Microcanonical analysis

To make it applicable to other transition types as well, it is attractive to extend this flatness idea by replacing the Maxwell construction with a more general principle, the principle of least sensitivity. This is a weaker condition, but it allows us to investigate first-and higher-order transitions by a microcanonical analysis more systematically and in much more detail [61],... 

This type of microcanonical analysis is very similar to Ehrenfest s classification scheme for phase transitions in the thermodynamic limit. In this scheme, the order of the transition is fixed by the smallest value of n, at which the nth-order derivative of the free energy with respect to an independent thermodynamic variable, e.g., d " F T, V,N)/dI " )vj, becomes discontinuous at any point. Obviously, first-order transitions are characterized by a discontinuity in the entropy as a function of temperature, S(T) = (dF(T, V,N)/dT)yjn, at the transition temperature T x- The discontinuity at the transition point h.S corresponds to a non-vanishing of the latent heat rtrA5= Ag > 0, In a second-order phase transition, the entropy is continuous, but the second-order derivative, which is related to the heat capacity, d F T, V,N)/dT )vjs[ Cy(T), is not. The heat capacity (or better the specific heat cy= Cy/N) possesses a discontinuity (often a divergence) at the critical temperature Tct-Although higher-order phase transitions are rather rare, Ehrenfest s scheme accommodates these transitions as well. 

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... 

Structural phase diagram for a flexible, elastic polymer with 90 monomers, parametrized by temperature T and nonbonded Interaction range 8. The transition lines (solid lines) were obtained by inflection-point microcanonical analysis. The crossover between collapse transition and nucleation cross is enlarged in the inset. The dashed vertical line separates solid phases that cannot be discriminated thermodynamically. The bottom figure shows for T = 02 the (canonical) probability that a conformation in these solid phases contains nic icosahedral cores, thereby separating fee or decahedral crystalline structures with Oje = 0 from Mackay icosahedral shapes (Ok > 1). From [136]. 

In Fig. 9.14, results from the canonical calculations (mean energy E) and specific heat per monomer cy) are shown as functions of the temperature. The specific heat exhibits a clear peak near T = 0.35 which is close to the folding temperature Tfou, as defined before in the microcanonical analysis. The loss of information by the canonical averaging process is apparent by comparing ( ) and the inverse, non-unique mapping of microcanonical... 

It is sometimes argued that proteins fold in solvent, where the solvent serves as a heat bath. This would provide a fixed canonical temperature such that the canonical interpretation is sufficient to imderstand the transition. However, the solvent-protein interaction is actually implicitly contained in the heteropolymer model and, nonetheless, the microcanonical analysis reveals this effect which is simply lost by integrating out the energetic fluctuations in the canonical ensemble (see Fig. 9.14). 

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,... 

Because of its unique perspective, we will explain the physical origin causing these differences by microcanonical analysis, which proves to be particnlarly suitable for this type of problem. 

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. 

As a microcanonical analysis shows, the adsorption transition of S1, for example, is rather second-order-like. For such small systems, energetic gaps are typically too tiny for first-order signals. In this context, see the general discussion in Section 6.4. 

---