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arXiv:cond-mat/0302378 · Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Stochastic collective dynamics of charged-particle beams in the stability regime

Nicola Cufaro Petroni, S. De Martino, Silvio De Siena, Fabrizio Illuminati

Published Dec 18, 2000• 19 citations

Abstract

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.

DOI: 10.1103/physreve.63.016501

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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.

In this paper we briefly review the main aspects of a recent proposal to simulate semiclassical corrections to classical dynamics by suitable classical stochastic fluctuations, and we apply it to the specific instance of charged beams 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, that is time--reversal invariant. Its diffusion coefficient p N c represents a semiclassical unit of emittance (here N is the number of particles in the beam, and c 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 Schro...

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

\sqrt{N}λ_{c}

represents a semiclassical unit of emittance (here

N

is the number of particles in the beam, and

λ_{c}

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 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.

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.

We present a modified version of Nelson's seminal paper on the derivation of the time-dependent Schrödinger equation which draws on the equation of motion of a particle that moves under the influence of a classical force field and additional stochastic forces. The emphasis of our elaboration is focused on the implication of allowing stochastic forces to occur, viz. that the energy E of the particle is no longer conserved on its trajectory in a conservative force field. We correlate this departure Δ E from its classical energy with the energy/time uncertainty relation where Δ t is the average time for Δ E to persist. The stability of atoms, the zero-point energy of oscillators, the tunneling effect and the diffraction at slits are shown to be directly connected with the occurrence of such energy fluctuations. We discuss and rederive Nelson's theory entirely from this point of view and generalize his approach to systems of N particles which interact via pair forces. Achieving reversibility in a description of particle motion that is akin to Brownian motion, represents a salient point of the derivation. We demonstrate that certain objections raised against Nelson's theory are without substance. We also try to put the particular worldview of this version of stochastic quantum mechanics into perspective with regard to the established Copenhagen interpretation.