Comment on "Abnormal Diffusion in Wind-tree Lattice Gasses"

The Boltzmann value of the diffusion coefficient in Gates's latti ce wind-tree model reported in reference [1] is corrected. The new expression is in agreement with low-density simulations. The low-density value of the diffusion coefficient calcula ted in reference [1] for a model of Gates [2J is incorrect . Here we obtain the correct expres sion, which agrees well in the relevant low-den sity regime with the molecular dynamics measurement of reference [IJ and wit h the new simulation data reported in table 1. T he model under con sid eration is a la ttice vers ion of t he Ehre nfest windt ree model, in which t rees are leftturning scatterers placed randomly at a fraction c of the sites of a square lat ti ce [2J. Let pi(n , t) represent the probabilit y of finding the wind particle a t time t at site n = {nx,ny} with arrival ( "precollisional") velocity e. , i = 1, 2, 3, 4 c D (Boltzmann) D (t = 128) D (t = 512) 0.1 2.25 2.24 2.22 0.2 1.00 1.00 0.94 0.3 0.58 0.58 0.55 0.4 0.38 0.35 0.32 0.5 0.25 0.21 0.18 Table 1: Scat terer concentration, D~x from equat ion (4) and diffusion coefficient (slope of th e meansquared displacement versus time) measured at 128 and 512 time steps. © 1989 Complex Syst ems P ublicat ions, Inc. 314 B. Th. Bouwens, M.H. Ernst, and P.M. Binder (mod 4) , equal to one of the nearest-neighbor lattice vectors. A configuration of scatterers is den ot ed by th e set of quenched random variables {cn }, where Cn takes the valu es 0 or 1 if site n is empty or occupi ed by a tree, resp ecti vely. T he Liouville equ at ion for the deterministic lattice Lorentz gas , intro du ced in reference [1J, is Pi(n + ei, t + 1) = (1 Cn)Pi(n ,t) +CnPi-l(n, t) (1.1) In the Boltzmann ap pro ximation all collisions in which the moving particle returns to a previously visited scatterer are neglect ed. In this approximat ion the average over the configurations of scatterers {Cn} can be direct ly performed by replacing (cn ) by its averag e value c. T his yields the model's Boltzmann equat ion. It reads, in the notation of reference [3], Pi(n + e., t + 1) = [(1 + cT)p(n , t)Ji (1.2 ) where Tij is a 4 x 4 collision matrix, formally written as T = b 1, with bpi = Pi-I . We note that , contrary to th e statement in reference [1] the giral scattering rules do not possess the full symmetry of the squ ar e lattice. T he diffusion tensor Det{3> where 0 , j3 denote cartesian components (x ,y) , is given by the long-time behavior of ~llt(ret(t)r,6(t )), where llta(t) = a(t + 1) a(t) is a forward t ime difference. It is therefore symmetric in 0 and j3 (Onsager symmetry) , and given by the Green-Kubo formula, 1 00 Det,6 = 2' L [cpet,6(T ) + CP,6et(T) CPet,6(O) ] 7=0 (1.3) where CP et,6 = (vet(t)v,6(t)) is the velocity correlation function . According to reference [3] it is given in the Boltzmann approximat ion by the symmetric part of ~ "e (cT):-le ',6 ~8 ,6 40 r o 'J J 4 c t (1.4) T he term involving ~CPet,6(O) = ~8et,6, is the "propagation diffusion" resulting from the discret e structure of space and time [4]. As it is of re lative order c, it was neglected in the low density limit of reference [3J. To evaluate equation (1.4) we calculate the rele vant eigenvectors and eigenvalues of Iij , defined through cTv = AV . The eigenvectors are vJ = ejx ± iejy, with corresponding eigenvalues A± = c(l ±i). Inserting these results in equ at ion (1.4) yields for the diffus ion tensor in Gates's model