New Approaches to Modern Physics/REVISION


118093 - New Approaches to Modern Physics
WINTER 2005

Advanced Statistical Mechanics


[Course Summary][Links to Lecture Notes and Homework][Announcements] [For Lecturers]
[JOAN ADLER's lectures]

REVISION

  1. First and second order phase transitions: if there is a finite discontinuity in one or more of the first derivatives of the appropriate thernodynamic potential the transition is termed first order. For a magnetic phase transition, the free energy (F=U-TS) is the appropriate potential (F= kTln Z, U - internal energy, T - temperature, S - entropy) with a discontinuity in the magnetization showing it is first order. If the first derivatives are continuous but higher derivatives are not it is a higher order transition. Here the magnetization would be continuous.
  2. Critical exponents describe the nature of the divergence in a second order transition. There are accepted names for the exponents of the common quantities in the Ising model which are:
    1. beta for the magnetization - not be confused with beta =1/kT in its Hamiltonian
    2. gamma for the susceptibility
    3. alpha for the specific heat
  3. Upper and lower critical dimensions.
    1. In mean field theory (alpha, beta, gamma) take the values (0(discontinuity),1/2,1) for the Ising model.
    2. In one dimension, the exact solution of the Ising model does not have a finite temperature transition.
    3. In the exact solution of the 2D Ising model the exponents take the values (0 (logarithm),1/8,7/4)
    4. In 3d the exponents also differ from the mean field ones and only in 4D do they take the mean field values (albeit with logarithmic correction terms).
    The lowest dimension which has a finite temperature transition is called the lower critical dimension and the dimension at which exponents become mean-field like is called the upper critical dimension. For the Ising model it is D=4.
This issues are rather deep, and a full explanation requires scaling theory and Renormalization Group theory. However the facts above are all thats needed to understand the topic of percolation. After all percolation is so easy you could explain it to your wife...........