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These values are sometimes called "classical," perhaps because they are very old. They are not consistent with experiments, as we can see from Table 16.2, though they are not far wrong. The mean-field critical exponents have a high degree of universality, for they are independent of the parameters a o and uo, and of the dimesionality of space. The only thing they could depend on is the number of components of the order parameter. Thus, they cannot satisfy Josephson's scaling law in general. (They do
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satisfy it for d = 4, the significance of which we shall discuss later.) The conclusion is that mean-field theory is not reliable quantitatively, and predicts too high a degree of universality. However, we do learn some important lessons from it: First, we learn that it is possible to understand qualitatively, and even to some extent quantitatively, how a perfectly regular function such as the Landau free energy can lead to a singular Gibbs free energy. The point is that, while 1/1(m, H, T) is a regular function of its arguments, the value iii that minimizes 1/1 may be a singular function of Hand T. The mean-field theory shows how such singularities can arise in a physically instructive way. Second, we learn about universality, even if mean-field theory overstates it. The fact that the critical exponents are independent of the parameters in the Landau free energy suggests the critical phenomena are not sensitive to the details of the microscopic Hamiltonian, but possibly only to general structures, such as the number of "slow" degrees of freedom (i.e., order parameters) and the dimensionality of space. These lessons have given impetus and direction to the more refined treatments that we shall study later.
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ab (mod N )
The Van der Waals equation of state can be regarded as a mean-field theory of gas-liquid transitions. That is, we can find a Landau free energy that gives the desired equation upon minimizing with ~espect to the order parameter. We do this by working backward. The Van der Waals equation in universal form is
6 1 . 5 = 305 = 68 (mod 79).
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The free energy of the solution can be obtained from the definition A = U - TS. To this end we first discuss the internal energy of the solution. It is a function of T, P, no, n 1. Further, it is assumed to be a homogeneous function of no and n 1. That is, if no and n 1 are simultaneously increased by a certain factor, U increases by the same factor. By making a Taylor series expansion we can write
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U(T, P, no, n 1) = U(T, P, no,O)
Montgomery multiplication is certainly more work than it is worth in the simple example considered above. However, suppose that instead of ~0x1puting ab (mod N ) , we want to compute ad (mod N). Then to use the
U(T, P, no, n 1) = nouo(T,
p) + n1u1(T, p)
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