×

Zero density of open paths in the Lorentz mirror model for arbitrary mirror probability. (English) Zbl 1302.82022

Summary: We show, incorporating results obtained from numerical simulations, that in the Lorentz mirror model, the density of open paths in any finite box tends to 0 as the box size tends to infinity, for any mirror probability.

MSC:

82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics
60J10 Markov chains (discrete-time Markov processes on discrete state spaces)
60K37 Processes in random environments
82D05 Statistical mechanics of gases
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Acedo, L., Santos, A.: Diffusion in lattice Lorentz gases with mixtures of point scatterers. Phys. Rev. E 50, 4577 (1994)
[2] Bianca, C.: On the existence of periodic orbits in nonequilibrium Ehrenfest gas. Int. Math. Forum 7, 221-232 (2012) · Zbl 1256.37016
[3] Bruin, C.: A computer experiment on diffusion in the Lorentz gas. Physica 72(2), 261-286 (1974)
[4] Bunimovich, L.A., Troubetzkoy, S.E.: Recurrence properties of Lorentz lattice gas cellular automata. J. Stat. Phys. 67(1), 289-302 (1992) · Zbl 0900.60103
[5] Bunimovich, L.A., Troubetzkoy, S.E.: Topological dynamics of flipping Lorentz lattice gas models. J. Stat. Phys. 72, 297 (1993) · Zbl 1099.82509
[6] Cohen, E.: New types of diffusion in lattice gas cellular automata. In: Mareschal, M., Holian, B. (eds.) Microscopic Simulations of Complex Hydrodynamic Phenomena, vol. 292. NATO ASI Series, pp. 137-152. Springer, US (1992) · Zbl 1084.82563
[7] Cohen, E., Wang, F.: New results for diffusion in Lorentz lattice gas cellular automata. J. Stat. Phys. 81, 445 (1995) · Zbl 1106.82349
[8] Cohen, E., Wang, F.: Diffusion and propagation in Lorentz lattice gases. Fields Inst. Commun. 6, 43 (1996) · Zbl 0885.60087
[9] Ernst, M.H., van Velzen, G.A.: Lattice Lorentz gas. J. Phys. A: Math. Gen. 22, 4611 (1989)
[10] Ernst, M.H., van Velzen, G.A.: Long-time tails in lattice Lorentz gases. J. Stat. Phys. 57, 455 (1989)
[11] Grimmett, G.: Percolation and Disordered Systems. Lectures on Probability Theory and Statistics, pp. 153-300. Springer, Berlin (1997) · Zbl 0884.60089
[12] Grimmett, G.: Percolation, 2nd edn. Springer, New York (1999) · Zbl 0926.60004
[13] Kong, X.P., Cohen, E.G.D.: Diffusion and propagation in triangular Lorentz lattice gas cellular automata. J. Stat. Phys. 62, 737 (1991)
[14] Kong, X.P., Cohen, E.G.D.: Lorentz lattice gases, abnormal diffusion, and polymer statistics. J. Stat. Phys. 62, 1153 (1991)
[15] Kong, X.P., Cohen, E.G.D.: A kinetic theorist’s look at lattice gas cellular automata. Physica D 47, 9-18 (1991)
[16] Kozma, G., Sidoravicius, V.: Lower bound for the escape probability in the Lorentz mirror model on the lattice. arXiv:1311.7437 [math.PR] pp. 1-2 (2013)
[17] Kraemer, A.S., Sanders, D.P.: Embedding quasicrystals in a periodic cell: dynamics in quasiperiodic structures. Phys. Rev. Lett. 111(12), 5501 (2013)
[18] Machta, J., Moore, S.M.: Diffusion and long-time tails in the overlapping Lorentz gas. Phys. Rev. A 32, 3164-3167 (1985)
[19] Machta, J., Zwanzig, R.: Diffusion in a periodic Lorentz gas. Phys. Rev. Lett. 50, 1959-1962 (1983) · Zbl 0974.82505
[20] Moran, B., Hoover, W.G.: Diffusion in a periodic Lorentz gas. J. Stat. Phys. 48, 709-726 (1987) · Zbl 1084.82563
[21] Quas, A.N.: Infinite paths in a Lorentz lattice gas model. Probab. Theory Relat. Fields 114(2), 229-244 (1999) · Zbl 0932.60087
[22] Ruijgrok, T.W., Cohen, E.: Deterministic lattice gas models. Phys. Lett. A 133(7), 415-418 (1988)
[23] Sahimi, M.: Applications of Percolation Theory. Taylor and Francis, London (2009)
[24] van Beijeren, H., Ernst, M.H.: Diffusion in Lorentz lattice gas automata with backscattering. J. Stat. Phys. 70, 793 (1993) · Zbl 0945.82540
[25] Wang, F., Cohen, E.: Diffusion in Lorentz lattice gas cellular automata: the honeycomb and quasi-lattices compared with the square and triangular lattices. J. Stat. Phys. 81, 467 (1995) · Zbl 1106.82362
[26] Ziff, R.M.: Hull-generation walks. Physica D 38, 377-383 (1989)
[27] Ziff, R.M., Kong, X., Cohen, E.: Lorentz lattice-gas and kinetic-walk model. Phys. Rev. A 44(4), 2410 (1991)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.