Critical phenomena and the Ising-mean field crossover

2 Critical phenomena and the Ising—mean field crossover 

In Section 7.2.4 it was shown that via a finite size scaling analysis a meaningful extrapolation of simulation data to the thermodynamic limit is possible, and in this way one can extract estimates for both critical exponents (/ , 7) and amplitudes C+(1V), C- N) and B N) of the collective scattering function above Tc (eq. [7.16]) and below Tc -ScoiK = 0) =C- N)r, t = I — T/Tc — 0 or the order parameter (/ )= B lf)t, respectively. While for short enough chains N 32), data both for the simple self-avoiding walk model of Fig. 7.3 and for the bond fluctuation model are nicely consistent with the expected critical exponents for the three-dimensional Ising models /3 0.325,7 Pi 1.241), for N 64 one rather finds effective exponents  

Here also the behavior of the amplitude ratio is included, which 

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