Entropy microcanonical

The configurational entropy per lattice site Sc, is related to the total (microcanonical) entropy Sc as... 

Standard procedures permit the evaluation of the entropy of a Fermi gas under the conditions of a grand canonical ensemble, which we will have to adjust to obtain the microcanonical entropy. For low excitation energies, E, the entropy is... 

The physical process of protein folding involves a phase transition from a statistical coiled state to a uniquely compact native state. A powerful approach to define these systems is to make use of the microcanonical entropy function [18-27]. The entropy function S(E) is related to the... 

An exact knowledge of the microcanonical entropy, or the density of the states, of a protein model in both the native and nonnative states is crucial for a precise characterization of the folding process of the model, such as whether the folding is first-order or gradual, whether the model can fold uniquely to the native structure, whether there is a discontinuity in the order parameter of the conformation in the folding transition, and so on. Once the microscopic entropy function is accurately determined, the statistical mechanics of the protein folding problem is solved. The accurate determination of the entropy function of protein models by the ESMC method requires a proper treatment of the computational problems discussed in Section IV. 

Figure 6.8 Microcanonical entropy of HOOH computed versus excitation energy above the zero point.

E = total energy h = Planck constant kg = Boltzmeinn constant k(B) = unimolecular micro canonical rate constant k(7) = canonical rate con stant Nq = initial number of ions formed M E - E< = number of states of the transition state up to - Elj above the critical energy E P E) = distribution of internal energies R(E,tj = rate of dissociation T = temperature Af = activation enthalpy A5 = activation entropy A5j = microcanonical entropy of activation Vr = reaction coordinate frequency p E) = density of states of the parent ion at internal energy E o= degeneracy of reaction path. 

The latter term is a constant and cancels, if only entropy differences are considered. This is what is typically of most interest (e g., in the microcanonical statistical analysis of structural transitions) and thus, one often simply writes for the microcanonical entropy... 

As it has already been discussed in Section 2.2.3, the microcanonical entropy is given by S E)=kQ i g E). In this context, the (microcanonical) temperature T E) becomes a defined quantity, which is associated to the entropic change caused by a variation of macrostate energy. We strictly introduce it via... 

The estimation of errors, as described in the previous section, is the only way to verify the quality of statistical data obtained by experiment or computer simulation. With the methods described, it is simple to obtain a reliable error estimate for a single quantity such as an expectation value O of a quantity O. However, it is often also desirable to interpret the curve behavior of a function in its argument space. The microcanonical analysis introduced in Section 2.7, for example, requires precise information about the monotony of energy-dependent quantities, such as the microcanonical entropy S E). We can, of course, measure the entropy for each energy bin and obtain and average over many samples or... 

In Section 2,7, we have discussed the enormous advantage of the microcanonical statistical analysis to identify phase transition points. We particularly emphasized the meaning of the relationship between the density of states, the microcanonical entropy, and the... 

We introduced the microcanonical analysis in Section 2.7 and found that the density of states g E) already contains all relevant information about the phases of the system. Alternatively, one can also use the phase space volume AG(E) of the energetic shell that represents the macrostate in the microcanonical ensemble in the energetic interval (E,E+ AE) with AE being sufficiently small to satisfy AG E)=g E)AE. In the limit AE —> 0, the total phase space volume up to the energy E can thus be expressed as G E) = dE g(E ). Since g E) is positive for all E, G(E) is a monotonically increasing function and this quantity is suitably related to the microcanonical entropy S(E) of the system. In the definition of Hertz,... 

We will now take a closer look at the adsorption transition in the phase diagram (Fig. 13.12) and we do this by a microcanonical analysis [307, 308]. As we have discussed in detail in Section 2,7, the microcanonical approach allows for a unique identification of transition points and a precise description of the energetic and entropic properties of structural transitions in finite systems. The transition bands in canonical pseudophase diagrams are replaced by transition lines. Figure 13.15 shows the microcanonical entropy per monomer s e)=N lng e) as a function of the energy per monomer e=EfN for a polymer with N=, 20 monomers and a surface attraction strength = 5, as obtained from multicanonical simulations of the model described in Section 13.6. 

---