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作 者：Joseph Rudnick. 著
出 版 社：Cambridge University Press
出版时间：2004 年1月
I S B N：0521828910
| Joseph Rudnick earned his PhD in 1970. He has held faculty positions at Tufts University and the University of California, Santa Cruz. He has held a visiting position at Harvard University. He is currently a Professor in the Department of Physics and Astronomy at the University of California, Los Angeles. .. << 查看详细 |
| preface 1 introduction to techniques 1.1 the simplest walk 1.2 some very elementary calculations on the simplest walk 1.3 back to the probability distribution 1.4 recursion relation for the one-dimensional walk 1.5 backing into the generating function for a random walk 1.6 supplement: method of steepest descents 2 generating functions i 2.1 general introduction to generating functions 2.2 supplement 1: gaussian integrals 2.3 supplement 2: fourier expansions on a lattice 2.4 supplement 3: asymptotic coefficients of power series 3 generating functions ii: recurrence, sites visited, and the role of dimensionality 3.1 recurrence 3.2 a new generating function 3.3 derivation of the new generating function 3.4 dimensionality and the probability of recurrence 3.5 recurrence in two dimensions 3.6 recurrence when the dimensionality, d, lies between 2 and 4 . 3.7 the probability of non-recurrence in walks on different cubic lattices in three dimensions 3.8 the number of sites visited by a random walk 4 boundary conditions, steady state, and the electrostatic analogy 4.1 the effects of spatial constraints on random walk statistics 4.2 random walk in the steady state 4.3 supplement: boundary conditions at an absorbing boundary 5 variations on the random walk 5.1 the biased random walk 5.2 the persistent random walk 5.3 the continuous time random walk 6 the shape of a random walk 6.1 the notion and quantification of shape 6.2 walks in d ]] 3 dimensions 6.3 final commentary 6.4 supplement 1: principal radii of gyration and rotational motion 6.5 supplement 2: calculations for the mean asphericity 6.6 supplement 3: derivation of (6.21) for the radius of gyration tensor, t, and the eigenvalues of the operator 7 path integrals and self-avoidance 7.1 the unrestricted random walk as a path integral 7.2 self-avoiding walks 8 properties of the random walk: introduction to scaling 8.1 universality 9 scaling of walks and critical phenomena 9.1 scaling and the random walk 9.2 critical points, scaling, and broken symmetries 9.3 ginzburg-landau-wilson effective hamiltonian 9.4 scaling and the mean end-to-end distance; (r2) 9.5 connection between the o(n) model and the self-avoiding walk 9.6 supplement: evaluation of gaussian integrals 10 walks and the o(n) model: mean field theory and spin waves 10.1 mean field theory and spin waves contributions 10.2 the mean field theory of the o(n) model 10.3 fluctuations: low order spin wave theory 10.4 the correlation hole 11 scaling, fractals, and renormalization 11.1 scale invariance in mathematics and nature 11.2 more on the renormalization group: the real space method 11.3 recursion relations: fixed points and critical exponents 12 more on the renormalization group 12.1 the momentum-shell method 12.2 the effective hamiltonian when there is fourth order interaction between the spin degrees of freedom …… references index |
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