Composition Triple Systems and Realization of Triple Products in Terms of Bilinear Algebras

We give a definition of new triple systems and study the correspondence between triple and bilinear products for many triple systems.     

Algebra of Effects in the Formalism of Quantum Mechanics on Phase Space as an M. V. and a Heyting Effect Algebra

We prove that the algebra of effects in the phase space formalism of quantum mechanics forms an M. V. effect algebra and moreover a Heyting effect algebra. It contains no nontrivial projections. We equip this algebra with certain nontrivial projections by passing to the limit of the quantum expectation with respect to any density operator. PACS: Primary 02.10.Gd, 03.65.Bz, Secondary 002.20.Qs This paper was a submission to the Sixth International Quantum Structure Association Conference (QS6), which took place in Vienna, Austria, July 1–7, 2002.     

Duality and Hidden Symmetries in Interacting Particle Systems

In the context of Markov processes, both in discrete and continuous setting, we show a general relation between duality functions and symmetries of the generator. If the generator can be written in the form of a Hamiltonian of a quantum spin system, then the “hidden” symmetries are easily derived. We illustrate our approach in processes of symmetric exclusion type, in which the symmetry is of SU(2) type, as well as for the Kipnis-Marchioro-Presutti (KMP) model for which we unveil its SU(1,1) symmetry. The KMP model is in turn an instantaneous thermalization limit of the energy process associated to a large family of models of interacting diffusions, which we call Brownian energy process (BEP) and which all possess the SU(1,1) symmetry. We treat in details the case where the system is in contact with reservoirs and the dual process becomes absorbing.     

Bounded Fluctuations and Translation Symmetry Breaking in One-Dimensional Particle Systems

We present general results for one-dimensional systems of point charges (signed point measures) on the line with a translation invariant distribution for which the variance of the total charge in an interval is uniformly bounded (instead of increasing with the interval length). When the charges are restricted to multiples of a common unit, and their average charge density does not vanish, then the boundedness of the variance implies translation-symmetry breaking—in the sense that there exists a function of the charge configuration that is nontrivially periodic under translations—and hence that is not mixing. Analogous results are formulated also for one dimensional lattice systems under some constraints on the values of the charges at the lattice sites and their averages. The general results apply to one-dimensional Coulomb systems, and to certain spin chains, putting on common grounds different instances of symmetry breaking encountered there.     

Conserved Mass Models and Particle Systems in One Dimension

In this paper we study analytically a simple one-dimensional model of mass transport. We introduce a parameter p that interpolates between continuous-time dynamics (p0 limit) and discrete parallel update dynamics (p=1). For each p, we study the model with (i) both continuous and discrete masses and (ii) both symmetric and asymmetric transport of masses. In the asymmetric continuous mass model, the two limits p=1 and p0 reduce respectively to the q-model of force fluctuations in bead packs [S. N. Coppersmith et al., Phys. Rev. E 53:4673 (1996)] and the recently studied asymmetric random average process [J. Krug and J. Garcia, cond-mat/9909034]. We calculate the steady-state mass distribution function P(m) assuming product measure and show that it has an algebraic tail for small m, P(m)m ^– , where the exponent depends continuously on p. For the asymmetric case we find (p)=(1–p)/(2–p) for 0p<1 and (1)=–1, and for the symmetric case, (p)=(2–p)²/(8–5p+p ²) for all 0p1. We discuss the conditions under which the product measure ansatz is exact. We also calculate exactly the steady-state mass–mass correlation function and show that while it decouples in the asymmetric model, in the symmetric case it has a nontrivial spatial oscillation with an amplitude decaying exponentially with distance.     

Discovering Noncritical Organization: Statistical Mechanical,Information Theoretic,and Computational Views of Patterns in One-Dimensional Spin Systems

We compare and contrast three different, but complementary views of “structure” and “pattern” in spatial processes. For definiteness and analytical clarity, we apply all three approaches to the simplest class of spatial processes: one-dimensional Ising spin systems with finite-range interactions. These noncritical systems are well-suited for this study since the change in structure as a function of system parameters is more subtle than that found in critical systems where, at a phase transition, many observables diverge, thereby making the detection of change in structure obvious. This survey demonstrates that the measures of pattern from information theory and computational mechanics differ from known thermodynamic and statistical mechanical functions. Moreover, they capture important structural features that are otherwise missed. In particular, a type of mutual information called the excess entropy—an information theoretic measure of memory—serves to detect ordered, low entropy density patterns. It is superior in several respects to other functions used to probe structure, such as magnetization and structure factors. ϵ-Machines—the main objects of computational mechanics—are seen to be the most direct approach to revealing the (group and semigroup) symmetries possessed by the spatial patterns and to estimating the minimum amount of memory required to reproduce the configuration ensemble, a quantity known as the statistical complexity. Finally, we argue that the information theoretic and computational mechanical analyses of spatial patterns capture the intrinsic computational capabilities embedded in spin systems—how they store, transmit, and manipulate configurational information to produce spatial structure.