Tamtam Proceedings - lamsin

36 Hauray et al.minimal time interval which the discrete dynamics can see. We fix this parameter fromthe beginning and somehow the main part of our work is to show that it is indeed correct,so takeε = R(0) . (7)N1/2dAt the initial time, we will choose our approximation so that the minimal distance betweentwo particles will be of order ε.The force term cannot be bounded at every time for the discrete dynamics (a quantitylike F ⋆ ρ N is not bounded even in the case of free transport), but we can expect that itsaverage on a short interval of time will be bounded. So we denote{ ∫ 1 t+ε}E(T ) = sup|E(X i (s))| ds , (8)t∈[0, T −ε],i=1,...,N εwith for T < εE(T ) =supi=1,...,Nt{ ∫ }1 T|E(X i (s))| ds , (9)ε 0thus obtaining a unique and consistant definition for all T > 0. Moreover we denote byE 0 the supremum over all i of |E(X i (0))|.This definition comes from the following intuition. The force is big when two particlesare close together. But if their speeds are different, they will not stay close for a long time.So we can expect the interaction force between these two particles to be integrable in timeeven if they "collide". There just remains the case of two close particles with almost thesame speed. To estimate the force created by them, we need an estimate on their number.One way of obtaining it is to have a bound onm(T ) =supt∈[0,T ],i≠jε|X i (t) − X j (t)| + |V i (t) − V j (t)| . (10)The control on m requires the use of a discretized derivative of E, more precisely, wedefine for any exponent β ∈ ] 1, d − α [, which also satisfies β < 2d − 3α (β = 1 wouldbe enough for short time estimates)∆E(T ) =supt∈[0, T −ε]with as for E, when T < ε∆E(T ) =supi,j=1,...,N,{ 1εsupi,j=1,...,N∫ t+εt|E(X i (s)) − E(X j (s))|ε β + |X i (s) − X j (s)|}ds , (11){ ∫ }1 T|E(X i (s)) − E(X j (s))|ε 0 ε β ds . (12)+ |X i (s) − X j (s)|Now, we introduce what we called the discrete infinite norm of the distribution of theparticle µ N . This quantity is the supremum over all the boxes of size ε of the total massthey contain divided by the size of the box. That is, for a measure µ we denoteTAMTAM –Tunis– 2005