PHASE TRANSITION (lectures)

Phase transitions

diverges (α > 0) for d < 4. This divergence is caused by the long-wavelength fluctuation which were neglected in the calculation of the free energy in the MF approximation.

In d = 3 ( = 1) we find α ≈ 0.17 and ν ≈ 0.58. The results of this paragraph are based on an expansion of d around d = 4 and are valid for small > 0 or for d ≥ 4. Nevertheless, we put = 1 when d = 3. That means that is not small and higher-order terms in must be included. This is then the −expansion, which gives good values for the critical exponents.

60

Phase transitions

Literature

[1]J.J.Binney, N.J.Dowrick, A.J.Fisher, and M.E.Newman, The Theory of Critical Phenomena, Oxford University Press (Oxford,1992).

[2]J.Cardy, Scaling and Renormalizationin Statistical Physics, Cambridge University Press (Cambridge,1996).

[3]А.З.Паташинский, В.Л.Покровский, Флуктуационная теория фазовых переходов (Москва, Наука,1982).

[4]K.Huang, Statistical Mechanics, J.Wiley (NewYork, 1987).

[5]Л.Д.Ландау, Е.М.Лифшиц, Статистическая физика, Наука (Москва, 1976).

[6]Ш.Ма, Современная теория критических явлений, Мир (Москва, 1980).

[7]I.Vilfan, Lecture in Statistical Mechanics, A.Salam, ICTP (Triest, Italy, 2002).

[8]М.А.Анисимов, Критические явления в жидкостях и жидких кристаллах, Наука (Москва, 1987).

[9]B.Simons, Phase transitions and collective phenomena, (Cambridge, 2001).

[10]Д.Ю.Иванов, Критическое поведение неидеализированных систем, ФИЗМАТЛИТ (Москва, 2003).

61