Figure 5
From: Calculation of π and Classification of Self-avoiding Lattices via DNA Configuration
Representative lattice configurations and analysis of self-avoiding random lattice growth generated by the self-avoiding walk algorithm. (a–d) Lattice configurations of self-avoiding random growth at a NS of 20. Open (a,b) and half-blocked configurations (growth blocked on either the right (c) or left (d) side of the lattices) are displayed. (e) Logarithmic numbers of lattice configurations (\(\mathrm{ln}\,{\rm{\Omega }}\,\) = S/k, where S is entropy and k is a constant) as a function of NS. \(\mathrm{ln}\,{\rm{\Omega }}\) obtained from the total numbers of available, open, and blocked (including half- and full-blocked) lattice configurations (\({{\rm{\Omega }}}_{{{\rm{N}}}_{{\rm{S}}}}={4}^{{{\rm{N}}}_{{\rm{s}}}}\), \({{\rm{\Omega }}}_{{\rm{O}}}\), and \({{\rm{\Omega }}}_{{\rm{B}}}\), respectively) at a given NS as well as from analytical evaluation of open lattice configuration (ΩA) are depicted. The intersection between ln ΩO and ln ΩB (occurred at 9.12 of NS) and the ratio of ΩB and ΩO are shown in the bottom and top insets, respectively. ΩO is larger and smaller than ΩB at below and above regions of the thin dotted line (marked at ΩB/ΩO = 1 in the graph of ΩB/ΩO), respectively. (f) A graph of difference of \(\mathrm{ln}\,{{\rm{\Omega }}}_{{\rm{O}}}\) and \(\mathrm{ln}\,{{\rm{\Omega }}}_{{\rm{B}}}\) (\({\rm{D}}\equiv \,\mathrm{ln}\,{{\rm{\Omega }}}_{{\rm{O}}}\mbox{--}\,\mathrm{ln}\,{{\rm{\Omega }}}_{{\rm{B}}}\)) as a function of NS. As mentioned, D becomes 0 at NS of 9.12 and the magnitude of D increases noticeably as NS increases or decreases from 9.12.
