Adiabatic-Nonadiabatic Transition in Warm Long-Range Interacting Systems: The Transport of Intense Inhomogeneous Beams

Abstract

We investigate the role of the temperature in the onset of singularities and the consequent breakdown in a macroscopic fluid model for long-range interacting systems. In particular, we consider an adiabatic fluid description for the transport of intense inhomogeneous charged particle beams. We find that there exists a critical temperature below which the fluid model always develops a singularity and breaks down as the system evolves. As the critical temperature is approached, however, the time for the occurrence of the singularity diverges. Therefore, the critical temperature separates two distinct dynamical phases: a nonadiabatic transport at lower temperatures and a completely adiabatic evolution at higher temperatures. These findings are verified with the aid of self-consistent N-particle simulations. For long-range self-interacting systems, it is generally very difficult to obtain a fully kinetic description of the dynamics. This is the case in plasmas, charged particle beams, and self-gravitating systems among others, where the collision duration time diverges in the thermodynamic limit [1][2][3][4][5]. A largely used tool to overcome this difficulty is the employment of macroscopic fluid models. In contrast to the kinetic description that requires the knowledge of the evolution of the distribution function in the full phase space, the fluid description is simpler because it is based on local macroscopic variables obtained by averaging over the momentum space. Moreover, because the fluid variables consist of readily understood macroscopic quantities, the physical interpretation of the phenomena under investigation is generally more direct. Nevertheless, except for very specific cases, the fluid description leads to an infinite hierarchy of equations which, in practice, have to be truncated to be analyzed. The truncation is obtained by assuming a certain characteristic for the system dynamics which is expressed by an equation of state. Perhaps, the simplest used approximation is the cold fluid, which completely neglects thermal effects by assuming a vanishing temperature [6][7][8][9][10][11][12]. Other examples of largely used equations of state are isothermal [13][14][15][16][17] and adiabatic [18,19]. At any rate, the validity of the fluid description resides not only on the choice of equation of state but also on the fact that the infinite hierarchy has to be convergent; i.e., the macroscopic fluid variables may not present singularities. In the case of cold fluids, a wellknown cause of singularities is the onset of the so-called wave breaking where the fluid description looses its validity due to a divergence in the particles density at a certain position and time. This phenomenon is associated with a filamentation in the phase space and may have relevant consequences such as temperature increase, energy redistribution, and particle acceleration, depending on the system.In this Letter, we investigate the role of the temperature in the onset of singularities in the macroscopic quantities and the consequen...

Mixing and space-charge effects in free-electron lasers

“…Deriving Eq. (10) with respect to h 0 , and defining @h j =@h 0 C as the compressibility, 14 we obtain an evolution equation with which one can identify the beginning of the phase-mixing process in the FEL, namely,…”

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“…In particular, a recent work on thermal effects suggests that even very small temperatures can be used to control the mixing time. 14 Solutions of Eq. (A4) for h 6 ¼ h 0 have exponential form and must be matched across h 0 .…”

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