- Email: [email protected]
Long time tails in a Lorentz gas
Physica A 153 (1988) 67-72 North-Holland, Amsterdam
LONG TIME TAILS IN A LORENTZ GAS Shankar P. DAS and Matthieu H. ERNST* Department of Physics, University of Florida, Gainesville, FL 32611, USA
Received 23 June 1988
To investigate the approach to the asymptotic long time behaviour in a d-dimensional Lorentz gas, we have calculated the ~7(t-2-dj2) correction to the long time tail t -1-d/2 in the velocity autocorrelation function, using low density kinetic theory. The results are compared with existing computer simulations.
In a large number of dynamical systems time correlation functions are found to decay for very long times as a power law, usually referred to as the long time tails. They were first observed in a computer simulation of the hard sphere fluid by Adler and Wainwright [1]. The authors found that the velocity autocorelation function (VACF) has a positive long time tail proportional to t -d/2 which is also typical for real fluids [2]. Furthermore, long time tails also occur in diffusive systems with static disorder, such as Lorentz gas and lattice percolation models [3]. Here the tails are negative and decay as t -~-d/2. These are caused by slow collective modes (hydrodynamic, diffusive) in the system. The existence of these tails is by now well understood on the basis of the fundamental kinetic theory for moderately dense gases [4-6] and also on the basis of more phenomenological theories such as mode coupling theories [7, 8] and fluctuating hydrodynamics [9]. Either method has its obvious advantages and drawbacks. In fluids, the comparison between the results from theory and computer simulations is good at low and moderate densities but not so good at higher densities [10]. In this work, however we will be interested in the asymptotic dynamics in a Lorentz gas [5] which consists of a diffusing particle in a random array of fixed scatterers usually hard spheres or hard disks. Ernst and Weyland [5] showed from a low density kinetic theory that the VACF in a Lorentz gas decays as - A t -(d/2+1). In computer simulations of the two dimensional Lorentz gas [11-13], a decay --t -2 was found but the * Permanent address: Institute for Theoretical Physics, State University, 3508 TA Utrecht, The Netherlands.
0378-4371 / 88 / $03.50 O Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
S.P. Das and M . H . Ernst / Long time tails in a Lorentz gas
68
coefficient A was almost a factor of four larger than predicted by the theory. However, by extrapolating their low density results towards zero density Alder and Alley [13] obtained numerical evidence that the coefficient A is fairly close to the value predicted by the low density kinetic theory. Unfortunately these low densities are not accessible in computer simulations. Among many others, two obvious reasons for the disagreement between theory and simulations might be: (i) the coefficient A(c) has a strong density dependence; (ii) the VACF has not yet reached its asymptotic tail in the time regimes where simulations have been carried out. The first point has been investigated by Machta et al. [8]. These authors have made an estimate of the coefficient A, as given by the mode coupling prediction. Their estimate gave A(c) - 1.8A(c = 0) for density c = 0.05, which suggests a strong density dependence. Here we explore the second point in order to obtain a better understanding of the crossover behaviour from intermediate to very long times. We were motivated to do so by recent results for long time tails of the VACF in site and bond percolation models. These models can be interpreted as a lattice Lorentz gas, where a blocked site or bond is interpreted as a hard scatterer. For these models kinetic theory calculations have been extended to one order higher in the concentration of the scatterers and to the first subleading asymptotic time tail [14]. Comparison of the theory with computer simulations at low densities show excellent agreement [15, 16]. At the concentrations c - 0 . 0 5 and in the time interval (10 ~< t ~<60) considered in the simulations both corrections are approximately of equal size. We carry out a similar calculation for the Lorentz gas, by extending the kinetic theory further to compute the subleading behaviour of the asymptotic tails. This is again done to lowest order in the density. We do obtain an effect qualitatively similar to the simulation results but the quantitative agreement is very poor, indicating that the higher order density corrections play an important role in the long time behaviour of the Lorentz gas. We start with the normalized VACF ~b(t)~ (z). ¢)(t)) where 13= v/v is a d-dimensional unit vector and ( . . . ) is a canonical average over an equilibrium ensemble in which v is fixed. The Laplace transform t~(z) of ~(t) can be expressed in terms of a memory function ~,(z) [6], d)(z) = f dt e-Zt~b(t) = (z + ~(z))-' .
(1)
0
In the ring approximation it is given by
~-~
z- nTo-ik.v
'
(2)
S.P. Das and M . H . Ernst I L o n g time tails in a Lorentz gas
69
where n is the density of scatterers and T k is the Fourier transform of the binary collision operator [5, 6],
T k = vo "d-1
f
d6-16 • (r I e -ik . . .[o~ . - 1),
(3)
v.o'<0
where the operator b,, is defined as b,,v = v - 2&(&- v) and o- = o-&. One may also define a time dependent diffusion coefficient as t
D(t) = (v2/d) f d r qJ(r).
(4)
0
The quantity % = ~,(0) in (2) is related to the Boltzmann value of the diffusion coefficient for a Lorentz gas; D O= v2/dTo, with D O= (3v/16mr) for d = 2 and D O= (v/3nTro "2) for d = 3 where or is the radius of a scatterer. To discuss the long time properties of the VACF at small density of scatterers, we need the leading and the subleading singularities in 4/(z) around z = 0. They are determined by the small k region in (2), where the denominator vanishes. This occurs where k - n ~ lo 1, 10 being the mean free path. Consequently [klcr-cr/lo¢l and the operators T k in (2) may be replaced by T 0. Following a procedure that is conceptually simple and straight forward, we first calculate the memory function T(t) in time language. We present the time evolution operator by its spectral decomposition: et(ik'"+"T°) : ~ ]
e"°t(k)Pt(k)
(5)
l
using eigenmodes and eigenvalues of the Boltzmann equation
(ik. v + nTo)q~(kO(f~) = oJt(k)q~°(~3).
(6)
Here Pt(k) projects an arbitary function f(fi) on the subspace spanned by the eigenfunction q~o(fi) such that ~o~, ~ok /.
(7)
The bracket denotes an average over a d-dimensional solid angle (f(~3)) = .f dfi f ( f i ) / f d~. For long times, only those eigenmodes are relevant that have a vanishing eigenvalue o~t(k) as k---~0. In a Lorentz gas there is only one slow mode (corresponding to mass conservation) which is the diffusive mode. Eigenmodes with nonvanishing eigen values at small k contribute terms to T(t) that decay exponentially fast at large t and are therefore neglected.
70
S.P. Das and M.H. Ernst / Long time tails in a Lorentz gas
For our purpose the eigenmodes and the eigenvalues are needed to ~(k 2) and 6(k 4) respectively. After lengthy calculations we obtain the eigenfunction and the eigenvalue from a perturbation expansion in k: ~k = 1 + ik
dD o v
k2dDo(
x -
v
x2
1) -
,
(8)
where x =/~- ft. The eigenvalue is 3
w°(k) = - D ° k 2
D_~ k4 + - . .
(9)
d +2 v
In principle one needs (~(k 3) terms in (8), but there contributions vanish due to parity. Once this is done the long lived part of (5) is inserted into the Laplace inverse y(t) of eq. (2), the k-integrals are performed and the resulting expression is Laplace transformed, yielding the dominant small z singularities in the form ~,(z) = 3'0 + azd/2 + bzd/2+l" Combination of this result with eq. (1) determines the dominant small z singularities of ~(z). Finally, using the Tauberian theorem that the small z behaviour of z ~ corresponds to a long time behaviour t - ~ - l / F ( - a), we find the leading and subleading terms in the long time behaviour of the VACF for a d-dimensional Lorentz gas to lowest order in the density as
~O(t) =
2~dD 2
1
v2 n
(47rOt)d/2+1
[ 1+
3(d + 1)(2d + 3) 1 ] 16 ~-~ ,
(10)
where v is the collision frequency given by [17]
d-tTf(d-1)/2
(11)
nvo" v = r [ ( d + 1)/2]
Furthermore, from (5) and (10) we get the long time tail for the diffusion coefficient [ 1 ( 1 ~d/2 3 ( d + l ) ( 2 d + 3 ) D(t) = D O 1 + ~ \4--~D~/ + 16n(d + 2)
( 1 ~d/2 1 ] \4--~-~! ~-~ .
(12)
For d = 2, ~0(t) can be written in the following dimensionless form: ~b(~-)= - ( n * / ~ - e ) [ 1 + 63/16~ + - " .]
(13)
S.P. Das and M.H. Ernst / Long time tails in a Lorentz gas
71
i
7.00 6.00 5.00 4.00 3.00
t
2.00 1 .00 0.00
5.00'
7100
9100
11t,00 "E
13t.00
15'.00 '
17t.00
19t.00 ' 2 1 . 0 0
Fig. 1. C o m pu t er results for the velocity autocorrelation function in two dimensions, C ( 7 ) =
-(Irr2/n*)[qJ(~")-exp(-4T/3)] for n* =0.05. The exp(-4~-/3) represents the contribution calculated from the Boltzman equation and is subtracted out to show only the memory effects. The solid line represents the result from eq. (13). and f o r d = 3
as
~0(~) = - n * 2 ( 3 ~ r ) 3 / 2 / ( 1 6 r s / 2 ) [ 1
+ 27/4~" + - . . ] ,
(14)
where ~- = ut and n* = nor d. Next, we compare our results with the computer simulations of Alder and Alley [13] as shown in fig. 1. The figure also shows our results for the normalized VACF written in a dimensionless form C(r) = - (~rrZ/n*) qJ(r). The simulation corresponds to a density n* = 0.05 and we have subtracted the short time part of the VACF, e -4z/3 c o m p u t e d from the Boltzmann equation. F r o m the data shown in fig. 1 we see that the quantitative agreement is rather p o o r when comparing the simulation results and the calculated leading plus subleading long time tails to lowest order in the density. H o w e v e r , our results for the VACF in eq. (13) explain in a qualitative fashion the slow approach to the dominant tail. The coefficient of the subleading term in eq. (13) also gives a rough estimate, namely % = 6 , for the crossover time f r o m intermediate to asymptotic dynamics. F u r t h e r m o r e we have to conclude that finite density corrections to the long time tail play an important role in the two-dimensional Lorentz gas.
Acknowledgement One of us (S.P.D.) was supported by National Science Foundation G r a n t C H E 8411932.
72
S.P. Das and M . H . Ernst / L o n g time tails in a Lorentz gas
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17]
B.J. Alder and T.E. Wainwright, Phys. Rev. A 1 (1970) 18. See Physics Today, January, 1984. M.H. Ernst, J. Machta, J.R. Dorfman and H. van Beijeren, J. Stat. Phys. 34 (1984) 477. R. Dorfman and E.G. Cohen, Phys. Rev. Lett. 25 (1970) 1257. M.H. Ernst and A. Weyland, Phys. Lett. A 34 (1971) 39. J.M.J. Van Leeuwen and A. Weyland, Physica 36 (1967) 457. A. Weyland and J.M.J. Van Leeuwen, Physica 38 (1968) 35. M.H. Ernst, E.H. Hauge and J.M.J. Van Leeuwen, Phys. Rev. Lett. 25 (1970) 1245. J. Machta, M.H. Ernst, J.R. Dorfman and H. van Beijeren, J. Stat. Phys. 35 (1984) 413. D. Forster, D.R. Nelson and M.J. Stephen, Phys. Rev. A 16 (1977) 732. T.T. Erpenbeck and W.W. Wood, Phys. Rev. A 32 (1985) 23. C. Bruin, Phys. Rev. Lett. 29 (1972) 1670; Physica 72 (1974) 261. J.C. Lewis and J.A. Tjon, Phys. Lett. A 66 (1978) 349. B.J. Alder and W.E. Alley, J. Stat. Phys. 19 (1978) 341; Physica A 121 (1983) 523. W.E. Alley, Ph.D. thesis, University of California/Livermore (1979). Th. M. Nieuwenhuizen, P.F.J. van Velthoven and M.H. Ernst, Phys. Rev. Lett. 57 (1986) 2477. D. Frenkel, Phys. Lett. A 121 (1985) 385. J.J. Brey and A. Santos, Phys. Lett. A 127 (1988) 5. M.H. Ernst and H. van Beijeren, J. Stat. Phys. 26 (1981) 1.
LONG TIME TAILS IN A LORENTZ GAS Shankar P. DAS and Matthieu H. ERNST* Department of Physics, University of Florida, Gainesville, FL 32611, USA
Received 23 June 1988
To investigate the approach to the asymptotic long time behaviour in a d-dimensional Lorentz gas, we have calculated the ~7(t-2-dj2) correction to the long time tail t -1-d/2 in the velocity autocorrelation function, using low density kinetic theory. The results are compared with existing computer simulations.
In a large number of dynamical systems time correlation functions are found to decay for very long times as a power law, usually referred to as the long time tails. They were first observed in a computer simulation of the hard sphere fluid by Adler and Wainwright [1]. The authors found that the velocity autocorelation function (VACF) has a positive long time tail proportional to t -d/2 which is also typical for real fluids [2]. Furthermore, long time tails also occur in diffusive systems with static disorder, such as Lorentz gas and lattice percolation models [3]. Here the tails are negative and decay as t -~-d/2. These are caused by slow collective modes (hydrodynamic, diffusive) in the system. The existence of these tails is by now well understood on the basis of the fundamental kinetic theory for moderately dense gases [4-6] and also on the basis of more phenomenological theories such as mode coupling theories [7, 8] and fluctuating hydrodynamics [9]. Either method has its obvious advantages and drawbacks. In fluids, the comparison between the results from theory and computer simulations is good at low and moderate densities but not so good at higher densities [10]. In this work, however we will be interested in the asymptotic dynamics in a Lorentz gas [5] which consists of a diffusing particle in a random array of fixed scatterers usually hard spheres or hard disks. Ernst and Weyland [5] showed from a low density kinetic theory that the VACF in a Lorentz gas decays as - A t -(d/2+1). In computer simulations of the two dimensional Lorentz gas [11-13], a decay --t -2 was found but the * Permanent address: Institute for Theoretical Physics, State University, 3508 TA Utrecht, The Netherlands.
0378-4371 / 88 / $03.50 O Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
S.P. Das and M . H . Ernst / Long time tails in a Lorentz gas
68
coefficient A was almost a factor of four larger than predicted by the theory. However, by extrapolating their low density results towards zero density Alder and Alley [13] obtained numerical evidence that the coefficient A is fairly close to the value predicted by the low density kinetic theory. Unfortunately these low densities are not accessible in computer simulations. Among many others, two obvious reasons for the disagreement between theory and simulations might be: (i) the coefficient A(c) has a strong density dependence; (ii) the VACF has not yet reached its asymptotic tail in the time regimes where simulations have been carried out. The first point has been investigated by Machta et al. [8]. These authors have made an estimate of the coefficient A, as given by the mode coupling prediction. Their estimate gave A(c) - 1.8A(c = 0) for density c = 0.05, which suggests a strong density dependence. Here we explore the second point in order to obtain a better understanding of the crossover behaviour from intermediate to very long times. We were motivated to do so by recent results for long time tails of the VACF in site and bond percolation models. These models can be interpreted as a lattice Lorentz gas, where a blocked site or bond is interpreted as a hard scatterer. For these models kinetic theory calculations have been extended to one order higher in the concentration of the scatterers and to the first subleading asymptotic time tail [14]. Comparison of the theory with computer simulations at low densities show excellent agreement [15, 16]. At the concentrations c - 0 . 0 5 and in the time interval (10 ~< t ~<60) considered in the simulations both corrections are approximately of equal size. We carry out a similar calculation for the Lorentz gas, by extending the kinetic theory further to compute the subleading behaviour of the asymptotic tails. This is again done to lowest order in the density. We do obtain an effect qualitatively similar to the simulation results but the quantitative agreement is very poor, indicating that the higher order density corrections play an important role in the long time behaviour of the Lorentz gas. We start with the normalized VACF ~b(t)~ (z). ¢)(t)) where 13= v/v is a d-dimensional unit vector and ( . . . ) is a canonical average over an equilibrium ensemble in which v is fixed. The Laplace transform t~(z) of ~(t) can be expressed in terms of a memory function ~,(z) [6], d)(z) = f dt e-Zt~b(t) = (z + ~(z))-' .
(1)
0
In the ring approximation it is given by
~-~
z- nTo-ik.v
'
(2)
S.P. Das and M . H . Ernst I L o n g time tails in a Lorentz gas
69
where n is the density of scatterers and T k is the Fourier transform of the binary collision operator [5, 6],
T k = vo "d-1
f
d6-16 • (r I e -ik . . .[o~ . - 1),
(3)
v.o'<0
where the operator b,, is defined as b,,v = v - 2&(&- v) and o- = o-&. One may also define a time dependent diffusion coefficient as t
D(t) = (v2/d) f d r qJ(r).
(4)
0
The quantity % = ~,(0) in (2) is related to the Boltzmann value of the diffusion coefficient for a Lorentz gas; D O= v2/dTo, with D O= (3v/16mr) for d = 2 and D O= (v/3nTro "2) for d = 3 where or is the radius of a scatterer. To discuss the long time properties of the VACF at small density of scatterers, we need the leading and the subleading singularities in 4/(z) around z = 0. They are determined by the small k region in (2), where the denominator vanishes. This occurs where k - n ~ lo 1, 10 being the mean free path. Consequently [klcr-cr/lo¢l and the operators T k in (2) may be replaced by T 0. Following a procedure that is conceptually simple and straight forward, we first calculate the memory function T(t) in time language. We present the time evolution operator by its spectral decomposition: et(ik'"+"T°) : ~ ]
e"°t(k)Pt(k)
(5)
l
using eigenmodes and eigenvalues of the Boltzmann equation
(ik. v + nTo)q~(kO(f~) = oJt(k)q~°(~3).
(6)
Here Pt(k) projects an arbitary function f(fi) on the subspace spanned by the eigenfunction q~o(fi) such that ~o~, ~ok /.
(7)
The bracket denotes an average over a d-dimensional solid angle (f(~3)) = .f dfi f ( f i ) / f d~. For long times, only those eigenmodes are relevant that have a vanishing eigenvalue o~t(k) as k---~0. In a Lorentz gas there is only one slow mode (corresponding to mass conservation) which is the diffusive mode. Eigenmodes with nonvanishing eigen values at small k contribute terms to T(t) that decay exponentially fast at large t and are therefore neglected.
70
S.P. Das and M.H. Ernst / Long time tails in a Lorentz gas
For our purpose the eigenmodes and the eigenvalues are needed to ~(k 2) and 6(k 4) respectively. After lengthy calculations we obtain the eigenfunction and the eigenvalue from a perturbation expansion in k: ~k = 1 + ik
dD o v
k2dDo(
x -
v
x2
1) -
,
(8)
where x =/~- ft. The eigenvalue is 3
w°(k) = - D ° k 2
D_~ k4 + - . .
(9)
d +2 v
In principle one needs (~(k 3) terms in (8), but there contributions vanish due to parity. Once this is done the long lived part of (5) is inserted into the Laplace inverse y(t) of eq. (2), the k-integrals are performed and the resulting expression is Laplace transformed, yielding the dominant small z singularities in the form ~,(z) = 3'0 + azd/2 + bzd/2+l" Combination of this result with eq. (1) determines the dominant small z singularities of ~(z). Finally, using the Tauberian theorem that the small z behaviour of z ~ corresponds to a long time behaviour t - ~ - l / F ( - a), we find the leading and subleading terms in the long time behaviour of the VACF for a d-dimensional Lorentz gas to lowest order in the density as
~O(t) =
2~dD 2
1
v2 n
(47rOt)d/2+1
[ 1+
3(d + 1)(2d + 3) 1 ] 16 ~-~ ,
(10)
where v is the collision frequency given by [17]
d-tTf(d-1)/2
(11)
nvo" v = r [ ( d + 1)/2]
Furthermore, from (5) and (10) we get the long time tail for the diffusion coefficient [ 1 ( 1 ~d/2 3 ( d + l ) ( 2 d + 3 ) D(t) = D O 1 + ~ \4--~D~/ + 16n(d + 2)
( 1 ~d/2 1 ] \4--~-~! ~-~ .
(12)
For d = 2, ~0(t) can be written in the following dimensionless form: ~b(~-)= - ( n * / ~ - e ) [ 1 + 63/16~ + - " .]
(13)
S.P. Das and M.H. Ernst / Long time tails in a Lorentz gas
71
i
7.00 6.00 5.00 4.00 3.00
t
2.00 1 .00 0.00
5.00'
7100
9100
11t,00 "E
13t.00
15'.00 '
17t.00
19t.00 ' 2 1 . 0 0
Fig. 1. C o m pu t er results for the velocity autocorrelation function in two dimensions, C ( 7 ) =
-(Irr2/n*)[qJ(~")-exp(-4T/3)] for n* =0.05. The exp(-4~-/3) represents the contribution calculated from the Boltzman equation and is subtracted out to show only the memory effects. The solid line represents the result from eq. (13). and f o r d = 3
as
~0(~) = - n * 2 ( 3 ~ r ) 3 / 2 / ( 1 6 r s / 2 ) [ 1
+ 27/4~" + - . . ] ,
(14)
where ~- = ut and n* = nor d. Next, we compare our results with the computer simulations of Alder and Alley [13] as shown in fig. 1. The figure also shows our results for the normalized VACF written in a dimensionless form C(r) = - (~rrZ/n*) qJ(r). The simulation corresponds to a density n* = 0.05 and we have subtracted the short time part of the VACF, e -4z/3 c o m p u t e d from the Boltzmann equation. F r o m the data shown in fig. 1 we see that the quantitative agreement is rather p o o r when comparing the simulation results and the calculated leading plus subleading long time tails to lowest order in the density. H o w e v e r , our results for the VACF in eq. (13) explain in a qualitative fashion the slow approach to the dominant tail. The coefficient of the subleading term in eq. (13) also gives a rough estimate, namely % = 6 , for the crossover time f r o m intermediate to asymptotic dynamics. F u r t h e r m o r e we have to conclude that finite density corrections to the long time tail play an important role in the two-dimensional Lorentz gas.
Acknowledgement One of us (S.P.D.) was supported by National Science Foundation G r a n t C H E 8411932.
72
S.P. Das and M . H . Ernst / L o n g time tails in a Lorentz gas
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17]
B.J. Alder and T.E. Wainwright, Phys. Rev. A 1 (1970) 18. See Physics Today, January, 1984. M.H. Ernst, J. Machta, J.R. Dorfman and H. van Beijeren, J. Stat. Phys. 34 (1984) 477. R. Dorfman and E.G. Cohen, Phys. Rev. Lett. 25 (1970) 1257. M.H. Ernst and A. Weyland, Phys. Lett. A 34 (1971) 39. J.M.J. Van Leeuwen and A. Weyland, Physica 36 (1967) 457. A. Weyland and J.M.J. Van Leeuwen, Physica 38 (1968) 35. M.H. Ernst, E.H. Hauge and J.M.J. Van Leeuwen, Phys. Rev. Lett. 25 (1970) 1245. J. Machta, M.H. Ernst, J.R. Dorfman and H. van Beijeren, J. Stat. Phys. 35 (1984) 413. D. Forster, D.R. Nelson and M.J. Stephen, Phys. Rev. A 16 (1977) 732. T.T. Erpenbeck and W.W. Wood, Phys. Rev. A 32 (1985) 23. C. Bruin, Phys. Rev. Lett. 29 (1972) 1670; Physica 72 (1974) 261. J.C. Lewis and J.A. Tjon, Phys. Lett. A 66 (1978) 349. B.J. Alder and W.E. Alley, J. Stat. Phys. 19 (1978) 341; Physica A 121 (1983) 523. W.E. Alley, Ph.D. thesis, University of California/Livermore (1979). Th. M. Nieuwenhuizen, P.F.J. van Velthoven and M.H. Ernst, Phys. Rev. Lett. 57 (1986) 2477. D. Frenkel, Phys. Lett. A 121 (1985) 385. J.J. Brey and A. Santos, Phys. Lett. A 127 (1988) 5. M.H. Ernst and H. van Beijeren, J. Stat. Phys. 26 (1981) 1.