This paper discusses methods for the construction of approximate real space renormalization transformations in statistical mechanics. In particular, it compares two methods of transformation: the "potential-moving" approach most used in the period 1975-1980 and the "rewiring method" as it has been developed in the last five years. These methods both employ a parameter, called χ or D in the recent literature, that measures the complexity of the localized stochastic variable forming the basis of the analysis. Both methods are here exemplified by calculations in terms of fixed points for the smallest possible values of χ. These calculations describe three models for two-dimensional systems: The Ising model solved by Onsager, the tricritical point of that model, and the three-state Potts model. The older method, often described as lower bound renormalization theory, provides a heuristic method giving reasonably accurate results for critical indices at the lowest degree of complexity, i.e. χ = 2. In contrast, the rewiring method, employing "singular value decomposition", does not perform as well for low χ values but offers an error that apparently decreases slowly toward zero as χ is increased. It appears likely that no such improvement occurs in the older approach. A detailed comparison of the two methods is performed, with a particular eye to describing the reasons why they are so different. For example, the older method is based on the analysis of spins, simple stochastic variables located at lattice sites. The new method uses "indices" describing linear combinations of different localized configurations. The old method quite naturally employed fixed points for its analysis; these are hard to use in the newer approach. A discussion is given of why the fixed point approach proves to be hard in this context. In the new approach the calculated the thermal critical indices are satisfactory for the smallest values of χ but hardly improve as χ is increased, while the magnetic critical indices do not agree well with the known theoretical values. * efrati@uchicago.edu arXiv:1301.6323v1 [cond-mat.stat-mech]
We present a theoretical discussion of the reversible parking problem, which appears to be one of the simplest systems exhibiting glassy behavior. The existence of slow relaxation, nontrivial fluctuations, and an annealing effect can all be understood by recognizing that two different time scales are present in the problem. One of these scales corresponds to the fast filling of existing voids, the other is associated with collective processes that overcome partial ergodicity breaking.The results of the theory are in a good agreement with simulation 1 data; they provide a simple qualitative picture for understanding recent granular compaction experiments and other glassy systems.
Positronium ͑Ps͒-He scattering presents one of the few opportunities for both theory and experiment to tackle the fundamental interactions of Ps with ordinary matter. Below the dissociation energy of 6.8 eV, experimental and theoretical work has struggled to find agreement on the strength of this interaction as measured by the momentum-transfer cross section ͑ m ͒. Here, we present work utilizing the Doppler broadening technique with an age-momentum correlation apparatus. This work demonstrates a strong energy dependence for this cross section at energies below 1 eV and is consistent with previous experimental results.
Billiard problems are simple examples of Harniltonian dynamical systems. These problems have been used as model systems to study the link betwen classical and quantum chaos. The heart of this linkage is provided by the periodic orbits in the classical system. In this article we will show that for an arbitrary right triangle, almost all trajectories that begin perpendicular to a side are periodic, that is, the set of points on the sides of a right triangle from which nonperiodic (perpendicular) trajectories begin is a set of measure zero. Our proof incorporates the previous result for rational right triangles (where the angles are rational multiples of~), while extending the result to nonrational right triangles. PACS number(s): 03.20. +i The game of billiards provides a profusion of interesting questions in classical mechanics. Coriolis [I] and Sommerfeld [2] were intrigued by questions concerning high and low shots, the causes and eFects of "English, " and the beautiful curved paths resulting from friction with the billiard cloth.
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