The Statistical Mechanics of Interacting Walks, Polygons, Animals and Vesicles
This book surveys and explains the mathematical methods and techniques used in the study of lattice models of polymers in solvents. The techniques include the self-avoiding walk and its related models including animal and tree graphs, surfaces and vesicles and directed models. The important feature in all these models is the contribution of conformational degrees of freedom to the free energy, which leads to the idea of a tricritical point. The book explores the theory of tricriticality...
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This book surveys and explains the mathematical methods and techniques used in the study of lattice models of polymers in solvents. The techniques include the self-avoiding walk and its related models including animal and tree graphs, surfaces and vesicles and directed models. The important feature in all these models is the contribution of conformational degrees of freedom to the free energy, which leads to the idea of a tricritical point. The book explores the theory of tricriticality showing how it can be used to interpret the limiting free energy and generating functions. Density function and pattern theorems are also discussed and are applied to models of collapsing and adsorbing walks, to composite polygons and crumpling surfaces.
1Introduction11.1Lattice models of polymers and vesicles11.2Walks and polygons41.3Scaling92Tricriticality152.1Interacting models of polygons152.2Classical tricriticality182.3Finite size scaling232.4Homogeneity of the generating function262.5Uniform asymptotics and the finite size scaling function283Density functions and free energies393.1The density function393.2Density functions and free energies503.3Properties of the density function533.4Examples574Exact models654.1Introduction654.2Partition polygons (Ferrers diagrams)674.3Stack polygons744.4Staircase polygons804.5The adsorption of staircase walks on the main diagonal984.6Partially directed walks with a contact activity1174.7Directed animals and directed percolation1225Interacting models of walks and polygons1265.1Walks and polygons1265.2The pattern theorem and interacting models of walks1435.3Polygons with curvature1575.4Polygons interacting with a surface: adsorption1675.5Torsion in polygons1945.6Dense walks and composite polygons2006Animals trees2276.1Lattice animals and trees2276.2Pattern theorems and interacting models of lattice animals2326.3Collapsing animals2376.4Adsorbing trees2506.5Embeddings of graphs with specified topologies2717Lattice vesicles and surfaces2857.1Introduction2857.2Square lattice vesicles2857.3Punctured disks in two dimensions2937.4Adsorbing disks in three dimensions3037.5Crumpling surfaces313App. ASubadditive functions333App. BConvex functions339App. CAsymptotics for q-factorials346App. DBond or edge percolation350Bibliography359Index377
