Configuration space, equilibrium phase

It may be evident (it should certainly not be surprising) that the extended MC framework needed to solve the phase-equilibrium problem entails exploration of a path that links the macrostates of the two competing phases, for the desired physical conditions ((i. The generic choices here are distinguished by the way in which the path is routed in relation to the key landmark in the configuration space, namely, the two-phase region which separates the macrostates of the two phases and which confers on them their (at least meta-) stability. Figure 2 depicts four conceptually different possibilities. 

Figure 2. Schematic representation of the four conceptually different paths (the heavy lines) one may utilize to attack the phase-coexistence problem. Each figure depicts a configuration space spanned by two macroscopic properties (such as energy, density. ..) the contours link macrostates of equal probability, for some given conditions c (such as temperature, pressure. ..). The two mountaintops locate the equilibrium macro states associated with the two competing phases, under these conditions. They are separated by a probability ravine (free-energy barrier). In case (a) the path comprises two disjoint sections confined to each of the two phases and terminating in appropriate reference macrostates. In (b) the path skirts the ravine. In (c) it passes through the ravine. In (d) it leaps the ravine.

It is not difficult to show that this function ip(R) satisfies the diffusion equation in Eq. (3.8). This establishes then a connection between the method of Kramers (an equivalent fictitious equilibrium problem in phase space) and the method of Kirkwood used here (a nonequilibrium problem in configuration space). 

An outstanding problem concerns itself with the structure of a hard sphere phase. This is a special instance of the more. general difficulty of the specification of the structure of infinitely extended random media. These questions will perhaps be the subject of a future mathematical discipline-stochastic geometry. The pair correlation function g(r), even if it is known, hardly suffices to specify uniqudy the stochastic metric properties of a random structure. For a finite N and V finite) system in equilibrium in thermal contact with a heat reservoir at temperature T, the density in the configuration space of the N particles [Eq. (2)]... 

Figure 1 Schematic behavior of the function f-AE) = v( )exp[—E/ gT] at a finite temperature T (see discussion about Eqs. [13]-[16]). For a large system, /ij.( ) has a sharp peak at the typical energy , and v( ) is very small, which means that the part of configurational space contributing significantly to this function is exceedingly small. In the case of a first-order phase transition, two peaks exist. A peptide can reside in several different stable states in thermodynamic equilibrium corresponding to several peaks of / j.( ) however because of the relatively small system size, the maxima of f (E) will not be sharp (compare with Figure 2).

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