Free energy singular part

If one assumes that tlie singular part A of the Helmholtz free energy is such a fimction... 

Free energy in the critical region is assumed to split into regular and singular parts (Gr and Gs respectivlely), only the latter of which obeys a scaling law,... 

The singular part of the free energy after n iterations Gsn has a value per spin which is a function of Kn and bn. Since Jo and Tc are constants the value per spin may be regarded as a function of (f )nt and (g )nB0. After n blockings the number of spins is n/(2d)n, where d is the number of dimensions. Thus... 

As surface corrugations grow, they will eventually touch each other and then folded parts will form on the surface. If the gel is sufficiently soft, the free energy can be lowered below the value in the homogeneous state by formation of a periodic pattern [20,21,89], The folded parts are singular surfaces like cracks and we encounter great mathematical difficulty. Nevertheless, we may analytically calculate the folded patterns in the highly compressible case 1... 

A similar homogeneity postulate can be written for the singular part of the free energy,... 

The free energy of a chain is the sum of a regular part which is proportional to S (to the number of links), and of a singular part F... 

From the free energy of the chain, we can deduce the internal energy and the specific heat of the chain. Let E(S, P) be the singular part of the internal energy and C(S, ft) the singular part of the specific heat. These quantities are given by... 

Griffiths has given a phenomenological (Landau) treatment of tricritical points which expresses the free energy as a sixth-order polynomial in an order parameter (which is some suitable linear combination of the physical densities , e.g. the mole fractions). The scaling properties of the singular part of the polynomial lead to four numbers = 5/6, 2 = 4/6 = 2/3, 3 = 3/6 = 1/2, = 2/6 = 1/3, in terms of which various critical exponents are expressed. Because this is an analytic (mean field) formulation, these exponents are classical , but it is believed that for experimental tricritical points in three dimensions they should be. ( Nonclassical logarithmic factors may exist, but these do not alter the exponents.)... 

The crossover exponent for the disorder is defined through the homogeneity of the singular part of the free energy in terms of the scaling fields /ui (temperature) and (j,2 (disorder). Under decimation, the free energy behaves... 

A different and usually more practical way of overcoming the difficulty is to look at differences in chemical potentials of the same solute in different phases. Since the singular part of the chemical potential is the same in the two phases, we can perform an expansion similar to (7.7) but for the difference of the Gibbs free energy of the two phases, in which case the coefficient of the linear term is finite. For instance, if we have two solvents Wi and W2, we can write... 

This is an extensive quantity and can be related to the free energy in thermal phase transitions. Since the specific heat is proportional to the second derivative of the free energy with respect to the temperature, the specific heat exponent a appears. Hence the formal analogy of the temperature and p. In the sum only the singular part has to be taken. 

---