States of matter

This is a written version of a popular science talk for school children given on India's National Science Day 2009 at Mumbai. I discuss what distinguishes solids, liquids and gases from each other. I discuss briefly granular matter that in some ways behave like solids, and in other ways like liquids.Comment: 5 pages, eps figures, written version of a public lecture given at Mumbai on Feb 28, 200

Self-avoiding random walks: Some exactly soluble cases

We use the exact renormalization group equations to determine the asymptotic behavior of long self‐avoiding random walks on some pseudolattices. The lattices considered are the truncated 3‐simplex, the truncated 4‐simplex, and the modified rectangular lattices. The total number of random walks C_n, the number of polygons P_n of perimeter n, and the mean square end to end distance 〈R^2_n〉 are assumed to be asymptotically proportional to μ^nn^(γ−1), μ^nn^(α−3), and n^(2ν) respectively for large n, where n is the total length of the walk. The exact values of the connectivity constant μ, and the critical exponents λ, α, ν are determined for the three lattices. We give an example of two lattice systems that have the same effective nonintegral dimensionality 3/2 but different values of the critical exponents γ, α, and ν

Fragmentation of a sheet by propagating, branching and merging cracks

We consider a model of fragmentation of sheet by cracks that move with a velocity in preferred direction, but undergo random transverse displacements as they move. There is a non-zero probability of crack-splitting, and the split cracks move independently. If two cracks meet, they merge, and move as a single crack. In the steady state, there is non-zero density of cracks, and the sheet left behind by the moving cracks is broken into a large number of fragments of different sizes. The evolution operator for this model reduces to the Hamiltonian of quantum XY spin chain, which is exactly integrable. This allows us to determine the steady state, and also the distribution of sizes of fragments.Comment: 7 pages, 3 figures, minot typos fixe

Entropy and phase transitions in partially ordered sets

We define the entropy function S (ρ) =lim_(n→∞)2n^(−2)ln N (n,ρ), where N (n,ρ) is the number of different partial order relations definable over a set of n distinct objects, such that of the possible n (n−1)/2 pairs of objects, a fraction ρ are comparable. Using rigorous upper and lower bounds for S (ρ), we show that there exist real numbers ρ_1 and ρ_2;.083<ρ_1⩽1/4 and 3/8⩽ρ_2<48/49; such that S (ρ) has a constant value (ln2)/2 in the interval ρ_1⩽ρ⩽ρ_2; but is strictly less than (ln2)/2 if ρ⩽.083 or if ρ⩾48/49. We point out that the function S (ρ) may be considered to be the entropy function of an interacting "lattice gas" with long‐range three‐body interaction, in which case, the lattice gas undergoes a first order phase transition as a function of the "chemical activity" of the gas molecules, the value of the chemical activity at the phase transition being 1. A variational calculation suggests that the system undergoes an infinite number of first order phase transitions at larger values of the chemical activity. We conjecture that our best lower bound to S (ρ) gives the exact value of S (ρ) for all

Lattices of effectively nonintegral dimensionality

We construct a class of lattice systems that have effectively nonintegral dimensionality. A reasonable definition of effective dimensionality applicable to lattice systems is proposed and the effective dimensionalities of these lattices are determined. The renormalization procedure is used to determine the critical behavior of the classical XY model and the Fortuin–Kasteleyn cluster model on the truncated tetrahedron lattice which is shown to have the effective dimensionality 2 log3 /log5. It is found that no phase transition occurs at any finite temperature

Studying Self-Organized Criticality with Exactly Solved Models

This is a somewhat expanded version of the notes of a series of lectures given at Lausanne and Stellenbosch in 1998-99. They are intended to provide a pedagogical introduction to the abelian sandpile model of self-organized criticality, and its related models : the q=0 state Potts model, Takayasu aggregation model, the voter model, spanning trees, Eulerian walkers model etc. It provides an overview of the known results, and explains the equivalence of these models. Some open questions are discussed in the concluding section.Comment: Latex with epsf, 47 pages, 14 figure