Comment on "Anomalous heat conduction and anomalous diffusion in one-dimensional systems"

0.06 MB·English

by  Ralf Metzler

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Comment on “Anomalous heat conduction and To explain this need for caution let us first address
anomalous diffusion in one-dimensional systems” subdiffusion, which corresponds to a long-tailed wait-
A relation between anomalous diffusion, in which the ing time distribution of the form ψ(t) ∼ (t/t0)−1−α/t0
meansquareddisplacementgrowsintimelikeh(∆x)2i= (0 < α < 1) [3]. In this case, it was shown in Ref. [4]
2D tα (0<α≤2), and anomalous heat conduction was thatthetemporaleigenfunctionsforafinitegeometryare
α
recently derived through a scaling approach by Li and given by Mittag-Leffler functions, and therefore the sur-
4 Wang [1] (hereafter LW). In this model it is assumed vival probability decays like t−α. Thus, the associated
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that heat transport in a 1D channel is solely due to the mean first passage time diverges: T = ∞, correspond-
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flowofnon-interactingparticles: thoseenteringthechan- ing to the dominance ofthe probabilityofnot movingin
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nel from the left have a different average kinetic energy subdiffusion[3]. (Weshouldnotethatinoneofthethree
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than those entering from the right. The energies of the references, Ref. [5], cited in LW to support their scaling
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particlesatbothendsofthe channelaredefinedthrough relation,the resultfor the firstpassagetime distribution
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the Boltzmanndistributionsthatcorrespondto thetem- is based on an integral expression, which diverges for a
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peratures of two heat baths coupled to either end. The waiting time distribution of long-tailed nature, and is

] authorsarecorrectinstatingthatdifferentbilliardmod- therefore wrong.) Without an externalbias, the conduc-
h elsdiscussedinliteraturebelongtothisclassofprocesses. tivity of a subdiffusive system vanishes [4, 6]. The other
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However, in this Comment we point out certain crucial case, in which the approach presented in LW fails are
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m inconsistenciesoftheLWmodelwiththephysicalpicture L´evyflights[3]. Theirmeanfirstpassagetime existsand
of random processes leading to normal and anomalous is finite, as can be shown using the methods described
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t diffusion. in Ref. [7]; however, their mean squared displacement
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Firstly, consider the collision-free heat transport be- diverges [8].
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s tween the two heat baths. This situation corresponds It must therefore be concluded that the model pro-
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t to ballistic transport, α = 2, and the mean first pas- posed in LW is by far less general than assumed there,
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m sagetime acquiresthe scaling T ∝L/v. In this case,the and due to the combination of two a priori unrelated
modelofLWreproducestheoriginalresult[2]fortheheat equations contains a crucialflaw in the foundations such
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d conductivity,κ∝L. Thefirstinconsistencybecomesap- that erroneous results ensue for both subdiffusion and
n parent already in this limiting case: since the typical L´evy flights. We also note that in a related context a
o velocities of particles entering the channel from the left model developed in Ref. [9] provides analytical and nu-
c and from the right are different, the corresponding left merical results for the heat conductivity consistent with
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and right mean first passage time necessarily differ, as our objections.

1 well. The equality of both first passage times invoked Finally,wepointoutthattheinterpretationintermsof
v in LW can only be fulfilled if the particles are thermal- the finite-time measurement in the case of subdiffusion,
9 izedwithin thechannel;however,underthisassumption, brought forth in the Reply of LW [10], would lead to
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theballisticnatureislostandthe wholemodelnolonger a correct result. However, it would cause an explicitly
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holds. Partially, this problem may be circumvented by cutofftime-dependentmeanfirstpassagetime,andwould
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taking the limiting transition T −T →0 in Eq. (4). therefore be different from the original model developed
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4 The crucial flaw in LW, as we are going to show now, in Ref. [1].
0 isthefactthatEq.(1)doesnotnecessarilyimplyEq.(2) We acknowledge discussions with A. Chechkin, S.
/ in the range 0 < α ≤ 2, and vice versa. Thus, although Denisov, J. Klafter and N. Korabel.
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a itistemptingtoarguethatifthetypicaldisplacementof Ralf Metzler
m the particlegrowslikeh(∆x)2i1/2 ∝tα/2 thenthe typical NORDITA – Nordic Institute for Theoretical Physics
- time for traveling a distance L will scale like τ ∝ L2/α, Blegdamsvej 17, 2100 Copenhagen Ø, Denmark
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one cannot conclude what exactly this time τ defines: it Igor M. Sokolov
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may well differ from the mean first passage time, T. In Institut fu¨r Physik, Humboldt-Universit¨at zu Berlin
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c particular, the latter may even diverge while τ exists. Newtonstraße 15, 12489 Berlin, Germany
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a [1] B. Li and J. Wang, Phys.Rev.Lett. 91, 044301 (2003) 64, 021107 (2001).
[2] Z.Rieder,J.L.LebowitzandE.Lieb,J. Math.Phys.8, [8] We note that for L´evy walks, which are often used to
1073 (1967). describe sub-ballistic superdiffusion [3], it is not known
[3] J. Klafter, A. Blumen and M. F. Shlesinger, Phys. Rev. what is theexact scaling of the mean first passage time.
A 35, 3081 (1987). Thus, also for superdiffusion it is not clear whether the
[4] R. Metzler and J. Klafter, Physica A 278, 107 (2000); result in LW holds.
compare to Phys.Rep. 339, 1 (2000). [9] S. Denisov, J. Klafter and M. Urbakh, eprint
[5] M. Gittermann, Phys. Rev.E 62, 6065 (2000). cond-mat/0306393. Phys. Rev.Lett., at press
[6] H. Scher and E. W. Montroll, Phys. Rev. B 12, 2455 [10] B. Li and J. Wang, Reply
(1975).
[7] I. M. Sokolov, J. Klafter and A. Blumen, Phys. Rev. E