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TITLE
DEDICATION
CERTIFICATE
DECLARATION
ACKNOWLEDGEMENT
GLOSSARY OF TERMS
CONTENTS
1. Introduction
1.1 Nature of polymer molecules in solution
Fig.1. Schematic representation of polymer molecules in dilute solutions.
1.2 Excluded volume
1.3 Theta point
1.4 Radius of gyration
Fig.1.2. Schematic representation of (C.G.) centre of gravity.
1.5 End-to-end distance
Fig.1.3. Schematic representation of skeletal atoms of a polymer chain and end-to-end distance.
2. Self-Avoiding Walk on Lattices: A Review
2.1 The random flight model
2.2 Freely rotating chains
2.3 Chains with restricted rotation and independent bond rotation potential
2.4 Restricted rotation with interdependent rotation potentials
Fig.2.2. Minimum energy conformations for (a) ethane (b) methanol.
Fig.2.3. The potential energy E (φ2) of rotation about the central C-C bond in butane.
2.5 Random walks, restricted walks and self-avoiding walks
2.6 Methods of investigating self-avoiding walks
2.6.1 Rigorous analysis
2.6.2 Monte Carlo method
2.6.3 Exact enumeration
2.6.4 Transition matrix
2.7 Properties of self-avoiding walks
2.7.1 Number of walks of n-steps, Cn
2.7.2 Number of polygonal closures of n-steps, qn
2.7.3 Mean square length ( (0, 2)
2.7.4 Probability distribution of the end points fn (x) dx
2.7.5 Mean square radius of Gyration (Sn)
2.8 Analogy with the Ising model
3. Random Walks: Basic Notions and Notations
Fig.3.1. IUustrating the positive paths in a one-dimensional random walk.
3.1 Returns to origin
3.2 Relation between fen and Urn
3.3 Theory of one dimensional random walk
3.3.1 Reflection principle
3.3.2 Statement of the reflection principle
Fig. 3.2. IUustmtion of the reflection principle
3.3.3 The Ballot theorem
Fig.3.3 Illustration of the refledion principle
3.3.4 The main Lemma
3.3.5 Probability of first return to origin-F2,
3.4 Multidimensional random walk
3.5 Probability as a measure
3.5.1 Rings and algebras
3.5.2 Measure on rings
3.5.3 A measurable space
3.5.4 Probability
3.6 The classification of random walk
3.6.1 Transition function
3.6.2 Recurrent and transient random walks
4. Method of Computation
4.1 Number systems
4.2 Representation of a number in different number systems
4.3 Generation of simple random walk
4.4 Returns to origin
4.5 Walks that avoid back-tracking
Table 4.1. Returns to origin in random walks which avoid back tracking in two dimension.
Table 4.2. Returns to origin in random walks which avoid back tracking in three dimension.
Table 4.3. Returns to origin in random walks which avoid back tracking in four dimension.
Table 4.4. Mean square end-to-end distance in random walk which avoid back tracking in various dimensions.
Fig.4.1. Plot of mean square length of walks that avoid back tracking against the number of steps in different dimensions.
Fig.4.2. Plot of the characteristic ratios of walks that avoid badc-tracking against the chain length for different dimensions.
Table 4.5. Lst of the characteristic ratios of walks that avoid back-tracking invarious dimensions.
4.6 Self-avoiding walk
Table 4.6. Number of walks with at Least one self-avoiding polygonal closure in two dimension.
Table 4.7. Number of walks with at least one self-avoiding polygonal closurein three dimension
Table 4.8. Number of walks with at least one polygonal closure in four dimension.
Table 4.9. Number of walks with at least one polygonal closure in five dimension.
4.7 Mean square end-to-end distance
Fig.4.3. Distribution of end-to-end distance square of 4-step simple random walks in different dimensions.
Fig.4.4. Distribution of end-to-end distance square of 5-step simple random walk in different dimensions.
Fig.4.5. Distribution of end-to-end distance square of 6-step simple random walks in different dimensions.
Fig.4.6. Distribution of end-to-end distance square of 4-step self-avoiding random waIks in different dimensions.
Fig.4.7. Distribution of end-to-end distance square of 5step self-avoiding randorn walks in different dimensions.
Fig.4.8. Distribution of end-to-end distance square of 6-step self-avoiding random walk in different dimensions.
Fig.4.9. Distribution of end-to-end distance square of 7-step simple and self avoiding random walks in different dimensions.
Fig.4.10. Dishibution of end-to-end distance square of 8-step simple and self avoiding randonr walks in different dimensions.
Fig.4.11. Distribution of end-to-end distance square of 9-step simple and self avoiding random walks in different dimensions.
Fig.4.12. Distribution of end-to-end distance square of 10-step simple and self avoiding random walks in different dimensions.
Fig.4.13. Distribution of end-to-end distance square of 11-step two dimensional simlple and self-avoiding random walks.
Fig. 4.14. Distribution of end-to-end distance square of 12-step two dimensional simple and self-avoiding random wak.
Fig.4.15. Distribution of end-to-end distance square of 13-step two dimensional simple and self-avoiding random walks.
Table 4.10. Number of self avoiding walks and its mean square lengths in two dimension.
Table 4.11. Number of self avoiding walk and its mean square lengths in three dimension.
Table 4.12. Number of self avoiding walks and its mean lengths in four dimension.
Table 4.13. Number of self avoiding walks and its mean lengths in five dimension.
Table 4.14. Number of self avoiding walks and its mean lengths in six dimension.
Table 4.15. Number of self avoiding walks and its mean lengths in seven dimension.
Table 4.16. Number of self avoiding walks and its mean lengths in eight dimension.
Table 4.17. Number of self avoiding walks and its mean square lengths in nine dimension.
Table 4.18. Number of self avoiding walks and its mean square lengths in ten dimension.
Fig. 4.16. Plot of mean square length of SAW against number of steps for 2-5 dimensions.
Fig.4.17. Plot of mean square length of SAW against number of steps for 6-10 dimensions.
Fig. 4.18. Plot of chamderstic ratios of SAW against number of steps for 2-5 dimensions.
Fig.4.19. Plot of characteristic ratios of SAW against number of steps for 6-10 dimensions.
4.8 The traps
Table 4.19. Number of walks in two dimension which will be trapped in thesucceeding step.
Fig.4.20. All possible walks of a 7-step trap in two dimensional SAW.
Fig. 4.21. The four different walks of an 8-step trap in a two-dimensional SAW
Fig. 4.22. An eleven step self-avoiding walk in three dimension which form a trap at the 11th step.
5. Results and Discussion
5.1. Introduction
5.2 Theory and notation
5.3 Evaluation of A; (for n up to 13), R, (for n up to 12) and P; (for 2n up to 14)
5.4 Comparison with exact enumeration data
Fig.5.1. Plot of the ratio R/A against the number of steps.
6. Conclusions and Scope for Further Work
6.1 Conclusions
6.2 Scope for further work
References
APPENDICES
APPENDIX 1 List of decimal numbers corresponding to walks with at least one SAP
APPENDIX 2 Exact enumeration data of the distribution of the end-toend distance square of simple random walks
APPENDIX 3 Exact enumeration data of the distribution of the end-to-end distance square of SAW
APPENDIX 4 List of decimal numbers corresponding to walks that give traps
APPENDIX 5 Algorithms