Transfer matrix method finite temperature

Another series study of this problem was given by Marenduzzo et al. [94], who found that the finite length polymer unfolds in multiple steps, as successive bonds break away from the surface. This becomes smoothed out as the length of the SAW tends to infinity. This phenomenon is discussed and illustrated further below. However, their geometry is restricted to a strip of width L, enabling them to use standard transfer matrix methods. As a consequence, they do not enumerate all SAW of a given length. For the two-dimensional lattice they find a reentrant force vs. temperature curve. 

The transfer matrix method for calculating the partition function of directed paths on the delta at a finite temperature is expressed in terms of a recurrence based on the fact that the Boltzmann weight of any path is the product of the Boltzmann weights of its parts. The method considers a directed path from the apex to any site on the delta as formed of two segments the last bond on the path and the rest of the path. For the simplicity of expression, the 1-1-1 dimensional case is described, referring to Figure 1. The partition function ZT(x,t) of directed paths from the apex (0,0) to a site (x,t) on the 1-1-1 dimensional delta is calculated recursively by the relation [25,26,28]... 

The above theoretical analysis was supplemented by a detailed numerical study of directed paths at finite temperatures [27] using the transfer matrix method described in the previous subsection. The thermal averages (x) and (x ) for the directed paths from the apex to the base of the delta were calculated using the partition function and the quantity g was evaluated as a function of the temperature T and path-length L in 1 -P 1, 2 -P 1 and 3 -P 1 dimensions [27]. The numerical data showed that in 1 -P 1 dimension... 

At finite temperatures the quenched and annealed averages over the configurations of disordered bond energies are not the same. Consequently ln P(x, L) lnP(x, L), where the former quantity is annealed-averaged and the latter is quenched-averaged. Theoretical studies [29,30] supplemented by numerical results obtained by the transfer matrix method [31] show that the disorder average of lnP(x, L) has the scaling form... 

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