## Results and discussion

From a heat capacity function, Cp(T), a wrong heat capacity function Cp(Le)(T) was obtained with the total entropy function of the macroscopic enthalpy. It clearly shows that the total entropy function of the macroscopic enthalpy cannot approximate the entropy of the microscopic enthalpy that was necessary to calculate the heat capacity function with Eq. (24).

Figure 3 shows the entropy as a function of enthalpy. All the entropy functions are represented as difference from N state. In this figure, the entropy S(De) is plotted as the function of the microscopic enthalpy, and the entropy S (Le) is shown as the function of the macroscopic enthalpy. The difference between S (Le) and S (De) at maximum in this case is 8 JK-1mol-1, which is comparable to the gas constant and causes the large difference in heat capacity as seen in Fig.1.

In Fig.4, the probability function is calculated from both entropy S(Le) and S(De). These functions were already displayed with a different method by A. Cooper which is more complicated than this paper [3]. The probability function at the midpoint temperature has two maximums and resemble to that of A. Cooper, while the incorrect probability function, /(Le), shows only one maxt mum. Mathematically it is obvious that /Le) has only one maximum because the following equation can be derived by partial derivative of both sides of Eq. (22)

-tOOO 0 1000

Enthalpy I kJ mol '

-tOOO 0 1000

Enthalpy I kJ mol '

Fig. 3 Entropy functions are shown as functions of enthalpy with reference to N state. ASn^S^-Sn, A5,N<De)=5<De)-5N and ASNd=SD-SN. S^ is the entropy function of the macroscopic enthalpy directly calculated from the total Gibbs energy and total entropy by Legendre transformation. SfDi) is the entropy function of the microscopic enthalpy composed from the entropy of N and D state (see text in detail)

Fig. 4 Probability function of enthalpy at the midpoint of thermal transition (320K)./<Le) is calculated from S(Le) in Figure 3 and/(De) is calculated from S<De) in the figure

where TH is the temperature where <h>(TH)=h satisfies. Because the <h> is a monotonously increasing function of T, the right hand side of Eq.(30) is positive when h is smaller than <h>(T), while it is negative when h is larger than <h>(T).

Therefore the function has one maximum where h=<h>(T). It shows that the entropy will cause only one maximum of probability function of enthalpy when the transformation is applied.

As shown above, the entropy function of the macroscopic enthalpy does not approximate that of the microscopic enthalpy around the thermal transition of proteins while it does very well for each thermodynamic state. It indicates that the thermal transition of proteins cannot be treated as one phase but treated as a phase transition in spite of the continuity of the thermodynamic functions.

Usually the first order phase transition requires the discontinuity of the first derivatives of Gibbs energy. Strictly speaking, however, the complete discontinuity will be achieved only for the infinite system. When the system becomes small to the size of proteins, the discontinuity cannot be observed. However the discrepancy between the two entropy functions may be observed in this system. It indicates that this discrepancy may become a good index for phase transition for such a system.

The deconvolution method was proposed for the thermal transition of biopolymers [1, 2]. This report clarifies the statistical mechanical meaning for the deconvolution method. If one thermodynamic state includes the thermal transition in it, the discrepancy of these two entropy functions can be obtained. Then the system can be deconvoluted to each thermodynamic state where the Legendre transformation becomes a good approximation to get the entropy function of the microscopic enthalpy.