Derivation of Mean Field Equations for Classical Systems


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1 Derivation of Mean Field Equations for Classical Systems Master Thesis iklas Boers
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3 Derivation of Mean Field Equations for Classical Systems Master Thesis iklas Boers Fakultät für Mathematik, Informatik und Statistik LudwigMaximiliansUniversität München München, January 2011 Supervisor: Prof. Dr. Detlef Dürr Second Supervisor: Prof. Dr. Peter Pickl
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5 Acknowledgements This thesis would have never been possible without the great support of Detlef Dürr and Peter Pickl. Their ingenuity as well as their patience are deeply appreciated. 3
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7 COTETS COTETS Contents 1 Introduction 7 2 The Counting Measure Construction of the Counting Measure Derivation of the Vlasov equation 27 4 Conclusion 37 5
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9 1 ITRODUCTIO 1 Introduction Consider a ewtonian system of identical particles interacting via a spherically symmetric pair potential φ C 2 b (Rd in d dimensions, which might be either attractive or repulsive. Here, C 2 b (Rd shall denote the set of functions on R d with bounded and continuous derivatives up to order 2. The Hamiltonian of such a system has the form H (X = i=1 p 2 i 2m + i<j φ(q i q j (1 with X = (x 1,..., x R 6, x i = (q i, p i R 6, leading to the equations of motion q i = H p i = ṗ i = H q i p i m = qi φ(q i q j. (2 ote that, since φ is symmetric and thus φ(0 = 0, we can omit the restriction i j in the sum. Substituting φ = 1 φ for the potential and setting m = 1, one obtains j=1 dq i dt dp i dt = p i = d2 q i dt 2 = 1 qi φ(q i q j. (3 j=1 These are the equations of motion for classical particles interacting 1 through the potential φ. It is this dynamics from which the Vlasov equation can be derived. ote that the coupling becomes weaker with growing. In a more general setting the Hamiltonian looks like where H β (X = i=1 p 2 i 2m + φ β (q i q j, (4 i<j φ β (q = 1+β φ( β/d q (5 7
10 1 ITRODUCTIO in dimension d for spherically symmetric φ Cb 2(Rd. Hence the support of φ β shrinks with for positive β. The case described above obviously corresponds to β = 0 and m = 1. Usually the potential energy is expected to scale in the same way in as the kinetic energy. This motivates the factor of 1 in the expression for φ β. In general, the ewtonian equations of motion for interacting particles are practically impossible to solve. One way to circumvent this problem and obtain at least some kind of solution is to change the level of description. Instead of the microscopic body problem, one considers an equation for a continuous mass density which effectively still describes the same situation but from a macroscopic point of view. It is intuitively clear that this coarse grained description gets more appropriate as the number of particles increases, becoming exact in the limit. The crucial observation in the case where the potential scales with 1/, i.e. the β = 0 case, is that the force q i = 1 qi φ(q i q j, i = 1,..., (6 j=1 looks like the empirical mean of the continuous function φ(q i of the random variables q j. In the limit, one might expect this to be equal to the expectation value of φ, given by the convolution 1 f t φ(q i = qi φ(q i qf t (q, pdqdp, (7 where f t (q, p for q, p R 3 denotes the mass density at q with momentum p at time t. In replacing the empirical mean by its expectation value, we switch from the body problem to an effective external force problem: Instead of summing up all interaction terms, one expects a single particle to feel only the mean field produced by all particles together. But if there is only an external force ṗ acting, then the time evolution of f t is dictated by the continuity equation on R 6 : t f t + q f t q + p f t ṗ = 0 (8 Inserting for ṗ the expectation value (7 yields t f t + q f t q p f t f t φ = 0. (9 1 Young s inequality assures that the convolution f φ exists for f L 1 (R 6 if φ is bounded: f t φ f t 1 φ 8
11 2 THE COUTIG MEASURE This equation is known as the Vlasov equation, a partial nonlinear differential equation which has a unique global solution in the space of probability densities if φ is bounded and Lipschitz continuous (see [3]. The question is how the replacement of (6 by (7 can be rigorously justified from microscopic dynamics. To put it differently, how well does a solution of the Vlasov equation approximate an actual configuration evolving in time according to ewton s equation of motion? 2 The Counting Measure We can add to the Hamiltonian an external potential V t (q: H β (X = i=1 p 2 i 2m + i<j φ β (q i q j + V t (q i (10 i=1 Since an external force has the same effect on all particles, independently of their distribution, this should not affect the derivation of a hydrodynamic equation. Since we will in the following only consider differences between exact ewtonian and mean field dynamics, this external potential will not appear anymore. For the mass density f t, one has the continuity equation on 1particle phase space Γ 1 = R 6 containing points x = (q, p t f t (x + q f t (x dq dt + pf t (x dp dt = 0. (11 From this one obtains the approximating mean field equation by replacing the interaction part of the force with its expectation value. Hence, instead of summing up all interaction terms, one expects a single particle to feel only the mean field produced by all particles together. It is clear that, for different scaling parameters 2 β, this procedure leads to different hydrodynamic equations. In the next section we will investigate the case β = 0 as an example of the method that we shall present in the following. If one looks at the mean field approximation from the perspective of a law of large numbers, one possibility to justify the replacement in the continuity equation would be to argue that the random variables x i R 6 are 2 Recall: φ β (q = 1+β φ( β/d q 9
12 2 THE COUTIG MEASURE distributed identically and independently (i.i.d.. Formally, this means that the particle density 3 F is a product of the 1particle densities: F(X = f(x i (12 for X = (x 1,..., x R 6 and x i = (q i, p i R 6. This situation is called molecular chaos. The question of justifying the mean field approximation can now be reformulated: If, initially, the random variables x i R 6 were distributed according to F 0 (X = i=1 f 0 (x i, (13 will this product structure survive the timeevolution? Moreover, in which sense will it survive, i.e. in which sense i=1 F t (X i=1 f t (x i (14 for solutions f t of the effective equation? Remark 2.1. It is crucial that the L 1 norm is not a suitable notion of distance in this context: Think of a situation in which some m out of particles are not i.i.d.. Then the distributing function would be of the form F t = g t (x 1,..., x m i=m+1 f t (x i. We would say that this is close to a product if m was small. The L 1  distance from a product distribution would be given by = Ft (x 1,..., x f t (x i dx 1... dx gt (x 1,..., x m i=1 m i=1 f t (x i dx 1... dx m, 3 Here, we will exclusively deal with situations where the distributing measure is absolutely continuous. 10
13 2 THE COUTIG MEASURE which does not tell us anything about the value of m. The m bad particles might be distributed according to a function whose support is disjoint m or has very little intersection with the support of f t : then supp(g t supp( f m t 0, Ft (x 1,..., x i=1 f t (x i dx 1... dx = 2. independently of m. Therefore the L 1 distance does not tell us what we want to know and we have to find a for our purposes more appropriate notion of distance. The strategy for answering the above questions consists of two steps: First, look for a measure α that tells us how many particles are nice, i.e. i.i.d., in a way that α 0 if most particles are nice and α 1 if most of them are bad. This will represent the desired notion of distance from product distribution. Then, secondly, try to prove a statement of the kind: If that measure was small in the beginning, then it will remain small for all times. This second step will most likely involve an application of Gronwall s lemma, which we will now state and prove: Theorem 2.1. (Gronwall s lemma Let f and c denote realvalued functions defined on [0, and let f be differentiable on (0,. If f satisfies then d f(t c(tf(t, dt Proof. Define a function g by for t [0,. Then f(t e t 0 c(sds f(0. g(t = e t 0 c(sds It holds d g(t = c(tg(t. dt 11
14 2 THE COUTIG MEASURE on (0, and therefore d f dt g = f g fg g 2 cfg cfg g 2 = 0 f(t g(t f(0 g(0 = f(0 since g(0 = 1 and g(t > 1 for all t (0,. For finite particle numbers, all our results will contain error terms depending on. We therefore need the following Corollary 2.1. Let f denote a continuous realvalued function on [0, and let f be differentiable on (0,. If f satisfies for positive constants c 1 and c 2 then d dt f(t c 1f(t + c 2, f(t e c 1t f(0 + ( e c 1t 1 c 2 c 1. Proof. Define then g(t := f(t + c 2 c 1, d dt g(t = d dt f(t c 1f(t + c 2 = c 1 g(t and thus by Gronwall s lemma which implies g(t e c 1t g(0, f(t = g(t c 2 c 1 e c 1t f(0 + ( e c 1t 1 c 2 c 1. 12
15 2 THE COUTIG MEASURE If a given physical system admits for a mean field approximation, then it is clear that a relation of the form α t cα t + o( (15 must holds for a constant c: At a given time t, the growth of α t will be caused by two different processes: There will be interactions within the i.i.d. part of the particles and there will be interactions of not i.i.d. particles with ones that were i.i.d., thereby infecting some of the good particles. Interactions among independently distributed particles occur statistically and are controllable by the law of large numbers. Interactions of bad particles with good ones on the other hand are what really causes the growth of α t beyond statistical fluctuations. To control them we need an inequality like (15. The measure α t, if reasonably defined, shall tell us the amount of particles which are not i.i.d. at time t. Then the subsequent growth of this amount has to be bounded by a multiple of this amount, if a mean field description is to make any sense. By Gronwall s lemma, it follows from (15 that α t e ct α 0 + o(. This is intuitively clear, since the amount of particles which are not i.i.d. should grow exponentially in time. Starting with a product distribution at time t = 0 corresponds to α 0 = 0. Then the timeevolved measure will be α t o( and after carrying out the limit, it would hold α t = 0 for all times. Then we would say that the product structure of F survives the timeevolution. In other words, we would have shown the propagation of molecular chaos. Still, there remains the technical difficulty to find the right measure α such that the constant c is as small as possible. ote that it is one advantage of the method to be able to give an error estimate of the mean field approximation, depending on the particle number. Of course, due to the pair interaction, the independence will be destroyed through timeevolution. More precisely, during each small time interval t, because we consider only pair interactions, not more than two particles will fall out of molecular chaos. 4 onetheless, for appropriate scaling β one hopes for this effect to be weak compared to the mixing, such that molecular chaos survives the better the larger the number of particles is. 4 The probability that two particles interact is of order t, thus the probability of having two interacting pairs is of order ( t 2 and is therefore negligible to first order in t 13
16 2.1 Construction of the Counting Measure 2 THE COUTIG MEASURE The proximity of F t to a product density can, of course, only become exact in the limit. In fact, F t itself cannot be expected to converge, since there will always be some few particles which fell out of the product distribution. Instead, one can hope that its sparticle correlation functions Ft s will converge to an sfold product of solutions of the mean field equation f t. As will turn out, their L 1 distance will be bounded by a multiple of α, thus tending to zero as. To obtain this result, the symmetry of F t under exchange of particles will be absolutely crucial. Another way of formulating the desired result is that the following diagram commutes: For f 0 L 1 (R 6, f t L 1 (R 6 and F t L 1 (R 6 f 0 MeanF ield f t s f s t L 1 (16 f 0 = F 0 ewton F treducing F s t 2.1 Construction of the Counting Measure At a given time t, the particle density F t can be decomposed in the following way 5 : Definition 2.1. For a solution f t : R 6 R of the Vlasov equation, an particle density F t : R 6 R and k {0,..., }, we define functions g k t : R 6 R by the following conditions (C: and F t = and g k t (X = g k t (x 1,..., x = χ k t (x 1,..., x ( g k t sym := 1! σ S n χ k t,σ(x 1,..., x i=k+1 i=k+1 f t (x i f t (x σ(i, (17 5 Such a decomposition always exists: Any function F can be written in such a way because the functions χ can be chosen to be such that (17 is true. 14
17 2 THE COUTIG MEASURE 2.1 Construction of the Counting Measure χ k t,σ(x 1,..., x dx σ(1... dx σ(k = c k > 0 such that c k = 1 The sets { } M k f t := gt k : R 6 R conditions (C are fulfilled we call different sectors for distinct k s. ote that M k f t M l f t for k < l, which shows that this decomposition of F t is not unique. Remark 2.2. In the following, we will make the symmetry property of F t explicit only where it is needed and ignore it in the rest of the cases, writing F t = gt k with gt k = χ k t (x 1,..., x f t (x i i=k+1 such that χ k t (x 1,..., x dx 1... dx k = c k and c k = 1 evertheless, F t will always be understood to be symmetric. The c k s are independent of x k+1,..., x by definition and independent of time due to Liouville s theorem. The function χ k t itself depends on x k+1,..., x, which reflects the idea that those particles which are not i.i.d. depend on the positions and momenta of those particles which are, but in such a way that integrating over the noni.i.d.coordinates gives a constant. The measure 6 α t is supposed to tell us the relative amount of particles that are not distributed independently. For this purpose, we take a weighted sum of the integral over the functions g k t : α = inf F t= g k t k g k t dx (18 6 Again we remark that we mean measure in the sense of a counting device and not in the sense of measure theory. 15
18 2.1 Construction of the Counting Measure 2 THE COUTIG MEASURE with the abbreviation dx = dx 1... dx. The decomposition of the  particle density F t = gk t is not unique, one could always put F t = χ t, yielding α = 1 although it should possibly be much smaller. We want to give F t as much product structure as possible. Therefore we have to put as much of its integral mass as possible into functions gt k with low kvalues. To achieve this, we take for α the infimum of the weighted sum over all possible decompositions. We take the integral over the gt k s because we want to benefit from the volume conservation of the Hamiltonian flow on phase space and from linearity of the integral in our derivation. This will become explicit when computing the time derivative of α below. For technical reasons, we will have to allow for negative parts of the functions χ k t : χ k t = (χ k t + + (χ k t in such a way that still c k = χ k t dx > 0. This is possible since we can always put F t = χ t with χ t dx = c = 1 > 0. But it is crucial that all functions over which we integrate in (18 are positive, otherwise we could get α = 0 for a nonproduct state: e.g. then we would find ft F t = f t + F t, } {{ } } {{ } g 0 g α = 0 + (F t f t dx = 0, the positive and negative parts of (F t f t exactly compensate due to Liouville s theorem. Obviously (gt k + dx gt k dx = c k. But, since we are dealing with probability densities, we want to have just the integral mass of the gt k s, given by the c k s, in the positive parts of the gt k s. Therefore we shift some integral mass of (gt k + over to the negative part, thereby constructing new functions: Definition 2.2. Let (gt k g be defined in such a way that it fulfills (gt k g 0 and (gt k g dx 1... dx = c k In addition 16
19 2 THE COUTIG MEASURE 2.1 Construction of the Counting Measure (g k t n := g k t (g k t g (19 such that (g k t n dx 1... dx = (g k t (g k t g dx = 0. (20 These conditions are realized by 7 (χ k t g := (χ k t + c k, (21 (χ k t + dx 1... dx k where the coordinates x k+1,..., x are kept fixed. From this we get which is obviously positive. (gt k g := (gt k + c k, (22 (χ k t + dx 1... dx k We are now in a position to define the counting device which shall tell us how good F t can be approximated by a product f t : Definition 2.3. Let F t L 1 (R 6 be the particle density and let f t L 1 (R 6 be a solution of the mean field equation in question. Then we define α(f t, f t := inf F t= g k t,gk t Mk f t [ k (gt k g dx + } {{ } =:c k ] (gt k n 1 (23 Suppose that at some time t the particle density is given by F t = g k t = (g k t g + (g k t n. (24 Then, developing these g k t s a time interval t further, we get F t+ t = F t Φ t = (g k t g Φ t + (g k t n Φ t. (25 The whole construction is such that one has to find a new decomposition at each time t. We will construct a decomposition of F t+ t by timeevolving the single g k t s and decomposing in such a way that the corresponding α is not too large: 7 We could have started with this, but we wanted to keep the definition as general as possible and what we really need is that (g k t g = c k. 17
20 2.1 Construction of the Counting Measure 2 THE COUTIG MEASURE Let Φ and Φ denote the Hamiltonian flows of the microscopic and the mean field dynamics. The exact dynamics develops (gt k g M k f t into (gt k g Φ t M k+2 f t to first order in t: (g k t g Φ t = (g k t g + i=1 qi (gt k g q i t + i=1 pi (gt k g p i t (26 and with p i = 1 qi φ(q i q j j=1 we see that pi (gt k g is multiplied with qi φ(q i q j, resulting in a loss of two i.i.d. particles for i, j {k + 1,..., }. But giving (gt k g Φ t a prefactor of k+2 in the counting device would be too coarse since there still remain parts for which only k coordinates are not i.i.d.. If we look at the exact time evolution as mean field evolution plus a perturbation, i.e. (g k t g Φ t = (g k t g Φ t + ( (g k t g Φ t, where the perturbation is defined by this equation, then it is clear that all parts of (g k t g Φ t for which k+2 particles are bad are in the perturbation ( (g k t g Φ t. Liouville s theorem implies that (gt k g Φ t dx = (gt k g Φ t dx = (gt k g dx = c k (27 and hence ( (g k t g Φ t dx = 0. This means that (g k t g Φ t has as much negative as positive integral mass. But in the definition of α we demand positivity of each term. Therefore we put for 0 < θ k < 1 (gt k g Φ t = ( (gt k g Φ t + θk (gt k g Φ t } {{ } M k+2 f t + (1 θ k (g k t g Φ t } {{ } M k f t (28 with θ k such that the first term, which will get a prefactor of k+2 in α, is positive for typical fluctuations of the exact dynamics around the mean field orbit. For stronger fluctuations this term can still be negative, which will be 18
21 2 THE COUTIG MEASURE 2.1 Construction of the Counting Measure dealt with by writing ( (g k t g Φ t + θk (g k t g Φ t = + ( ( (g kt g Φ t + θk (g kt g Φ t g ( ( (g kt g Φ t + θk (g kt g Φ t n and absorbing the ( n part into the second sum in α (together with (g k t n Φ t. We could have shifted all the negative parts of the perturbation over to the second sum in α by writing ( (g k t g Φ t = ( (g k t g Φ t g + ( (g k t g Φ t n, but then the estimates for this term would become too bad. Heuristically, the introduction of θ k allows us to draw a line between typical and untypical fluctuations. Remark 2.3. At this point we can already identify a constraint of the new strategy: If (g k t g Φ t = 0 on some set, we cannot take anything from it by writing a convex combination as in (28 and the procedure fails. Eventually, this will translate into the condition that p f t c(tf t for some function c(t which is bounded on compact time intervals, as will become more explicit in due course. At time t, α is given by α(f t, f t = k inf F t= gt k,gk t Mk f t (g k t g dx + (g k t n 1 (29 Suppose the decomposition is such that 8 ( t 2 + α(f t, f t = k (g k t g dx + (g k t n 1 (30 for some (small t. At time t + t, because it includes the infimum over the decompositions at this time, α(f t+ t, f t+ t will be smaller than the α we constructed by timeevolving and suitably decomposing the right hand side of (30, which is given by 8 We do not know if the infimum is actually attained. The decomposition here is chosen to be such that its corresponding α (without an infimum in front is by ( t 2 larger than the infimum of it over all decompositions. 19
22 2.1 Construction of the Counting Measure 2 THE COUTIG MEASURE α(f t+ t, f t+ t := + + = + + k + 2 ( ( (gt k g Φ t + θk (gt k g Φ g t dx k (1 θ k (gt k g Φ t dx ( ( (gt k g Φ t + θk (gt k g Φ n t + (g k t n 1 Φ t ( k + 2 θ kc k + k (1 θ kc k (31 ( ( (gt k g Φ t + θk (gt k g Φ n t + (g k t n 1 Φ t k c k + (gt k n θ k c k ( ( (gt k g Φ t + θk (gt k g Φ n t + (g k t n 1 Φ t It follows that α(f t+ t, f t+ t α(f t, f t (32 [ t t θ k c k + t ( + ( ] (gt k g Φ t + θk (gt k g Φ n 1 t and hence we have α(t + t α(t α(t lim (33 t 0 [ t 1 2 lim θ k c k + t 0 t + ( ( (gt k g Φ t + θk (gt k g Φ n 1 t ]. 20
23 2 THE COUTIG MEASURE 2.1 Construction of the Counting Measure Remark 2.4. Observe that this derivation heavily relies on volume conservation: Throughout the computation we use line (27, implying the timeindependence of the c k s. In a way, also the introduction of θ reflects the idea that volume is preserved by the Hamiltonian flow: θ k (gt k g dx = θ k c k stands for that amount of integral mass which typically moves from a sector M k f t to M k+2 f t, see line (31 above. This is as far as we can get without specifying θ k which depends on the scaling behavior of the interaction. Later we will give the right expression for θ k for an interaction potential with scaling property φ = 1 φ and then derive the Vlasov equation. But first we will show that propagation of molecular chaos implies convergence of the marginals of the particle density towards products of solutions of the mean field equation. From that, we will also derive convergence of empirical densities towards solutions of the mean field equation. But for the sake of argument, assume ɛ k was such that for α(t c 1 α(t + c 2, (34 where c 2 stands for the error term that should go to zero as goes to infinity. Then an application of Gronwall s lemma would imply α(t e c 1t α(0 + (e c 1t 1 c 2 c 1 (35 and since α(0 = 0 if we assume all particles to be i.i.d. initially, it would follow α(t o( for all t, leading to α(t = 0 in the limit. For finite, the estimation for α will include an error term decreasing with. One advantage of the presented method is thus that the result is not just a statement for the limit, but it is actually possible to establish the speed of convergence in the particle number. If one succeeds in proving an equation of the type (15, one can show that the smarginal of the symmetric particle density F t, defined as Ft s := F t dx s+1... dx (36 converges in L 1 norm to an sfold product of solutions f t of the mean field equation: Theorem 2.2. Let F t : R 6 R + denote the timedependent probability density of particles evolving according to ewton s equations of motion with interaction potential φ β as above and let f t : R 6 R + be a solution of the mean field equation in question. If for α as defined above holds 21
24 2.1 Construction of the Counting Measure 2 THE COUTIG MEASURE then the marginals of F t converge: α(f t, f t 0, (37 as. F s t L 1 s f t (x i t R + s (38 i=1 Remark 2.5. By definition it is possible to choose for each a decomposition of F t such that α(f t, f t + 1 = k c k + (g k t n 1. To see this, observe that the integral mass of F t can be freely distributed over the g k t s (yielding the values of the c k s as long as the right hand side stays larger than α, which carries the infimum over all decompositions of F t. Since lim α = 0, sending on both sides yields lim ( k c k + (gt k n 1 = 0. This means that in the limit there exists a decomposition of F t which minimizes k c k + (gt k n 1 if the counting measure vanishes. We will need this for the following proof. Proof. Pick a decomposition of F t such that α(f t, f t + 1 = k c k + (g k t n 1. We make use of the symmetry of F t and reorder the permutations σ into permutations σ for which the first s particles are distributed according to f t and permutations σ for which this is not the case: 22
25 2 THE COUTIG MEASURE 2.1 Construction of the Counting Measure F s t := = + 1! + F t dx s+1... dx [ s 1 f t (x 1 f t (x s χ k t,σ! (x 1,..., x σ f(x σ (idx s+1... dx i=k+1,σ (i 1,...,s σ k= s+1 1! χ k t,σ (x 1,..., x σ i=k+1 χ k t,σ(x 1,..., x f t (x i dx s+1... dx ] i=k+1 The relative frequency for the permutations σ is k k s + 1 s + 1 = ( s!! f t (x i dx s+1... dx. ( k! ( k s! < 1, whereas for the permutations σ the relative frequency can be estimated from above as 9 ( s!! s 1 i=0 ( s i k! ( k! (k s + i! ( k i! for c(s = s 1 ( s i=0 i. We therefore have s 1 ( ( s i ( s k k i s i=0 ( s k c(s s i Ft s f t (x 1 f t (x s (40 ( s k + c(s χ k t,σ s (x 1,..., x f t (x i dx s+1... dx i=k χ k! t,σ(x 1,..., x f t (x i dx s+1... dx k= s+1 σ 9 ote that they sum up to one: ( s! s ( s k!! i (k s + i! i=0 i=k+1 ( k! ( k i! = 1 (39 23
26 2.1 Construction of the Counting Measure 2 THE COUTIG MEASURE With k= s+1 s + 1 k k 1! σ k 1! σ χ k t,σ(x 1,..., x χ k t,σ(x 1,..., x in the last line above it follows that i=k+1 i=k+1 f t (x i dx s+1... dx f t (x i dx s+1... dx (41 Ft s f t (x 1 f t (x s 1 by assumption and hence ( s k c(s s c k + s + 1 [( s ] c(s + s s + 1 } {{ } c(s+1 0 k c k ( α(f t, f t + 1 F s t L 1 f t (x 1 f t (x s. Remark 2.6. Under the conditions of the theorem, the L 1 convergence F s t L 1 f t (x 1 f t (x s (42 for all t and s implies w convergence of the corresponding measures: Ft s dx w s f t (x i dx i (43 i=1 Weak convergence of the empirical density µ X(t := 1 i=1 δ xi (t towards weak solutions of the Vlasov equation µ(t was shown, among others, by eunzert and Braun and Hepp (see [1],[2],[3]. The key equation in eunzert s proof is d BL (µ X(t, µ(t ect d BL (µ X(0, µ(0 (44 24
27 2 THE COUTIG MEASURE 2.1 Construction of the Counting Measure for the bounded Lipschitz distance d BL (see e.g. [3]. We will show that for absolutely continuous measures, this result follows from the L 1  convergence of the marginals towards products of solutions of the Vlasov equation: Corollary 2.2. Let µ X(t describe the positions and momenta of particles and let µ ft (dx = f t (xdx for f : R R 6 R. Then the L 1 convergence implies F 1 t f t and F 2 t f t f t (45 in mean with respect to P Ft. µ X(t d BL µ f t (46 Proof. The crucial observation is that the metric d BL is defined for functions g which are Lipschitz continuous with constant 1: D := {g : R [0, 1], g(x g(y x y }. A partition of phase space Γ 1 = R 6 into cells i of sidelength δ is very useful, because the test functions g vary at most by δ on such a cell: d BL (µ X(t, µft := sup g D = sup g D i i sup g D µ X(t (dxg(x i µ X(t f t (xg(xdx (dxg(x f t (xg(xdx i (dxg(x f t (xg(xdx i i µ X(t ow g D and hence, as already explained, there exists δ > 0 such that g(x g(y δ for all x, y i and, in addition, g(x 1 for all x: d BL (µ X(t, µft i (1 + δ 2δ + i (dx f t (xdx i i µ X(t µ X(t ( i µ ft ( i Integrating this with respect to P F and noting that 25
28 2.1 Construction of the Counting Measure 2 THE COUTIG MEASURE F(Xd BL (µ X(t, µft dx = F t (Xd BL (µ X, µ ft dx since F(X = F t (X(t by definition, we obtain by dominated convergence F t (Xd BL (µ X, µ ft dx i F t (X µ X ( i µ ft ( i dx + 2δ. (47 } {{ } =:λ Using CauchySchwarz s inequality, we get λ ( F t (XdX ( µ X ( i µ ft ( i 1/2 2 ( = F t (XdX ( 1 = Ft 1 (xdx + i 2 Ft 1 (xdx i ( + ( (µ X ( i 2 2µ X ( i µ ft ( i + ( µ ft ( i 2 1/2 ( 1 2 Ft 2 (x, ydxdy (48 i i ( 2 1/2 f t (xdx + f t (xdx i i 1 Ft 1 (xdx 1 Ft 2 (x, ydxdy i i i Ft 2 (x, y f t (xf t (y dxdy i + 2 i i Ft 1 (x f t (x dx f t (xdx i 1/2 0 and since the left hand side of (47 is independent of δ, sending δ to zero implies lim F t (Xd BL (µ X, µ ft dx = 0. (49 ote that to get line (48, we used the symmetry of F t under particle exchange. 26
29 3 DERIVATIO OF THE VLASOV EQUATIO Remark 2.7. This obviously implies for all κ > 0 since P ({X = (x 1,..., x d BL (µ X(t, µft > κ } 0 (50 F t χ { d BL (µ X(t,µf t κ >1} dx1... dx 1 κ which tends to zero for all κ. F t d BL (µ X(t, µft dx 1... dx, (51 3 Derivation of the Vlasov equation In the last section we explained how one can in principle derive an effective mean field equation from microscopic dynamics using the counting measure α. However, we did this in very general terms and stopped at the point where a specification of the parameters θ k was inevitable. θ k was introduced to compensate for the negative parts in the difference (gt k g Φ t (gt k g Φ t at least as long as the dynamics stays within typical fluctuations. At this stage it is quite clear that we expect these typical fluctuations to be of order in total, i.e. 1/ per particle, since we have the law of large numbers at our disposal and the fluctuations from the mean field orbit are the deviations from the expectation value. Hence, θ k (gt k g Φ t should be of the typical size of the difference between 10 and (g k t g Φ t = (g k t g + qi (gt k g q i + i=1 pi (gt k g p i (52 i=1 (g k t g Φ t = (g k t g + where we defined qi (gt k g q i + i=1 pi (gt k g π i, (53 i=1 10 ote that the following equations are only true to first order in t. 27
30 3 DERIVATIO OF THE VLASOV EQUATIO and π i = p i = 1 qi φ(q i q j t for i = 1,..., (54 j=1 { 1 j=1 q i φ(q i q j t, for i = 1,..., k f t (q, p qi φ(q i qdqdp t, for i = k + 1,..., (55 for a solution f t of the Vlasov equation. ote that we ignored the part of the force coming from an external potential, namely qi V t (q i, because it appears in both p i and π i and we only consider the difference p π. The splitting of the mean field momenta π i is for technical reasons. We could have let the k bad particles evolve according to the expectation value of the force, but it doesn t matter so much since they are not i.i.d. and can thus not be expected to act according to mean field dynamics. We will in the following use the abbreviation δk(q i q j := qi φ(q i q j qi φ(q i qf t (q, pdqdp. (56 For the difference between the two time evolutions we then have p (g k t g ( p π = i=k+1 [ k pi (gt k g 1 ( δk(q i q j (57 j=1 ] + ( δk(q i q j t, j=k+1 where one should keep in mind that the first k particles are not distributed according to a product, whereas the last k particles are, therefore the splitting into two sums. We need a bound for the set in particle phase space Γ on which (g k t g Φ t (g k t g Φ t +θ k (g k t g Φ t is still negative. We will now define θ k and then show that this set is in fact exponentially small in. Definition 3.1. For an interaction potential scaling as φ (q = 1 φ(q with φ such that φ is bounded, the parameter θ k shall be defined for k {0,..., } and 0 < κ < 1 as 2 θ k := pf t φ k t + pf t ( k 1/2+κ t (58 f t f t 28
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