Abstract
Systems representable as a time-dependent Ginzburg-Landau model with nonconserved order parameter are considered in a block (V=L d) geometry with periodic boundary conditions, both for space dimensionalitiesd≧4 andd=4−ε. A systematic approach for studying finite size effects on dynamic critical behavior is developed. The method consists in constructing an effective reduced dynamics for the lowest-energy (q=0) mode by integrating out the remaining degrees of freedom, and generalizes recent analytic approaches for studying static finite size effects to dynamics. Above four dimensions, the coupling to the other (q≠0) modes is irrelevant and the probability densityP(Φ,t) for the normalized order parameterΦ=∫dd xϕ(x,t)/V satisfies a Fokker-Planck equation. The dynamics is equivalently described by the Langevin equation for a particle moving in a |Φ|4 potential or by a supersymmetric quantum mechanical Hamiltonian. Dynamic finite size scaling is found to be broken, e.g. the order parameter relaxation rate varies at the bulk critical temperatureT c,∞ as ωυ(T c,∞ L)∼L −d/2 asL→∞. By contrast, ford<4, the coupling to the other (q≠0) modes cannot be ignored and dynamic finite size scaling is valid. The asymptotic behavior of correlation and response functions can be studied within the framework of an expansion in powers of ɛ1/2. The scaling function associated with ωυ is computed to one-loop order. Finally, the many component (n→∞) limit is briefly considered.
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References
Fisher, M.E.: In: Critical Phenomena. Proceedings of the 1970 Enrico Fermi Summer School, Course 51. Green, M.S. (ed.). New York: Academic Press 1972
Barber, M.N.: In: Phase transitions and critical phenomena. Domb, C., Lebowitz, J.L. (eds.), Vol.8. New York: Academic Press 1983
Binder, K. (ed.): In: Topics in Current Physics. Vol. 7: Monte Carlo methods in statistical physics. Berlin, Heidelberg, New York: Springer 1979 and 1986
Brézin, E., Zinn-Justin, J.: Nucl. Phys. B257 [FS 14], 867 (1985)
Rudnick, J., Guo, H., Jasnow, D.: J. Stat. Phys.41, 353 (1985)
Eisenriegler, E., Tomaschitz, R.: (to be published)
Halperin, B.I., Hohenberg, P.C., Ma, S.: Phys. Rev. Lett.29, 1548 (1972); Phys. Rev. B10, 139 (1974)
Eisenriegler, E.: Z. Phys. B-Condensed Matter61, 299 (1985)
Diehl, H.W.: In: Phase transitions and critical phenomena. Domb, C., Lebowitz, J.L. (eds.), Vol. 10. New York: Academic Press 1986
Privman, V., Fisher, M.E.: J. Stat. Phys.33, 385 (1983)
Brézin, E.: J. Phys. (Paris)43, 15 (1982)
Janssen, H.K.: Z. Phys. B-Condensed Matter and Quanta23, 377 (1976)
Langouche, F., Roekaerts, D., Tirapegui, E.: Physica95A, 252 (1979)
Symanzik, K.: Nucl. Phys. B190 [FS3], 1 (1981)
Dietrich, S., Diehl, H.W.: Z. Phys. B-Condensed Matter51, 343 (1983)
Suzuki, N.: Progr. Theor. Phys.58, 1142 (1977)
Fisher, M.E., Racz, Z.: Phys. Rev. B13, 5039 (1976)
Bausch, R., Janssen, H.K.: Z. Phys. B-Condensed Matter and Quanta25, 275 (1976)
Sancho, J.M., San Miguel, M., Gunton, J.D.: J. Phys. A13, L443 (1980)
Bausch, R., Janssen, H.K., Wagner, H.: Z. Phys. B-Condensed Matter and Quanta24, 113 (1976)
see e.g. Risken, H.: In: The Fokker-Planck equation. Methods of solution and applications. Berlin, Heidelberg, New York, Tokyo: Springer 1984
Bernstein, M., Brown, L.S.: Phys. Rev. Lett.52, 1933 (1984)
Witten, E.: Nucl. Phys. B185, 513 (1981)
Bender, C.M., Cooper, F., Freedman, B.: Nucl. Phys. B219, 61 (1983)
Dekker, H., van Kampen, N.G.: Phys. Lett.73A, 374 (1979)
Goldschmidt, Y.: (to appear)
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Diehl, H.W. Finite size effects in critical dynamics and the renormalization group. Z. Physik B - Condensed Matter 66, 211–218 (1987). https://doi.org/10.1007/BF01311657
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DOI: https://doi.org/10.1007/BF01311657