Abstract
The equilibrium and non-equilibrium properties of regular harmonic crystals are well understood (Montroll, 1956, Lanford and Lebowitz, 1974). For them it is possible to define an infinite volume equilibrium state for dimensionality three and higher. This state is a Gaussian measure. For one (resp. two) dimensions it is not possible to define such a state because the mean square displacement of any particle diverges as |A| (resp. ln|A|) as |A| →∞, where |A|) is the volume of the system. In all dimensions, however, it is possible to define a state on the algebra generated by the difference variables (Lanford and Lebowitz, 1974).
Chapter PDF
References
Hi. Brascamp and E.H. Lieb (1974), Some inequalities for Gaussian Measures, International Conference on Functional Integration and its Applications, Cumberland Lodge, England, April 1–4, 1974.
H.J. Brascamp and. E.H. Lieb (1975), On extensions of the Brunn-Minkowski and
With H.J. Brascamp and J.L. Lebowitz in Bull. Int. Statist. Inst. 46, 393–404 (1975)
P.C. Hohenberg (1967), Phys. Rev. 158, 383.
O.E. Lanford III and J.L. Lebowitz (1974), Time Evolution and Ergodic Properties of Harmonic System, 1974 Battelle Rencontres,Springer-Verlag, New York-Heidelberg-Berlin (in press).
L. Leindler (1972), Acta Sci. Math. Szeged 33, 217.
N.D. Mermin (1967), J. Math. Phys. 8, 1061.
N.D. Mermin (1968), Phys. Rev. 176, 250.
N.D. Mermin and H. Wagner (1966), Phys. Rev. Letters 17, 1133.
E.W. Montroll (1956), in: Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, Vol. III, Univ. of Calif. Press, Berkeley, page 209.
A. Prékopa (1971), Acta Sci. Math. Szeged 32, 301.
A. Prékopa (1973), Acta Sci. Math. Szeged 34, 335.
Y. Rinott (1973), Thesis, Weizmann Institute, Rehovot, Israel.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1975 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Brascamp, H.J., Lieb, E.H., Lebowitz, J.L. (1975). The Statistical Mechanics of Anharmonic Lattices. In: Nachtergaele, B., Solovej, J.P., Yngvason, J. (eds) Statistical Mechanics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-10018-9_22
Download citation
DOI: https://doi.org/10.1007/978-3-662-10018-9_22
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-06092-2
Online ISBN: 978-3-662-10018-9
eBook Packages: Springer Book Archive