Singular Free Energy

We will now show that the form of Sdis ( ) in the gap has no relevance to the physics of the problem. Therefore, the entropy extension over the gap can be arbitrary as long as Sdis ( ) is continuous and concave. We consider quantities associated with communal entropies in this section, but will not show comm for simplicity of notation. The argument can be easily extended to configurational entropies. We compare Zdis(T) and Z T). They only differ in terms containing E Ek- For E k, they use the same function Sdis( ). Thus, for T TKOr( dis(T) k), the two PFs are identical. Consider T = Tk and write 

---

At this stage finite-size scaling theory [155, 193-197, 199] is applicable for the description of the specific heat maximum and of the onset of the superfluid density (see the beginning of Section 11), which characterize the rounded-off X transition. The singular free energy density, /, of the finite system (in the absence of external fields) can be described in terms of a universal function (T()) in the form [194—197] f = where is a metric factor,... 

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

Beutler, T. C., Mark, A. E., van Schaik, R. C., Gerber, P. R., van Gunsteren, W. F. Avoiding singularities and numerical instabilities in free energy calculations based on molecular simulations. Chem. Phys. Letters 222 (1994) 529-539... 

Buetler T C, A E Mark, R C van Schaik, P R Gerber and W F van Gunsteren 1994. Avoiding Singularities and Numerical Instabilities in Free Energy Calculations Based on Molecular Simulations. Chemical Physics Letters 222 529-539. 

When A = 0 one recovers the Lennard-Jones potential. When A = 1, the atom is annihilated smoothly and the singularity disappears progressively. The parameter a can be chosen to increase the smoothness of the free energy. A small a results in a near singularity around A = 1 while a large a results in a near singularity around A = 0. See article by Beutler et al. [55] for an algorithm to calculate an appropriate value of a. 

Simonson, T., Free energy of particle insertion. An exact analysis of the origin singularity for simple liquids, Molec. Phys. 1993, 80, 441447... 

Avoiding singularities and numerical instabilities in free energy calculations based on molecular simulations, Chem. Phys. Lett. 222 529 (1994). 

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... 

Crystals grow from their supersaturated vapor by the addition of vapor atoms at their free surfaces. In this process, the surface is subjected to an effective pressure due to the difference in free energy between the solid and vapor. The interface moves outward toward the vapor as it acts as a sink for the incoming flux of atoms. The mechanism by which atoms leave the vapor phase and eventually become permanently incorporated in the crystal is often relatively complex, and the kinetics of the process depends upon the type of surface involved (i.e., singular, vicinal,... 

An initially singular and atomically flat surface advances only if ledges, possessing kinks, are nucleated and formed on the surface. This can occur if adatoms cluster together on the surface and nucleate new ledges at small pillbox-shaped clusters of adatoms as indicated in Fig. 12.1. The free energy to form such a cluster... 

---