Boundary Row Research Articles

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AbstractDisinfection Alternatives for Safe Drinking Water, by E. A. Bryant, G. P. Fultan, and G. C. Budd. Published by van Nostrand Reinhold, 115 Fifth Avenue, New York 10003, USAGroundwater Treatment Technology (2nd Edition), E. K. Nyer. Published by van Nostrand Reinhold, 115 Fifth Avenue, New York 10003, USAWater and Wastewater Engineering Hydraulics, by T. J. Casey. Published by Oxford University Press, Walton Street, Oxford, OX2 6DPControl of Eutrophication in Inland Waters, by H. Klapper. Published by Ellis Horwood Ltd., Market Cross House, Cooper Street, Chichester, West Sussex, PO19 1EBDynamics of Gravel‐bed Rivers. Edited by P. Billi, R.D. Hey, C.R. Thorne & P. Tacconi. Published by John Wiley & Sons Ltd, Baffins Lane, Chichester, West Sussex, PO19 1UDBiology of Wastewater Treatment, by N. F. Gray. Published by Oxford University Press, Walton Street, Oxford OX2 6DPWater Quality Assessments ‐ A Guide to the use of Biota, Sediments and Water in Enviromental Monitoring. Edited by Deborah Chapman. Published by Chapman and Hall. 2‐6 Boundary Row, London SE1 8HNWater and Gas Mains Corrosion, Degradation and Protection, edited by Clauda Basalo. Published by Ellis Horwood Ltd, Market Cross House, Cooper Street, Chichester, West Sussex PO19 1EBThe Valley of the Stour from Weston Green to Harwich, by N.G. Rogers. Published by Ian Henry Publications Ltd., 20 Park Drive, Romford, Essex, RM1 4LHA Green History of the World, by Clive Ponting. Published by Penguin Books Ltd, 27 Wrights Lane, London W8 5TZ

Theoretical and Computational Dynamics of a Compressible Flow S-I. Pai and S. Luo Van Nostrand Reinhold, 115 Fifth Avenue, New York, NY 1003, USA. 1991. Distributed by Chapman and Hall, 2-6 Boundary Row, London, SE1 8HN. 1991. 699pp. Illustrated. £76.

Theoretical and Computational Dynamics of a Compressible Flow S-I. Pai and S. Luo Van Nostrand Reinhold, 115 Fifth Avenue, New York, NY 1003, USA. 1991. Distributed by Chapman and Hall, 2-6 Boundary Row, London, SE1 8HN. 1991. 699pp. Illustrated. £76. - Volume 96 Issue 959

Thermodynamics of an anisotropic boundary of a two-dimensional Ising model

We consider a semi-infinite two-dimensional Ising model with its spins on the boundary row having a different interaction energy E′1 from the ferromagnetic bulk. We find that the boundary specific heat has two divergent terms: one of which diverges linearly at the bulk critical temperature Tc, and the other, logaritmically. The linearly divergent term is independent of E′1, and the coefficient of the logaritmically divergent term is a decreasing function of E′1. There is a boundary latent heat at Tc, which is identical to McCoy and Wu's result. The boundary spins, which can be either ferromagnetic or antiferromagnetic, are aligned for temperature lower than Tc. The boundary spontaneous magnetization approaches zero in the form of A(E′1)(1−T/Tc)1/2, and the boundary zero field magnetic susceptibility diverges at Tc in the form −B (E′1)ln|1−T/Tc|, where A(E′1) and B(E′) are increasing functions of E′1.

Theory of a Two-Dimensional Ising Model with Random Impurities. III. Boundary Effects

We study the effects which immobile random impurities may have on magnetic properties and spin-spin correlation functions of a ferromagnetic system properties the Curie temperature. This is done within the context of the model introduced in the first paper of this series by computing the spin correlation functions on the boundary of a half-plane of random Ising spins, where the boundary row is allowed to intereact with a magnetic field $\mathfrak{H}$. We find that as $T\ensuremath{\rightarrow}{T}_{c}\ensuremath{-}$ the average boundary spontaneous magnetization vanishes as ${T}_{c}\ensuremath{-}T$. However, the average boundary magnetic susceptibility is shown not to exist for a finite range of temperature about ${T}_{c}$. Furthermore, at ${T}_{c}$ the average boundary magnetization behaves as $\ensuremath{-}{N}^{\ensuremath{-}1}$ sign $(\mathfrak{H}){[\mathrm{ln} N{\ensuremath{\beta}}_{c}|\mathfrak{H}|]}^{\ensuremath{-}1}$, where ${N}^{\ensuremath{-}1}$ is a measure of the width of the distribution of random bonds. Whenever $T\ensuremath{-}{T}_{c}=O({N}^{\ensuremath{-}2})$, the average spin-spin correlation function for two spins on the boundary is shown to approach its limit at infinite separation as some inverse power of the separation instead of as an exponential. At ${T}_{c}$ this average correlation function behaves asympotically as ${N}^{\ensuremath{-}2} {(\mathrm{ln}m{N}^{\ensuremath{-}2})}^{\ensuremath{-}1}$ when $m$ the separation between boundary spins, is large. Finally, we make the probabilistic nature of the boundary spontaneous magnetization more precise by computing its probability distribution function.

Theory of Toeplitz Determinants and the Spin Correlations of the Two-Dimensional Ising Model. I

We consider the rectangular Ising model on a half-plane of infinite extent and study some of the consequences connected with the presence of the boundary. Only the spins on the boundary row are allowed to interact with a magnetic field $\mathfrak{H}$. The method of Pfaffians is employed to obtain exact expressions for the partition function. It is found that the free energy is the sum of two terms, one of which is independent of $\mathfrak{H}$ and proportional to the total number of lattice sites, while the other depends on $\mathfrak{H}$ and is proportional to the number of lattice sites on the boundary. This separation makes it possible to define various thermodynamic quantities associated with the boundary. In particular, the boundary magnetization is shown to be discontinuous, in the ferromagnetic case, at zero magnetic field for temperatures below the bulk critical temperature ${T}_{c}$. This discontinuity, which is the spontaneous boundary magnetization, goes to zero as ${(1\ensuremath{-}\frac{T}{{T}_{c}})}^{\frac{1}{2}}$ as $T\ensuremath{\rightarrow}{T}_{c}\ensuremath{-}$. For $T={T}_{c}$, the discontinuity is of course absent, and the boundary magnetization behaves as $\ensuremath{-}\mathfrak{H}\mathrm{ln}\mathfrak{H}$ for small $\mathfrak{H}$. The boundary susceptibility at zero magnetic field in the ferromagnetic case exhibits a logarithmic singularity at $T={T}_{c}$, both above and below transition. An interesting feature is that the ferromagnetic boundary magnetization, although discontinuous for $Tl{T}_{c}$, may be analytically continued beyond the point $\mathfrak{H}=0$. We interpret this as a hystersis phenomenon which we study in detail by computing the probability distribution function for the average boundary spin. The correlation function for two spins, both on the boundary row, is also obtained exactly and its asymptotic behavior is given. Finally, we derive an expression for the magnetization in any row and explicitly evaluate it for the second row, i.e., the row next to the boundary.