On an Extension Problem for Density Matrices

We investigate the problem of the existence of a density matrix rho on the product of three Hilbert spaces with given marginals on the pair (1,2) and the pair (2,3). While we do not solve this problem completely we offer partial results in the form of some necessary and some sufficient conditions on the two marginals. The quantum case differs markedly from the classical (commutative) case, where the obvious necessary compatibility condition suffices, namely, trace_1 (rho_{12}) = \trace_3 (rho_{23}).Comment: 12 pages late

The promoter of the human interleukin-2 gene contains two octamer-binding sites and is partially activated by the expression of Oct-2

The gene encoding interleukin-2 (IL-2) contains a sequence 52 to 326 nucleotides upstream of its transcriptional initiation site that promotes transcription in T cells that have been activated by costimulation with tetradecanoyl phorbol myristyl acetate (TPA) and phytohemagglutinin (PHA). We found that the ubiquitous transcription factor, Oct-1, bound to two previously identified motifs within the human IL-2 enhancer, centered at nucleotides -74 and -251. Each site in the IL-2 enhancer that bound Oct-1 in vitro was also required to achieve a maximal transcriptional response to TPA plus PHA in vivo. Point mutations within either the proximal or distal octamer sequences reduced the response of the enhancer to activation by 54 and 34%, respectively. Because the murine T-cell line EL4 constitutively expresses Oct-2 and requires only TPA to induce transcription of the IL-2 gene, the effect of Oct-2 expression on activation of the IL-2 promoter in Jurkat T cells was determined. Expression of Oct-2 potentiated transcription 13-fold in response to TPA plus PHA and permitted the enhancer to respond to the single stimulus of TPA. Therefore, both the signal requirements and the magnitude of the transcription response of the IL-2 promoter can be modulated by Oct-2

Percolation in the Harmonic Crystal and Voter Model in three dimensions

We investigate the site percolation transition in two strongly correlated systems in three dimensions: the massless harmonic crystal and the voter model. In the first case we start with a Gibbs measure for the potential, $U=\frac{J}{2} \sum_{} (\phi(x) - \phi(y))^2$, $x,y \in \mathbb{Z}^3$, $J > 0$ and $\phi(x) \in \mathbb{R}$, a scalar height variable, and define occupation variables $\rho_h(x) =1,(0)$ for $\phi(x) > h (<h)$. The probability $p$ of a site being occupied, is then a function of $h$. In the voter model we consider the stationary measure, in which each site is either occupied or empty, with probability $p$. In both cases the truncated pair correlation of the occupation variables, $G(x-y)$, decays asymptotically like $|x-y|^{-1}$. Using some novel Monte Carlo simulation methods and finite size scaling we find accurate values of $p_c$ as well as the critical exponents for these systems. The latter are different from that of independent percolation in $d=3$, as expected from the work of Weinrib and Halperin [WH] for the percolation transition of systems with $G(r) \sim r^{-a}$ [A. Weinrib and B. Halperin, Phys. Rev. B 27, 413 (1983)]. In particular the correlation length exponent $\nu$ is very close to the predicted value of 2 supporting the conjecture by WH that $\nu= \frac{2}{a}$ is exact.Comment: 8 figures. new version significantly different from the old one, includes new results, figures et

A Fresh Look at Entropy and the Second Law of Thermodynamics

This paper is a non-technical, informal presentation of our theory of the second law of thermodynamics as a law that is independent of statistical mechanics and that is derivable solely from certain simple assumptions about adiabatic processes for macroscopic systems. It is not necessary to assume a-priori concepts such as "heat", "hot and cold", "temperature". These are derivable from entropy, whose existence we derive from the basic assumptions. See cond-mat/9708200 and math-ph/9805005.Comment: LaTex file. To appear in the April 2000 issue of PHYSICS TODA

The Effect Of Microscopic Correlations On The Information Geometric Complexity Of Gaussian Statistical Models

We present an analytical computation of the asymptotic temporal behavior of the information geometric complexity (IGC) of finite-dimensional Gaussian statistical manifolds in the presence of microcorrelations (correlations between microvariables). We observe a power law decay of the IGC at a rate determined by the correlation coefficient. It is found that microcorrelations lead to the emergence of an asymptotic information geometric compression of the statistical macrostates explored by the system at a faster rate than that observed in absence of microcorrelations. This finding uncovers an important connection between (micro)-correlations and (macro)-complexity in Gaussian statistical dynamical systems.Comment: 12 pages; article in press, Physica A (2010)

Location of the Lee-Yang zeros and absence of phase transitions in some Ising spin systems

We consider a class of Ising spin systems on a set \Lambda of sites. The sites are grouped into units with the property that each site belongs to either one or two units, and the total internal energy of the system is the sum of the energies of the individual units, which in turn depend only on the number of up spins in the unit. We show that under suitable conditions on these interactions none of the |\Lambda| Lee-Yang zeros in the complex z = exp{2\beta h} plane, where \beta is the inverse temperature and h the uniform magnetic field, touch the positive real axis, at least for large values of \beta. In some cases one obtains, in an appropriately taken \beta to infinity limit, a gas of hard objects on a set \Lambda'; the fugacity for the limiting system is a rescaling of z and the Lee-Yang zeros of the new partition function also avoid the positive real axis. For certain forms of the energies of the individual units the Lee-Yang zeros of both the finite- and zero-temperature systems lie on the negative real axis for all \beta. One zero-temperature limit of this type, for example, is a monomer-dimer system; our results thus generalize, to finite \beta, a well-known result of Heilmann and Lieb that the Lee-Yang zeros of monomer-dimer systems are real and negative.Comment: Plain TeX. Seventeen pages, five figures from .eps files. Version 2 corrects minor errors in version

Partially ordered models

We provide a formal definition and study the basic properties of partially ordered chains (POC). These systems were proposed to model textures in image processing and to represent independence relations between random variables in statistics (in the later case they are known as Bayesian networks). Our chains are a generalization of probabilistic cellular automata (PCA) and their theory has features intermediate between that of discrete-time processes and the theory of statistical mechanical lattice fields. Its proper definition is based on the notion of partially ordered specification (POS), in close analogy to the theory of Gibbs measure. This paper contains two types of results. First, we present the basic elements of the general theory of POCs: basic geometrical issues, definition in terms of conditional probability kernels, extremal decomposition, extremality and triviality, reconstruction starting from single-site kernels, relations between POM and Gibbs fields. Second, we prove three uniqueness criteria that correspond to the criteria known as bounded uniformity, Dobrushin and disagreement percolation in the theory of Gibbs measures.Comment: 54 pages, 11 figures, 6 simulations. Submited to Journal of Stat. Phy

Modulated phases of a 1D sharp interface model in a magnetic field

We investigate the ground states of 1D continuum models having short-range ferromagnetic type interactions and a wide class of competing longer-range antiferromagnetic type interactions. The model is defined in terms of an energy functional, which can be thought of as the Hamiltonian of a coarse-grained microscopic system or as a mesoscopic free energy functional describing various materials. We prove that the ground state is simple periodic whatever the prescribed total magnetization might be. Previous studies of this model of frustrated systems assumed this simple periodicity but, as in many examples in condensed matter physics, it is neither obvious nor always true that ground states do not have a more complicated, or even chaotic structure.Comment: 12 pages, 3 figure

Pattern formation in systems with competing interactions

There is a growing interest, inspired by advances in technology, in the low temperature physics of thin films. These quasi-2D systems show a wide range of ordering effects including formation of striped states, reorientation transitions, bubble formation in strong magnetic fields, etc. The origins of these phenomena are, in many cases, traced to competition between short ranged exchange ferromagnetic interactions, favoring a homogeneous ordered state, and the long ranged dipole-dipole interaction, which opposes such ordering on the scale of the whole sample. The present theoretical understanding of these phenomena is based on a combination of variational methods and a variety of approximations, e.g., mean-field and spin-wave theory. The comparison between the predictions of these approximate methods and the results of MonteCarlo simulations are often difficult because of the slow relaxation dynamics associated with the long-range nature of the dipole-dipole interactions. In this note we will review recent work where we prove existence of periodic structures in some lattice and continuum model systems with competing interactions. The continuum models have also been used to describe micromagnets, diblock polymers, etc.Comment: 11 pages, 1 figure, to appear in the AIP conference proceedings of the 10th Granada Seminar on Computational Physics, Sept. 15-19, 2008. (v2) Updated reference