First-order phase transitions, finite-size scaling

Using the finite-size scaling method, study of the analytical behavior of the energy near the critical point shows that the open-shell system, such as the lithium-like atoms, is completely different from that of a closed-shell system, such as the helium-like atoms. The transition in the closed-shell systems from a bound state to a continuum resemble a first-order phase transition, while for the open-shell system the transition of the valence electron to the continuum is a continuous phase transition [9]. 

Molecular systems are challenging from the critical phenomenon point of view. In this section we review the finite-size scaling calculations to obtain critical parameters for simple molecular systems. As an example, we give detailed calculations for the critical parameters for Hj-like molecules without making use of the Born-Oppenheimer approximation. The system exhibits a critical point and dissociates through a first-order phase transition [11],... 

In general, the temperatures at which the heat capacity and the compressibility reach their respective maxima in finite systems are different. From the finite size scaling theory it follows that in the case of a second-order phase transition (e.g. at the critical point) Qm = a/t and 7 = 7/j/, while for any first-order phase transition ocm — Im — d, where d is the dimensionality of the system. The above predictions are often used to determine the nature of phase transitions studied by computer simulation methods [77]. The finite size scaling theory implies also that near the critical point the system free energy is given by... 

Aeeording to the finite-size scaling theory of the first-order phase transitions, the finite size of the erystaUite eauses a rounding of singularities in the behavior of isothermal compressibility and heat eapacity. The boundary of the two-phase region formed on finite crystallites has length proportional to its one-dimensional size (L). This introduces an additional term to the chemical potential of the surface film [336] ... 

For each system, there is a critical temperature that separates two well-defined regions the first one, where first-order phase transitions exist (abmpt jumps, at low temperatures), and the second one, where one observes smooth isotherms (at high temperatures). The exact calculation of that critical temperature is beyond the scope of this work because it would involve a deeper and computationally demanding study with the aid of finite-size scaling techniques. 

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