Inconsistencies are pointed out in a recent proposal [L. Diosi, Phys. Rev. A
80, 064104 (2009); arXiv:0905.3908v1] for a quantum version of the classical
linear Boltzmann equation.Comment: 3 pages; v3: corresponds to published versio

Bassano Vacchini

J. R. Taylor

K. Huang

Klaus Hornberger

English

arXiv

Hornberger, K.

Vacchini, B.

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Comment on "Quantum linear Boltzmann equation with finite intercollision time"

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Comment on “Quantum linear Boltzmann equation with finite intercollision time”

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Inconsistencies are pointed out in a recent proposal [L. Diosi, Phys. Rev. A 80, 064104 (2009)] for a quantum version of the classical linear Boltzmann equation

K. Hornberger

B. Vacchini

AIR Universita degli studi di Milano

PHYSICAL REVIEW A 82, 036101 (2010)Comment on “Quantum linear Boltzmann equation with finite intercollision time”Klaus HornbergerMax Planck Institute for the Physics of Complex Systems, No¨thnitzer Straße 38, 01187 Dresden, GermanyBassano VacchiniUniversita` degli Studi di Milano, Dipartimento di Fisica, Via Celoria 16, I-20133 Milano, Italy andINFN Sezione di Milano, Via Celoria 16, I-20133 Milano, Italy(Received 23 December 2009; published 27 September 2010)Inconsistencies are pointed out in a recent proposal [L. Dio´si, Phys. Rev. A 80, 064104 (2009)] for a quantumversion of the classical linear Boltzmann equation.DOI: 10.1103/PhysRevA.82.036101 PACS number(s): 03.65.Yz, 05.60.Gg, 03.65.Ta, 05.20.DdI. INTRODUCTIONIn a recent Brief Report [1] Dio´si proposes a quantumversion of the classical linear Boltzmann equation. As ex-plained below, we think that this proposed equation has anumber of unfavorable properties, both from a physical andfrom a conceptual point of view, which cast serious doubtson its validity. All these problems are nonexistent in what weconsider the appropriate quantum linear Boltzmann equation(QLBE) [2–4], while the proposed equation has no physicallymeaningful advantage over the QLBE. In particular, thepossible wish for linearity in the gas momentum distribution,which serves as the motivation of [1], is consistently fulfilledwithin the QLBE by an approximation that is much less drasticthan the modifications entailed in [1].We are concerned with a Markovian quantum masterequation for the motion of a single, distinguished test particle(mass M) due to collisions with a stationary, homogeneous,and ideal background gas of distinguishable particles (mass m,density ngas). The latter is characterized by the momentumdistribution function µgas( p) = 〈 p|ρgas| p〉, which may or maynot be given by the Maxwell distribution. The equation issupposed to be valid as long as a Markovian description of thereduced quantum state of the test particle is appropriate.II. DISCUSSION OF THE MASTER EQUATIONDio´si’s proposal involves an “intercollision time”τ = 1ngasσ√πmkBT, (1)and his master equation is constructed such that in the limitof vanishing gas density ngas → 0, i.e., for τ → ∞, it reducesto the classical linear Boltzmann equation for the diagonalmomentum matrix elements. In the following, we do notdiscuss our reservations about the derivation of the proposedequation and the implied violation of energy conservationin individual scattering interactions, but we consider onlyproblems of the final master equation at finite τ . We startby exposing conceptual deficiencies and then move on toinconsistencies in the predicted physical behavior.A. Conceptual deficiencies1. Off-shell extension ill definedThe equation is not well defined because it involvesthe elastic scattering amplitude f ( pf , pi) at off-shell values| pi | = | pf |. While it is common in scattering theory towork with off-shell extensions of the scattering operators,this is only done for computational convenience [5]. Allphysically relevant properties of the elastic scattering processdepend solely on the on-shell values, not least because ofthe arbitrariness of the off-shell extension [5]. Even if onewere to agree on a definite choice of the off-shell extension,the proposed equation would remain ill defined becausethe value of the off-shell energy parameter, which is anindependent variable, remains unspecified.From a physical point of view, it seems implausible that thesupposed “energy uncertainty” related to finite intercollisiontimes (a property involving the state of motion of gas and testparticle) has anything to do with a possible off-shell extensionof the elastic scattering amplitude (which is a function of theinteraction potential only).2. Dependence on representation of δ functionThe proposed equation depends on a particular choice ofthe representation of the δ function. It has the property thatits square can be related to another representation of theδ function. One can envisage representations of the δ functionquite different from the one used in the proposed equation,which share the same property, e.g., a Gaussian function. Thosewould yield manifestly different equations (e.g., leading toa different prediction for the constant Dxx). This highlightsthe arbitrariness of using a particular representation. From aphysical point of view, it seems implausible that one form of“smoothing” should be favored over another.3. Nonlinearity in the gas densityThe proposed equation is nonlinear with respect to thegas density since the definition (1) of τ includes ngas. Froma physical point of view, it seems implausible that theLiouvillian for the Markovian dynamics should be a nonlinearfunction of ngas. The reason is that the background gasis noninteracting and nondegenerate, implying that the gasparticles are uncorrelated, while three-particle collisions areexcluded by assumption. Therefore, each gas particle affectsthe test particle equally, so that its effect is described by thesame mapping. This implies that the Liouvillian for the effectof the total gas is proportional to the number of gas particles,rendering the master equation linear in ngas.4. Definition of τIf the quantity τ is to correctly describe the time elapsingbetween collisions among gas and test particle, it should not1050-2947/2010/82(3)/036101(3) 036101-1 ©2010 The American Physical SocietyCOMMENTS PHYSICAL REVIEW A 82, 036101 (2010)depend only on the state of the gas [via the temperature in(1) or more generally via µgas( p) = 〈 p|ρgas| p〉], but it mustalso depend on the motional state of the test particle. Thedefinition (1) is therefore inappropriate, in particular if the testparticle is much faster than the gas particles. In addition, (1) isnot well defined because the energy dependence of σtot is notspecified. The physically meaningful mean intercollision timeis given by the expression [6]τ−1phys[ρ] =∫d P〈P |ρ|P〉ngas∫dpµgas( p)× vrel( p,P)σtot[Erel = m∗v2rel( p,P)/2], (2)with vrel( p,P) = | p/m − P/M| the relative velocity, andm∗ the reduced mass. However, the use of the appropriatedefinition (2) in [1] would yield a nonlinear time evolutionequation for ρ, thus violating the basic requirement of a linearquantum state evolution.From a physical point of view, the dependence of thecollision rate on the state of the test particle is rather important,not least because the total cross section may depend strongly onErel for large test particle velocities. Moreover, this dependenceis necessary to grant the approach to the stationary solution.We note that the QLBE discussed in [2–4] incorporates thisdependence of the collision rate on ρ [by means of a rateoperator with expectation value τ−1phys[ρ]; see Eq. (12) in [3]],while the required linearity in ρ is a result of the use of conceptsfrom the theory of generalized quantum measurements [7].5. Dynamics of momentum populationsAs a natural consistency requirement, one expects that oncethe test particle state is indistinguishable from a classicalphase space distribution, its dynamics should be governedby the classical linear Boltzmann equation (with a quantummechanical scattering cross section). This is not fulfilledin the proposed equation since at finite ngas the dynamicsof the populations in the momentum representation differsfrom the one predicted by the classical equation, implying adiscontinuous transition to the classical description at a finitegas density.6. No canonical stationary solutionAs a consequence of Sec. II A 5, the canonical thermal stateof the test particle, ρ ∝ exp(−βP2/2M), is not a stationarysolution of the proposed master equation in a Maxwell-Boltzmann gas. In particular, the collision kernel does notsatisfy the detailed balance condition. From a physical pointof view, it seems quite important that the state of maximalentropy corresponds to the stationary solution. This is the caseeven in extensions of the QLBE which account for quantumdegeneracies in the gas [4].7. Infinite “position diffusion” for ngas → 0The limit of quantum Brownian motion of the proposedmaster equation predicts a “position diffusion” coefficientDxx [given in Eq. (23) of [1]] that grows above all boundsas one decreases the gas density ngas → 0. However, oneexpects the free Schro¨dinger equation to be obtained in thelimit of vanishing gas density, so that the predicted behavioris obviously unphysical.B. Supposed linearity in the gasmomentum distributionLet us now comment on the supposed linearity in the gasmomentum distribution discussed in [1]. First, we note that (1)is a function of the gas temperature and therefore in generala functional of the gas momentum distribution, τ = τ [µgas].Writing the gas state dependence of the proposed equation ina consistent fashion, one thus finds that the proposed equationis a manifestly nonlinear expression in µgas, in stark contrastto the claim in [1]. The advertised advantage of the proposedequation is therefore unfulfilled.At the same time, it seems doubtful whether one can orshould expect linearity in µgas within a Markovian description.A linear behavior of the reduced time evolution with respect toan initial, uncorrelatedρgas(0) can be expected only based on anexact solution for system plus environment, while a Markoviandescription necessarily involves approximations, required toobtain a Lindblad structure of the generator, granting completepositivity of the quantum evolution [8]. The linearity in µgasis recovered easily in the framework of the QLBE once thetest particle momentum operator can be approximated by acharacteristic C-number in the argument of µgas.C. Momentum decoherenceWe now consider the “apparently overlooked” “surprisingcollisional decoherence effect” advocated in [1]. It is importantto stress that, far from being overlooked, the effect ofthe momentum exchange due to a single collision is fullyaccounted for in the QLBE, as already discussed in [4,9,10].The decoherence rate for a hypothetical superposition of twodifferent momentum states is predicted by the QLBE to bethe average of the corresponding total collision rates. Thisis indeed required on physical grounds since any collisionchanges the momentum by definition, and it is immediatelyseen by looking at the momentum representation of theequation, Eq. (2.7) in [4]: Considering the off-diagonal elementthat characterizes the coherence, one notes that the “gainterms” do not affect its temporal change because for an initialsuperposition of momentum eigenstates there are no otheroff-diagonal elements, so that the initial decoherence rate isgiven by the “loss terms” arising from the anticommutator,which amount to the arithmetic mean of the total collisionrates. Since there are no stronger physical decoherencemechanisms available, we consider any prediction beyond thisas unphysical.We emphasize that the detailed derivation of the QLBEapplies dynamic scattering theory to a wave packet decom-position of the relative motion [3]. As such, all the details ofthe dynamic quantum scattering process are incorporated byconstruction, including the particles’ energy uncertainty andthe finite interaction time. There is no room for additionaleffects in the framework of the two-particle Schro¨dingerequation. The limits of this treatment are met precisely ifthe test particle can never be considered asymptotically freebetween two collisions because it interacts with more than onegas particle all the time. The notion of an intercollision time ismeaningless in such situations, and it would be wrong to usescattering theory altogether.036101-2COMMENTS PHYSICAL REVIEW A 82, 036101 (2010)D. Non-Markovian extensionsAs for non-Markovian extensions, it is clear that the QLBEloses its validity at densities and temperatures where thecollisions can no longer be taken as independent two-particleevents, since the interactions between the gas particles start toplay a role and the collisions cannot be considered as complete.The related problem of the derivation of non-Markovian quan-tum kinetic equations for the description of self-interactingdense gases has been considered, e.g., in [11,12], relying onadvanced many-body techniques. In such frameworks non-Markovian quantum extensions of the nonlinear Boltzmannequation have been obtained, indicating that memory effectsin dense gases beyond the Born approximation generallyinvolve nontrivial memory kernels. The difficulties listedabove in Secs. II A 1–II A 7 might be avoidable by pursuingan analogous microscopic approach for the QLBE, rather thanintroducing arbitrary ad hoc modifications. It thus appearsthat entirely new theoretical approaches are required at highdensities, not least since non-Markovian effects or correctionscannot possibly be described by a master equation in Lindbladform with time-independent coefficients [13]. In any case, itseems reasonable to use the properties of the QLBE, includingits various limiting forms and its experimental confirmationsdiscussed in [4], as a benchmark consistency check for anyproposed extension of that master equation.III. CONCLUSIONIn summary, the equation proposed in [1] provides nophysically meaningful advantage over established results,while displaying conceptual inconsistencies, as well as leadingto unphysical predictions. We conclude that, rather thanbringing “a dramatic change in our understanding of thequantum behavior of the test particle,” the equation in [1]is a step in the wrong direction.ACKNOWLEDGMENTSB.V. acknowledges financial support by Ministerodell’Istruzione, dell’Universita` e della Ricerca (MIUR), underPRIN2008.[1] L. Dio´si, Phys. Rev. A 80, 064104 (2009).[2] K. Hornberger, Phys. Rev. Lett. 97, 060601 (2006).[3] K. Hornberger and B. Vacchini, Phys. Rev. A 77, 022112(2008).[4] B. Vacchini and K. Hornberger, Phys. Rep. 478, 71 (2009).[5] J. R. Taylor, Scattering Theory (John Wiley & Sons, New York,1972).[6] K. Huang, Statistical Mechanics (John Wiley & Sons, New York,1987).[7] K. Hornberger, Europhys. Lett. 77, 50007 (2007).[8] B. Vacchini, Phys. Rev. E 63, 066115 (2001).[9] H.-P. Breuer and B. Vacchini, Phys. Rev. E 76, 036706(2007).[10] B. Vacchini and K. Hornberger, Eur. Phys. J. Special Topics 151,59 (2007).[11] D. Kremp, M. Bonitz, W. D. Kraeft, and M. Schlanges, Ann.Phys. 258, 320 (1997).[12] V. G. Morozov and G. Ro¨pke, Ann. Phys. 278, 127 (1999).[13] H.-P. Breuer, E.-M. Laine, and J. Piilo, Phys. Rev. Lett. 103,210401 (2009).036101-3

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