Random walk approach to the d-dimensional disordered Lorentz gas
RRandom walk approach to the d -dimensional disordered Lorentz gas Artur B. Adib ∗ Laboratory of Chemical Physics, NIDDK, National Institutes of Health, Bethesda, Maryland 20892-0520, USA (Dated: October 29, 2018)A correlated random walk approach to diffusion is applied to the disordered non-overlappingLorentz gas. By invoking the Lu-Torquato theory for chord-length distributions in random media[J. Chem. Phys. , 6472 (1993)], an analytic expression for the diffusion constant in arbitrarynumber of dimensions d is obtained. The result corresponds to an Enskog-like correction to theBoltzmann prediction, being exact in the dilute limit, and better or nearly exact in comparison torenormalized kinetic theory predictions for all allowed densities in d = 2 ,
3. Extensive numericalsimulations were also performed to elucidate the role of the approximations involved.
Keywords: Lorentz model; Boltzmann-Lorentz model; Lorentzian gas; Transport
I. INTRODUCTION
The problem of computing the transport coefficientsof collisional models from first-principles can be ap-proached with a variety of methods, each with differingscope and complexity. At the most elementary level, ki-netic arguments based on the concept of mean-free-pathyield estimates with the correct dependence on the rele-vant parameters, but as one might expect miss the cor-rect prefactors even at low densities [1]. Progress canbe made using the Boltzmann equation and the asso-ciated Chapman-Enskog method, which give the exactlow-density limit, though at the expense of substantiallygreater effort [2]. At higher densities, renormalized ki-netic theory accounts for the peculiar divergences onefaces when expanding the coefficients in powers of den-sity [3], though here too the improvement comes at thecost of increased complexity in the calculation.For the case of the periodic Lorentz gas – i.e. thesimple model of a point particle colliding with a fixedarray of spherical scatterers – an alternate approach ofgreat simplicity that bypasses the above kinetic routehas been proposed by Machta and Zwanzig [4]. Theseauthors recognize the fact that for small enough spac-ings between the scatterers (i.e., in the opposite limit ofthe dilute approximation), the diffusing particle spendsmost of its time trapped inside effective “cages” beforefinding its way into an adjacent trapping region. Theparticle motion can thus be approximated as a randomwalk on a lattice, where the rate of transition is given bythe cage escape rate, and consequently a simple analyticformula for the diffusion constant D can be derived. Themethod has been subsequently extended and refined (seee.g. [5, 6, 7]), but its success remains contingent uponthe prevalence of such trapping events.The approach adopted in this paper—though concep-tually different from that of Machta and Zwanzig—is anattempt to arrive at a result of similar simplicity andaccuracy for the disordered case, without relying on the ∗ Electronic address: [email protected]
FIG. 1: (Color) A typical trajectory of the non-overlappingLorentz gas in d = 2 at packing fraction φ = 0 .
4. The pointparticle propagates freely at constant velocity v until it suffersspecular collisions with the scatterers, each of radius R . Inset:Representative segment of the trajectory trapped in an effec-tive “cage,” an event that is ultimately responsible for corre-lations in the displacements ∆ x between collisions. In highernumber of dimensions or at lower densities, where greatervoids exist between neighboring scatterers, such events areeffectively suppressed. existence of kinetic traps. The method is a simple appli-cation of correlated random walks (CRW), whose mainresult is the ability to express D very simply in terms ofthe correlation and moments of free-paths [8]. For thepresent problem, these can be easily computed by invok-ing the scattering properties of point particles by hyper-spheres (Eq. (4)) and the distribution of chord-lenghts inrandom media [9], respectively. A closely related appli-cation of CRWs has been used in the study of Knudsendiffusion in porous media [10, 11]. In the following, thecorrelated random walk method for the computation ofthe diffusion constant will be briefly reviewed and appliedto the d -dimensional Lorentz gas. All calculations will bereported for the disordered non-overlapping case, wherethe scatterers are distributed like a liquid in equilibrium. a r X i v : . [ c ond - m a t . s t a t - m ec h ] F e b II. THEORY
To begin the derivation, note that the net displacementof a particle starting from the origin consists of discretesteps ∆ x i between successive collisions i − i (seeFig. 1), so that x ( n c ) = (cid:80) n c i =1 ∆ x i , where n c is the num-ber of collisions. Thus, using the fact that n c is relatedto the continuous time t through t = (cid:80) n c i =1 ∆ t i = n c (cid:104) ∆ t (cid:105) (the last equality being valid in the large n c limit) andthat ∆ t = ∆ x/v where v ≡ | v | is a constant, routinemanipulations of the mean square displacement (cid:10) x ( n c ) (cid:11) give an exact formula for D expressed solely in terms ofthe statistics of the collision vectors, namely D = v d (cid:104) ∆ x (cid:105) (cid:34)(cid:10) ∆ x (cid:11) + 2 ∞ (cid:88) l =1 (cid:104) ∆ x · ∆ x l (cid:105) (cid:35) , (1)where ∆ x is the displacement of the particle before itsfirst collision. This expression can be seen as the discreteanalog of the so-called Green-Kubo formula for the dif-fusion constant in terms of the continuous velocity corre-lation function (curiously, the latter was already derivedby Taylor himself decades before Green and Kubo [8],suggesting the more appropriate nomenclature Taylor-Green-Kubo for that result [7]).The assumptions specific to the Lorentz gas that willbe made (and tested) in the remainder of the paper canbe cast directly in terms of the statistical properties ofthe collision vectors ∆ x = ∆ x ˆ v , where ˆ v is the unitvelocity vector. To wit, it will be explicitly assumedthat (a) their magnitudes ∆ x are uncorrelated, and that(b) their orientations ˆ v change in a Markovian fashion.The last assumption is the simplest condition consistentwith the dynamics of the gas particles, as the underlyingspecular collision rules manifestly introduce angular cor-relations between successive velocity vectors, and hencethese cannot be assumed to be completely uncorrelated.As such assumptions are essentially equivalent to Boltz-mann’s molecular chaos hypothesis, they are satisfied fora random arrangement of scatterers at sufficiently lowdensities, and thus the ensuing predictions for D shouldbe exact in this limit.With assumption (a), the infinite sum in Eq. (1) sim-plifies to (cid:104) ∆ x (cid:105) (cid:80) ∞ l =1 (cid:104) ˆ v · ˆ v l (cid:105) , and in accordance with(b), we can invoke the Chapman-Kolmogorov relation toexpress the conditional probability P (ˆ v l | ˆ v ) in terms of P (ˆ v l | ˆ v l − ) and hence obtain the recursion relation (cid:104) ˆ v · ˆ v l (cid:105) = (cid:104) cos θ (cid:105) (cid:104) ˆ v · ˆ v l − (cid:105) , (2)where (cid:104) cos θ (cid:105) ≡ (cid:104) ˆ v l · ˆ v l − (cid:105) (this result assumes that P (ˆ v l | ˆ v l − ) has azimuthal symmetry about ˆ v l − , as inthe case of scattering by hyperspheres). This relationcan be iterated to yield the simple result [19] (cid:104) ˆ v · ˆ v l (cid:105) = (cid:104) cos θ (cid:105) l . (3)Now, from the specular collision rules of the gas, a directgeometric computation assuming a uniform flow of inci-dent particles towards a d -dimensional spherical scatterer d =2 d =3 l ! % E rr o r d =2 d =3 ! ˆ v · ˆ v l " FIG. 2: (Color) Discrete velocity correlation function of theLorentz gas at packing fraction φ = 0 . d = 2 and d = 3,according to simulation results (main panel, symbols) andthe Markovian prediction given by Eqs. (3) and (4) (mainpanel, lines). Inset: Relative error between simulation andthe Markovian prediction for the integrated autocorrelationfunction P ∞ l =0 (cid:104) ˆ v · ˆ v l (cid:105) . gives the following expression for the average scatteringcosine: (cid:104) cos θ (cid:105) = d − d + 1 , (4)a result that can be easily checked against the usual dif-ferential cross-sections of hard disks and hard spheres(compare with the analogous result for the case of porousmedia in d = 3 [10, 11]). In Fig. 2 the above predictionsfor the discrete velocity correlation function (Eqs. (3) and(4)) are compared with simulation results, illustrating inparticular the robustness of the Markovian approxima-tion even at high densities.Use of the foregoing in Eq. (1) reduces the dependenceof D on the series of collision vectors to their first twomarginal moments, namely D = v (cid:104) ∆ x (cid:105) d (cid:34)(cid:32) (cid:10) ∆ x (cid:11) (cid:104) ∆ x (cid:105) − (cid:33) + d + 14 (cid:35) . (5)This result is a straightforward consequence of assump-tion (b) in correlated random walks (analogous to whatwas done originally in one dimension [8]), where themain additional ingredient is the explicit computationof the degree of correlation between adjacent ˆ v , i.e. (cid:104) ˆ v l · ˆ v l − (cid:105) = (cid:104) cos θ (cid:105) , given by Eq. (4) for the presentproblem. Generalization to non-spherical obstacles iseasily performed, provided one knows their scatteringproperties so that a formula similar to Eq. (4) can beobtained. ! =0.3 ! =0.4 ! =0.6 ! =0.8 d =2 d =3 ! =0.3 ! =0.4 ! =0.5 ! =0.6 l og [ p ( ∆ x / R ) ] l og [ p ( ∆ x / R ) ] ∆ x / R ∆ x / R FIG. 3: (Color) Distribution of collision distances p (∆ x/R )in the Lorentz gas for d = 2 and d = 3. The distributionsare arbitrarily normalized for clarity of presentation. Thesymbols are simulation results at the indicated packing frac-tions, while the lines are exponential distributions with theexponents λ predicted by the Lu-Torquato theory for chord-lengths (Eq. (7)). A. Low-density behavior
To check the above prediction for D against knownresults in the dilute (Boltzmann) limit, first note thatin this case the ∆ x are exactly exponentially distributed[1], and hence the term in parenthesis in Eq. (5) vanishes.The first moment of this distribution is given by 1 / ( σρ ),where σ is the total cross section of the obstacles and ρ = N/V is their number density. For d = 2 and d = 3,this yields, respectively, D B vR = 3 π
16 1 φ , and D B vR = 49 1 φ , (6)where φ is the packing fraction of the obstacles, and thesubscript emphasizes that these results are only valid inthe Boltzmann limit. These are precisely the same pre-dictions obtained via more sophisticated methods basedon, e.g., the Chapman-Enskog method or Zwanzig’s op-erator expansion scheme [12, 13, 14]. In accordance with ! D / ( v R ) Elementary kineticsBoltzmannWeijland-van Leeuwen d= ! max ! D / ( v R ) Present workSimulation d =3 ! max FIG. 4: (Color) The diffusion constant of the Lorentz gasin d = 2 (top) and d = 3 (bottom) according to the vari-ous theories mentioned in the paper. Legends are the samefor both figures. The elementary kinetics predictions are D/ ( vR ) = ( π/ φ − for d = 2 and D/ ( vR ) = (4 / φ − for d = 3, the latter being identical to the Boltzmann predic-tion and hence not shown. The long-dashed lines are theBoltzmann results given by Eq. (6), while the red curves arefrom renormalized kinetic theory [12, 13]. The analytic re-sults of this work (blue curves, closer to simulation results)are given by Eq. (9), and are visually indistinguishable fromwhat one obtains by using Eq. (5) with the required momentsmeasured from the simulation itself (not shown). The arrowsindicate the maximum packing fractions φ max ≈ .
907 and φ max ≈ .
740 for d = 2 ,
3, respectively [9]. the general rule [1], such predictions differ from those ofelementary kinetic theory—namely, D K = v (cid:104) ∆ x (cid:105) /d —by factors of order unity. (An accidental exception inthe present problem is the d = 3 case, due to the ef-fectively randomized nature of its scattering, as can beseen by the vanishing of Eq. (4)). In terms of the presentframework, it can be easily seen that the source of errorin elementary kinetic theory lies in its complete neglectof correlations in ˆ v ; indeed, had we made this assump-tion in our derivation, our expression for D would havebeen coincident with D K . ! C o rr e l a t i on t i m e o f ! x ( un i t s o f c o lli s i on s ) d =3 d =2 Number of collisions A u t o c o rr e l a t i on o f ! x ! =0.4, 0.6, 0.8 d =2 FIG. 5: (Color) Correlation times of the collision distances∆ x for d = 2 , x over all possible lags ( l = 1 , , . . . , ∞ ). Inset: Theautocorrelation function itself for three densities in d = 2. B. High-density extension
To go beyond the Boltzmann limit, a theory for thedistribution of ∆ x at high densities is necessary. Herewe invoke the Lu-Torquato theory for chord-length dis-tributions in random media [15, 16]. This distribution isdefined as the probability of finding an unobstructed linesegment of length ∆ x connecting the surfaces of any twoobstacles [9]. The Lu-Torquato approach is conceptuallysimple and elegant, and uses the scaled-particle methodof liquid theory to relate chord-length distributions to theproblem of finding a thin cylindrical cavity in the liquid.The central prediction for non-overlapping spheres is thatthe distribution is exponential, viz. p (∆ x ) ≡ λe − λ ∆ x ,with exponent given by [9] λ = ω d − ω d φ − φ R , (7)where ω d ≡ π d/ / Γ(1 + d/
2) is the volume of a d -dimensional unit sphere. A comparison of these predic-tions with the present Lorentz gas simulations is shown inFig. 3. With the exception of the salient non-exponentialbehavior for φ = 0 . d = 2 (due to excessive trapping),the theory is in excellent agreement with the numericalresults. Finally, with Eqs. (5) and (7) we arrive at our resultfor the diffusion constant of the d -dimensional Lorentzgas: DvR = d + 14 d ω d ω d − − φφ , (8)which for the particular cases of d = 2 and d = 3, reads,respectively, DvR = 3 π
16 1 − φφ , and DvR = 49 1 − φφ . (9)Note that, incidentally, this result can be obtainedthrough an ad hoc excluded-volume (i.e. Enskog-like [17])correction to the Boltzmann collision frequency, whichultimately adds an extra factor of 1 − φ to Eq. (6).The above predictions are compared with simulationas well as with the renormalized kinetic theory of Weij-land and van Leeuwen [12, 13] in Fig. 4. As immediatelyapparent, the theory is in excellent agreement with sim-ulation for d = 3, while the agreement for d = 2 is lesssatisfactory; this can be traced back to the sensitivity ofassumption (a) to the dimensionality of the problem (seeFig. 5), as already anticipated in Fig. 1. III. CONCLUSIONS
To summarize, in this paper a simple analytic expres-sion for the diffusion constant of the disordered non-overlapping Lorentz gas in arbitrary number of dimen-sions was derived (Eq. (8)). This result was obtainedthrough a simple application of correlated random walks,combining the analytic predictions of Lu and Torquato(Eq. (7)) with the exact scattering properties of pointparticles by hard hyperspheres (Eq. (4)). The simplic-ity and accuracy of the ensuing theory in comparison toprevious renormalized kinetic methods [12, 13] are re-markable (cf. Fig. 4), and in this regard the goal set outin the introduction was achieved [20]. It is hoped thatthese encouraging results will foster the application ofsimilar ideas to more general models of diffusion.
Acknowledgments
The author is indebted to Attila Szabo for numer-ous discussions and suggestions. This research was sup-ported by the Intramural Research Program of the NIH,NIDDK. [1] D. A. McQuarrie,
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Polymer Physics (Ox-ford, Oxford, 2003).[19] As the reader might have noticed, this is a simple gen-eralization for the case of non-fixed angles of the wellknown result for the correlation between bond vectors ina freely rotating ideal polymer [18].[20] It should be noted, however, that the present approachis not a low-density expansion, and hence it is not sur-prising that it does not capture the subtle logarithmicdivergences of D as φ → d = 2 ,,