On phase transition and the critical size in spatially restricted systems

Abstract

We study the finite-size effects on the critical temperature in spatially restricted systems with bulk second-order phase transition using the Fokker–Planck equation approach. It is established that the dependence of the transition temperature on system size is characterized by the competition of two length scales. The first scale is similar to the correlation length, determining the critical behavior in sufficiently large samples. The second scale appears as a consequence of the stochastic nature of the order parameter and controls the transitional features in small samples, particularly in the vicinity of the critical size. It is also found that the rate of the critical slowdown of relaxation of the order parameter fluctuations increases as the volume of the system decreases.

Dynamic heat capacity in spatially restricted systems with phase transition

“…We will associate these temperatures with finite sample (pseudo) critical temperatures T c . From figure 4 it follows, that the temperature T c decreases with a decrease of the volume, resembling experimental data [45][46][47] and our previous results concerning the order parameter relaxation [40]. For sufficiently large systems we obtain ν = 0.719 (for α = 1) and ν = 0.665 (for α = 10), which is close to the prediction of the hyperscaling relation ν ≈ 2 3 in the three-dimensional case [22].…”

“…Therefore, for the bulk dynamic heat capacity the leading timescale is given by 2α|T ∞ c −T | above and below T ∞ c . On the other hand, in case of bulk susceptibility the corresponding timescales are different above and below [40]. Consequently, the 'law of two' [22] is valid for the order parameter relaxation, but is violated for the energy relaxation.…”

“…Namely, the critical temperature behaves in sufficiently small systems as T c ∼ L p , where p ≈ 3. The approximate relation for the transition temperature was derived in [40]…”

“…The crucial role of the correlation length ξ in large systems has to be devolved to the length ρ in small samples. The length ρ also determines the leading length scale for the transitional behavior in the vicinity of the critical size (see, for details, [40]).…”