George Gaspari

The Shapes of Random Walks

A theoretical description of the shape of a random object is presented that is analytically simple in application but quantitatively accurate. The asymmetry of the object is characterized in terms of the invariants of a tensor, analogous to the moment-of-inertia tensor, whose eigenvalues are the squares of the principal radii of gyration. The complications accompanying ensemble averaging because of random processes are greatly reduced when the object is embedded in a space of high dimensionality, d. Exact analytical expressions are presented in the case of infinite spatial dimensions, and a procedure for developing an expansion in powers of l/d is discussed for linear chain and ring-type random walks. The first two terms in such an expansion lead to results for various shape parameters that agree remarkably well with those calculated by computer simulation. The method can be extended to yield an approximate, but extremely accurate, expression for the probability distribution function directly. The theoretical approach discussed here can, in principle, be used to describe the shape of other random fractal objects as well.

Elements of the Random Walk: An Introduction for Advanced Students and Researchers

Preface 1. Introduction to techniques 2. Generating functions I 3. Generating functions II: recurrence, sites visited, and the role of dimensionality 4. Boundary conditions, steady state, and the electrostatic analogy 5. Variations on the random walk 6. The shape of a random walk 7. Path integrals and self-avoidance 8. Properties of the random walk: introduction to scaling 9. Scaling of walks and critical phenomena 10. Walks and the O(n) model: mean field theory and spin waves 11. Scaling, fractals, and renormalization 12. More on the renormalization group References Index.

The shapes and sizes of closed, pressurized random walks.

Two-dimensional cell-like membranes acted on by osmotic pressure differentials are represented by closed, unrestricted random walks. The treatment omits excluded-volume effects, and the pressure that is imposed thus favors an oriented area, so that the shriveled configuration of a vesicle with excess external pressure is inaccessible in this model. Nevertheless, the approach has the decided advantage of yielding analytic expressions in a complete statistical analysis. Results are presented for the average square of the radius of gyration, the asphericity, and the probability distribution of the principal components of the radius of gyration tensor. The analysis is done in both the constant-pressure and constant-area ensembles.

CLUSTER DISTRIBUTION IN MEAN-FIELD PERCOLATION : SCALING AND UNIVERSALITY

The partition function of the finite

A management overview methodology for technology assessment

1+\epsilon

n-vector model in the limit n-->0 and the statistics of linear polymer systems: A Ginzburg-Landau theory.

state Potts model is shown to yield a closed form for the distribution of clusters in the immediate vicinity of the percolation transition. Various important properties of the transition are manifest, including scaling behavior and the emergence of the spanning cluster. The predictions are compared with simulations. Agreement is found to be good, although convergence between theory and numerical results as the system size is increased is, in some cases, unaccountably slow.

Elements of the Random Walk: Boundary conditions, steady state, and the electrostatic analogy

Abstract The authors present an explicit, albeit simplified methodology for the management of technology assessment. A checklist is provided for the formulation and testing of this methodology that is applicable to the private sector as well as to government agencies.