Random N-Particle Klimontovich-Maxwell System: Probabilistic Analysis, Fluctuations from Mean and Ecker Hierarchy James Ellison University of New Mexico Mathematics and Statistics We analyze the relativistic NN-particle motion, with random initial conditions, coupled to the microscopic Maxwell equations by the Lorentz force. We use the Klimontovich density, KK, where K(r,p,t;w0)=(1/N)∑N1Ki(r,p,t;w0)K(r,p,t;w0)=(1/N)∑1NKi(r,p,t;w0), and Ki(r,p,t;w0)=δ(r−Ri(t;w0))δ(p−Pi(t;w0)).Ki(r,p,t;w0)=δ(r−Ri(t;w0))δ(p−Pi(t;w0)). Here the initial phase space position of the ithith particle is (Ri(0;w0),Pi(0;w0))=w0i(Ri(0;w0),Pi(0;w0))=w0i. These are identically distributed six dimensional random vectors and thus w0=(w01,…,w0N)∈R6Nw0=(w01,…,w0N)∈R6N. The notion of the KK-density is old, but to our limited knowledge, our probabilistic framework using random initial conditions is new. The associated NN-particle probability density function (pdf) is denoted by ΨNΨN and is assumed to have permutation symmetry. We refer to this set up as the Random Klimontovich-Maxwell System (RKMS). The primary focus in our talk will be on the expected value K¯K¯ of KK and on the fluctuations K~=K−K¯K~=K−K¯ where K¯≡Ψ1K¯≡Ψ1 is the single particle pdf related to ΨN(w,t)ΨN(w,t) by integration. However, the associated macroscopic Maxwell fields are important as well. Taking expected values in the RKMS, by multiplying by ΨN(w0,0)ΨN(w0,0) and integrating over w0w0, we obtain a new system denoted by RKMS¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯RKMS¯. It contains an evolution law which must be satisfied by the coupled Ψ1Ψ1 and averaged Maxwell Fields. However this evolution law is not closed unless the fields and particles are uncorrelated in a certain way, in which case RKMS¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯RKMS¯ is the well-known Vlasov-Maxwell system. To develop an approximate closed system, we introduce what we call the Ecker hierarchy, a relativistic modification of the BBGKY hierarchy. The Ecker hierarchy may be useful for understanding FEL fluctuations, it is old but our interpretation using random initial conditions is new. As expected, the first level is not closed. However, linearizing and assuming the particles are uncorrelated gives a closed system. Here, by uncorrelated we mean that Ψ2(r,p,r′,p′,t)=Ψ1(r,p,t)Ψ1(r′,p′,t)Ψ2(r,p,r′,p′,t)=Ψ1(r,p,t)Ψ1(r′,p′,t), where Ψ2Ψ2 is the two particle pdf, again related to ΨN(w,t)ΨN(w,t) by integration. Note that RKMS¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯RKMS¯ has only one level and that in the first level Ψ1Ψ1 is the pdf for \overline{RKMS} and for the Ecker hierarchy. However on the first level the fields for RKMS¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯RKMS¯ and for the Ecker hierarchy are different. Note also that on every level the Ecker hierarchy gives generalized Maxwell’s equations. In addition, we present the system which evolves the fluctuations K¯–Ψ1K¯–Ψ1. When this system is combined with the RKMS¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯RKMS¯ and linearized we find a closed system which may also shed light on FEL dynamics. We are guided here by the approach of Kim and Lindberg to a comparatively simpler FEL model discussed in the FEL2011 Proceedings, and hope to understand their distinction between collective and individual aspects of fluctuations in our context. Finally, we compare KK and K¯K¯, however, since KK is zero except at the particle positions where it is infinite, we find it useful to introduce a coarse graining, by integrating over an (r,p)(r,p) phase space region AA. We denote these integrals by KA(t;w0)KA(t;w0) and K¯A(t;w0)K¯A(t;w0) and find that KAKA is (1/N)(1/N) times a sum of identically distributed Bernoulli random variables. Calculating, as in the proof of the weak law of large numbers, we find that the L2L2 difference is (KA−K¯A)2≈(1/N)Ψ1A(t)[1−Ψ1A(t)]+[Ψ2A(t)−Ψ21A(t)](KA−K¯A)2≈(1/N)Ψ1A(t)[1−Ψ1A(t)]+[Ψ2A(t)−Ψ1A2(t)] and is only small if the particles are nearly uncorrelated, i.e.i.e., if Ψ2≈Ψ1Ψ1Ψ2≈Ψ1Ψ1 . We anticipate this to be true in many beam dynamics contexts, since otherwise the commonly used Vlasov Maxwell system would be suspect. Since we don’t have independence, probabilistic analysis, such as that used in the weak and strong laws of large numbers, the central limit theorems and large deviation theory, needs to include some form of dependence. Likely some kind of mixing or ergodic behavior, e.g.e.g., related to chaos, must be present in RKMS making the L2L2 difference small. We hope to convince workshop participants that RKMS is both a rich and important dynamical system for study. We believe there are significant open problems that can lead to useful collaborations between accelerator scientists and researchers in mathematics (pure and applied), dynamical systems (perturbation theory, and ergodic and chaos theory), and numerical analysis and computational science. GENERAL COMMENT In general, beam dynamics in modern particle accelerators is a rich source of deep and interesting problems in Applied Mathematics including its usual subdisciplines, as well as probability, stochastic processes and mathematical statistics (e.g.e.g., density estimation). It is an exciting area of applied mathematics. There are problems which are as difficult and sophisticated as those found in Mathematical Physics. An example are problems in spin dynamics that use ideas from topological dynamics and group actions, and from the Zimmer-Feres bundle approach to ergodic theory. There is work in progress being spearheaded by Heinemann. But also, there is low lying fruit because there are many problems just at the boundary of what we know, problems that are important and yet simple enough for immediate progress. Examples from our work include rigorous long time perturbation theory, such as the method of averaging for PDEs (see related poster), and the QED of synchrotron radiation for spin-orbit dynamics including a white noise approximation which is suspect. This is joint work by G. Bassi (BNL), J. A. Ellison (UNM), and K. A. Heinemann (UNM).