Statistical Mechanics - Physics at Oregon State University

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200 CHAPTER 9. GENERAL METHODS: CRITICAL EXPONENTS. approximation, the answers in the Bethe approximation and in mean-field theory will be different. If, on the other hand, the correlation length ξ is much larger than the size of the cluster, the size of the cluster is not important and the problem is essentially solved in a mean-field approach. Since ξ diverges near Tc, the Bethe approximation is equivalent to a mean field theory in this limit. Therefore, the Bethe approximation will give the same critical exponents as mean-field theory. In order to make a real improvement, we need to take into account correlations of infinite range. We can improve the analysis, however. We will use the susceptibility as an example, but the analysis woks for all critical exponents. Near the critical temperature we have χ ≈ A±|T −Tc| γ , plus higher order terms. This means that near the critical temperature we have log(χ) =≈ log(A±) + γ log |T − Tc|. The second term will dominate if we are close enough to the critical temperature. Hence we have log(χ) log |T − Tc| → γ (9.57) near the critical temperature. Hence if we plot the ratio on the left as a function of temperature, we should be able to extract the value of γ. Very simply stated, so what can be wrong. First of all, we assume that we know the value of the critical temperature. That might be true or not. What happens if we are wrong? If we do not use the exact value of the critical temperature a plot of the ratio of the logarithms will show two peaks. The ratio will go to zero when the denominator goes to infinity, or when the temperature reaches the erroneous value of the critical temperature. The ratio will go to infinity when the enumerator diverges, at the true value of the critical temperature. For example, see figure 9.1 where we have used χ = 1 |t−1| (1 + 0.1|t − 1|) and moved Tc by a small amount to Tc = 1.01. We can use this behavior to find the critical temperature. If we have sufficient data near the phase transition we try different values of Tc and see when the peak and zero merge. But what is near the critical temperature? One could guess that the correction in χ is a factor of the form 1 + B±|T − Tc|. This gives an extra term in the denominator equal to the logarithm of this expression, and near the critical temperature we now find log(χ) log |T − Tc| |T − Tc| → γ + B± log |T − Tc| (9.58) The second term goes to zero, indeed. But how fast? We do not know. See figure 9.2 for a typical case. We used the same form as before, but now with the correct value of the critical temperature. In addition, the previous argument assumes that the behavior near the critical temperature is given by powers only. There are theoretical results where near the critical temperature we have logarithmic factors, χ ∝ |T − Tc| γ log |T − Tc|. Because the logarithm goes to infinity slower than any inverse power, such a factor does not change the value of the critical exponent, but it certainly changes the behavior near the phase transition. We
9.3. CRITICAL EXPONENTS. 201 −0.75 −1.0 −1.25 −1.5 −1.75 −2.0 0.9 0.95 Figure 9.1: Ideal case to find critical exponent, with wrong guess of critical temperature. need to get very close to the phase transition to get rid of such terms, see figure 9.3. Unfortunately, reality is even more complicated. The best way to discuss this is to think about the correlation length ξ, the length over which spins act coherently. This length diverges near the critical temperature. Using the correlation length we can define the range of temperatures needed to find the values of the critical exponents. If the range of the interactions in the material is Rint we need temperatures close enough to Tc so that ξ(T − Tc) ≫ Rint. But in real life the correlation length cannot increase beyond the size of the sample. Suppose the sample has similar dimensions in all directions, given by a number Rs. If we have ξ(T − Tc) > Rs the sample will act as a large molecule with a given susceptibility, which will remain the same. There is no divergence compensating the denominator, and the ratio of the logarithms goes to zero, see figure 9.4. This shows that we can still find the critical exponent for an infinite sample by connecting both sides of the curve, but the precision does suffer. Also, if the sample dimensions are not the same in all directions we will see a transition from three to two to one to zero dimensional behavior! Finally, we can use this to analyze cluster data. If the size of the cluster is larger than the effective range of the interactions the ratio of the logarithms will be close to the correct value, but if the correlation length becomes larger than the cluster size we will get the mean field results. See figure 9.5, where we changed the model so the exact results for γ is two. The previous discussion shows the importance of the thermodynamic limit. For a finite sample the behavior right at the critical temperature is always normal or analytic. Nothing diverges. But close to the critical temperature we see the characteristics of an infinite sample. The larger the sample, the closer we can get to the critical temperature an still see the behavior of an infinite x 1.0 1.05 1.1
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Statistical Mechanics Henri J.F. Ja
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IV CONTENTS 4.3 Gas of poly-atomic
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VI CONTENTS
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VIII INTRODUCTION Introduction. The
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X INTRODUCTION In chapter nine we d
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2 CHAPTER 1. FOUNDATION OF STATISTI
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