The classical treatment of the Ising model makes no distinction between systems of different dimensionality, so, if it fails so badly for d= 2, one might have expected that it would also fail for

Conformational changes in biopolymers are commonly described by a model that has been derived by an application of the one-dimensional Ising model to the problem of cooperative transitions from random coil states into ordered mostly helical conformations of (homo)biopolymers (see e.g. Cantor and Schimmel, 1980). Although the threshold is mostly of the cooperative transition type, landscapes can be constructed for which the threshold corresponds to a first order phase transition. [Pg.196]

Another model which includes interaction and for which partial results are available on the decay of initial correlations is that of the one dimensional time-dependent Ising model. This model was first suggested by Glauber,18 and analyzed by him for one-dimensional Ising lattices. Let us consider a one-dimensional lattice, each of whose sites contain a spin. The spin on site,/ will be denoted by s/t) where Sj(t) can take on values + 1, and transitions are made randomly between the two states due to interactions with an external heat reservoir. The state of the system is specified by the spin vector s(t) = (..., s- f), s0(t), Ji(0>---)- A- full description of the system is provided by the probability P(s t), but of more immediate interest are the reduced probabilities [Pg.212]

As a contribution to the study of these problems, stochastic models are here developed for two cases a freely-jointed chain in any number of dimensions, and a one-dimensional chain with nearest-neighbor correlations. Our work has been directly inspired by two different sources the Monte Carlo studies by Verdier23,24 of the dynamics of chains confined to simple cubic lattices, and the analytical treatment by Glauber25 of the dynamics of linear Ising models. No attempt is made in this work to introduce the effects of excluded volume or hydrodynamic interactions. [Pg.306]

The coefficients a and b (see Table 3.2) take into account the restrictions in spin dimensionality. For a = b= 1, the Heisenberg model with isotropic exchange interaction and isotropic susceptibility results. The combination of a = 1 and b = 0 yields the strictly anisotropic Ising model, in which the orientation of the spins is restricted to the z-axis. Consequently, the susceptibility is strongly orientation dependent and one needs to differentiate between x" in the direction of the z-axis ( easy axis ) and x perpendicular to z. The molar susceptibilities are then related as [Pg.90]

As is apparent from Figure 10.1, an a-helical structure imposes fairly rigid constraints on the relative positions of successive residues in a peptide chain. Thus there is a loss of entropy that must be overcome energetically in order for an a-helix to form. To explain the underlying biophysics of this system, John Schellman introduced a theory of helix-coil transitions that is motivated by the Ising model for one-dimensional spin system in physics [180, 170], [Pg.242]

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