Abstract
We study 1-dimensional continuum fields of Ginzburg-Landau type under the presence of an external and a long-range pair interaction potentials. The corresponding Gibbs states are formulated as Gibbs measures relative to Brownian motion [17]. In this context we prove the existence of Gibbs measures for a wide class of potentials including a singular external potential as hard-wall ones, as well as a non-convex interaction. Our basic methods are: (i) to derive moment estimates via integration by parts; and (ii) in its finite-volume construction, to represent the hard-wall Gibbs measure on C(ℝ;ℝ+) in terms of a certain rotationally invariant Gibbs measure on C(ℝ;ℝ3).
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Betz, V.: Existence of Gibbs measures relative to Brownian motion. Markov Process. Related Fields 9, 85–102 (2003)
Betz, V., Lőrinczi, J.: A Gibbsian description of P(φ)1-processes, preprint (1999)
Bolthausen, E., Deuschel, J.-D., Zeitouni, O.: Entropic repulsion of the lattice free field. Commun. Math. Phys. 170, 417–443 (1995)
Bolthausen, E., Deuschel, J.-D., Zeitouni, O.: Absence of a wetting transition for a pinned harmonic crystal in dimensions three and larger. J. Math. Phys. 41, 1211–1223 (2000)
Caputo, A., Velenik, Y.: A note on wetting transition for gradient fields. Stoch. Proc. Appl. 87, 107–113 (2000)
Davies, E.B.: Spectral Theory and Differential Operators. Cambridge Studies in Advanced Mathematics no.42: Cambridge University Press, 1995
Funaki, T.: The reversible measures of multi-dimensional Ginzburg-Landau type continuum model. Osaka J. Math. 28, 463–494 (1991)
Hohenberg, P.C., Halperin, B.I.: Theory of Dynamic critical Phenomena. Rev. Mod. Phys. 49, 435–479 (1977)
Iwata, K.: Reversible measures of a P(φ)1-time evolution, in ``Prob. Meth. in Math. Phys. (eds. K. Itô and N. Ikeda), Proc. of Taniguchi Symp.'', 195–209 (1985)
Knight, F.B.: Essentials of Brownian Motion and Diffusion. Mathematical Surveys no.18: AMS 1981
Hariya, Y., Osada, H.: Diffusion processes on path spaces with interactions. Rev. Math. Phys. 13, 199–220 (2001)
Lebowitz, J.L., Maes, C.: The effect of an external field on an interface, entropy repulsion. J. Statist. Phys. 46, 39–49 (1987)
Lebowitz, J.L., Presutti, E.: Statistical mechanics of systems of unbounded spins. Commun. Math. Phys. 50, 195–218 (1976)
Lőrinczi, J., Minlos, R.A.: Gibbs measures for Brownian paths under the effect of an external and a small pair potential. J. Statist. Phys. 105, 605–647 (2001)
Lőrinczi, J., Minlos, R.A., Spohn, H.: The infrared behaviour in Nelson's model of a quantum particle coupled to a massless scalar field. Ann. Henri Poincaré 3, 269–295 (2002)
Nelson, E.: Interaction of nonrelativistic particles with a quantized scalar field. J. Math. Phys. 5, 1190–1197 (1964)
Osada, H., Spohn, H.: Gibbs measures relative to Brownian motion. Ann. Probab. 27, 1183–1207 (1999)
Reed, M., Simon, B.: Methods of Modern Mathematical Physics, Vol.IV: Analysis of Operators. New York: Academic Press, 1978
Revuz, D., Yor, M.: Continuous Martingales and Brownian Motion. 2nd ed. Berlin: Springer-Verlag, 1994
Rosen, J., Simon, B.: Fluctuations in P(φ)1 processes. Ann. Probab. 4, 155–174 (1976)
Ruelle, D.: Superstable interactions in classical statistical mechanics. Commun. Math. Phys. 18, 127–159 (1970)
Simon, B.: Functional Integration and Quantum Physics. New York: Academic Press, 1979
Spohn, H.: Large Scale Dynamics of Interacting Particles. Berlin: Springer-Verlag, 1991
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Hariya, Y. Construction of Gibbs measures for 1-dimensional continuum fields. Probab. Theory Relat. Fields 136, 157–170 (2006). https://doi.org/10.1007/s00440-005-0481-0
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00440-005-0481-0