Phenomenological Renormalization Group Methods

Some renormalization group approaches have been proposed during the last few years which are close in spirit to the Nightingale phenomenological procedure. In essence, by exploiting the finite size scaling hypothesis, the approximate critical behavior of the model on infinite lattice is obtained through the exact computation of some thermal quantities of the model on finite clusters. In this work some of these methods are reviewed, namely the mean field renormalization group, the effective field renormalization group and the finite size scaling renormalization group procedures. Although special emphasis is given to the mean field renormalization group (since it has been, up to now, much more applied an extended to study a wide variety of different systems) a discussion of their potentialities and interrelations to other methods is also addressed.Comment: Review Articl

Some Critical Thoughts on Computational Materials Science

1. A Model is a Model is a Model is a Model The title of this report is of course meant to provoke. Why? Because there always exists a menace of confusing models with reality. Does anyone now refer to “first principles simulations”? This point is well taken. However, practically all of the current predictions in this domain are based on simulating electron dynamics using local density functional theory. These simulations, though providing a deep insight into materials ground states, are not exact but approximate solutions of the Schrödinger equation, which - not to forget - is a model itself [1]. Does someone now refer to “finite element simulations”? This point is also well taken. However, also in this case one has to admit that approximate solutions to large sets of non-linear differential equations formulated for a (non-existing) continuum under idealized boundary conditions is what it is: a model of nature but not reality. But us let calm down and render the discussion a bit more serious: current methods of ground state calculations are definitely among the cutting-edge disciplines in computational materials science and the community has learnt much from it during the last years. Similar aspects apply for some continuum-based finite element simulations. After all this report is meant to attract readers into this exciting field and not to repulse them. And for this reason I feel obliged to first make a point in underscoring that any interpretation of a research result obtained by computer simulation should be accompanied by scrutinizing the model ingredients and boundary conditions of that calculation in the same critical way as an experimentalist would check his experimental set-up

Universality classes in nonequilibrium lattice systems

This work is designed to overview our present knowledge about universality classes occurring in nonequilibrium systems defined on regular lattices. In the first section I summarize the most important critical exponents, relations and the field theoretical formalism used in the text. In the second section I briefly address the question of scaling behavior at first order phase transitions. In section three I review dynamical extensions of basic static classes, show the effect of mixing dynamics and the percolation behavior. The main body of this work is given in section four where genuine, dynamical universality classes specific to nonequilibrium systems are introduced. In section five I continue overviewing such nonequilibrium classes but in coupled, multi-component systems. Most of the known nonequilibrium transition classes are explored in low dimensions between active and absorbing states of reaction-diffusion type of systems. However by mapping they can be related to universal behavior of interface growth models, which I overview in section six. Finally in section seven I summarize families of absorbing state system classes, mean-field classes and give an outlook for further directions of research.Comment: Updated comprehensive review, 62 pages (two column), 29 figs included. Scheduled for publication in Reviews of Modern Physics in April 200

Universality classes in nonequilibrium lattice systems

This work is designed to overview our present knowledge about universality classes occurring in nonequilibrium systems defined on regular lattices. In the first section I summarize the most important critical exponents, relations and the field theoretical formalism used in the text. In the second section I briefly address the question of scaling behavior at first order phase transitions. In section three I review dynamical extensions of basic static classes, show the effect of mixing dynamics and the percolation behavior. The main body of this work is given in section four where genuine, dynamical universality classes specific to nonequilibrium systems are introduced. In section five I continue overviewing such nonequilibrium classes but in coupled, multi-component systems. Most of the known nonequilibrium transition classes are explored in low dimensions between active and absorbing states of reaction-diffusion type of systems. However by mapping they can be related to universal behavior of interface growth models, which I overview in section six. Finally in section seven I summarize families of absorbing state system classes, mean-field classes and give an outlook for further directions of research.Comment: Updated comprehensive review, 62 pages (two column), 29 figs included. Scheduled for publication in Reviews of Modern Physics in April 200

Diagonalizing transfer matrices and matrix product operators: a medley of exact and computational methods

Transfer matrices and matrix product operators play an ubiquitous role in the field of many body physics. This paper gives an ideosyncratic overview of applications, exact results and computational aspects of diagonalizing transfer matrices and matrix product operators. The results in this paper are a mixture of classic results, presented from the point of view of tensor networks, and of new results. Topics discussed are exact solutions of transfer matrices in equilibrium and non-equilibrium statistical physics, tensor network states, matrix product operator algebras, and numerical matrix product state methods for finding extremal eigenvectors of matrix product operators.Comment: Lecture notes from a course at Vienna Universit

Nonequilibrium Critical Phenomena and Phase Transitions into Absorbing States

This review addresses recent developments in nonequilibrium statistical physics. Focusing on phase transitions from fluctuating phases into absorbing states, the universality class of directed percolation is investigated in detail. The survey gives a general introduction to various lattice models of directed percolation and studies their scaling properties, field-theoretic aspects, numerical techniques, as well as possible experimental realizations. In addition, several examples of absorbing-state transitions which do not belong to the directed percolation universality class will be discussed. As a closely related technique, we investigate the concept of damage spreading. It is shown that this technique is ambiguous to some extent, making it impossible to define chaotic and regular phases in stochastic nonequilibrium systems. Finally, we discuss various classes of depinning transitions in models for interface growth which are related to phase transitions into absorbing states.Comment: Review article, revised version, LaTeX, 153 pages, 63 encapsulated postscript figure

Physical modelling of epithelia: reverse engineering cell competition in silico

Cell competition is a phenomenon in which less fit cells are removed from a tissue for optimal survival of the host. Competition has been observed in many physiological and pathophysiological conditions, especially in the prevention of tumor development. While there have been extensive population-scale experimental studies of competition, the competitive strategies and their underlying mechanisms in single cells are poorly understood. To date, two main mechanisms of cell competition have been described. Mechanical competition arises when the two competing cell types have different sensitivities to crowding. In contrast, during biochemical competition, signaling occurs at the interface between cell types leading to apoptosis of the loser cells. However, rigorously testing these hypotheses remains challenging due to the difficulty of obtaining sufficient single cell level information to bridge scales to the whole tissue. In this thesis, I present metrics aimed at characterising competition at the single cell level. Then, I demonstrate the development of a multi-layered, cell-scale computational model that I use to gain understanding on the single cell mechanisms that govern mechanical competition and decipher the "rules of the cellular game". After benchmarking cell growth and homeostasis in pure populations, I show that competition emerges when both cell types are included in simulations. I then investigate the impact of each computational parameter on the outcome of cell competition. Intriguingly, the outcome of biochemical competition is controlled by topological entropy between cell types, whereas the outcome of mechanical cell competition is exclusively controlled by differences in energetic potential between cell types. As 90% of cancers arise from epithelia and a number of genetic diseases present symptoms of epithelial fragility, I anticipate that my model of realistic implementation of epithelia will be of use to the biophysics and computational modelling community