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Mean field theory gives sensible results as long as we are able to neglect fluctuations in the system under consideration. The Ginzburg Criterion tells us quantitatively when mean field theory is valid. It also gives the idea of an upper critical dimension, a dimensionality of the system above which mean field theory gives proper results, and the critical exponents predicted by mean field theory match exactly with those obtained by numerical methods. ==Example: Ising Model== If is the order parameter of the system, then mean field theory requires that the fluctuations in the order parameter are much smaller than the actual value of the order parameter near the critical point. Quantitatively, this means that : Using this in the Landau theory, which is identical to the mean field theory for the Ising model, the value of the upper critical dimension comes out to be 4. If the dimension of the space is greater than 4, the mean-field results are good and self-consistent. But for dimensions less than 4, the predictions are less accurate. For instance, in one dimension, the mean field approximation predicts a phase transition at finite temperatures for the Ising model, whereas we know that there is none, except at T=0 or , from the exact analytic solution that can be evaluated in one dimension. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Ginzburg criterion」の詳細全文を読む スポンサード リンク
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