Some Results and a Conjecture for Manna's Stochastic Sandpile Model

We present some analytical results for the stochastic sandpile model, studied earlier by Manna. In this model, the operators corresponding to particle addition at different sites commute. The eigenvalues of operators satisfy a system of coupled polynomial equations. For an L X L square, we construct a nontrivial toppling invariant, and hence a ladder operator which acting on eigenvectors of evolution operator gives new eigenvectors with different eigenvalues. For periodic boundary conditions in one direction, one more toppling invariant can be constructed. We show that there are many forbidden subconfigurations, and only an exponentially small fraction of all stable configurations are recurrent. We obtain rigorous lower and upper bounds for the minimum number of particles in a recurrent configuration, and conjecture a formula for its exact value for finite-size rectangles.Comment: 12 pages. 3 eps figures. Minor revision of text. Some typographical errors fixed. Talk given at StatPhys-Calcutta III, Jan. 1999. To appear in Physica

Playing with sandpiles

The Bak-Tang-Wiesenfeld sandpile model provdes a simple and elegant system with which to demonstate self-organized criticality. This model has rather remarkable mathematical properties first elucidated by Dhar. I demonstrate some of these properties graphically with a simple computer simulation.Comment: Contribution to the Niels Bohr Summer Institute on Complexity and Criticality; to appear in a Per Bak Memorial Issue of PHYSICA A; 6 pages 3 figure

Branched Polymers on the Given-Mandelbrot family of fractals

We study the average number A_n per site of the number of different configurations of a branched polymer of n bonds on the Given-Mandelbrot family of fractals using exact real-space renormalization. Different members of the family are characterized by an integer parameter b, b > 1. The fractal dimension varies from $log_{_2} 3$ to 2 as b is varied from 2 to infinity. We find that for all b > 2, A_n varies as $\lambda^n exp(b n ^{\psi})$, where $\lambda$ and $b$ are some constants, and $0 < \psi <1$. We determine the exponent $\psi$, and the size exponent $\nu$ (average diameter of polymer varies as $n^\nu$), exactly for all b > 2. This generalizes the earlier results of Knezevic and Vannimenus for b = 3 [Phys. Rev {\bf B 35} (1987) 4988].Comment: 24 pages, 8 figure

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

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

How BRCA1 deficiency affects emergency granulopoeisis in cells

BRCA1 mutation carriers are predisposed to breast and ovarian cancer. Chemotherapy is a common treatment used in breast cancer patients. However, chemotherapy can cause damage to bone marrow. Bone marrow is responsible for the production of white blood cells, namely neutrophils, which are the first line of defense in the innate immune system2. When an infectious or inflammatory challenge presents itself, neutrophils are used up in large quantities, and the hematopoietic system in the body has to rapidly adapt to increased demands by switching from the process of steady-state granulopoeisis to emergency granulopoeisis3. Evidence has shown that BRCA1 mutation carriers who have undergone chemotherapy treatment experience low counts of neutrophils1. Additional evidence has shown that the Fanconi gene pathway contributes to genomic stability during emergency granulopoeisis, and increased Fanconi C (Fancc) gene expression contributes to emergency granulopoeisis4. Since the BRCA1 gene is downstream of the FANCC gene, a myeloid leukemia cell line (U937) was tested to determine whether BRCA1 deficiency contributes to emergency granulopoeisis as well. Different concentrations of the protein IL-1Beta was added to the cells in order to mimic the emergency granulopoeisis response, and both FANCC and BRCA1 gene expression was measured. The general trend for the expression of both genes was found to be different than has previously been reported4. Shanley S et al. Clin Cancer Res. 2006; 12(23): 7033-7038. Kolaczkowzka E et al. Nat Rev Immunol. 2013; 13(3): 159-175. Manz M et al. Nature Reviews Immunology. 2014; 14: 302-314. Hu L et al. J Clin Invest. 2013; 123(9): 3952-3966

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