Coherent-anomaly method in self-avoiding walk problems |
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| Institution: | 1. Emergency Department, Hospital of Merano (SABES-ASDAA), Merano-Meran, Italy;2. Emergency Department, Nuovo Santa Chiara Hospital, Azienda Ospedaliero-Universitaria Pisana, Pisa, Italy;3. Emergency Department, Hospital of San Paolo (ASL N°2 Savonese), Savona, Italy;4. Emergency Department, University of Verona, Verona, Italy;5. Department of Clinical and Experimental Medicine, University of Pisa, Pisa, Italy;1. Department of Biology, Washington University, St. Louis, MO 63130, USA;2. Center for Solar Energy and Energy Storage, Washington University, St. Louis, MO 63130, USA;3. Department of Energy, Environmental & Chemical Engineering, Washington University, St. Louis, MO 63130, USA;1. School of Environmental Science and Engineering, Key Laboratory of Subsurface Hydrology and Ecological Effects in Arid Region of Ministry of Education, Key Laboratory of Environmental Protection & Pollution and Remediation of Water and Soil of Shaanxi Province, Chang''an University, Xi''an 710054, PR China;2. The School of Earth Science and Resources, Chang''an University, Xi''an 710054, PR China |
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| Abstract: | Self-avoiding walk (SAW), being a nonequilibrium cooperative phenomenon, is investigated with a finite-order-restricted-walk (finite-ORW or FORW) coherent-anomaly method (CAM). The coefficient β1r in the asymptotic form Cnr≅ β1r λn1r for the total number Cnr of r- ORW's with respect to the step number n is investigated for the first time. An asymptotic form for SAW's is thus obtained form the series of FORW approximants, Cnr≅ brgμn(1 + a/r)n, as the envelope curve Cn≅b(ae/g)gμnng. Numerical results are given by Cn≅1.424n0.27884.1507n and Cn≅1.179n0.158710.005n for the plane triangular lattice and f.c.c. lattice, respectively. A good coincidence of the total numbers estimated from the above simple formulae with exact enumerations for finite-step SAW's implies that the essential nature of SAW (non-Markov process) can be understood from FORW (Markov process) in the CAM framework. |
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