# Download A Modern Approach to Critical Phenomena by Igor Herbut PDF By Igor Herbut

Read Online or Download A Modern Approach to Critical Phenomena PDF

Similar thermodynamics and statistical mechanics books

thermodynamics of nonequilibrium processes

Textual content: English, Polish (translation)

Extra info for A Modern Approach to Critical Phenomena

Example text

At a fixed particle number μ(T ) evidently also has to be a decreasing function. Assume that at some temperature TBEC , μ(TBEC ) = 0. The temperature dependence of the chemical potential will be then as in Fig. 3. 612 . 29) 2 (2π)d e 2mkB2TkBEC h2 −1 Since at all temperatures below TBEC the chemical potential of the noninteracting system stays zero, the value of the integral for total number of N = V 30 Ginzburg–Landau–Wilson theory particles in Eq. 28) is in fact at its finite maximum at T = TBEC .

36) This partition function defines the Ginzburg–Landau–Wilson theory. e. it has two real components. One can consider a more general field with N real components, with the action of the same form as in Eq. 36). For N = 1 such an action would be symmetric only with respect to an overall change of sign of the fields, (r ) → − (r ). This is the same global Z 2 symmetry as present in the Ising model, and we expect that the two models will lie in the same universality class. The action in Eq. 36) with N = 1 real field should describe then the ferromagnetic transition in the Ising model.

At T = Tc and p = pc , therefore, the equation of state simplifies into (v − vc )3 = 0. When compared with the general equation this determines vc = 3b, pc = a/(27b2 ), and kB Tc = 8a/(27b). Rescaling the parameters as p/ pc → p, v/vc → v, and T /Tc → T , the van der Waals equation may be cast into the universal form p+ 3 (3v − 1) = 8T, v2 also known as the law of corresponding states. Near the critical point one may expand 3 p − 1 = 4(T − 1) − 6(T − 1)(v − 1) − (v − 1)3 2 + O((T − 1)(v − 1)2 , (v − 1)4 ) and determine the actual liquid and gas densities from the Maxwell construction uG uL u dp = uG uL 9 u −6(T − 1) − u 2 du = 0, 2 where u = v − 1.