Entropy basin

From a thermodynamic perspective, Stillinger and Weber demonstrated that the total entropy of the liquid can similarly be divided into two additive terms, a configurational and a vibrational contribution.5,6 The configurational part Sc measures the number of structurally distinct basins of attraction on the PEL that the configuration point accesses at a given temperature, whereas the vibrational contribution Svib characterizes the number of states associated with intra-basin fluctuations. Thus, the AG relationship, when viewed from the PEL perspective, suggests that it is the thermodynamic availability of basins on the landscape that dominates the rate of liquid-state diffusive processes. 

The next step when computing configurational entropy is to calculate the vibrational contribution to the entropy Sv b- The most commonly employed technique used to accomplish this calculation is to assume that the configuration point of the liquid executes harmonic vibrations around its inherent structures (i.e., Svib Sharmb which is a description that can be expected to be accurate at low temperatures. The quantity Sharm for a given basin is then computed as117... 

Thus, the configurational entropy. S, (7. p) expresses the number of local potential energy minima and hence can be evaluated by potential energy landscape and thermodynamic integration methodology [53]. The vibration entropy was calculated within the framework of a harmonic approximation to each basin [165,166], an approximation that is valid at low temperatures [53]. Three notable results were obtained by Sastry [53] ... 

Figure 16. The entropy Sst relevant to the size of four potential basins in phase space. Say( ) represents entropy in terms of the accumulated residence time 1 (E). The dotted curve without a mark symbol represents a Lindeman index.

Here A and B are positive constants, expected to depend on which of frei or r] is involved, and. Vconf is the molar configurational entropy to be obtained from calorimetric measurements for crystal, liquid, and glass, i conf is a measure of the landscape s ruggedness and is related to the basin enumeration function introduced in Section II,... 

Thus, all thermodynamic (equilibrium) properties of a macroscopic system with a Gaussian landscape can be calculated from knowledge of ,-<> (p) (the density dependence of the basin depth per particle in the high-temperature limit), rrdensity dependence of the configurational entropy... 

The communal entropy is defined as the difference between the configurational entropy S(T) of the system and the entropy Sb(T) when the particles are confined in their cells [40], that is, the basin (we suppress showing the V dependence) ... 

The communal entropy is the entropy due to the deconfinement ofthe system from the basin. Accordingly, it vanishes when the system is confined in a basin such as in Figure 10.1a. Let Tr > 0 denote the temperature when this happens for SCL ... 

By a careful analysis, Goldstein [23] has shown that much of the excess entropy arise from nonbasin contributions, such as vibrational differences between glassy and crystalline phases, and is reflected most prominently in boson peaks. Thus, it is obvious that one cannot take the excess entropy as a genuine measure of the basin entropy or the entropy associated with the number of cell representations. 

What distinguishes experimentally prepared SCL is that the drop in its extrapolated basin entropy actually becomes negative below Tk > 0 (5, 6, 9, 25]. 

At low enough temperatures, the communal entropy vanishes when SCL or CR is confined to a single basin. 

At present, it is merely a hypothesis, which needs to be verified [75]. We first identify the communal entropy in a rigorous manner. We introduce the canonical PF (T) ofthejth basin obtained by restricting the integrationin (10.5) over points that belong to the basin. Let Fj(T) = — TlnZ J T) be the corresponding basin free energy and Sj its entropy. We follow [38] and express Z(T) (we do not show V... 

Let Nx Fh, T) denote the number of basins of free energy [, (b basin) in and Sx Fh,T) = In Nx Fy T) the corresponding IS class entropy. (Note that basins with the same free energy need not all belong to the same basin class.) At a given T, Zn x is dominated by those basins in Sx for which the class free energy "PxiFh, T) = Py,—TSx is minimum as a function of Pb- The minimum occurs at Fb = Fx, where... 

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