Thermodynamic Properties of Binary Solid Solutions on the Basis of the Nearest Neighbor Approximation
TL;DR: In this article, an approximate method of computing the partition function of a binary solid solution is formulated, and it is shown that no metastable phase is predicted if the only constraints are on the mean energy and mean composition.
Abstract: An approximate method of computing the partition function of a binary solid solution is formulated. If the only constraints are on the mean energy and mean composition, it is shown that no metastable phase is predicted. The partition function is shown to be obtainable from the largest eigenvalue of a quadratic form and the condition for a phase transition is shown to be related to the degeneracy of the largest eigenvalue. The physical interpretation of the eigenfunction is shown to be related to the probability of surface configuration, while the square of the eigenfunction is related to the probability of a configuration on the interior of the crystal. Some simple examples are discussed which are related to the effect of coordination number on phase transitions.
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TL;DR: In this paper, the transition between the helical and randomly coiled forms of a polypeptide chain is discussed by reference to a simple model that allows bonding only between each group and the third preceding.
Abstract: The transition between the helical and randomly coiled forms of a polypeptide chain is discussed by reference to a simple model that allows bonding only between each group and the third preceding Two principal parameters are introduced, a statistical parameter that is essentially an equilibrium constant for the bonding of segments to a portion of the chain that is already in helical form, and a special correction factor for the initiation of a helix A third parameter which specifies the minimum number of segments in a random section between two helical portions has only a minor effect on the results The partition function for this model is handled in two alternative ways, either as a summation suitable for short chains, or in terms of the eigenvalues and eigenvectors of a characteristic matrix; the latter is more suitable for long chains A transition from the random to the helical form is encountered as either the bonding parameter or the chain length is increased The critical value of the bonding pa
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TL;DR: In the absence of an external magnetic field, the Onsager method has been shown to be exactly soluble and shows a phase transition as discussed by the authors, which has attracted a lot of interest in the last few decades.
Abstract: The two-dimensional Ising model for a system of interacting spins (or for the ordering of an AB alloy) on a square lattice is one of the very few nontrivial many-body problems that is exactly soluble and shows a phase transition. Although the exact solution in the absence of an external magnetic field was first given almost twenty years ago in a famous paper by Onsager1 using the theory of Lie algebras, the flow of papers on both approximate and exact methods has remained strong to this day.2 One reason for this has been the interest in testing approximate methods on an exactly soluble problem. A second reason, no doubt, has been the considerable formidability of the Onsager method. The simplification achieved by Bruria Kaufman3 using the theory of spinor representations has diminished, but not removed, the reputation of the Onsager approach for incomprehensibility, while the subsequent application of this method by Yang4 to the calculation of the spontaneous magnetization has, if anything, helped to restore this reputation.
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TL;DR: In this paper, the theory of cooperative phenomena in crystals is studied and the authors propose a method to solve the problem of cooperative phenomenon in crystals, which is based on the concept of cooperation.
Abstract: (1960). On the theory of cooperative phenomena in crystals. Advances in Physics: Vol. 9, No. 35, pp. 245-361.
532 citations
TL;DR: In this article, the existence of persistent states in neural networks was shown to occur only if a certain transfer matrix has degenerate maximum eigenvalues, and it was suggested that these persistent states are associated with short term memory while the eigenvectors of the transfer matrix are a representation of long term memory.
Abstract: We show that given certain plausible assumptions the existence of persistent states in a neural network can occur only if a certain transfer matrix has degenerate maximum eigenvalues. The existence of such states of persistent order is directly analogous to the existence of long range order in an Ising spin system; while the transition to the state of persistent order is analogous to the transition to the ordered phase of the spin system. It is shown that the persistent state is also characterized by correlations between neurons throughout the brain. It is suggested that these persistent states are associated with short term memory while the eigenvectors of the transfer matrix are a representation of long term memory. A numerical example is given that illustrates certain of these features.
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2,983 citations
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01 Jan 1924
TL;DR: In this paper, Courant et al. describe an approach for the klassical Ideal einer gewermassen atomistischen Auffassung der Mathematik verlangt, den Stoff in Form of Voraussetzungen, Satzen and Beweisen zu kondensieren.
Abstract: VIII uber den Inhalt im einzelnen unterrichtet das ausfuhrliche Ver zeichnis. Zur Form ist etwas Grundsatzliches zu sagen: Das klassische Ideal einer gewissermassen atomistischen Auffassung der Mathematik ver langt, den Stoff in Form von Voraussetzungen, Satzen und Beweisen zu kondensieren. Dabei ist der innere Zusammenhang und die Motivierung der Theorie nicht unmittelbar Gegenstand der Darstellung. In kom plementarer Weise kann man ein mathematisches Gebiet als stetiges Gewebe von Zusammenhangen betrachten, bei dessen Beschreibung die Methode und die Motivierung in den Vordergrund treten und die Kri stallisierung der Einsichten in isolierte scharf umrissene Satze erst eine sekundare Rolle spielt. Wo eine Synthese beider Auffassungen untunlich schien, habe ich den zweiten Gesichtspunkt bevorzugt. New Rochelle, New York, 24. Oktober 1937. R. Courant. Inhaltsverzeichnis. Erstes Kapitel. Vorbereitung. - Grundbegriffe. I. Orientierung uber die Mannigfaltigkeit der Losungen 2 1. Beispiele S. 2. - 2. Differentialgleichungen zu gegebenen Funk tionenscharen und -familien S. 7. 2. Systeme von Differentialgleichungen 10 1. Problem der Aquivalenz von Systemen und einzelnen Differential 2. Bestimmte, uberbestimmte, unterbestimmte gleichungen S. 10. - Systeme S. 12. J. Integrationsmethoden bei speziellen Differentialgleichungen. . . . . . 14 1. Separation der Variablen S. 14. - 2. Erzeugung weiterer Losungen durch Superposition. Grundlosung der Warmeleitung. Poissons Integral S.16. 4. Geometrische Deutung einer partiellen Differentialgleichung erster Ord nung mit zwei unabhangigen Variablen. Das vollstandige Integral . . 18 1. Die geometrische Deutung einer partiellen Differentialgleichung erster Ordnung S. 18. - 2. Das vollstandige Integral S. 19. - 3. Singulare Integrale S. 20."
1,282 citations
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TL;DR: In this paper, it was shown how this set of laws of perfect solutions could be deduced by thermodynamic reasoning from certain assumptions about the additivity of energies and volumes on mixing, but these assumptions were not of a very simple form.
Abstract: 1. Introduction and Definitions .—In a previous paper the author discussed the laws of dilute and of perfect solutions. It was pointed out that the laws of dilute solutions take different forms according to the concentration scale used, these forms becoming identical only at infinite dilution. Of these various sets of laws that corresponding to the mole-fraction scale of concentration has in certain respects simpler properties than the others and is more symmetrical between solvent and solute. In particular only in this form is it possible for the laws of dilute solutions to hold at all concentrations, in which case they become the laws of perfect solutions. It was shown how this set of laws of perfect solutions could be deduced by thermodynamic reasoning from certain assumptions about the additivity of energies and volumes on mixing, but these assumptions were not of a very simple form. Nor was any reason found why the laws of dilute solutions should take the particular form corresponding to the mole-fraction scale of concentration, except analogy with the laws of perfect solutions. In the present paper an attempt will be made to remedy this omission by considerations of statistical mechanics. The method used will be that of partition functions described in Fowler’s text-book. This method is more elegant than Gibbs’ method of the canonical ensemble, does not suffer from the logical inconsistencies of Boltzmann’s method of “thermodynamic probability,” and is more powerful than either of these.
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