Running Head: The Ising Model The Ising Model University Name Subject Name Instructor Name The Ising Model Introduction This essay is aimed at explaining what the Ising Model is to a beginner in the field. The essay follows a report format with an introduction to the topic…
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Finally, the paper culminates at a viable conclusion. The Ising Model, which has been named after the German physicist E. Ising, is a mathematically exploitable model pertaining to the realm of statistical physics or statistical mechanics. Analysing ferromagnetism has been the primary application area of the Model. However, the Ising Model can be used to explain and/or predict various physical phenomena in different fields like molecular biology, crystal chemistry, solid state, etc. as well. The aim of using this model is to find out more on the physics of phase transitions and it serves as a simplified framework to analyse phase transitions in the real substances. The square lattice Ising Model in two-dimensions is one of the most relevant and useful models to examine and understand the phenomenon of a phase transition. (Gallavotti, 1999; Ising, 1925; Lenz, 1920) Background and Historical Significance Although the Ising Model has been named after E. Ising, the inventor of the Model is W. Lenz. Lenz gave this model as a problem to solve to his disciple Ising. In Beitrage zum Verstandnis der magnetischen Eigenschaften in festen Korpern, Lenz (1920) put forward the idea of a systematic physical-statistical model to comprehend the magnetic properties in solids. A few years later, in Beitrag zur Theorie des Ferromagnetismus, Ising (1925) solved the Ising Model in one dimension which has no phase transition. In explaining the Model, Cipra (1987) focuses on the formation of binary alloys and the process of ferromagnetism with special reference to spontaneous magnetisation as the original application areas of the work of Ising (1925). “The latter is also of interest historically: an understanding of ferromagnetism – and especially “spontaneous magnetization” – was the original purpose of the Ising model and the subject of Ising’s doctoral dissertation.” (Cipra, 1985, p. 937) Generally because of this historical importance, ferromagnetism is widely used to interpret and explain the various characteristics of the Ising Model. After Ising solved the Model in one dimension, no significant achievement could be made in the following years. However, much later in the year 1944, L. Onsager managed to solve the square lattice variety of Ising Model in two-dimensions through an analytical description. In the context of crystal statistics, Onsager (1944) described the phenomenon of phase change as “an order-disorder transition” (Onsager, 1944, p. 117). Almost a decade later, Yang (1952) explained spontaneous magnetisation with the help of two-dimensional Ising Model. In this way, study of higher dimensional varieties of Ising Model became feasible and the scope of the Model expanded beyond the realm of statistical physics. The Model was extensively used to study the inter-particle interactions to understand the behaviours of atoms and molecules of real substance in the course of phase transitions. (Brush, 1967) Explanation In order to explain the Ising Model to a beginner, we must first understand the subject of statistical physics since Ising Model has mostly been used to explain certain critical concepts that are covered in this realm. With reference to the Model under discussion, we will have to emphasise the mechanistic aspects and approaches of the subject. Therefore, more precisely, Ising Model is much related to statistical mechanics. According to Catterall (2012): “
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