Statistical physics and stromatolite growth: new perspectives on an ancient dilemma

This paper outlines our recent attempts to model the growth and form of microbialites from the perspective of the statistical physics of evolving surfaces. Microbialites arise from the environmental interactions of microbial communities (microbial mats). The mats evolve over time to form internally laminated organosedimentary structures (stromatolites). Modern day stromatolites exist in only a few locations, whereas ancient stromatolitic microbialites were the only form of life for much of the Earth's history. They existed in a wide variety of growth forms, ranging from almost perfect cones to branched columnar structures. The coniform structures are central to the heated debate on the oldest evidence of life. We proposed a biotic model which considers the relationship between upward growth of a phototropic or phototactic biofilm and mineral accretion normal to the surface. These processes are sufficient to account for the growth and form of many ancient stromatolities. These include domical stromatolites and coniform structures with thickened apical zones typical of Conophyton. More angular coniform structures, similar to the stromatolites claimed as the oldest macroscopic evidence of life, form when the photic effects dominate over mineral accretion.Comment: 8 pages, 3 figures. To be published in Proceedings of StatPhys-Taiwan 2004: Biologically Motivated Statistical Physics and Related Problems, 22-26 June 200

Random walks on finite lattice tubes

Exact results are obtained for random walks on finite lattice tubes with a single source and absorbing lattice sites at the ends. Explicit formulae are derived for the absorption probabilities at the ends and for the expectations that a random walk will visit a particular lattice site before being absorbed. Results are obtained for lattice tubes of arbitrary size and each of the regular lattice types; square, triangular and honeycomb. The results include an adjustable parameter to model the effects of strain, such as surface curvature, on the surface diffusion. Results for the triangular lattice tubes and the honeycomb lattice tubes model diffusion of adatoms on single walled zig-zag carbon nano-tubes with open ends.Comment: 22 pages, 4 figure

Magnetization Plateaux in Bethe Ansatz Solvable Spin-S Ladders

We examine the properties of the Bethe Ansatz solvable two- and three-leg spin-$S$ ladders. These models include Heisenberg rung interactions of arbitrary strength and thus capture the physics of the spin-$S$ Heisenberg ladders for strong rung coupling. The discrete values derived for the magnetization plateaux are seen to fit with the general prediction based on the Lieb-Schultz- Mattis theorem. We examine the magnetic phase diagram of the spin-1 ladder in detail and find an extended magnetization plateau at the fractional value $= {1/2}$ in agreement with the experimental observation for the spin-1 ladder compound BIP-TENO.Comment: 11 pages, 1 figur

Integrable O(n) model on the honeycomb lattice via reflection matrices : Surface critical behaviour

We study the $O(n)$ loop model on the honeycomb lattice with open boundary conditions. Reflection matrices for the underlying Izergin-Korepin $R$-matrix lead to three inequivalent sets of integrable boundary weights. One set, which has previously been considered, gives rise to the ordinary surface transition. The other two sets correspond respectively to the special surface transition and the mixed ordinary-special transition. We analyse the Bethe ansatz equations derived for these integrable cases and obtain the surface energies together with the central charges and scaling dimensions characterizing the corresponding phase transitions.Comment: LaTeX, 29 pages, with 5 PostScript figure

Intense Arctic Ozone Depletion in the Spring of 2011

Observations of record-breaking ozone depletion during the Arctic spring of 2011 were made at 76˚ N in Thule, Greenland. The ozone total column amount of 290 DU measured on 18 March 2011 is the lowest value from the 12-year observation record and represents an ozone depletion of up to 48% of a typical March column. The unique 2010 – 11 vortex was characterized by sustained low stratospheric temperatures and stability that resisted breakup through March. Simultaneous observations of O3, HF, HCl, HNO3, and ClONO2 demonstrate strong subsidence and substantial conversion of chlorine from its normal reservoirs.Au printemps 2011, des observations d’appauvrissement record de l’ozone ont été faites dans l’Arctique à 76˚ N à Thule, au Groenland. Le 18 mars 2011, la colonne d’ozone total a été mesurée à 290 DU, ce qui représente la valeur la plus faible depuis que les observations ont commencé à être consignées il y a 12 ans. Cela constitue un appauvrissement de l’ozone allant jusqu’à 48 % de la colonne typiquement enregistrée en mars. Le vortex unique dénoté en 2010-2011 était caractérisé par des températures stratosphériques faibles et soutenues ainsi que par une stabilité ayant résisté à la dissipation jusqu’en mars. Des observations simultanées de O3, HF, HCl, HNO3 et ClONO2 ont démontré une forte subsidence et une conversion substan­tielle du chlore à partir des réservoirs normaux

Mean field analysis of Williams-Bjerknes type growth

We investigate a class of stochastic growth models involving competition between two phases in which one of the phases has a competitive advantage. The equilibrium populations of the competing phases are calculated using a mean field analysis. Regression probabilities for the extinction of the advantaged phase are calculated in a leading order approximation. The results of the calculations are in good agreement with simulations carried out on a square lattice with periodic boundaries. The class of models are variants of the Williams- Bjerknes model for the growth of tumours in the basal layer of an epithelium. In the limit in which only one of the phases is unstable the class of models reduces to the well known variants of the Eden model.Comment: 21 pages, Latex2e, Elsevier style, 5 figure

Correlation lengths and E_8 mass spectrum of the dilute A_3 lattice model

The exact perturbation approach is used to derive the elementary correlation lengths $\xi_i$ and related mass gaps $m_i$ of the two-dimensional dilute A_L lattice model in regimes 1 and 2 for L odd from the Bethe Ansatz solution. In regime 2 the A_3 model is the E_8 lattice realisation of the two-dimensional Ising model in a magnetic field at T=T_c. The calculations for the A_3 model in regime 2 start from the eight thermodynamically significant string types found in previous numerical studies. These string types are seen to be consistent in the ordered high field limit. The eight masses obtained reduce with the approach to criticality to the E_8 masses predicted by Zamolodchikov, thus providing a further direct lattice determination of the E_8 mass spectrum.Comment: 57 pages, Latex, Elsevier style file

The critical fugacity for surface adsorption of self-avoiding walks on the honeycomb lattice is 1+21+\sqrt{2}

In 2010, Duminil-Copin and Smirnov proved a long-standing conjecture of Nienhuis, made in 1982, that the growth constant of self-avoiding walks on the hexagonal (a.k.a. honeycomb) lattice is $\mu=\sqrt{2+\sqrt{2}}.$ A key identity used in that proof was later generalised by Smirnov so as to apply to a general O(n) loop model with $n\in [-2,2]$ (the case $n=0$ corresponding to SAWs). We modify this model by restricting to a half-plane and introducing a surface fugacity $y$ associated with boundary sites (also called surface sites), and obtain a generalisation of Smirnov's identity. The critical value of the surface fugacity was conjectured by Batchelor and Yung in 1995 to be $y_{\rm c}=1+2/\sqrt{2-n}.$ This value plays a crucial role in our generalized identity, just as the value of growth constant did in Smirnov's identity. For the case $n=0$, corresponding to \saws\ interacting with a surface, we prove the conjectured value of the critical surface fugacity. A crucial part of the proof involves demonstrating that the generating function of self-avoiding bridges of height $T$, taken at its critical point $1/\mu$, tends to 0 as $T$ increases, as predicted from SLE theory.Comment: Major revision, references updated, 25 pages, 13 figure

Integrable vertex and loop models on the square lattice with open boundaries via reflection matrices

The procedure for obtaining integrable vertex models via reflection matrices on the square lattice with open boundaries is reviewed and explicitly carried out for a number of two- and three-state vertex models. These models include the six-vertex model, the 15-vertex $A_2^{(1)}$ model and the 19-vertex models of Izergin-Korepin and Zamolodchikov-Fateev. In each case the eigenspectra is determined by application of either the algebraic or the analytic Bethe ansatz with inhomeogeneities. With suitable choices of reflection matrices, these vertex models can be associated with integrable loop models on the same lattice. In general, the required choices {\em do not} coincide with those which lead to quantum group-invariant spin chains. The exact solution of the integrable loop models -- including an $O(n)$ model on the square lattice with open boundaries -- is of relevance to the surface critical behaviour of two-dimensional polymers.Comment: 35 pages, LaTeX with PostScript figures; minor corrections, version to appear in Nucl. Phys.

Critical behaviour of the dilute O(n), Izergin-Korepin and dilute ALA_L face models: Bulk properties

The analytic, nonlinear integral equation approach is used to calculate the finite-size corrections to the transfer matrix eigen-spectra of the critical dilute O(n) model on the square periodic lattice. The resulting bulk conformal weights extend previous exact results obtained in the honeycomb limit and include the negative spectral parameter regimes. The results give the operator content of the 19-vertex Izergin-Korepin model along with the conformal weights of the dilute $A_L$ face models in all four regimes.Comment: 23 pages, no ps figures, latex file, to appear in NP