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- Statistical Mechanics Topic Please Don T Write Wrong Answer 1 (61.5 KiB) Viewed 43 times
statistical mechanics topic. please don't write wrong answer..
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statistical mechanics topic. please don't write wrong answer..
statistical mechanics topic. please don't write wrong
answer..
4.6. Consider the linearized RG transformation (14.3.12), with (a11 a12) a21 a22 (3) such that (a11922-a12a21) #0. We now introduce the "generalized coordinates" u₁ and u₂ through equations (14.3.13); clearly, u₁ and u₂ are certain linear combinations of the system parameters ki and k2. (a) Show that the slopes of the lines u₁ = 0 and u₂ = 0, in the (k₁, k₂)-plane, are a21 21-a11 a21 λ2 - a22 m1 = λ₂-a11 a12 and m2 λ1-922 a12 respectively; here, ₁ and 2 are the eigenvalues of the matrix A. Verify that the product 1m2 21/12 and hence the two lines are mutually perpendicular if and only if a12 = 21. (b) Check that, in the special case when a12 = 0 but a21 #0, the above slopes assume the simple form m₁ = and m₂ = a21/(a11-a22) whereas, in the special case when a21 = 0 but a12 #0, they become m1 = (a22-a11)/a12 and m2 = 0; note that Figure 14.7 pertains to the latter case. (c) Examine as well the cases for which either an or a22 is zero; Figure 14.6 pertains to the latter of these cases. 4.7. Check that the critical exponents (14.4.38) through (14.4.40), in the limit n→∞, agree with the corresponding exponents for the spherical model of Section 13.5 with d < 4.
answer..
4.6. Consider the linearized RG transformation (14.3.12), with (a11 a12) a21 a22 (3) such that (a11922-a12a21) #0. We now introduce the "generalized coordinates" u₁ and u₂ through equations (14.3.13); clearly, u₁ and u₂ are certain linear combinations of the system parameters ki and k2. (a) Show that the slopes of the lines u₁ = 0 and u₂ = 0, in the (k₁, k₂)-plane, are a21 21-a11 a21 λ2 - a22 m1 = λ₂-a11 a12 and m2 λ1-922 a12 respectively; here, ₁ and 2 are the eigenvalues of the matrix A. Verify that the product 1m2 21/12 and hence the two lines are mutually perpendicular if and only if a12 = 21. (b) Check that, in the special case when a12 = 0 but a21 #0, the above slopes assume the simple form m₁ = and m₂ = a21/(a11-a22) whereas, in the special case when a21 = 0 but a12 #0, they become m1 = (a22-a11)/a12 and m2 = 0; note that Figure 14.7 pertains to the latter case. (c) Examine as well the cases for which either an or a22 is zero; Figure 14.6 pertains to the latter of these cases. 4.7. Check that the critical exponents (14.4.38) through (14.4.40), in the limit n→∞, agree with the corresponding exponents for the spherical model of Section 13.5 with d < 4.
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