The Transverse Field Ising Model Investigation Of The Correlation Properties Of Cooperative Phenomena

The term cooperative phenomena is used to describe all phenomena caused by the interactions of large numbers of elementary particles such as the electrons in atoms. The simultaneous collective action by great numbers of these particles brings about several different physical property changes in systems, which are in general termed cooperative phenomena. Phase transition is the most fundamental of cooperative phenomena. Common examples of phase transition-types of cooperative phenomena include ferromagnetism, superconductivity, and order-disorder transformations etc.The study of cooperative phenomena demands the use of mathematical applications which can be quite challenging and problematic to deal with. Therefore to overcome the challenges involved in the calculations, there is the need to introduce models in which there are overwhelming simplifications in the inter-particle interactions. This also requires that these simplifications do not compromise the cooperative aspect of the problem which it seeks to tackle. The hope then is that a theoretical study of these simplified models(nonetheless they could still be mathematically difficult to solve) will reproduce the most fundamental features of the phenomena exhibited by actual physical systems. The Ising Model is the simplest of these models and has a venerable traditional of being used to understand the microscopic origins of phenomenaThe Ising model is a simple system which was initially created to study magnetism but has gradually developed to be used to examine several natural phenomena beyond the realm of physics. Indeed, the model is so simple and also has grown to become better understood mathematically and lends itself very well to computer applications. Therefore it has become an excellent framework for theoretical studies. By examining very simple two-dimensional interactions, the Ising model has become particularly useful for studying several problems involving pair wise interactions in diverse fields. Presently, the Ising model is commonly used to study any type of aggregate behavior, be it a microscopic or a macroscopic scale.In this research work, phase diagrams, density curves and global phase diagrams are developed with the aid of the 2D transverse field Ising model with temperature-dependent parameters. By supposing the novel simple dependent relations of the interaction parameters on temperature, the diagrams are straightforwardly obtained, which may be used to describe the closed-loop behavior for the phase transition of the systems. By the use of this novel theory, and with the aid MathCAD and MATLAB software, the diagrams were calculated.In fact, in the first category of phase diagrams and the associated density curves, the reentrant phase behaviors of closed-loop curves were first obtained. This model and theory combines well to show good coincident reentrant behavior with variations of temperature and density most commonly exhibited in some colloids and complex fluid mixtures as well as proteins. The phase diagrams showcases the microscopic interaction parameters of the exchange and transverse field coupling with temperature and analysis are carried out on each diagram?s properties.In the second category of diagrams, the phase diagrams are obtained by the same formalism but this time the plots are presented in polarization-temperature space. The diagrams shows the topological changes brought on phase transitions when the microscopic interaction parameters are modified. Several shapes of phase diagrams, corresponding to those commonly obtained both experimentally and theoretically, are obtained and described.The third category of microscopic interaction and phase transitions are presented by way of global phase diagrams(GPDs). Here global phase diagrams with the closed-loop behavior are calculated for the phase transition of systems by means of the 2D transverse field Ising model with nearest neighbor interaction. The 3D graph plotted by the various physical parameters gives a clear appreciation and qualitative understanding of the reentrant phase behavior of the system. Here also the diagrams show the close correlation between experimental phenomena and theoretical calculation for the closed-loop behavior for the phase transition of the systems.In all this research work, the transition properties of the transverse Ising model(TIM) have been investigated by assuming simple temperature dependent relations among the interaction parameters. This temperature dependence is called the effective-model, has enabled a deeper insight into more complicated systems such as complex fluids, whose modeling had been otherwise difficult to achieve by other mathematical calculations. Using this method the reentrant behavior of these complex systems have been obtained graphically and analyzed by way of phase diagrams, density curves and global phase diagrams. Hence the topologies of these graphs apart from showing the interdependence of interaction parameters and assumed exponent parameters, also gives us a new approach to calculate common and exotic phase diagrams obtained experimentally.This research basically develops a different method to calculate the reentrant behavior of some complex fluids and other exotic phase diagrams on the basis of the transverse Ising model which dwells on the cooperativity of interaction parameters and coupling to the transverse field. The results show that the exchange interaction parameter greatly influences the size of the ordered phase. Hence the larger the value of this constant, the larger the size of the ordered phase for a given value of the transverse field. In general this means that the higher values of the exchange parameter bring about phase transitions that straddle a wider range of polarizations and temperatures.