Order parameter distribution

For a second-order transition, this problem is conveniently studied in terms of the order parameter distribution function, Pi,( ). Finite size scaling theory implies that near the critical point P/.( ) longer depends on the three variables L, 1 — T/T separately but rather is a scaled function of two variables (1 — only where v is the critical exponent of... 

Equation (10) shows that we can always accomplish our objective if we can measure the full canonical distribution of an appropriate order parameter. By full we mean that the contributions of both phases must be established and calibrated on the same scale. Of course it is the last bit that is the problem. (It is always straightforward to determine the two separately normalized distributions associated with the two phases, by conventional sampling in each phase in turn.) The reason that it is a problem is that the full canonical distribution of the (an) order parameter is typically vanishingly small at values intermediate between those characteristic of the two individual phases. The vanishingly small values provide a real, even quantitative, measure of the ergodic barrier between the phases. If the full -order parameter distribution is to be determined by a direct approach (as distinct from the circuitous approach of Section IV.B, or the off the map approach to be discussed in Section IV.D), these low-probability macrostates must be visited. 

What of the classical critical points The classical order parameter distribution function has not yet been determined, and so we have no choice but to employ the Ising distribution in the MFFSS analysis. We therefore have only very rough estimates of the critical parameters from MFFSS analyses of the systems with a = 4 [Tc = 1.3724(1), p = 0.2993(1)] and a = 3.1... 

Once a suitable weight function has been determined, a long simulation is performed in the course of which both phases are visited many times. During this run, the biased form of the order parameter distribution P(M 7) is accumulated in the form of a histogram. The applied bias can be unfolded from this distribution in the usual fashion to yield the true equilibrium distribution P(M), from which the relative free energy difference of the two phases can be read off as the logarithm of the ratio of peak weights in P(M). 

Fig. 13. In an umbrella sampling simulation for pathways configuration space is covered with a sequence of overlapping windows W[i]. For each window a separate transition path sampling simulation for paths with starting points in A and endpoints in >V[ ] is carried out. From the order parameter distributions matched where the windows overlap the distribution PA X,t) can be calculated...

In Figs. 65-67 we have shown the fourfold, sixfold, and twelvefold bond orientational order parameter distribution functions for DRPs, calculated using only intact bonds. For comparison, we have also shown the corresponding random distributions, calculated using randomly chosen bond angles and the same (intact bond) coordination number statistics as the DRP system. As for the WCA system, the distributions for the DRP system have sharp peaks corresponding to particular vertex types (see Table III), features which do not appear in the random distributions. These observations support the tiling description we have developed. 

The gradual change of the order-parameter distribution function with temperature in the critical region from a single peak structure above the critical temperature Tc to a double-peak structure below Tc already indicates that the finite size of the simulation box severely distorts the singular behavior that is expected at the phase transition in the thermodynamic limit, L-> oo. As is... 

A still more recent method [5] concentrates on the critical region for fluids. It seeks to match the order-parameter distribution at the critical point to that of the Ising-system (presumed to belong to the same universality class), thereby allowing evaluation of some nonuniversal critical parameters. This is an exciting idea. Its application requires the adjustment of several fitting parameters one hopes that it will prove to be robust and precise. 

Fig. 14 (a) Order parameter (fraction of detached monomers) versus surface potential e for different pulling forces. The chain is of length N = 128. (b) Order parameter distribution for pulling force fa/k T = 6.0 and different adhesion energy e/keT. The critical surface potential for this force is Ed = 6.095 0.03 so that the values eJk T = 6.09 and 6.10 are on each side of the detachment line. Reprinted with permission from [60]... 

P. A. Penz. Voltage-Induced Velocity and Optical Focusing in Liquid Crystals. Rhys. Rev. Lett., 24, p. 1405 (1970) Order Parameter Distribution for the Electrohydrodynamic Mode of a Nematic Liquid Crystal, Mol. Cryst. Liq. Cryst, 15, p. 141 (1971). 

A rather interesting method, motivated by successful applications to studies of liquid-gas phase separation of Lennard-Jones fluids and other fluids in the canonical ensemble, " starts from the observation that the temperature variation of the order parameter distribution can be analyzed in subsystems of linear dimension 1. Although in the total system (of linear dimension L) the order parameter is conserved, for different types of monomers (and chains). Of course, subsystems must be large enough, so that they still contain many chains. 

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