω ( k ) − k × " β k − βmq ln (cid:0) − e − β ( ω ( q ) − ω ( k )) (cid:1) (cid:0) − e − β ( ω ( k )+ ω ( q )) (cid:1) (1 − e − βω ( q ) ) +4 m Li (cid:0) e − β ( ω ( q ) − ω ( k )) (cid:1) + 4 m Li (cid:0) e − β ( ω ( k )+ ω ( q )) (cid:1) − m Li (cid:0) e − βω ( q ) (cid:1)(cid:3) θ ( q − k ) . (A.5)The form of other kernels J >l ( k, q ) with l = 0 are similar. If we substitute the asymptoticexpansion of Li n ( x ) near x = 1,Li ( x ) ∼ π − x ) ∞ X n =1 (1 − x ) n n − ∞ X n =1 (1 − x ) n n , x → , (A.6)we obtain J < ( k, q ) ∼ θ ( k − q ) 64 πn c ma β k ln (cid:18) − q k (cid:19) , k, q → , (A.7) J > ( k, q ) ∼ θ ( q − k ) 64 πn c ma β k ln q + k q − k , k, q → . (A.8)These results yield the expression of H ( x ) given in Eq. (38), once the factor 64 πn c ma /β × p β/m × β/ m is extracted to define the time scale τ in Eq. (36).Near zero temperature the expression for U ( k , k ′ ) reads U ( k , k ′ ) = (cid:20) k − k ′ ) θ ( k − k ′ )32 π mn n ( ω )[1 + n ( ω ′ )][1 + n ( ω − ω ′ )]+( k ↔ k ′ )] δ ( u − − k + k ′ ) π mn n ( ω + ω ′ )[1 + n ( ω )][1 + n ( ω ′ )] δ ( u − . (A.9)We note that the δ ( u − ω = ck + αk is not strictlylinear and, since α > 0, the angle between the wave vectors is close to zero. The aboveexpression leads directly to the limiting forms given in Eqs. (72) and (112).5