Derivation of Kinetic Equations

Transcription

1 CHAPTER 2 Derivation of Kinetic Equations As we said, the mathematical object that we consider in Kinetic Theory is the distribution function 0 apple f(t, x, v). We will now be a bit more precise about the link between microscopic and mesoscopic descriptions of asystem. Consideragas(ormoregenerallyasystem)ofN particles (or more generally bodies) indistinguable, and in particular of the same mass m =1. Fornowforsimplificationwewillconsiderthemtobepointparticles (i.e. zero volume), an assumption that we will come back to later. 1. Newtonian Viewpoint We assume some binary interaction through a potential, depending only on the distance between two interacting bodies and in addition an external force with some potential (t, position). If we label each particle with an index 1 apple i apple N, andcallthei-th particle s position x i and velocity v i, then Hamiltonian s formulation of Newtonian mechanics gives us (recall that the masses of particles are normalized to one) that (1.1) where H = dx i dt = H v i NX vi 2 2 i=1 {z } kinetic energy + X i<j dv i dt = (x i x j ) + {z } interaction energy This corresponds to the following ODE s H x i NX i=1 (t, x i ). {z } potential energy 8 1 apple i apple N, ẋ i = v i, v i = X i6=j r x (x i x j ) r x (x i ). For now, we are assuming that and are regular i.e. su ciently smooth to ignore any reasonable di erentiability/convergence issue. As a general convention in the sequel: 15

2 16 2. DERIVATION OF KINETIC EQUATIONS Note that when we write a vector on the bottom of the derivative, it really means the directional derivative in that direction. Lowercase letters refer to one i-th particle whereas uppercase letters refer to the vector of all particles X =(x 1,...,x N )andv = (v 1,...,v d ). If each x i 2 R 3,thisisactuallyavectorofvector (matrix). Discuss the signs of the di erent part of the energy functional. 2. The notion of dynamical system (S) Here introduce the hamiltonian formulation, and some key elementary properties. Exercise 2. Define for any t 2 R S(t) :(R d R d ) N! (R d R d ) N, (X, V ) 7! S t (X, V )=(X t,v t ) given by the evolution (1.1) between time 0 and t. Prove that (1) H is constant along S, and(2)s(t) has Jacobian 1 for a fixed t (this is one form of the Liouville theorem). 3. Statistical Viewpoint 3.1. The N-body Liouville equation. Now, instead of considering the above microscopic viewpoint of the trajectories, we will shift to a more statistical viewpoint. As such, we consider a joint distribution function for all of the particles F N (t, X, V ). We claim that F N evolves as such: (3.1) F N t + NX i=1 H v i F N x i H F N =0 x i v i We shall call this equation the N-particle Liouville equation. Letusexplain heuristically how we derived (i.e. guessed ) from the ODE s defining the trajectories. Since we want an equation on the distribution function of particles, it is clear that this distribution should be preserved when following the trajectories of each particle: 8 t 0, F(t, S t (X, V )) = F (0,X,V). In other words, we consider the distribution of particles following their trajectories. FIGURE TO BE ADDED HERE Let us now di erentiate in time the equation with the help of the chainrule (assuming that every terms are smooth):

3 t F (t, X t,v t )+ 4. THE CHARACTERISTICS METHOD (S) 17 t X t X F (t, X t,v t ) + t V t which means, using the equations on X t and V t : t F (t, X t,v t )+ V H X F (t, X t,v t ) X H which is the desired equation at the point (t, X t,v t ). V F V F 4. The characteristics method (S) (t, X t,v t )=0 (t, X t,v t )=0 Here, we have turned a nonlinear ODE into a linear PDE, which is a common trade-of in one way or another! We have traded nonlinearity for a linear problem in a much bigger space, i.e. the space of probability distributions on (R d R d ) N which is infinite dimensional. If we can solve (3.1) in general, then we can recover a solution to (1.1) by starting with initial data (X 0,V 0 ), which will have the solution (Xt,V t) (see the next exercise). Observe therefore that the Liouville equation contains as much information as the set of coupled ODE s of the trajectories. It can be seen understood as the evolution of the distribution of particles following all possible trajectories, according to their initial distribution in the phase space. Exercise 3. Show that F N (t, X, V )=F N (0,S t (X, V )). Deduce formally that the solution starting with initial data (X0,V 0 ) is given by (Xt,V t) (although we assumed regularity so far, it happens that the previous formula is still valid for non-regular solutions). This formula is natural and also justifies equation (3.1). This is a first instance of the so-called characteristics method. Exercise 4. Prove (as before, using any regularity desired) that if at initial time F N (0,, )isaprobabilitydensity,thenforallt 0wehaveF N (t,, ) 0and F N (t, X, V )dx dv = F N (0,X,V)dX dv. (R d R d ) N (R d R d ) N In other words, show that F N remains a probability distribution as well as preserving the total mass of the system.

4 18 2. DERIVATION OF KINETIC EQUATIONS Explain how the Liouville theorem translates in this statistical viewpoint. 5. The many-particle limit 5.1. The One Particle Distribution. Observe that the N-particle Liouville equation, although it allows for considering superpositions of all trajectories at the same time, still contains exactly the same amount of information as the original Newton equations. And in most situations this is much more that we need, want and can experimentally measure (do not forget we have to enter initial data in our equation for it to be useful!). We will now simplify our description of the system by throwing away information. (Hopefully) the system is described by a one particle distribution, i.e. a typical distribution of one particle, for which we can obtain adistributionfromf N by taking the first marginal: f(t, x, v) := F N (t, X, V )dx 2 dx 3,... dx N dv 2... dv N. (R d R d ) N 1 Why the marginal according to the first variable without loss of generality? We consider F N to be symmetric under permutations of the particle s indexes due to the indistinguability of the particles, so choosing i =1is not important, as we would obtain the same answer for any other choice of index. How can we obtain an equation for how f evolves? The natural thing to do is of course to integrate (3.1) in the x 2,v 2,...,x N,v N variables. For notational simplicity we write X (1) =(x 2,x 3,...,x N ) 2 R d(n 1) and similarly for V (1). The first term in (3.1) is F N (R d R d ) t N 1 dx (1) dv (1) = t apple (R d R d ) N 1 F N dx (1) dv (1) = f t. The integral of the second term in (3.1) is the sum over i of H F N dx (1) dv (1) (R d R d ) v N 1 i x i = = (R d ) N 1 v i (R d R d ) N 1 v i F N x i dx (1) dv (1) F N dx (1) (R d ) x N 1 i 8 >< f v 1 i =1 dv (1) = x 1 >: 0 i 2.

5 (R d R d ) N 1 j6=i 5. THE MANY-PARTICLE LIMIT 19 Here, if i = 1 we can pull the x 1 derivative out of the integral, and it follows as above. If i 2, then the x i integral vanishes by Green s Theorem because it is the integral of the derivative of f (which we are again assuming su ciently regular). Finally, the integral of the third term is the (negative of) the sum over i of H F N dx (1) dv (1) (R d R d ) x N 1 i v i " # X = ( (x i x j )) F N + ( (t, x i )) F N dx (1) dv (1). x i v i x i v i Calling in the RHS the first term A and the second B, we have that when i =1 NX A = ( (x 1 x j ) F N dx (1) dv (1) j=2 (R d R d ) x N 1 1 v 1 =(N 1) (x 1 x 2 ) F N dx (1) dv (1) x 1 v 1 where in the last equality we used symmetry of F N under permutation of indices. Now, defining the second marginal of F N f (2) := F N dx (2) dv (2) (R d R d ) N 2 where X (k) omits the first k coordinates, and similarly with V (k),wemay further simplify our calculation of A with i =1to A =(N 1) ( (x 1 x 2 )) f(2) dx 2 dv 2. x 1 v 1 R d R d When i 6= 1,theusualargumentshowsthatA =0,sowehavethat 8 >< (N 1) ( (x 1 x 2 )) f(2) dx 2 dv 2 i =1 A = R >: d R x d 1 v 1 0 i 2. Now, we simplify the expression for B. Fori =1 B = (t, x 1 ) F N dx (1) dv (1) (R d R d ) x N 1 1 v 1 F N = (t, x 1 ) dx (1) dv (1) x 1 (R d R d ) v N 1 1

6 20 2. DERIVATION OF KINETIC EQUATIONS Thus = (t, x 1 ) x 1 v 1 F N dx (1) dv (1) (R d R d ) N 1 8 >< f i =1 B = x 1 v 1 >: 0 i 2. = x 1 f v 1. Thus, combining these into our integrated (3.1) gives the equation for the one marginal distribution (5.1) f t + v f x f x v (N 1) x ( (x y))f(2) (x, y, v, w)dydw =0 v (where we have made the substitutions x 1! x, x 2! y, v 1! v, v 2! w). How can we interpret this equation? Binary collisions mean that the evolution of the first marginal (f) dependsonthesecondone. Similarly we could show that f (2) s evolution depends on f (3) and so on. Doing so, we could write down the BBGKY hierarchy (Bogoliubov, Born, Green, Kirkwood, Yvon hierarchy, sometimes called Bogoliubov hierarchy) for f,f (2),...,f (N) = F N The Many-particle or Thermodynamic Limit. The goal of the thermodynamical limit is to take N!1and recover an equation for only the first marginal. The rough idea is that we would very much like to write f (2) = f f := f(x, v)f(y, w) in order to get a closed equation on the one particle marginal. This is the most natural guess if the particles are not too much coupled. However there are two complications: This independence assumption is easily seen to be incorrect, precisely due to the interactions in the system which create correlations between particles. What Boltzmann understood (and Kac [37] formulated mathematically) was that (maybe) as N! 1 this could be true in some sense: f (2) f f as N! +1. This is the idea of molecular chaos of Boltzmann. There should some kind of scaling in the interaction (here represented by the interaction potential ), if not the factor (N 1) would blow-up in our equation (5.1).

7 6. MEAN FIELD MODELS 21 In general one has to use more complicated models in which the particles have non-zero radius r and some assumptions of the form (1) N 1(orevenN!1). (2) There is a fixed volume V, sobecausen!1we must also take r = r(n)! 0. (3) Some version of molecular chaos. (4) Some way of scaling the interaction, i.e. = N. Given di erent choices in how to make these assumptions we arrive at different models. This is how the kinetic equations are derived and obtained from physics, at least formally. 6. Mean Field Models 6.1. The mean-field limit. In this approach, we do not try to describe each binary interaction, but only their collective e ect. It is a good approach when the interaction potential is not too sensitive to the precise position of each particle. Often this assumption is called long range interaction. The extreme opposite case is when there are only contact interactions by collisions, like for the case of hard balls / spheres that we shall consider in the next section. Mathematically, we let N (z) = 1 (z). We take r = r(n)! 0such N that Nr3 1. In other words, this is an assumption of dilute gas. Notice V that the force between two particles is O(1/N ), but there are N 1other particles, so a particular particle feels a force of O( N 1)=O(1). N This model is well adapted to electromagnetic and gravitational forces. What we hope for is that under these assumptions, following our previous calculations we could get the following Vlasov equation for f: (6.1) f t + v f x f x v x ( (x y))f (x, v)f(y, w)dydw =0 v This equation is obtained from (5.1) by plugging N = /N, usingthe decorelation assumption f (2) f f, andtakingthelimitn! The main mean-field kinetic PDEs. In the case where there is no external force = 0(thisplaysnoessentialroleinthediscussion here) and we start microscopically from the Coulomb interaction between electrons 1 (x y) = x y

8 22 2. DERIVATION OF KINETIC EQUATIONS ions we obtain the Vlasov-Poisson for plasmas (assuming here that the electric charge of an electron is normalized to 1andnormalizedallphysical constants): f t + v r xf + F m r vf =0 where F = E with E the electric field from the other particles, and E = r where (themean-fieldpotential)satisfiestheso-calledpoisson equation apple = f dv {z } charge density where ions is a fixed density of ions in the plasma. We assume there is no overall charge, which gives us that ions dx = f dx dv. This is the main equation in plasma physics. This force field F is therefore self induced and is responsible for a quadratic nonlinear term in the equation. This nonlinearity results in oscillations in the plasma. In the case of Newton gravitation forces between stars microscopically (assuming that all stars have mass normalized to 1 and normalizing all physical constants) 1 (x y) = x y we obtain the gravitational Vlasov-Poisson equation, inwhichf describes the distribution of starts in the galaxy: F = E, whereagaine = r is the gravitational field from the other stars, and (the mean-field potential) satisfies the Poisson equation = where = f dv is the local matter density. (6.2) Summing up, what we call the Vlasov-Poisson equations are: f t + v r xf + F r v f =0 Plasma: F = r x x = ions Galaxies: F = r x x = f dv. f dv Note that the plasma case is repulsive and the galaxy case is attractive.

9 6. MEAN FIELD MODELS Structure of the Vlasov-Poisson equations. Let us discuss more in details the formal structure of the equation. The first crucial remark is that this is a reversible PDE, which is in contrast to the Boltzmann equation coming next. To see this, one checks that if f = f(t, x, v) isa solution, then g(t, x, v) :=f( t, x, v) is also a solution, but reversing time and velocity (just like for the particle systems we started from). Hence the Vlasov-Poisson equations have inherited the reversibility feature of the Newton equations in the mean-field limit. Observe also that this equation is now nonlinear due to the mean-field term, but it shares a similar structure with the previous Liouville equation we started from f t + Hf v f x H f x f =0 v with now the following microscopic mean-field Hamiltonian function H f (x, v) = v f(t, x). Finally we show in the next exercise that there is a mesoscopic Hamiltonian preserved along the evolution of the PDE, corresponding to the average of the microscopic Hamiltonian against the particle distribution. Exercise 5. Show that the following equality is conserved with time (as usual, show this formally without worrying about convergence/smoothness issues): H(f) := R d R d f v f dx dv = Rd Rd f v 2 2 dx dv ± Rd F 2 2 dx where the + corresponds to plasmas and the to galaxies. Remark that this is one hint that the plasma case might be easier, because its energy term is written as the sum of two positive quantities Extensions. There are two main features that one may want to incorporate into these equations: (1) First in the of plasmas an important aspect is the magnetic field for so-called magnetic plasmas. The correct model is then the the Vlasov- Maxwell equations (including the full Lorentz force with a magnetic field) which are f t + v r xf e m (E + v B) r vf =0

10 24 2. DERIVATION OF KINETIC EQUATIONS where E,B satisfy Maxwell s equations r E =, r B =0, r E = t B, rb = J + t E with J =cst vf dv. Onecouldevenincludespecialrelativitye ectwhich yields mathematically a compactification of the variable v. Almost no existence/uniqueness results are known for these equations, the best result so far is the conditionnal result of Glassey and Strauss [26]. (2) Second in the case of gravitational interaction, one may want to incorporate the general relativity and the curved structure of the spacetime. This would result in the so-called Vlasov-Einstein, where the Vlasov equations for the transport of matter are coupled with the Einstein equation for the metric of the space-time structure. There are very few for this equation as well, we refer to the book [52] byanexpertinthisfield. 7. Boltzmann-Grad Limit and Collisional Models In contrast to the mean field models, here we consider short range interactions. Typically the model is elastic collisions by hard spheres. Notice that this is extremely sensitive to the position of the particles, i.e. up to the scale of their size. For example, if two particles of diameter d are a distance d apart, and if they are on a collision course, then moving one of them a distance of 2d will make it so that they do not collide. From the scaling (as N!1), requiring the mean free path `(N) = v1/3 N the radius of the particles, and r(n) Nr(N) 2 = O(1) as N!1, along with some version of molecular chaos f (2) f f onecanobtain the Boltzmann equation (with no external force) 1 (7.1) f t + v r xf = Q(f,f) where Q(f,f) is a bilinear integral operator acting on v only (so it is local in t and x), representing interactions between particles. Q is defined as (7.2) Q(f,f) := v 2R 3 [f(v )f(v 0 0 )!2S {z } 2 appearing f(v)f(v ) ] B(v v {z },w) d! dv {z } dissapearing collision kernel, ( 0) 1 Note that this is not reversible, in contrast to the Vlasov-Poisson equation!

11 where 7. BOLTZMANN-GRAD LIMIT AND COLLISIONAL MODELS 25 v 0 := v hv v,!i! v 0 := v + hv v,!i! Here, the terms marked appearing are there because they represent two particles colliding and then having velocities v 0 and v 0,andsimilarlywith dissapearing. We assume that B is even in the first coordinate. Let us only sketch the idea of the formal derivation, which turns out to be much more complicated than the mean-field limit. We start from the N particle Liouville equation t F N + V r X F N =0 but which is now posed on the domain N := {8i 6= j, x i x j 2r(N)}. We then consider again the one-particle distribution f(t, x, v) := F N (t, X, V )dx 2 dx 3,... dx N dv 2... dv N (R d R d ) N 1 and we search for an evolution equation on it: t f + v 1 r x1 f = NX j=2 h = (N 1) + X (1),V (1) 2 (1) N v j r xj F N f (2) (x 1,x 2,v 1,v 2 ) (v 1 v 2 ) n 12 d 12 dv 2 f (2) (x 1,x 2,v 1,v 2 ) (v 1 v 2 ) n 12 d 12 i +cancellingornegligeableterms by Green s theorem, where n 12 is the outer normal to the sphere x 1 x 2 = 2r(N), d 12 is the surface element on the same sphere, and f (2) is as before the two-particle distribution, and where denotes the surface term for + outgoing collisions (v 1 v 2 ) n 12 0and denotes the surface term for ingoing collisions (v 1 v 2 ) n 12 apple 0. We have here neglected multiple collisions (more than binary) which have zero measure in the limit, considered a domain without boundary use the cancellation of the surface terms not involving x 1 (thanks to the reversibility of the collisions).

12 26 2. DERIVATION OF KINETIC EQUATIONS Then in the surface integrals and one has to express outgoing + velocities in + in terms of the ingoing velocities in. This is where atimearrowisintroduced. Thischoiceseemsinnocentandarbitraryat the microscopic level but cannot be reversed after the limit N! +1 has been taken 2. Under the previous assumptions (scaling, molecular chaos) we then formally obtain the Boltzmann equation with the collision kernel B(v v,!)= (v v )!. Exercise 6. Show that (7.3) (7.4) v 0 + v 0 = v + v v v 0 2 = v 2 + v 2 i.e. we can think of this as two particles colliding with initial velocities (v, v )andthenleavingwithvelocities(v 0,v ). 0 The above shows us that we have energy and momentum conservation (which we expected, because this is an elastic collision). Exercise 7. (1) For a fixed! 2 S 2 show that the map (v, v ) 7! (v 0,v 0 ) has Jacobian 1. (2) Deduce formally that for a test function (v) Q(f,f) (v)dv = 1 [f 0 f ff ]B(v v,!)( + 4 )d! dv dv v2r 3 v 2R 3!2S 2 where the 0 and after and f signify evaluating the function at v 0, v respectively, i.e. f 0 = f(v ). 0 (3) Given (2), what can you deduce from the following choices of : =1 = v i i =1, 2, 3 = v 2 =logf Note that everything we have done above does not depend on the dimension, so we will refer instead to a general dimension d instead of d =3from now on. For the integration by parts argument, we used some decay of at infinity, in order to integrate by parts, so the assignments (v) =1,v i,etc 2 The other choice at the microscopic level (expressing pre-collisional velocities in terms of post-collisional ones) would lead to a backward Boltzmann equation, with a minus in front of the collision operator.

13 7. BOLTZMANN-GRAD LIMIT AND COLLISIONAL MODELS 27 might not seem justified, but by multiplying by a cuto function and then letting it tend to 1, it is not hard to make these arguments more rigorous. Thus, we have obtained a priori estimates, which we can write compactly as 0 Q(f,f) 1 1 v A dv =0 R d v 2 From the Boltzmann equation (7.1), it is not hard to deduce a priori that 0 d f 1 1 v A dv dx =0 dt R d R d v 2 and d dt H(f) = d f log f dv dx = D(f) apple 0. dt R d R d Also, as we remarked above, notice that if we have equality in the entropy a priori estimate, i.e. then we have that Q(f,f)logf =0 (7.5) ff = f 0 f 0 holds. One of the reasons Boltzmann was convinced that his equation was correct is that this condition implies that f must be a gaussian. Exercise 8. If (7.5) holds for f 2 Cc 1 (R t R d x R d t ; R), show that f is a gaussian. (More hints in the example sheet). As remarked above, there has been few progress on the Cauchy problem for the full Boltzmann equation, (7.1). Thus, simplified models are sometimes studied. For example, the BGK collision model replaces Q(f,f) with M f f where M f = (2 T) d/2 e v u 2 /2T is the gaussian with the same parameters v, and T as f (as defined in (1.2), (1.1) and (1.3)). Exercise 9. Check that the BGK model satisfies the same formal properties as the original Boltzmann equation (conservation laws, H-theorem).

14 28 2. DERIVATION OF KINETIC EQUATIONS 8. A priori estimates and nonlinear PDEs (S) Show the naive Gronwall estimate at work on the two previous nonlinear equations, the associated fixed point iteration method (cf. Picard-Lindëlof) and its limit for long-time results. Then explain what is an a priori estimates and why it is crucial in order to go beyond the short-time results obtained by the naive approach. Then explain the contradiction between controlling the nonlinear term and the controls provided by the known a priori estimates given by physics. Illustrate with VP, BE, but also NS. 9. Bibliographical and historical notes Alargeamountoftheinitialdraftversionofthischapterisindebtedto the nice lecture notes [54] oflauresaint-raymond. The reformulation of the mechanics of Newton in terms of the (now called) Hamiltonian canonical forms was discovered by the irish mathematician Hamilton in The so-called Liouville equation for the evolution of the probability density in phase space associated to a dynamical system goes back to a paper of the french mathematician Liouville of 1838, as well as the so-called Liouville theorem. The BBGKY hierarchy obtained from the many-particle Liouville equation was discovered in independent works published in 1946 and 1947 by Bogoliubov in USSR, Kirkwood in USA, Born in Germany and his former student scottish student Green. Concerning the Vlasov-Poisson, there have been important progresses on the Cauchy problem. As suggested in the introduction, in a small space of solutions (with high regularity and decay at infinity) local existence and uniqueness is not too hard and have been proven [2] andglobalexistence and uniqueness for small initial data [6], but global existence in general has been unknown for a while, while in a very large space (like L 1 functions satisfying bounds on the macroscopic Hamiltonian H and the entropy) it is possible to get global existence [3], but uniqueness is then unknown. Pfa elmoser [51] andthenlionsandperthame[45] showedtheexistence and uniqueness of global solutions in the all space x 2 R d. Their settings are di erent: Pfa elmoser assumes that the initial data is C k with compact support (thus building classical smooth solutions), and Lions and Perthame consider initial data which are L 1 with some velocity moments and some initial mild regularity estimate on the initial density

15 10. EXERCISES 29 in = f in dx. However it is hard to make any interesting qualitative observations about these solutions. These proofs have later been digested, optimized, extended and improved: see [55], [7], [34], [49]. Concerning the mean-field limit and the derivation of the Vlasov-Poisson, only partial progresses have been and the problem remains open for the Coulomb and Newton interactions: we refer to [9] and[22] inthecaseof smooth interactions, and to [31] forapartialresultinthecaseofsingular interactions. Concerning the Boltzmann equation the situation is much less advanced. There has been only partial progress on the Cauchy problem. For high regularity (i.e. a small space) perturbative solutions (close to equilibrium or close to vacuum) have be built, see for instance [56] and[36]. For low regularity, DiPerna and Lions [21] showedin1989thatthereisglobal existence of renormalized solutions which is a type of (very) weak solution. However, nothing is known in between these two. With further invariances we have better theories: in the case of spatially homogeneous solutions or one-dimensional (in space) solutions. Concerning the Boltzmann-Grad limit this remains an important open problem. The best (and by far most important) result so far is due to Lanford [41] andprovesthelimitforagasofhardspheresbutonlyfor averyshorttime(lessthanthemean-freetimebetweencollisionsthata particle encounters). This theorem has been extended to a global in time limit theorem in the case of solutions close to vacuum in [35]. Among existing books, two nice references are [25], more oriented towards Vlasov equations, and [16], more oriented towards Boltzmann collisional equations. In particular the first chapters the latter book explained in great details the Boltzmann-Grad limit and Lanford s Theorem. 10. Exercises

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