The Ising Model - Essay Example

?Running Head: The Ising Model The Ising Model 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. Next, background and historical significance of the Model is explained so that the reader can understand that why the Ising Model should be studied and what are its contributions to modern physics. Next, explanations regarding the Model are furnished to elucidate the basic concepts. 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): “Statistical Physics attempts to predict the properties of complex systems containing many interacting components undergoing random thermal motions. These might be molecules in a gas, atoms in a magnet, polymers in solution or a host of other interesting physical systems. One way in which physicists can gain insight into these difficult systems is by constructing idealised models which, it is hoped, will exhibit some of the interesting features of real systems but are simpler to study.” The Ising Model is a tool of statistical physics which deals with the physical and mechanical aspects of phase transitions of real substance. These transitions extensively involve understanding of particle behaviour inside the substance. Such transitions occur when there is a small parametric change like that of pressure or temperature which culminates at a qualitative, large-scale change in a system’s physical state. These sorts of phase transitions are highly come across in physics, and application of Ising Model is aimed at refining the mathematical understanding of such phenomena. However, phase transitions are yet not completely understood. “One purpose of the Ising Model explain how short-range interactions between, say, molecules in a crystal give rise to long-range, correlative behaviour, and to predict in some sense the potential for a phase transition.” (Cipra, 1987, p. 937) The most famous variety of a simplified Ising Model is the Ising Model in two-dimensions that can be used to analyse the behaviour of simple magnetic substances. A set of magnetic orientations or spins comprise this model. This set is arranged across a regular, graphical square lattice. Every spin or orientation can exist in one of the two states, which one can imagine as ‘up’ and ‘down’. Energy of such a system is determined by the total of the elementary interactions that one can find between a spin or orientation and its neighbours (viz. neighbouring sites) across the lattice. (Catterall, 2012; Yang, 1952; Gallavotti, 1999) In furtherance, Catterall (2012) states: “At very low temperatures the system will be found in its lowest energy state in which all the spins are, for example, up. As the system is heated the spins start to jiggle around and complicated motions result. The goal of statistical physics is to not to predict all of these detailed motions but only to calculate certain average properties of these motions, for example, how many spins on average are pointing up, what is the mean energy, etc.” Hence, Ising Model serves as one of the main foundations of statistical mechanics. In a graphical representation of the square lattice, each site can possess only two values (say up and down or + and --), and the other sites in neighbourhood prefer similar energy values. In the form of a system of up and down or + and – spins, the model can help to analyse magnetism. At high temperatures, the system looses magnetism; while at low temperatures, the system attains magnetised state. (Sethna, 1997) Again, if we think that the sites in the lattice are either vacant or occupied (say they have values 0 and 1 respectively), the model can explain liquid – gas transitions too. (Sethna, 1997) When a graph ? (e.g., an n-dimensional lattice) is given, per every site of the lattice y ? ? there is a distinct variable ?y which can be either -1 or +1. The configuration of the spin, ? = (?y)y?? is an assignment to the spin value of each site in the lattice. For any pair of adjacent sites, x, y ?? one would have an interaction denoted by Yxy, and for any x ? ? one would have an external field hx. Each configuration’s energy E (?) is described by the following equation: Where the first summation is over the pairs of adjoining spins (we count every pair only once). The configuration’s probability PT(?) is given by Boltzmann distribution according as the inverse temperature T?0: And the constant of normalisation is given by the following equation: The above equation serves as the partition function which is instrumental in describing and analysing the spins. (Adapted from the works of Gallavotti, 1999; Ising, 1925 and Lenz, 1920) In the beginning of 20th century, some experts believed that these mathematical deductions cannot explain phase transitions in one-dimension because of the discrepancies described below: The derivations are based on the summation value of e –TE over all the configurations. The exponential function is all over analytical as a function of T. The sum of analytical things is also analytical. Given that the logarithm of the partition function is not analytical as the function of temperature T near the instant of phase transition, the theory cannot work and fails to explain phase changes in real substances. (Brush, 1967) However, these arguments work for a finite summation of exponentials and precisely establish that singularities are absent only in the free energy contained in a system of finite extent. For systems that are in thermodynamic limit (i.e., for the infinite systems), the infinite summation can clearly lead to precise singularities. Convergences to the thermodynamic limit are fast, which elucidate phase behaviour on a comparatively small lattice, and hence phase transitions too can be studied in real substances that are multidimensional. R. Pierels was the first physicist who discovered this extremely useful characteristic of the Ising Model. (Palmer, 2007; Brush, 1967) Conclusion Nowadays, Ising Model has become more relevant in the age of advanced quantum mechanics and grid computing. Researchers like Cipra (1987) have aptly remarked that this statistical-physical model has been extensively analysed by several scholars in the fields of physics and mathematics. The main reason behind this high importance of the Ising Model is that it is mathematically operable and it also facilitates probabilistic modelling of magnetic behaviour, phase behaviour, temperature response, etc. of the different substances we find in nature. List of References Brush, S.G. (1967), History of the Lenz-Ising Model, Reviews of Modern Physics 39, pp. 883–893. Catterall, S. (2012), Ising applet, The Department of Physics at Syracuse University, Syracuse NY. Available: http://www.phy.syr.edu/courses/ijmp_c/Ising.html. Last accessed on 11th May, 2012. Cipra, B.A. (1987), An introduction to the Ising model, American Mathematical Monthly, 94, pp. 937-959. Gallavotti, G. (1999), Texts and Monographs in Physics, Berlin: Springer-Verlag Ising, E. (1925), Beitrag zur Theorie des Ferromagnetismus, Physikalische Zeitschrift  31, pp. 253–258. Lenz, W. (1920), Beitrage zum Verstandnis der magnetischen Eigenschaften in festen Korpern, Physikalische Zeitschrift 21, pp. 613–615. Onsager, L. (1944), Crystal statistics. I. A two-dimensional model with an order-disorder transition, Phys. Rev. (2) 65 (3–4), pp. 117–149. Palmer, J. (2007), Planar Ising Correlations, Boston: Birkhauser Sethna, J.P. (1997), Introduction to the Ising Model, Cornell University, Ithaca NY. Available: http://pages.physics.cornell.edu/~sethna/teaching/sss/ising/intro.htm. Last accessed on 11th May, 2012. Yang, C. N. (1952), The spontaneous magnetization of a two-dimensional Ising model, Physical Rev. (2), 85, pp. 808–816. Read More