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Problem Statement: The use of pulsed electric fields (PEFs) as a tumor treatment modality is receiving increased traction. A typical clinical procedure involves insertion of a pair of electrodes into the tumor and administration of PEFs (amplitude: ~1 kV/cm; pulse-width: 100 μs). This leaves a zone of complete cell death and a sub-lethal zone where a fraction of the cells survive. There is substantial evidence of an anti-tumor systemic immune profile in animal patients treated with PEFs. However, the mechanism behind such immune profile alterations remains unknown, and the effect of PEFs on cell signaling within sub-lethal zones remains largely unexplored. Moreover, different values of a PEF pulse parameter, for e.g. the pulse-widths of 100 μs and 100 ns, may have different effects on cell signaling. Thus, the challenge of answering the mechanistic questions is compounded by the large PEF parameter space consisting of different combinations of pulse-widths, amplitudes, and exposure times. Intellectual merit: This Ph.D. research provides proof that sub-lethal PEFs can enhance anti-tumor signaling in triple negative breast cancer cells by abrogating thymic stromal lymphopoietin signaling and enhancing stimulatory proteins such as the tumor necrosis factor. Furthermore, experimental evidence produced during this Ph.D. research demonstrates that PEFs may not directly impact the intracellular mitochondrial membrane at clinically relevant field amplitudes. As demonstrated in this work, PEFs may influence the mitochondria via an indirect route such as disruption of the actin cytoskeleton and/or alteration of ionic environment in the cytoplasm due to cell membrane permeabilization. Thus, a reductionist approach to understanding the influence of PEFs on cell signaling is proposed by limiting the study to membrane dynamics. To overcome the problem of investigating the entire PEF parameter space, this Ph.D. research proposes a first-principle thermodynamic approach of scaling the PEF parameter space such that an understanding developed in one regime of PEF pulse parameter values can be used to understand other regimes of the parameter space. Demonstration of the validity of this scaling model is provided by coupling Monte-Carlo methods for density-of-states with the steepest-entropy-ascent quantum thermodynamic framework for the non-equilibrium prediction of the lipid membrane dynamics. / Ph. D. / A complete cure for cancer is still far from being realized despite very promising developments on the front of molecular drug therapy. One promising conceptual approach would be to achieve the ability to re-tune the cancerous signals that drive disease progression. To overcome current challenges in tuning cancerous signaling a paradigm change in cancer treatment is necessary. For example, a treatment strategy to alter cell signaling which leverages both the physical and chemical properties that accompany malignancy may be required. Electric fields, be it in the form of low-amplitude steady state fields or high-amplitude pulsed electric fields (PEFs), can induce distinct physical and chemical effects on cells. Hence, the use of electric fields as a clinical tumor treatment modality is receiving increased traction. However, the effect of these electric fields on cell signaling and cell behavior remains largely unexplored. This Ph.D. work provides experimental evidence that PEFs can directly impact cancerous cell signaling towards a less inflammatory and possibly less cancerous state. Although a noteworthy finding, the data poses another challenging question, i.e., how does the electric field impact cell behavior? Answering this mechanistic question is essential for FDA approval and a broader clinical use of the electric field modalities. An impediment to answering this question is the vast parameter space of electric fields (e.g., amplitude, pulse width, and number of pulses), which makes performing experimental mechanistic studies untenable. It is argued via experimental evidence gathered during this work that applying scaling laws applicable to lipid membranes may provide a solution to reducing the candidate PEF parameters to a manageable number. A non-equilibrium thermodynamic model is proposed that allows studying the behavior of lipid species using scaled electric field parameters. Thus, the v understanding gained via the proposed model can direct the next level of extensive biological assays and animal studies and eventually lead to effective cancer treatments.

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Here, we propose a method for generating a hierarchy of mean-field approximations to study the properties of the driven diffusive Ising model at nonequilibrium steady state. In addition, the present study offers a demonstration of the practical application of the information theoretic methods to a simple interacting nonequilibrium system. The application of maximum entropy principle to the system, which is in contact with a heat reservoir, leads to a minimization principle for the generalized Helmholtz free energy. At every level of approximation the latter is expressed in terms of the corresponding mean—field variables. These play the role of variational parameters. The rate equations for the mean-field variables, which incorporate the dynamics of the system, serve as constraints to the minimization procedure. The method is applicable to high temperatures as well to the low temperature phase coexistence regime and also has the potential for dealing with first-order phase transitions. At low temperatures the free energy is nonconvex and we use a Maxwell construction to find the relevant information for the system. To test the method we carry out numerical calculations at the pair level of approximation for the 2-dimensional driven diffusive Ising model on a square lattice with attractive interactions. The results reproduce quite well all the basic properties of the system as reported from Monte Carlo simulations. / Ph. D.

HTML clipboard Statistical physics seeks to explain macroscopic properties of matter in terms of microscopic interactions. Of particular interest is the phenomenon of phase transition: the sudden changes in macroscopic properties as external conditions are varied. Two models in particular are of great interest to mathematicians, namely the Ising model of a magnet and the percolation model of a porous solid. These models in turn are part of the unifying framework of the random-cluster representation, a model for random graphs which was first studied by Fortuin and Kasteleyn in the 1970’s. The random-cluster representation has proved extremely useful in proving important facts about the Ising model and similar models. In this work we study the corresponding graphical framework for two related models. The first model is the transverse field quantum Ising model, an extension of the original Ising model which was introduced by Lieb, Schultz and Mattis in the 1960’s. The second model is the space–time percolation process, which is closely related to the contact model for the spread of disease. In Chapter 2 we define the appropriate space–time random-cluster model and explore a range of useful probabilistic techniques for studying it. The space– time Potts model emerges as a natural generalization of the quantum Ising model. The basic properties of the phase transitions in these models are treated in this chapter, such as the fact that there is at most one unbounded fk-cluster, and the resulting lower bound on the critical value in <img src="http://upload.wikimedia.org/math/a/b/8/ab820da891078a8245d7f4f3252aee4f.png" />. In Chapter 3 we develop an alternative graphical representation of the quantum Ising model, called the random-parity representation. This representation is based on the random-current representation of the classical Ising model, and allows us to study in much greater detail the phase transition and critical behaviour. A major aim of this chapter is to prove sharpness of the phase transition in the quantum Ising model—a central issue in the theory— and to establish bounds on some critical exponents. We address these issues by using the random-parity representation to establish certain differential inequalities, integration of which gives the results. In Chapter 4 we explore some consequences and possible extensions of the results established in Chapters 2 and 3. For example, we determine the critical point for the quantum Ising model in <img src="http://upload.wikimedia.org/math/a/b/8/ab820da891078a8245d7f4f3252aee4f.png" /> and in ‘star-like’ geometries. / HTML clipboard Statistisk fysik syftar till att förklara ett materials makroskopiska egenskaper i termer av dess mikroskopiska struktur. En särskilt intressant egenskap är är fenomenet fasövergång, det vill säga en plötslig förändring i de makroskopiska egenskaperna när externa förutsättningar varieras. Två modeller är särskilt intressanta för en matematiker, nämligen Ising-modellen av en magnet och perkolationsmodellen av ett poröst material. Dessa två modeller sammanförs av den så-kallade fk-modellen, en slumpgrafsmodell som först studerades av Fortuin och Kasteleyn på 1970-talet. fk-modellen har sedermera visat sig vara extremt användbar för att bevisa viktiga resultat om Ising-modellen och liknande modeller. I den här avhandlingen studeras den motsvarande grafiska strukturen hos två näraliggande modeller. Den första av dessa är den kvantteoretiska Isingmodellen med transverst fält, vilken är en utveckling av den klassiska Isingmodellen och först studerades av Lieb, Schultz och Mattis på 1960-talet. Den andra modellen är rumtid-perkolation, som är nära besläktad med kontaktmodellen av infektionsspridning. I Kapitel 2 definieras rumtid-fk-modellen, och flera probabilistiska verktyg utforskas för att studera dess grundläggande egenskaper. Vi möter rumtid-Potts-modellen, som uppenbarar sig som en naturlig generalisering av den kvantteoretiska Ising-modellen. De viktigaste egenskaperna hos fasövergången i dessa modeller behandlas i detta kapitel, exempelvis det faktum att det i fk-modellen finns högst en obegränsad komponent, samt den undre gräns för det kritiska värdet som detta innebär. I Kapitel 3 utvecklas en alternativ grafisk framställning av den kvantteoretiska Ising-modellen, den så-kallade slumpparitetsframställningen. Denna är baserad på slumpflödesframställningen av den klassiska Ising-modellen, och är ett verktyg som låter oss studera fasövergången och gränsbeteendet mycket närmare. Huvudsyftet med detta kapitel är att bevisa att fasövergången är skarp—en central egenskap—samt att fastslå olikheter för vissa kritiska exponenter. Metoden består i att använda slumpparitetsframställningen för att härleda vissa differentialolikheter, vilka sedan kan integreras för att lägga fast att gränsen är skarp. I Kapitel 4 utforskas några konsekvenser, samt möjliga vidareutvecklingar, av resultaten i de tidigare kapitlen. Exempelvis bestäms det kritiska värdet hos den kvantteoretiska Ising-modellen på <img src="http://upload.wikimedia.org/math/a/b/8/ab820da891078a8245d7f4f3252aee4f.png" /> , samt i ‘stjärnliknankde’ geometrier. / QC 20100705

This thesis begins with the fundamentals of conformal field theory in three dimensions. The general properties of the conformal bootstrap are then reviewed. The three-dimensional Ising model is presented from the perspective of the renormalization group, after which the conformal field theory aspect at the critical point is discussed. Finally, the bootstrap programme is applied to the three-dimensional Ising model using numerical techniques, and the results analysed.

Neste trabalho estudamos alguns modelos estocásticos reversíveis e irreversíveis por meio de varias técnicas que incluem expansões em serie, simulações numéricas e métodos analíticos. Primeiramente, construímos uma expansão supercrítica para a densidade do processo de contato em uma dimensão, que fornece a taxa crítica e o expoente crítico pelo método de aproximantes de Pad´e. Depois, examinamos um modelo de pilha de areia com restrição de altura assimétrico que apresenta fluxo de partículas não-nulo no estado estacionário e suas propriedades criticas são determinadas em função do parâmetro de assimetria p. Finalmente, estudamos de forma analítica o modelo de Glauber linear, que é idêntico, em uma dimensão, ao modelo de Glauber. Em qualquer numero de dimensões, e possível obter uma expressão para a susceptibilidade do modelo de Glauber linear a partir da expansao em s´erie perturbativa, cujos coeficientes são determinados em todas as ordens. Também discutimos como generalizar esse método para obter expansões em série para o modelo de Glauber em duas dimensões. / In this work we study some reversible and irreversible stochastic models using various techniques that include series expansions, numerical simulations and analytical methods. Firstly we write a supercritical series expansion of the particle density of the one-dimensional contact process, which gives us the critical annihilation rate and critical exponent ¯ after using the Pad´e approximants method. Secondly we examine the assimetric height restricted sandpile model, which presents a non-zero particle flux at the stationary active state and its critical properties are determined as a function of the assimetry parameter p. Finally we study analitically the linear Glauber model, which is identical in one dimension to the Glauber model. It is possible in any dimension to obtain an expression of the susceptibility of the linear Glauber model from a perturbative series expansion in which the coefficients can be determined at all orders. We also discuss how to generalize the method in order to obtain a series expansion for the Glauber model in two dimensions.

Nosso objeto de estudo são os sistemas de spins com interações de longo alcance; em particular, estamos interessados em sistemas cuja probabilidade invariante é o modelo de Ising em A^S, onde A = {-1, 1} é o espaço de spins e S = Z^d é o espaço de sítios. Apresentamos dois resultados originais que são consequências da aplicação de algoritmos de simulação perfeita e de acoplamento no contexto da construção deste tipo de sistemas e de suas respectivas probabilidades invariantes. / Our object of interest are spin systems with long-range interactions. As a special case, we are interested in systems whose invariant measure is the Ising model on A^S, where A = {-1, 1} is the space of spins and S = Z^d is the space of sites. We present two original results that are byproducts of the application of Perfect Simulation and Coupling algorithms in the context of the construction of these spin systems and their respective invariant measures.

Estudamos o comportamento de modelos para sistemas modulados magnéticos e estruturais. A primeira parte deste trabalho e dedicada ao modelo de Ising com interações competitivas numa rede de Bethe, no limite de coordenação infinita, num campo magnético. Focalizamos nossa atenção no comportamento das fases comensuráveis na presença de campo. Obtivemos vários diagramas T H utilizando algoritmos numéricos muito mais eficientes do que a simples iteração do mapeamento associado ao modelo. Na segunda parte estudamos o modelo de FrenkelKontorova com primeiro e segundo harmônicos no potencial externo. Encontramos e investigamos transições de segunda ordem no interior das fases comensuráveis de período ímpar. Essas transições, denominadas simétricaassimétricas, estão associadas à quebra da simetria por reflexão que ocorre para potenciais suficientemente fortes. / We studied the behavior of models for magnetically and structurally modulated systems. The first part of this work is dedicated to the study of the Ising model with competing interactions on a Bethe lattice, in the infinite coordination limit, in a magnetic field. We focused our attention on the behavior of the commensurate phases in the presence of a field. We obtained various T H phase diagrams using numerical methods far more efficient than simple iteration of the mapping associated to the model. In the second part we studied the FrenkelKontorova model with first and second harmonics in the external potential. We found and investigated the second order transitions within the commensurate phases of odd periodicity. These transitions, called symmetricasymmetric transitions, are related to the breaking of reflection symmetry which occurs at high potentials.

This thesis explores different aspects of a surprising field of research: the conformally invariant scaling limits of planar statistical mechanics models. The aspects developed here include the proof of convergence of certain interfaces in the critical Ising magnetization model (joint work with Hugo Duminil-Copin and Clement Hongler), a study of the near-critical behavior of the uniform spanning tree in the scaling limit (joint work with Laure Dumaz and Wendelin Werner), the construction of an interesting measure on continuous loops satisfying a certain stability property under deformation (joint work with Julien Dubedat) as well as some related algebraic considerations, and finally, notes on a paper of Sheffield, that studies a certain coupling of the scaling limits of discrete interfaces - SLE curves - together with random surfaces obtained from the Gaussian free field.