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International Journal of Modern Physics B

MESOSCOPIC DESCRIPTION OF DYNAMICAL SYSTEMS: HIERARCHY OF RECURSIVE EQUATIONS FOR THE MOMENTS OF THE DENSITY DISTRIBUTION WITH AN APPLICATION TO CHARGED PARTICLE BEAM DYNAMICS

S. De Martino, Silvio De Siena, Renato Fedele, Stephan I. Tzenov

Published Feb 20, 2004• 0 citations
Publisher

Abstract

The quantumlike formalism in the form of Madelung fluid has been applied to describe the collective dynamics of a mesoscopic aggregate of particles in a potential field. The scheme has been specialized to the one-degree-of-freedom dynamics of a charged particle beam in an accelerator in the presence of high-order multipole nonlinearities. A hierarchy of recursive equations satisfied by the moments of the density distribution has been obtained. It has been shown that the recursion relations can be in principle solved if a finite number of the first several moments is known. Beam dynamics in the presence of octupole and decapole nonlinearities has been studied as an example, demonstrating the above scheme. It has been shown that an appropriate mechanism can be introduced to control stability and coherence of states when high-order nonlinearities have to be taken into account.

DOI
10.1142/s0217979204024276

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The quantumlike formalism in the form of Madelung fluid has been applied to describe the collective dynamics of a mesoscopic aggregate of particles in a potential field. The scheme has been specialized to the one-degree-of-freedom dynamics of a charged particle beam in an accelerator in the presence of high-order multipole nonlinearities. A hierarchy of recursive equations satisfied by the moments of the density distribution has been obtained. It has been shown that the recursion relations can be in principle solved if a finite number of the first several moments is known. Beam dynamics in the presence of octupole and decapole nonlinearities has been studied as an example, demonstrating the above scheme. It has been shown that an appropriate mechanism can be introduced to control stability and coherence of states when high-order nonlinearities have to be taken into account.
A recent proposal (see quant-ph/9803068) to simulate semiclassical corrections to classical dynamics by suitable classical stochastic fluctuations is applied to the specific instance of charged beam dynamics in particle accelerators. The resulting picture is that the collective beam dynamics, at the leading semiclassical order in Planck constant can be described by a particular diffusion process, the Nelson process, which is time-reversal invariant. Its diffusion coefficient Nλc\sqrt{N}λ_{c}N​λc​ represents a semiclassical unit of emittance (here NNN is the number of particles in the beam, and λcλ_{c}λc​ is the Compton wavelength). The stochastic dynamics of the Nelson type can be easily recast in the form of a Schroedinger equation, with the semiclassical unit of emittance replacing Planck constant. Therefore we provide a physical foundation to the several quantum-like models of beam dynamics proposed in recent years. We also briefly touch upon applications of the Nelson and Schroedinger formalisms to incorporate the description of collective coherent effects.
A classical description of the dynamics of a dissipative charged-particle fluid in a quadrupole-like device is developed. It is shown that the set of the classical fluid equations contains the same information as a complex function satisfying a Schrodinger-like equation in which Planck's constant is replaced by the time-varying emittance, which is related to the time- varying temperature of the fluid. The squared modulus and the gradient of the phase of this complex function are proportional to the fluid density and to the current velocity, respectively. Within this framework, the dynamics of an electron bunch in a storage ring in the presence of radiation damping and quantum-excitation is recovered. Furthermore, both standard and generalized (including dissipation) coherent states that may be associated with the classical particle fluids are fully described in terms of the above formalism.
A classical description of the dynamics of a dissipative charged-particle fluid in a quadrupole-like device is developed. It is shown that the set of the classical fluid equations contains the same information as a complex function satisfying a Schrödinger-like equation in which Planck's constant is replaced by the time-varying emittance, which is related to the time-varying temperature of the fluid. The squared modulus and the gradient of the phase of this complex function are proportional to the fluid density and to the current velocity, respectively. Within this framework, the dynamics of an electron bunch in a storage ring in the presence of radiation damping and quantum-excitation is recovered. Furthermore, both standard and generalized (including dissipation) coherent states that may be associated with the classical particle fluids are fully described in terms of the above formalism.
We introduce a description of the collective transverse dynamics of charged (proton) beams in the stability regime by suitable classical stochastic fluctuations. In this scheme, the collective beam dynamics is described by time-reversal invariant diffusion processes deduced by stochastic variational principles (Nelson processes). By general arguments, we show that the diffusion coefficient, expressed in units of length, is given by lambda(c)sqrt[N], where N is the number of particles in the beam and lambda(c) the Compton wavelength of a single constituent. This diffusion coefficient represents an effective unit of beam emittance. The hydrodynamic equations of the stochastic dynamics can be easily recast in the form of a Schrödinger equation, with the unit of emittance replacing the Planck action constant. This fact provides a natural connection to the so-called "quantum-like approaches" to beam dynamics. The transition probabilities associated to Nelson processes can be exploited to model evolutions suitable to control the transverse beam dynamics. In particular we show how to control, in the quadrupole approximation to the beam-field interaction, both the focusing and the transverse oscillations of the beam, either together or independently.
We apply a recursive approach to the continuity equation of the Madelung fluid resulting in a propagation equation for particle probability densities. This propagation equation can be used to propagate particle distributions in the presence of a Madelung pressure field. We show that the derived propagation equation goes over into the guidance equation of the de Broglie-Bohm theory in the limit of well located single particles. As an example, we propagate particles that enter the lower slit of a double-slit experiment, while the Madelung fluid enters both slits.