Stability of steady states in kinetic Fokker-Planck equations for Bosons and Fermions

Transcription

1 Stability of steay states in kinetic Fokker-Planck equations for Bosons an Fermions L. Neumann, C. Sarber RICAM-Reort

2 STABILITY OF STEADY STATES IN KINETIC FOKKER-PLANCK EQUATIONS FOR BOSONS AND FERMIONS LUKAS NEUMANN AND CHRISTOF SPARBER Abstract. We stuy a class of nonlinear kinetic Fokker-Planck tye equations moeling quantum articles which obey the Bose-Einstein an Fermi- Dirac statistics, resectively. We establish the existence of classical solutions in the erturbative regime an rove exonential convergence towars the equilibrium. Version: November 6th, Introuction an main results In recent years the rigorous mathematical stuy of kinetic equations has been enlarge to a class of moels which take into account also quantum effects, cf. [25] for a general overview. This so-calle quantum kinetic theory can be seen as an attemt to incororate certain roerties of an unerlying quantum systems into the framework of classical statistical mechanics. One might hoe that these hybri moels on the one han allow for a somewhat simler escrition of the article ynamics, while maintaining, on the other han, some urely quantum mechanical features such as generalize statistics for Bosons an Fermions. Clearly such moels can only be justifie in a semiclassical regime, resectively, in situations where the transort roerties of the articles are mainly governe by Newtonian mechanics. Inee this oint of view has alreay been aote in the classical aer by Uehling an Uhlenbeck [24], in which they erive their celebrate nonlinear Boltzmann tye equation for quantum articles. Following their sirit most of the quantum kinetic moels stuie so far invoke nonlinear collision oerators of Boltzmann tye, see, e.g., [5, 6, 16, 17]. For these kin of moels, a articular focus of interest is the long time behavior of their solutions, in articular the convergence towars steay states, which generalize the classical Maxwellian istribution, cf. [8, 16, 18, 20]. Very often though, the simlifie case of a satially homogeneous gas is consiere. In what follows, we shall also be intereste in such kin of relaxation-to-equilibrium henomena, in the satially inhomogeneous case. We shall not eal with a Boltzmann tye equation, but rather stuy a nonlinear Fokker-Planck tye moel FP). More recisely, we consier the following kinetic equation 1.1) t f + x f + κf 2 ) = iv f + f1 + κf)), where, for any t 0, f = ft, x, ) 0 enotes the article istribution on hase sace Ω x R. In what follows, the satial omain is chosen to be Ω x = T, the 2000 Mathematics Subject Classification. 82C40, 82C10, 76R50, 82D37. Key wors an hrases. Fokker-Planck oerator, Bose-Einstein statistics, Fermi-Dirac statistics, long time behavior. C.S. has been suorte by the APART research grant fune by the Austrian Acaemy of Sciences. 1

3 2 L. NEUMANN AND C. SPARBER -imensional torus. This setting can be seen as a convenient an mathematically simler relacement for the incororation of confining otentials V x), neee to guarantee the existence of nontrivial steay states in the whole sace. In 1.1) we set κ = 1 for Fermions an κ = 1 for Bosons. Remark 1.1. Obviously, for κ = 0 equation 1.1) simlifies to the classical linear Fokker-Planck equation or Kramer s equation) on hase sace, i.e. t f lin + x f lin = f lin + iv f lin ). For this linear moel, the convergence to equilibrium has been recently stuie in [4, 11, 19], using several ifferent aroaches. The FP tye moel 1.1) has been introuce in [14] an a formal erivation from the satially homogeneous) Uehling-Uhlenbeck equation is given in [21]. Different hysical alications can be foun in [9, 12, 13, 15] ealing with, both, the satially homogeneous as well as the inhomogeneous case see also [10] an the references given therein). More recently, a similar but somewhat simler FP tye moel has been roose in [22, 23] to escribe self-gravitating articles an the formation of Bose-Einstein conensates in a kinetic framework. The authors consier 1.2) t f + x f = iv f + f1 + κf)), where, in contrast to 1.1), only the iffusive art of the equation inclues a nonlinearity. As we will see both equation however share the same steay states. In orer to eal with both moels, we stuy from now on the following initial value roblem { t f + x f + σκf 2 ) = iv f + f1 + κf)), 1.3) f t=0 = f 0 x, ), with σ = 1 or σ = 0, corresoning to the case 1.1) an 1.2), resectively. We note that the long time behavior of these moels in the satially homogeneous case an for = 1) has been rigorously investigate quite recently in [3] via an entroy-issiation aroach. In what follows, the initial hase sace istribution f 0 L 1 T x R ) is assume to be normalize accoring to 1.4) f 0 x, ) x = M, T R for some given mass M > 0. This normalization is conserve by the evolution. Moreover in the fermionic case, i.e. κ = 1, we require f 0 x, ) < 1, x, ) T R, as usual in the hysics literature [9]. In articular the latter is neee to efine the associate quantum mechanical entroy functionals, i.e. ) 2 H[f] := f + f ln f κ1 + κf) ln1 + κf) x, 2 T R which obviously requires ft, x, ) < 1, if κ = 1. It is now straightforwar to verify that ineenent of the articular choice of σ) the unique steay state of 1.3) is given by 1 1.5) f = ), ex θ κ where the constant θ is use to ensure that f satisfies the mass constraint 1.4). In the bosonic case θ R +, whereas in the fermionic case we can allow for θ R,

4 FOKKER-PLANCK EQUATION FOR BOSONS AND FERMIONS 3 cf. [3, 7] for more etails. In the latter situation the istribution 1.5) is the well known Fermi-Dirac equilibrium istribution. On the other han, for κ = 1, f is the so-calle regular Bose-Einstein istribution. Finally, if κ = 0, formula 1.5) simlifies to the classical Maxwellian, i.e. f lin = M e 2 /2, where log M = θ. Note that in any case the equilibrium state is ineenent of x since we have chosen T as our satial omain. In the bosonic case there is an aitional ifficulty, at least for 3, since one can not associate an arbitrarily large M > 0 to the steay state. More recisely, the maximum amount of mass comrise by f is etermine via 1 T R e 2 /2 1 x =: M crit <, i.e. for θ = 0. Due to mass conservation this inuces a threshol on M. This roblem, which oes not aear in imensions = 1, 2, has le to the introuction of more general bosonic steay states, where an aitional δ-istribution aroriately normalize) is ae to f, cf. [6, 8]. This singular measure can then be interrete as a so-calle Bose-Einstein conensate BEC). The formation of a δ-measure in finite or infinite time is a task of extensive research in quantum kinetic theory, see, e.g., [2, 8]. For our nonlinear moel though, incluing such generalize solutions on a rigorous mathematical level seems to be out of reach so far an we thus have to imose θ > 0 in the bosonic case. Inee, as we shall see below, we also require θ > 0 in the fermionic case, although for ifferent an rather technical reasons.) Our main task here is the escrition of the convergence for solutions to 1.3) towars the steay state 1.5). To this en we shall concetually follow the aroach given in [19] where the tren to equilibrium is stuie for a wie class of kinetic moels close to equilibrium. The ifference in our case being mainly that we are ealing with local nonlinearities in the Boltzmann tye moels stuie in [19] the nonlinearities usually aear only within an integral kernel) which moreover are also allowe to enter in the transort art of the consiere equation. We thus linearize the solution f of 1.3) aroun the steay state f in the form 1.6) f = f + g µ. where the new unknown gt, x, ) R can be interrete as a erturbation of the equilibrium state such that T R g µ x = 0. In 1.6) we use the aitional time-ineenent) scaling factor µ := f + κf 2, which allows for an easier escrition of the functional framework given below. Plugging 1.6) into 1.3) straightforwar calculations formally yiel the following equation for g 1.7) t g σκf ) x g = Lg) + Qg),

5 4 L. NEUMANN AND C. SPARBER Above the linearize collision oerator L is efine by 1.8) Lg) = 1 µ iv g µ ) + η g µ ) )) 1 = g + g 2 η κµ, where we use the short han notation η := 1 + 2κf. The quaratic remainer Q is given by 1.9) Qg) = κ µ iv µ g 2) σµ x g 2 ) ). The main result is as follows. Theorem 1.2. Let f 0 be of the form 0 f 0 = f + g 0 µ, with θ > 0. Moreover if κ = σ = 1, i.e. the bosonic case with nonlinear transort, assume that in aition be θ > θ, for a certain θ > 0. There exists an ɛ 0 > 0, such that for all f 0 with g H k ɛ < ɛ 0, for N k > 1+/2, the equation 1.3) amits a unique solution 0 f C[0, ); H k T x R )). Moreover ) µ 1/2 H ft) f Cɛ 0e τt, k where C, τ are ositive constants. The roof of this result is given in Section 3. To this en we first collect several reliminary results in the Section 2. Remark 1.3. Note that the assumtion θ > 0 is imose for both cases, the bosonic as well as the fermionic one. Physically seaking this means that we have to consier initial ata f 0 with sufficiently small mass M. For Bosons this is crucial in orer to avoi formation of a BEC in 3. In the fermionic case the reason to imose θ > 0 is to guarantee η > 0, which we will make use of several times. It might be ossible to overcome this restriction, cf. [3], albeigth the authors follow a ifferent aroach. The aitional restriction θ > θ is not neee for the moel 1.2) where only a nonlinear iffusion oerator is resent. The reason for this aitional constraint when ealing with 1.1) is that we have to maintain a funamental regularizing roerty in x of the transort art, cf. Section 3 for more etails. Since θ is then etermine by a transcenental nonlinear equation, we can not give an exact value for it but only erform numerical exeriments which inicate that θ Formal calculations given in [23] inicate that an analogous theorem as above oes not hol if one inclues the ossibility of a BEC, i.e. if one allows θ = 0 for κ = 1, in imension 3. In articular it seems that in general one can not exect exonential convergence towars a BEC equilibirum state, cf. [23] or [8] where a ifferent kinetic moel is use. The theorem an its roof can be seen as a variant of a rather new tye of mathematical aroach in the stuy of kinetic moels, base on the socalle hyo-coercivity roerty of the linearize equation. Here the hrase hyo- has to be unerstoo as in the istinction between hyo-ellitic

6 FOKKER-PLANCK EQUATION FOR BOSONS AND FERMIONS 5 an classical) ellitic oerators. For more etails on that we refer to [26]. The theorem irectly yiels the following result for the quantum mechanical entroies. Corollary 1.4. Uner the same assumtions as above H[ft)] H[f ] Ce τt, i.e. we have exonential ecay in relative entroy. Proof. Inserting f = f + µ g, where µ g = 0, erforming a Taylor exansion aroun the steay state f, an finally using Theorem 1.2. yiels the assertion of the corollary. Remark 1.5. One might think of several extensions of our result. As alreay iscusse in [19] one coul take into account weak external otentials, i.e. otentials V x) such that V C 2 T ) small enough, or self-consistent otentials, which stem from a couling to Poisson s equation. The latter case might be articularly interesting in semiconuctor moeling, where the fermionic FP tye equation coul be use to escribe the ynamical behavior of charge carries obeying the hysically correct equilibrium statistics. 2. Stuy of the linearize collision oerator We shall now erive several roerties on the linearize collision oerator L to be use in the roof of the main result. First note that L is self ajoint on L 2 R ) an, by artial integration, one obtains 2.1) Lg), g L 2 R ) = R g + 2 η g 2 = Thus the kernel of the non-ositive oerator L is given by KerL) = san{ µ }. R g µ ) 2 µ. Let us efine the orthogonal rojection in L 2 R ) onto this kernel via 1 Πf) := f ) µ µ, ρ R where we set ) ρ = µ > 0 R for reasons of normalization. Note that this is only a rojection in the momentum variable R. Motivate by 2.1) we introuce the following weighte sace where Λ := { f L 2 R ) : f Λ < }, f 2 Λ := f 2 L 2 + η f 2 L 2, Here, an in what follows, we write L 2 L 2 R ) for simlicity. Moreover we enote for fx, ) Λ L2 f Λ := f Λ, T x ) the inuce norm on hase sace. Obviously the Λ norm controls the L 2 norm for κ nonnegative. In the fermionic case κ = 1) however this is not true in general,

7 6 L. NEUMANN AND C. SPARBER since η may change sign. For our functional aroach the control of L 2 via Λ is crucial an thus we have to guarantee that η > 0 by assumtion. This imlies that we nee to imose for κ = 1 that f < 1/2, R, or equivalently θ > 0. This certainly is more restrictive than the usual boun i.e. f < 1, use in the hysics literature. In summary we require θ > 0 in the bosonic case to revent BEC an in the fermionic case to ensure that η is globally boune away from zero. First we obtain a Poincaré inequality for the steay state of the linearize moel. Lemma 2.1. If θ > 0 then the strictly ositive an normalize) measure µ /ρ satisfies a Poincaré inequality on R, i.e. R g 2 µ 1 ρ ) 2 gµ C g 2 µ, C > 0. R R Note that in the fermionic case κ = 1 this lemma inee hols more generally for any θ R. Proof. Let A be efine by A = ln µ. Having in min the results of [1] it is enough to rove that A can be ecomose as as A = A 1 + A 2, where A 1 ) is uniformly convex an A 2 ) is a local erturbation, such that a, b > 0, s.t. R : a e A2) b. To this en note that A is given by ) ) A = log e 2 /2+θ = 2 e 2 /2+θ κ) θ 2 log e 2 /2+θ e 2 /2+θ κ ) = θ + 2 log 1 1 κe 2 /2+θ. We now ick A 1 = 2 /2 an A 2 to be the rest of the terms aearing on the r.h.s.. Since 2 /2 is strictly convex it is larger than some ositive constant outsie a ball of finite raius. Outsie this ball the function A 2 is naturally boune. On the other han, insie the ball A 2 is boune because A, as well as A 1 are naturally boune if κ = 1, or if κ = 1 an θ > 0. With the above lemma in han we can now establish the coercivity of the linearize collision oerator. Lemma 2.2. For θ > 0 there exists a λ > 0 such that Lg), g L 2 λ g Πg) 2 Λ, g Λ. The coercivity roerty in R ) of the oerator L is inee an essential requirement to establish our main result. It woul not hol for examle in L 2 R ) or H 1 R ) an inuces the articular choice of Λ an its corresoning norm.

8 FOKKER-PLANCK EQUATION FOR BOSONS AND FERMIONS 7 Proof. We start with Lg), g L 2 = ρ g µ ) 2 µ C g 2 1 C g µ ρ ρ ρ µ g ) 2 ) µ g ) 2 = C g Πg) 2 L 2 for some C > 0, where we use the fact that the measure µ /ρ satisfies a Poincaré inequality ue to the revious lemma. Now to imrove on the L 2 norm we use Lg), g L 2 = g Πg)) + ) 2 2 η g Πg)) K 1 g Πg) 2 Λ + K 2 g Πg) 2 L 2. Aing the two inequalities above multilie by aroriate constants) finishes the roof. We get a similar result for the erivatives w.r.t. R. Lemma 2.3. Let θ > 0, then there exist ositive constants C 1 an C 2, such that for any g L 2 with g Λ Lg), g L 2 C 1 g 2 Λ + C 2 g 2 L. 2, Proof. A lengthy calculation yiels ) Lg), g L 2 = g) 2 + g 2 2 η 2 4 2κµ 2 g κµ + ) κµ κ g 2 2 µ + 10)η 2 2 ) µ η. 2 The last integral on the r.h.s. is ominate by the L 2 norm, since µ ecays exonentially fast as. We also have that 1/4 + 2κµ C > 0, R. This obviously hols true for the bosonic case but is also guarantee in the fermionic situation where f < 1/2. Thus we can estimate Lg), g L 2 C 1 g 2 Λ + 2 η g 2 + C 2 g 2 L 2 an a classical interolation argument alie to the secon term on the r.h.s. yiels the assertion of the lemma with ifferent constants C 1, C 2 ). Finally we nee the following technical lemma. Lemma 2.4. For g, h Λ it hols that Lh), g L 2 C g Λ h Λ, C > 0.

9 8 L. NEUMANN AND C. SPARBER Proof. We first note that ) 1/2 g Λ h Λ = g 2 L + η 2 g 2 L 2 C g 2 L ) η g 2 2 h 2 L + η 2 h 2 L 2 ) 1/2 ) 1/2 h 2 L ) η h ) 1/2 for some C > 0 since the Λ norm ominates the L 2 norm. Using the following simle algebraic estimate a 2 + b 2 )c ) ) 1/2 ac + b, with a, b, c, R) we further obtain 1/2 C g Λ h Λ g L 2 h L ) η 2 g ) η h ) 2 2. The roof then follows by alying the Cauchy Schwarz inequality to both terms on the right han sie an integrating by arts in the first one. 3. Convergence for the linear moel an roof of Theorem 1.2 Now we are able to establish the long time asymtotics for the linearize equation, which eventually will be translate also to the nonlinear moel 1.3) in the erturbative setting. Proosition 3.1. Consier the linearize Fokker Planck tye equation 3.1) t g σκf ) x g = Lg), with L given by 1.8) an θ > 0. Moreover if κ = σ = 1, assume in aition that θ > θ, for a certain θ > 0. Then, if the intial ata g 0 H k T R ), for k N, the solution gt) exists globally in time an gt) g H k Ce τt, with C, τ > 0, where the global equilibrium g is given by ) 1 µ g = g 0 µ x. ρ T R Proof. The results of Section 2 show that our linearize collision oerator L fits into the class of moels stuie in [19]. We will sketch the roof an stress the ifferences which occur ue to the changes in the transort oerator. Note that, since the equation is linear, we can w.r.o.g. consier the case where g 0. This can always be achieve by subtracting initially the rojection onto the global equilibrium, i.e. by consiering initial ata g 0 = g 0 g. We start with k = 1. The iea of the roof is to stuy the time evolution of a combination of erivatives w.r.t. x an. More recisely we consier the following functional F[gt)] := α g 2 + β x g 2 + γ g 2 + δ x g, g, where enotes the stanar norm on L 2 T x R ) an α, β, γ, δ are some ositive constants. We will not go into etails about the choice these coefficients but note that δ has to be small enough in comarison to β an γ such that F is ositive an controlle from above an below by the square of the usual H 1 T x R ) norm of

10 FOKKER-PLANCK EQUATION FOR BOSONS AND FERMIONS 9 g. On the other han δ has to be strictly ositive, since we nee it in orer to close the argument see blow). We aim to rove that ) 3.2) g t F[gt)] C 2 Λ + x, g 2 Λ. To this en we calculate the time erivatives of the various summans in F. First, for the L 2 norm we have 3.3) t g 2 = 2 Lg), g) λ g Πg) 2 Λ, where we have use that the transort art oes not contribute ue to its ivergence form) an the assertion of Lemma 2.2. Next the satial erivatives evolve accoring to t x g 2 = 2 x Lg), x g λ x g Π x g) 2 Λ, where we again use that the contribution from the transort term vanishes an the fact that L commutes with x thus allowing us to aly Lemma 2.2 also on x g. For the erivatives w.r.t. we get some aitional terms by the coefficient of the transort oerator t g 2 = 2 Lg), g + 4κσµ 2 2η 1 + 2κσf ) ) x g, g 2 Lg), g + K 1 x g 2 + K 2 g 2. To estimate the term invoking the mixe erivative we use that f as well as 2 µ are uniformly boune in L. Now the first term on the r.h.s. is estimate by Lemma 2.3, which yiels some aming for g in Λ. However this comes at the rice of an aitional term g, with a ositive sign. We slit g = g Πg)) + Πg) an estimate g 2 L 2 g Πg) 2 L 2 + Πg) 2 L 2 g Πg) 2 L 2 + C T x g 2 L 2, C T > 0. For the secon inequality we use the classical Poincare inequality w.r.t x T an the fact that Πg) has zero mean on the torus, since g = 0. Note that this is an imrovement in the sense that the term g Πg) 2 can be cancele by ajusting α, an having in min 3.3). Finally we look at the mixe erivatives w.r.t x an, which evolve acoring to t xg, g = 2 L x g), g x g, 1 + 2σκf 2σκµ 2 ) ) x g. For the first term on the r.h.s. we invoke Lemma 2.4, which together with the Cauchy-Schwarz inequality in x imlies L x g), g C x g Π x g) 2 Λ + C g 2 Λ. The secon term on the r.h.s., which stems from the transort art, generates a aming for x g which the oerator L can not rovie as it only acts in ), rovie that 3.4) 1 + 2κσf 2κσµ 2 K > 0. From here we can extract a aming effect in the Λ norm, ue to Poincaré s inequality an the fact that Π x g) is well behave in x g 2 K x g 2 Λ + C g Πg) 2 Λ + C x g Π x g) 2 Λ.

11 10 L. NEUMANN AND C. SPARBER Assuming for the moment that 3.4) is true it remains to a all the above terms an to fin coefficients α, β, γ, δ in F, such such that all ba terms in the above given estimates i.e. those which come with the wrong or without sign) can be controlle an the ifferential inequality 3.2) hols true. This can be one similarly to [19], although it is a somewhat lengthy roceure an we will not elaborate further on it. The functional F clearly inuces a new norm on hase sace, equivalent to H 1 R T ), via 2 H1 := F[ ] an similarly we can efine higher orer norms H k, k N, cf. [19]. To retain the funamental) aming roerty in the satial erivatives coming from the evolution of the mixe term it remains to show that the constraint 3.4) hols true. We enote Ψ κ ; θ) := 1 + 2κf 2κµ 2. If κ = 1 an since θ > 0, it is clear that Ψ 1 ; θ) K > 0 because η 0 in this case. For Bosons, i.e. κ = 1, however the situation is more ifficult. Note that lim Ψ 1 ; θ) = lim η 2µ 2 ) = 1, θ > 0, an thus by continuity it is enough to make sure that Ψ 1 ; θ) 0, R, θ > θ. Straightforwar calculations show that Ψ 1 ; θ) can only be zero if e 2 /2+θ ) e 2 /2+θ 1 = 0, which inee imlies e 2 /2+θ = Obviously this equality can not be true for θ larger than some critical value θ. Numerical exeriments suggest that this critical value is aroximately θ In summary one obtains the final estimate 3.2), which finishes the roof for k = 1. To rocee to higher orer estimates in the Sobolev inex k N, we observe that the roof of Lemma 2.3) can be generalize in a straightforwar way to obtain xl j Lg), xl j g L 2 C 1 xl j g 2 Λ + C 2 g 2 H k 1 for any multi-inices j, l such that k = j + l, j 1. An inuction argument in k N, analogously to the one given in [19] then yiels the corresoning statement in H k. We o not run into aitional roblems ue to the coefficient of the transort oerator, since the terms containing the highest orer erivatives of g can be treate as before an the lower orer terms which contain erivatives of Ψ κ ; θ)) can be hanle by interolation. This finishes the roof. Now we aly the result for the linearize equation the nonlinear roblem. Proof of Theorem 1.2. We have to show that the quaratic nonlinearity oes not change the estimates obtaine for the linearize equation, as long as the eviation from the equilibrium is small. The function g = f f )µ 1/2 solves 1.7) from which we euce t g 2 H k = 2 T g, g H k + 2 Qg), g H k,,

12 FOKKER-PLANCK EQUATION FOR BOSONS AND FERMIONS 11 where T := L 1 + 2σκf ) x an L, Q are given in 1.8), 1.9), resectively. From the roof of Proosition 3.1 we know that T g, g H k C xl j g 2. Λ j + l k Thus, if we can rove the following roerty for the nonlinear art Qg), g H k C Q g 2 3.5) H k xl j g Λ, j + l k it follows, since H k H k, that t g 2 H C k xl j g 2 + C Λ Q g 2 H k j + l k j + l k xl j g Λ, which conclues the roof of Theorem 1.2 by a Gronwall tye argument. In orer to rove 3.5), we recall that Qg) is given by Qg) = κ iv µ g 2 ) ) κ σµ x g 2 ) ) Q 1 g) Q 2 g), µ µ an note that µ 1/2 xl j µ ) L T x R ), for all multi-inices l, j N. We shall now treat Q 1 g) an Q 2 g) searately, using that H k R ) L R ), for k > /2, together with Leibniz formula to ifferentiate Qg). It is then relatively easy to see that 3.5) hols for Q 1 g) by Sobolev imbeing, as soon as N k > /2, since the Λ norm incororates an aitional erivate w.r.t R. The estimate for Q 2 g), g H k is more comlicate though, since Q 2 contains a erivate w.r.t. x T which is not taken into account for by the Λ norm. Thus we have to estimate terms of the form xl j µ x g 2 ), xl j g L 2, j + l k. Since µ an all of its erivatives are well behave the highest orer terms are µ x xl j g 2 ), xl j g L 2, j + l = k, an moreover, because of the aitional erivative w.r.t. in the Λ norm, the most roblematic terms are those where l = k. Denoting l xl we comute µ x l g 2 ), l g = 2 µ L 2 g x l g, l g L 2 + µ i i=1 0 r l+δ i 0< r < l +1 ) l + δi r r g l+δi r g, l g, L 2 where δ i enotes the i -th stanar basis vector in R. Using ivergence theorem, we obtain µ x l g 2 ), l g L 2 = µ x g) l g, l g L 2 + µ i i=1 0 r l+δ i 1 r < l ) l + δi r r g l+δi r g, l g. L 2 We only estimate the first term on the r.h.s. sie since those aearing in the summation can be treate similarly. This yiels remember l = k) µ x l g 2 ), l g c L 2 1 l g 2 L 2 x g L c 2 g 2 H g k H k,

13 12 L. NEUMANN AND C. SPARBER as soon as N k > 1 + /2 an since g H k the esire estimate 3.5). j + l k x l j g we obtain Λ Remark 3.2. The above given roof shows that we can not rocee comletely analogous to [19] since the nonlinearity stemming from Q 2 g) has to be carefully integrate by arts. Acknowlegement. L. N. thanks M. Escobeo for fruitful iscussions on similar quantum kinetic moels. C. S. is greatful for the kin hositality of the Johann Raon Institute for Alie Mathematics RICAM). References 1. A. Arnol, P. Markowich, G. Toscani, an A. Unterreiter, On convex Sobolev inequalities an the rate of convergence to equilibrium for Fokker-Planck tye equations. Comm. Part. Diff. Equ. 26/ ), W. Bao, L. Pareschi, an P. Markowich, Quantum kinetic theory: moeling an numerics for Bose-Einstein conensation. In: Moeling an Comutational Methos for Kinetic Equations, , Birkhäuser, J. A. Carrillo, J. Rosao an F. Salvarani, 1D Nonlinear Fokker-Planck equations for Fermions an Bosons, to aear in Alie Mathematics Letters. 4. L. Desvillettes an C. Villani, On the tren to global equilibrium in satially inhomogeneous entroy-issiating systems: the linear Fokker-Planck equation. Comm. Pure Al. Math. 541) 2001), J. Dolbeault, Kinetic moels an quantum effects: a moifie Boltzmann equation for Fermi-Dirac articles. Arch. Rational Mech. Anal ), no. 2, M. Escobeo, S. Mischler, an M. A. Valle, Homogeneous Boltzmann equation in quantum relativistic kinetic theory. Electronic Journal of Differential Equations. Monograh, M. Escobeo, S. Mischler, an M. A. Valle, Entroy maximisation roblem for quantum an relativistic articles. Bull. Soc. Math. France ), no. 1, M. Escobeo, S. Mischler, an J. J. L. Velazquez, Asymtotic escrition of Dirac mass formation in kinetic equations for quantum articles. J. Diff. Equ ), no. 2, T. D. Frank, Classical Langevin equations for the free electron gas an blackboy raiation. J. Phys. A ), no. 11, T. D. Frank, Nonlinear Fokker-Planck equations. Sringer Series in Synergetics, Sringer, F. Herau an F. Nier, Isotroic hyoelliticity an tren to the equilibrium for the Fokker- Planck equation with high egree otential. Arch. Rat. Mech. Anal ), no. 2, G. Kaniaakis, Generalize Boltzmann equation escribing the ynamics of bosons an fermions. Phys. Lett. A ), G. Kaniaakis, H-theorem an generalize entroies within the framework of nonlinear kinetics. Phys. Lett. A ), no. 5-6, G. Kaniaakis an P. Quarati, Kinetic equation for classical articles obeying an exclusion rincile. Phys. Rev. E ), no. 6, G. Laenta, G. Kaniaakis, an P. Quarati, Stochastic evolution of systems of articles obeying an exclusion rincile. Phys. A ), no. 3-4, X. Lu, A moifie Boltzmann equation for Bose-Einstein articles: isotroic solutions an long-time behavior. J. Stat. Phys ), no. 5-6, X. Lu, On satially homogeneous solutions of a moifie Boltzmann equation for Fermi- Dirac articles. J. Stat. Phys ), no. 1-2, X. Lu an B. Wennberg, On stability an strong convergence for the satially homogeneous Boltzmann equation for Fermi-Dirac articles. Arch. Ration. Mech. Anal ), no. 1, C. Mouhot an L. Neumann, Quantitative erturbative stuy of convergence to equlibrium for collisional kinetic moels in the torus. Nonlinearity to aear. 20. L. Neumann an C. Schmeiser, Convergence to global equilibrium for a kinetic moel for fermions. SIAM J. Math. Anal ),

14 FOKKER-PLANCK EQUATION FOR BOSONS AND FERMIONS A. Rossani an G. Kaniaakis, A generalize quasi-classical Boltzmann equation. Physica A ), J. Soik, C. Sire, an P. H. Chavanis, Self-gravitating Brownian systems an bacterial oulations with two or more tyes of articles. Phys. Rev. E ), J. Soik, C. Sire, an P. H. Chavanis, Dynamics of the Bose-Einstein conensation: analogy with the collase ynamics of a classical self-gravitating Brownian gas. Prerint, arxiv: con-mat/ E. A. Uehling an G. E. Uhlenbeck, Transort henomena in Einstein-Bose an Fermi- Dirac gases. Phys. Rev ), C. Villani, A review of mathematical toics in collisional kinetic theory. In: Hanbook of Mathematical Flui Dynamics, S. Frielaner an D. Serre es.), Elsevier Science, C. Villani Hyocoercivity. Prerint, arxiv: htt://lanl.arxiv.org/abs/math.ap/ Johann Raon Institute for Comutational an Alie Mathematics, Altenbergerstraße 69, A-4040 Linz, Austria aress: lukas.neumann@oeaw.ac.at Wolfgang Pauli Institute Vienna & Faculty of Mathematics, Vienna University, Norbergstraße 15, A-1090 Vienna, Austria aress: christof.sarber@univie.ac.at