Abstract
We present a brief tutorial introduction into the quantum Hamiltonian formalism for stochastic many-body systems defined in terms of a master equation for their time evolution. These models describe interacting classical particle systems where particles hop on a lattice and may undergo reactions such as A+A→0. The quantum Hamiltonian formalism for the master equation provides a convenient general framework for the treatment of such models which, by various mappings, are capable of describing a wide variety of phenomena in non-equilibrium physics and in random media. The formalism is particularly useful if the quantum Hamiltonian has continuous global symmetries or if it is integrable, i.e. has an infinite set of conservation laws. This is demonstrated in the case of the exclusion process and for a toy model of tumor growth. Experimental applications of other integrable reaction-diffusion models in various areas of polymer physics (gel electrophoresis of DNA, exciton dynamics on polymers and the kinetics of biopolymerization on RNA) are pointed out.
This is a preview of subscription content, log in via an institution to check access.
Preview
Unable to display preview. Download preview PDF.
References
Alcaraz, F. C., Rittenberg, V. (1993), Phys. Lett. B 314, 377.
Alcaraz, F. C., Droz, M., M. Henkel and V. Rittenberg (1994), Ann. Phys. (USA) 230, 250.
Alexander, S., Holstein, T. (1978), Phys. Rev. B 18, 301.
Barkema, G. T., Marko, J., Widom, B. (1994), Phys. Rev. E 49, 5303.
Barkema, G. T., Caron, C., Marko, J.F. (1996), Biopolymers 38 665.
Barkema, G. T., Schütz, G. M. (1996), Europhys. Lett. 35 139.
Bethe, H. (1931), Z. Phys. 71, 205.
Bramson, M., Griffeath, D. (1980), Z. Wahrsch. Verw. Geb. 53, 183.
Cardy, J. (1997), in Proceedings of Mathematical Beauty of Physics, J.-B. Zuber, ed., Advanced Series in Mathematical Physics 24, 113.
de Gennes, P.G. (1971), J. Chem. Phys. 55, 572.
Dahmen, S.R. (1995), J. Phys. A 28, 905.
Derrida, B., Domany, E., Mukamel, D. (1992), J. Stat. Phys. 69, 667.
Derrida, B., Evans, M.R., Hakim, V., Pasquier, V. (1993a), J. Phys. A 26, 1493.
Derrida, B., Janowsky, S. A., Lebowitz, J. L., Speer, E. R. (1993b), Europhys. Lett. 22, 651.
Derrida, B., Evans, M. R. (1994), in Probability and Phase Transition, G. Grimmet, ed. (Kluwer Academic, Dordrecht).
de Vega, H.-J., Gonzalez-Ruiz, A. (1994), J. Phys. A 27, 6129.
Dieterich, W., Peschel, I. (1983), J. Phys. C 16, 3841.
Doi, M. (1976), J. Phys. A 9, 1465 (1976).
Drasdo, D. (1996) in Self-organization of Complex Structures: From Individual to Collective Dynamics, F. Schweitzer (ed.) (Gordon and Breach, London).
Duke, T. A. J. (1989), Phys. Rev. Lett. 62, 2877.
Edwards, S.F. (1967), Proc. Phys. Soc., London 92, 9.
Family, F., Amar, J. G. (1991), J. Stat. Phys. 65, 1235.
Felderhof, B. U. (1971), Rep. Math. Phys. 1, 215.
Glauber, R. J. (1963), J. Math. Phys. 4, 294.
Gradshteyn, I. S., Ryzhik, I.M. (1980), Tables of Integrals, Series and Products, fourth edition (Academic Press, London), Ch. 15
Grassberger, P., Scheunert, M. (1980), Fortschr. Phys. 28, 547 (1980).
Grynberg, M. D., Newman, T. J., Stinchcombe, R.B. (1994), Phys. Rev. E 50, 957.
Gwa, L.-H., Spohn, H. (1992), Phys. Rev. A 46, 844.
Jordan, P., Wigner, E. (1928), Z. Phys. 47, 631.
Kadanoff, L. P., Swift, J. (1968), Phys. Rev. 165, 310 (1968).
Kim, D. (1995), Phys. Rev. E 52, 3512.
Krebs, K., Pfannmüller, M. P., Wehefritz, B., Hinrichsen, H. (1995), J. Stat. Phys. 78, 1429.
Kroon, R., Fleurent, H., Sprik, R. (1993), Phys. Rev. E 47, 2462.
Lieb, E., Schultz, T., Mattis, D. (1961), Ann. Phys. (N.Y.) 16, 407.
Ligget, T. (1985), Interacting Particle Systems, (Springer, New York).
Lindenberg, K., Agyrakis, P., Kopelman, P. (1995), J. Phys. Chem. 99, 7542.
Lushnikov, A. A. (1987), Phys. Lett. A 120, 135.
MacDonald, J. T., Gibbs, J. H., Pipkin, A. C. (1968), Biopolymers 6, 1.
MacDonald, J. T., Gibbs, J. H. (1969), Biopolymers 7, 707.
Mattis, D. C., (1965), The Theory of Magnetism, (Harper and Row, New York).
Peschel, I., Rittenberg, V., Schulze, U. (1994), Nucl. Phys. B 430, 633.
Prähofer, M., Spohn, H. (1997), Physica A 233, 191.
Privman, V. (1994), Phys. Rev. E 50, 50.
Privman, V. (ed.) (1997), Nonequilibrium Statistical Mechanics in One Dimension (Cambridge University Press, Cambridge, U.K.).
Rácz, Z. (1985), Phys. Rev. Lett. 55, 1707.
Rubinstein, M. (1987), Phys. Rev. Lett. 59, 1946.
Sandow, S., Trimper, S. (1993), Europhys. Lett. 21, 799.
Sandow, S., Schütz, G. (1994), Europhys. Lett. 26, 7.
Santos, J. E., Schütz, G. M., Stinchcombe, R. B. (1996), J. Chem. Phys. 105, 2399.
Santos, J. E. (1997), J. Phys. A 30, 3249.
Schmittmann, B., Zia, R. K. P., (1996) in Phase Transitions and Critical Phenomena, eds. C. Domb and J. Lebowitz (Academic Press, London).
Schulz, M., Trimper, S. (1996), Phys. Lett. A 216, 235.
Schütz, G. (1993), Phys. Rev. E 47, 4265.
Schütz, G., Domany, E. (1993), J. Stat. Phys. 72, 277.
Schütz, G., Sandow, S. (1994), Phys. Rev. E 49, 2726.
Schütz, G. M. (1995a), J. Stat. Phys. 79, 243.
Schütz, G. M. (1995b), J. Phys. A 28, 3405.
Schütz, G. M. (1997), J. Stat. Phys. 88, 427.
Siggia, E. (1977), Phys. Rev. B 16, 2319.
Spitzer, F. (1970), Adv. Math. 5, 246.
Spohn, H. (1991), Large Scale Dynamics of Interacting Particle Systems (Springer, New York).
Spouge, J.L. (1988), Phys. Rev. Lett. 60, 871.
Stinchcombe, R. B., Grynberg, M. D., Barma, M. (1993), Phys. Rev. E 47, 4018.
Stinchcombe, R. B., Schütz, G. M. Phys. (1995), Phys. Rev. Lett. 75, 140.
Thacker, H. B. (1981), Rev. Mod. Phys. 53, 253.
Torney, D. C., McConnell, H. M. (1983), J. Phys. Chem. 87, 1941.
van Leeuwen, J. M. J. (1991), J. Phys. I 1. 1675.
van Leeuwen, J. M. J., Kooiman, A. (1992), Physica A 184, 79.
van Kampen, N. G. (1981) Stochastic methods in Physics and Chemistry (North Holland, Amsterdam).
von Smoluchovsky, M. (1917), Z. Phys. Chem. 92, 129.
Widom, B., Viovy, J. L. Défontaines A. D. (1991), J. Phys. I France 1, 1759.
Droz, M., Rácz, Z., Schmidt, J. (1989), Phys. Rev. A 39, 2141.
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1998 Springer-Verlag
About this paper
Cite this paper
Schütz, G.M. (1998). Reaction-diffusion mechanisms and quantum spin systems. In: Meyer-Ortmanns, H., Klümper, A. (eds) Field Theoretical Tools for Polymer and Particle Physics. Lecture Notes in Physics, vol 508. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0106878
Download citation
DOI: https://doi.org/10.1007/BFb0106878
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-64308-1
Online ISBN: 978-3-540-69747-3
eBook Packages: Springer Book Archive