Specific heat microcanonical

Microcanonical specific heat Cy (f) for the 2 x FI complex. Note the negativity in the backbending regions. From [254]. 

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. 

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... 

To construct the TPE for a specific process one has to use the appropriate distributions of initial conditions and short time transition probabihties. For an equilibrium system in contact with a heat bath, the distribution of initial conditions is canonical, while for an isolated equilibrium system at constant energy the initial conditions are distributed microcanonically. 

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