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This chapter considers lattice models with gauge symmetry and discusses their properties from the point of view of phase transitions and spectrum structure. It focuses on pure lattice gauge theories (without fermions) and studies them mainly with lattice methods. The chapter first constructs lattice models with gauge symmetry. It shows that they provide a lattice regularization of the continuum gauge theories studied in Chapters18-21: the low temperature or small coupling expansion of the lattice model is a regularized continuum perturbation theory. It then discusses pure gauge theories (without matter fields) on the lattice. It discovers that gauge theories have properties quite different from the ferromagnetic systems studied so far. In particular, the absence of a local order parameter will force us to examine the behaviour of a non-local quantity, a functional of loops called hereafter Wilson's loop to distinguish between the confined and deconfined phases. Results will be obtained in the high temperature or strong coupling limit and in the mean field approximation.

This chapter discusses the non-linear s-model, a field theory characterized by an orthogonal O(N) symmetry acting non-linearly on the fields. The study has several motivations. From the viewpoint of statistical physics, the model appears in the study of the large-distance properties, in the ordered phase at low temperature, of lattice spin models with O(N) symmetry and short-range interactions. Indeed, in the case of continuous symmetries, the whole low-temperature phase has a non-trivial large-distance physics due to the presence of Goldstone modes with vanishing mass or infinite correlation length. Moreover, the model possesses, in two dimensions, the property of asymptotic freedom (the Gaussian fixed point is marginally stable for the large-momentum or short-distance behaviour) and the spectrum is non-perturbative. These properties are shared, in dimension 4, by quantum chromodynamics (QCD), a non-Abelian gauge theory and a piece of the Standard Model of fundamental interactions describing physics at the microscopic scale.

As an introduction to the physics of phase transitions and critical phenomena, this chapter explains a number of basic and fundamental ideas such as phases, phase transitions, phase diagrams, universality, and critical phenomena. Especially important is the concept of order parameter, a quantity that measures the degree of asymmetry in the broken symmetry phase. Intuitive accounts are given to the concepts of coarse-graining, and scale and renormalization group transformations, which are powerful, systematic tools to analyze critical behaviour of macroscopic systems. Also explained are several spin and lattice gas model systems, on the basis of which phase transitions and critical phenomena will be studied.

This chapter considers the example of the N-vector model, that is, the O(N) symmetric (φ²)² field theory. It presents results for critical exponents, the equation of state and some amplitude ratios. It discusses more thoroughly critical exponents because they allow the most detailed and accurate comparison between Field Theory, lattice models, and experiments.

This chapter provides an introduction to a set of theoretical and numerical models that have been developed for this purpose. Spatial dynamic models are presented in three integral parts: modelling core, context, and method. Modelling cores are dynamic models for describing population demography and spread. Lagrangian models of random walks and step-selection functions that aim to portray and explain individual movement, and Eulerian models of reaction-diffusion and integrodifference equations that aim to capture the spatial dynamics of populations are introduced. Modelling context defines the arena for implementing the core. The basics of species distribution models to provide the context of suitable habitat are presented, leaving aspects of hybrid models, biotic interactions, and non-equilibrium dynamics to other chapters. Modelling methods are techniques for implementing cores in context. Different agent-based models, including individual-based models, cellular automata, and gravity (network) models, are explained.

Soft matter solutions are made of large solute dissolved in small solvent. The contrast in the size of solute and solvent gives special features to soft matter solutions. This chapter discusses these special features from the general solution theory. It first explains general solution theory for incompressible solutions (an assumption generally made for soft matter) and then considers the relation between free energy of mixing, osmotic pressure, and chemical potentials, followed by a discussion of the conditions for having uniform solutions or phase separated solutions. A model free energy is derived for the lattice model of solutions. Next, two typical soft matter solutions – the polymer solutions and colloidal solutions – are explained. The effect of size difference on osmotic pressure and phase diagrams, and the reason why inter-surface potential is important in colloidal solutions are discussed.

In the earliest days of science researchers were arguing philosophically what might be the reasonable explanation for an observed phenomenon. The majority of the contemporary scientific community claims that these arguments are useless because they do not add anything to our understanding of nature. The current consensus on the aim of science is that science collects facts (data) and discerns the order that exists between and among the various facts (e.g., Feynman 1985). According to this approach the mission of science is over when the phenomenon under investigation has been described. It is left to the philosophers to answer the question what is the governing physical process behind the observed physical phenomenon. Quantum mechanics is a good example of this approach, “It works, so we just have to accept it.” The consequence is that nearly 90 years after the development of quantum theory, there is still no consensus in the scientific community regarding the interpretation of the theory’s foundational building blocks (Schlosshauer et al. 2013). I believe that identifying the physical process governing a natural phenomenon is the responsibility of science. Dutailly (2013) expressed this quite well: A “black box” in the “cloud” which answers our questions correctly is not a scientific theory, if we have no knowledge of the basis upon which it has been designed. A scientific theory should provide a set of concepts and a formalism which can be easily and indisputably understood and used by the workers in the field. In this study the main unifying principle in chemistry, the periodic system of the chemical elements (PSCE) is investigated. The aim of the study is not only the description of the periodicity but also the understanding of the underlying physics resulting in the PSCE. By 1860 about 60 elements had been identified, and this initiated a quest to find their systematic arrangement. Based on similarities, Dobereiner (1829) in Germany suggested grouping the elements into triads.