Critical behavior, finite-size scaling

E. V. Albano. The critical behavior of dimer-dimer surface reaction models. Monte Carlo and finite-size scaling investigation. J Stat Phys 69 643-666,1992. 

The relevant questions to be addressed at this point are related to the calculation of Xc, the determination of the leading term in the asymptotic behavior of E(k) near and the existence of a square-integrable eigenfunction at the threshold. In the forthcoming sections we will develop the finite-size scaling as a powerful method to obtain accurate numerical estimations of critical parameters. Here we will present some exact results. 

The third method is a direct finite-size scaling approach to study the critical behavior of the quantum Hamiltonian without the need to make any explicit analogy to classical statistical mechanics [54,88]. The truncated wave function that approximate the eigenfunction Eq. (52) is given by... 

The convergence law of the results of the PR method is related to the corrections to the finite-size scaling. From Eq. (55) we expect that at the critical value of nuclear charge the correlation length is linear in N. In Fig. 9 we plot the correlation length of the finite pseudosystem (evaluated at the exact critical point Xc) as a function of the order N. The linear behavior shows that the asymptotic equation [Eq. (60)] for the correlation length holds for very low values of N [87]. 

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]. 

We studied the critical behavior of the eigenfunctions and resonances of the Hamiltonian equation, Eq. (101), using the FSS method described in Section IV. As a basis function for the finite-size scaling procedure, we used the orthonormalized eigenfunctions of the harmonic oscillator with mass equal to 1 and frequency equal to a ... 

J. E. Hunter and W. P. Reinhardt (1995) Finite-size-scaling behavior of the free-energy barrier between coexisting phases - determination of the critical-temperature and interfacial-tension of the Lennard-Jones fluid. J. Chem. Phys. 103, pp. 8627-8637... 

Computer simulation is invariably conducted on a model system whose size is small on the thermodynamic scale one typically has in mind when one refers to phase diagrams. Any simulation-based study of phase behavior thus necessarily requires careful consideration of finite-size effects. The nature of these effects is significantly different according to whether one is concerned with behavior close to or remote from a critical point. The distinction reflects the relative sizes of the linear dimension L of the system—the edge of the simulation cube, and the correlation length —the distance over which the local configurational variables are correlated. By noncritical we mean a system for which L E, by critical we mean one for which L [Pg.46]

Let us briefly review the well-studied subject of phase transitions and critical phenomena [39], Examples of critical points include a magnet at the onset of ordering, a liquid-vapor system at the critical temperature and pressure, and a binary liquid system that is about to phase-separate. The key point is that the fluctuations in a system at its critical point occur at all scales, and the system is exquisitely sensitive to tiny perturbations. Even though sharp phase transitions can occur only in infinitely large systems, behavior akin to that at a phase transition is observed for systems of finite size as well. Indeed, for a system near a critical point, the largest scale over which fluctuations occur is determined either by how far away one is from the critical point or by the finite size of the system. 

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