Abstract
We study a class of stochastic Ising (or interacting particle) systems that exhibit a spatial distribution of impurities that change with time. It may model, for instance, steady nonequilibrium conditions of the kind that may be induced by diffusion in some disordered materials. Different assumptions for the degree of coupling between the spin and the impurity configurations are considered. Two interesting well-defined limits for impurities that behave autonomously are (i) the standard (i.e., quenched) bond-diluted, random-field, random-exchange, and spin-glass Ising models, and (ii) kinetic variations of these standard cases in which conflicting kinetics simulate fast and random diffusion of impurities. A generalization of the Mattis model with disorder that describes a crossover from the equilibrium case (i) to the nonequilibrium case (ii) and the microscopic structure of a generalized heat bath are explicitly worked out as specific realizations of our class of models. We sketch a simple classification of transition rates for the time evolution of the spin configuration based on the critical behavior that is exhibited by the models in case (ii). The latter are shown to have an exact solution for any lattice dimension for some special choice of rates.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
C. J. Thompson,Mathematical Statistical Mechanics (Princeton University Press, Princeton, New Jersey, 1972), and references therein.
R. J. Glauber,J. Math. Phys. 4:294 (1963).
K. Kawasaki,Phys. Rev. 145:224 (1966); see also K. Kawasaki, inPhase Transitions and Critical Phenomena, Vol.4, C. Domb and M. S. Green eds. (Academic Press, London, 1972).
N. G. van Kampen,Srochastic Processes in Physics and Chemistry (North-Holland, Amsterdam, 1981).
T. M. Liggett,Interacting Particle Systems (Springer-Verlag, New York, 1985).
J. Ziman,Models of Disorder (Cambridge University Press, Cambridge, 1979).
G. Grinstein, inFundamental Problems in Statistical Mechanics VI, E. G. D. Cohen, ed. (North-Holland, Amsterdam, 1985).
M. Mézard, G. Parisi, and M. A. Virasoro,Spin Glass Theory and Beyond (World Scientific, Singapore, 1987).
D. Amit, H. Gutfreund, and H. Sompolinsky,Ann. Phys. (N.Y.)173:30 (1987), and references therein.
S. F. Edwards and P. W. Anderson,J. Phys. F 5:965 (1975);6:1927 (1976); see also D. Sherrington and S. Kirkpatrick,Phys. Rev. Lett. 35:1972 (1975).
R. Leibman,Statistical Mechanics of Periodic Frustrated Ising Systems (Springer-Verlag, Berlin, 1986).
G. Parisi,Phys. Rep. 67:97 (1980).
K. Binder and A. P. Young,Rev. Mod. Phys. 58:801 (1986).
Y. Imry and S. Ma,Phys. Rev. Lett. 35:1399 (1975); J. Z. Imbrie,Phys. Rev. Lett. 53:1747 (1984); D. S. Fisher,Phys. Rev. Lett. 56:419 (1986); J. Bricmont and A. Kupiainen,Phys. Rev. Lett. 59:1829 (1987).
R. Pynn and A. Skjeltorp, eds.,Time Dependent Effects in Disordered Systems (Plenum Press, New York, 1987).
J. O. Andersson, J. Mattsson, and P. Svedlindh,Phys. Rev. B 46:8297 (1992).
M. F. Thorpe and D. Beeman,Phys. Rev. B 14:188 (1976).
P. L. Garrido and J. Marro,Europhys. Lett. 15:375 (1991).
J. J. Alonso and J. Marro,Phys. Rev. B 45:10408 (1992).
P. L. Garrido and J. Marro,J. Phys. A: Math. Gen. 25:1453 (1992).
A. I. López-Lacomba and J. Marro,Phys. Rev. B 46:8244 (1992).
J. J. Alonso and J. Marro,J. Phys.: Cond. Matter 4:1 (1992).
P. L. Garrido and J. Marro, inArtificial Neural Networks, A. Prieto ed. (Springer-Verlag, Berlin, 1991).
D. C. Mattis,Phys. Lett. A 56:421 (1976).
J. T. Cushing,Applied Analytical Mathematics for Physical Scientists (Wiley, New York, 1975), Chapter 5.
J. M. González-Miranda, A. Labarta, M. Puma, J. F. Fernández, P. L. Garrido, and J. Marro,Phys. Rev. E, in press.
P. L. Garrido and J. Marro,Phys. Rev. Lett. 62:1929 (1989).
A. I. López-Lacomba, P. L. Garrido, and J. Marro,J. Phys. A: Math. Gen. 23:3809 (1990).
J. Marro, J. F. Fernández, J. M. González-Miranda, and M. Puma, private communication.
A. Aharony,Phys. Rev. B 18:3318 (1978).
R. Bruinsma,Phys. Rev. B 30:289 (1984).
T. Yokota,Phys. Rev. B 38:11669 (1988).
G. Grinstein, C. Jayaprakash, and Y. He,Phys. Rev. Lett. 55:2527 (1985).
M. A. Muñoz and P. L. Garrido, private communication.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Garrido, P.L., Marro, J. Kinetic lattice models of disorder. J Stat Phys 74, 663–686 (1994). https://doi.org/10.1007/BF02188575
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/BF02188575