We analyze the relativistic \(N\)-particle motion, with random initial conditions, coupled to the microscopic Maxwell equations by the Lorentz force. We use the Klimontovich density, \(K\), where \( K(r, p, t; w_0) = (1/N ) \sum_{1}^{N} K_i(r, p, t; w_0)\), and \(K_i(r, p, t; w_0)= \delta (r - R_i(t; w_0)) \delta(p - P_i(t; w0)).\) Here the initial phase space position of the \(i^{th}\) particle is \((R_i(0; w_0), P_i(0; w_0)) = w_{0i}\). These are identically distributed six dimensional random vectors and thus \(w_0 = (w_{01}, \dots , w_{0N} ) \in \mathbb{R}^{6N}\). The notion of the \(K\)-density is old, but to our limited knowledge, our probabilistic framework using random initial conditions is new. The associated \(N\)-particle probability density function (pdf) is denoted by \(\Psi_N\) 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 \(\bar{K}\) of \(K\) and on the fluctuations \(\tilde{K} = K - \bar{K}\) where \(\bar{K} \equiv \Psi_1\) is the single particle pdf related to \(\Psi_N (w, t)\) by integration. However, the associated macroscopic Maxwell fields are important as well.
Taking expected values in the RKMS, by multiplying by \(\Psi_N (w_0, 0)\) and integrating over \(w_0\), we obtain a new system denoted by \(\overline{RKMS}\). It contains an evolution law which must be satisfied by the coupled \(\Psi_1\) 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 \(\overline{RKMS}\) 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 \(\Psi_2 (r, p,r',p' , t)= \Psi_1 (r, p, t) \Psi_1( r',p', t)\), where \(\Psi_2\) is the two particle pdf, again related to \(\Psi_N (w, t)\) by integration. Note that \(\overline{RKMS}\) has only one level and that in the first level \(\Psi_1\) is the pdf for \overline{RKMS} and for the Ecker hierarchy. However on the first level the fields for \( \overline{RKMS} \) 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 \(\bar{K} – \Psi_1\). When this system is combined with the \(\overline{RKMS}\) 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 \(K\) and \(\bar{K}\), however, since \(K\) is zero except at the particle positions where it is infinite, we find it useful to introduce a coarse graining, by integrating over an \((r, p)\) phase space region \(A\). We denote these integrals by \(K_A(t; w_0)\) and \(\bar{K}_A(t; w_0)\) and find that \(K_A\) is \((1/N)\) 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 \(L_2\) difference is \((K_A - \bar{K}_A)^2 \approx (1/N ) \Psi_{1A}(t)[1 - \Psi_{1A}(t)] + [\Psi_{2A}(t) - \Psi_{1A}^2(t)] \) and is only small if the particles are nearly uncorrelated, \({\it i.e.}\), if \(\Psi_2 \approx \Psi_1 \Psi_1\) . 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, \(\it{e.g.}\), related to chaos, must be present in RKMS making the \(L_2\) 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 (\({\it e.g.} \), 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).
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