Fluctuating hydrodynamics of the classical electron gas

P.-M. Binder

doi:10.1103/PhysRevE.49.R3565

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Fluctuating hydrodynamics of the classical electron gas

P.-M. Binder

Wolfson College, Oxford OX2 6UD, United Kingdom
Center for Nonlinear Studies, Los Alamos National Laboratory, Los Alamos, New Mexico 87545

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Phys. Rev. E 49, R3565(R) – Published 1 May, 1994

DOI: https://doi.org/10.1103/PhysRevE.49.R3565

Abstract

Fluctuations of the diffusion tensor in a lattice Lorentz gas model are calculated numerically from a sample of mesoscopic-size systems, and used as input in a mode-coupling expression that predicts long-time tail amplitudes. The results are in good agreement with previously published direct measurements of the tail amplitude over a wide range of scatterer concentrations. We argue that the previous, long-standing disagreement between theory and simulations comes from concentration-independent configurational fluctuations in the diffusion tensor, which we have measured separately and which are not taken into account by existing theories.

References (19)

H. A. Lorentz, Proc. K. Ned. Akad. Wet. 7, 438, (1905); ibid. 7, 585 (1905); ibid. 7, 684 (1905); P. Ehrenfest, Collected Scientific Papers (North Holland, Amsterdam, 1959), p. 229.
J. M. J. van Leeuwen and A. Weyland, Physica 36, 456 (1967); J. M. J. van Leeuwen, ibid. 38, 35 (1968).
M. H. Ernst and A. Weyland, Phys. Lett. 34A, 39 (1971).
C. Bruin, Phys. Rev. Lett. 29, 1670 (1972); Physica 72, 261 (1974).
B. J. Alder and W. E. Alley, J. Stat. Phys. 19, 341 (1978).
J. C. Lewis and J. A. Tjon, Phys. Lett. 66A, 349 (1978).
W. Götze, E. Leutheusser and S. Yip, Phys. Rev. A 23, 2634 (1981); ibid. 24, 100 (1981).
T. Keyes and J. Mercer, Physica A 95, 473 (1979); A. Masters and T. Keyes, Phys. Rev. A 26, 2129 (1982).
For recent work see P. M. Bleher, J. Stat. Phys. 67, 461 (1992); L. A. Bunimovich and S. E. Troubetzkoy, ibid. 67, 289 (1992); A. S. Cukrowski, Chem. Phys. 159, 39 (1992); J. Kortus and C. Olesky, J. Phys. A: Math. Gen. 25, 1093 (1992); H. Sumi, Solid State Commun. 85, 1 (1993); A. Baranyai, D. J. Evans, and E. G. D. Cohen, J. Stat. Phys. 70, 1085 (1993), and references therein.
P. M. Binder and D. Frenkel, Phys. Rev. A 42, 2463 (1990).
D. Frenkel, F. van Luijn and P. M. Binder, Europhys. Lett. 20, 7 (1992).
C. P. Lowe and A. J. Masters, Physica A 195, 149 (1993).
M. H. Ernst, J. Machta, J. R. Dorfman and H. van Beijeren, J. Stat. Phys. 34, 477 (1984); J. Machta, M. H. Ernst, H. van Beijeren and J. R. Dorfman, ibid. 35, 413 (1984).
We have excluded scatterer configurations allowing infinite free paths, as these would give rise to an infinite diffusion tensor; such configurations have zero probability of occurring in infinite systems. We have also checked our configurations, generated with the Berkeley random number generator, for correlations such as those found by A. M. Ferrenberg, D. P. Landau and Y. J. Wong, Phys. Rev. Lett. 69, 3382 (1992). We found that the scatterer to scatterer distance, n, followed closely the expected distribution, Prob(n) sim c (1-c $)^{n}$ , and therefore that no serious correlation effects appear in our systems. Finally, we note that the spectrum of decay rates of simple diffusive modes becomes discrete in this case and as a consequence Eqs. (1) and (2) will hold but only up to a cutoff time t sim $L^{2}$ $D^{- 1}$ , where L is the system size: see L. F. Perondi and P. M. Binder, Phys. Rev. B 48, 4136 (1993). Finally, the finite size corrections for D in a system of size L are of order (1/ $L^{2}$ ), and therefore are negligible for the system sizes considered here: see M. J. A. M. Brummelhuis and H. J. Hilhorst, J. Stat. Phys. L. F. Perondi and P. M. Binder53, 249 (1988) and , 47, 14,221 (1993).
M. H. Ernst and P. M. Binder, J. Stat. Phys. 51, 981 (1988).
We have chosen the triangular lattice to avoid the effects of staggered invariants, as described in Ref. [9] and in P. M. Binder and M. H. Ernst, Physica A 164, 91 (1990).
We estimate the error e from truncating the simulation after 400 time steps as follows: since the long time tails have already set in Ref. [10], the neglected contribution to the diffusion tensor measurement is of order a ${int}_{400}^{\infty}$ $t^{- 2}$ dt sim a/400. As a < D, $e/D < 1
H. van Beijeren and M. H. Ernst, J. Stat. Phys. 70, 793 (1993).
R. Bongratz and Ch. Morkel, J. Non Cryst. Sol. 156, 205 (1993).

[c1] H. A. Lorentz, Proc. K. Ned. Akad. Wet. 7, 438, (1905); ibid. 7, 585 (1905); ibid. 7, 684 (1905); P. Ehrenfest, Collected Scientific Papers (North Holland, Amsterdam, 1959), p. 229.

[c2] J. M. J. van Leeuwen and A. Weyland, Physica 36, 456 (1967); J. M. J. van Leeuwen, ibid. 38, 35 (1968).

[c3] M. H. Ernst and A. Weyland, Phys. Lett. 34A, 39 (1971).

[c4] C. Bruin, Phys. Rev. Lett. 29, 1670 (1972); Physica 72, 261 (1974).

[c5] B. J. Alder and W. E. Alley, J. Stat. Phys. 19, 341 (1978).

[c6] J. C. Lewis and J. A. Tjon, Phys. Lett. 66A, 349 (1978).

[c7] W. Götze, E. Leutheusser and S. Yip, Phys. Rev. A 23, 2634 (1981); ibid. 24, 100 (1981).

[c8] T. Keyes and J. Mercer, Physica A 95, 473 (1979); A. Masters and T. Keyes, Phys. Rev. A 26, 2129 (1982).

[c9] For recent work see P. M. Bleher, J. Stat. Phys. 67, 461 (1992); L. A. Bunimovich and S. E. Troubetzkoy, ibid. 67, 289 (1992); A. S. Cukrowski, Chem. Phys. 159, 39 (1992); J. Kortus and C. Olesky, J. Phys. A: Math. Gen. 25, 1093 (1992); H. Sumi, Solid State Commun. 85, 1 (1993); A. Baranyai, D. J. Evans, and E. G. D. Cohen, J. Stat. Phys. 70, 1085 (1993), and references therein.

[c10] P. M. Binder and D. Frenkel, Phys. Rev. A 42, 2463 (1990).

[c11] D. Frenkel, F. van Luijn and P. M. Binder, Europhys. Lett. 20, 7 (1992).

[c12] C. P. Lowe and A. J. Masters, Physica A 195, 149 (1993).

[c13] M. H. Ernst, J. Machta, J. R. Dorfman and H. van Beijeren, J. Stat. Phys. 34, 477 (1984); J. Machta, M. H. Ernst, H. van Beijeren and J. R. Dorfman, ibid. 35, 413 (1984).

[c14] We have excluded scatterer configurations allowing infinite free paths, as these would give rise to an infinite diffusion tensor; such configurations have zero probability of occurring in infinite systems. We have also checked our configurations, generated with the Berkeley random number generator, for correlations such as those found by A. M. Ferrenberg, D. P. Landau and Y. J. Wong, Phys. Rev. Lett. 69, 3382 (1992). We found that the scatterer to scatterer distance, n, followed closely the expected distribution, Prob(n) sim c (1-c $)^{n}$ , and therefore that no serious correlation effects appear in our systems. Finally, we note that the spectrum of decay rates of simple diffusive modes becomes discrete in this case and as a consequence Eqs. (1) and (2) will hold but only up to a cutoff time t sim $L^{2}$ $D^{- 1}$ , where L is the system size: see L. F. Perondi and P. M. Binder, Phys. Rev. B 48, 4136 (1993). Finally, the finite size corrections for D in a system of size L are of order (1/ $L^{2}$ ), and therefore are negligible for the system sizes considered here: see M. J. A. M. Brummelhuis and H. J. Hilhorst, J. Stat. Phys. L. F. Perondi and P. M. Binder53, 249 (1988) and , 47, 14,221 (1993).

[c15] M. H. Ernst and P. M. Binder, J. Stat. Phys. 51, 981 (1988).

[c16] We have chosen the triangular lattice to avoid the effects of staggered invariants, as described in Ref. [9] and in P. M. Binder and M. H. Ernst, Physica A 164, 91 (1990).

[c17] We estimate the error e from truncating the simulation after 400 time steps as follows: since the long time tails have already set in Ref. [10], the neglected contribution to the diffusion tensor measurement is of order a ${int}_{400}^{\infty}$ $t^{- 2}$ dt sim a/400. As a < D, $e/D < 1

[c18] H. van Beijeren and M. H. Ernst, J. Stat. Phys. 70, 793 (1993).

[c19] R. Bongratz and Ch. Morkel, J. Non Cryst. Sol. 156, 205 (1993).

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Fluctuating hydrodynamics of the classical electron gas

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