Hirshfield and Park reply

Abstract

Hirshfield and Park Reply:We are grateful for the opportunity Larson [1] provides for us to clarify our published Letter. As shall be shown, there is no violation of the second law of thermodynamics implied by our results, since the total phase space for the system does not decrease.Equation (1) of our Letter, from which shrinkage in the energy spread of an electron beam was calculated, was derived from the Vlasov equation, where the electrons were subjected to assigned cavity fields. For noninteracting particles, it is well known [2] that the quantity f(p,r,t)d 3 pd 3 r is conserved, where /(p,r,f) is the timedependent distribution function in the sixfold momentum (p) and configuration (r) space. For an electron beam, this phase-space volume is related to the emittance. The entropy -A:J"^/ 3 /7^/ 3 ry*(p,r,^ )ln[y(p,r,/)] can be shown to be constant in this case as well, although it is customary in beam physics applications to deal with the phase-space volume directly.Larson has suggested that somehow, in discussing the phase-averaged energizing or deenergizing of groups of electrons with energies lower or higher than a mean value, we have obscured what is in fact a broadening in the energy distribution. It is indeed true that a particle's initial phase with respect to the external field will determine whether or not it initially experiences an increase or decrease in its energy. A detailed examination of the phase-dependent dynamical equations reveals that most particles with energy higher than the mean experience a decrease in energy, while most with energy lower experience an increase [3]. However, when the particle energies are distributed narrowly about the mean energy, the energies will eventually all coalesce to the mean. Near the discrete energies identified in our Letter, the rate of coalescence was predicted to be more rapid than for other energies. We (perhaps intemperately) labeled this coalescence process "cooling" in our Letter, but the quotes were prominently displayed.So how is the phase space conserved? Clearly, while the momentum part is shrinking, the configuration part is swelling. This phenomenon has been exhibited for single-particle orbits by Humphries [4], and shown in parti-

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