Quantum models of classical mechanics: maximum entropy packets

In a previous paper, a project of constructing quantum models of classical properties has been started. The present paper concludes the project by turning to classical mechanics. The quantum states that maximize entropy for given averages and variances of coordinates and momenta are called ME packets. They generalize the Gaussian wave packets. A non-trivial extension of the partition-function method of probability calculus to quantum mechanics is given. Non-commutativity of quantum variables limits its usefulness. Still, the general form of the state operators of ME packets is obtained with its help. The diagonal representation of the operators is found. A general way of calculating averages that can replace the partition function method is described. Classical mechanics is reinterpreted as a statistical theory. Classical trajectories are replaced by classical ME packets. Quantum states approximate classical ones if the product of the coordinate and momentum variances is much larger than Planck constant. Thus, ME packets with large variances follow their classical counterparts better than Gaussian wave packets.Comment: 26 pages, no figure. Introduction and the section on classical limit are extended, new references added. Definitive version accepted by Found. Phy

Cryogenic and vacuum sectorisation of the LHC arcs

Following the recommendation of the LHC TC of June 20th, 1995 to introduce a separate cryogenic distribution line (QRL), which opened the possibility to have a finer cryogenic and vacuum sectorisation of the LHC machine than the original 8 arcs scheme, a working group was set up to study the implications: technical feasibility, advantages and drawbacks as well as cost of such a sectorisation (DG/DI/LE/dl, 26 July 1995). This report presents the conclusions of the Working Group. In the LHC Conceptual Design Report, ref. CERN/AC/95-05 (LHC), 20 October 1995, the so-called "Yellow Book", a complete cryostat arc (~ 2.9 km) would have to be warmed up in order to replace a defective cryomagnet. Even by coupling the two large refrigerators feeding adjacent arcs at even points to speed up the warm-up and cool down of one arc, the minimum down-time of the machine needed to replace a cryomagnet would be more than a full month (and even 52 days with only one cryoplant). Cryogenic and vacuum sectorisation of an arc into smaller sectors is technically feasible and would allow to reduce the down-times considerably (by one to three weeks with four sectors of 750 m in length, with respectively two or one cryoplants). In addition, sectorisation of the arcs may permit a more flexible quality control and commissioning of the main machine systems, including cold testing of small magnet strings. Sectorisation, described in detail in the following paragraphs, consists essentially of installing several additional cryogenic and vacuum valves as well as some insulation vacuum barriers. Additional cryogenic valves are needed in the return lines of the circuits feeding each half-cell in order to complete the isolation of the cryoline QRL from the machine, allowing intervention (i.e. venting to atmospheric pressure) on machine sectors without affecting the rest of an arc. Secondly, and for the same purpose, special vacuum and cryogenic valves must be installed, at the boundaries of machine sectors, for the circuits not passing through the cryoline QRL. Finally, some additional vacuum barriers must be installed around the magnet cold masses to divide the insulation vacuum of the magnet cryostats into independent sub-sectors, permitting to keep under insulating vacuum the cryogenically floating cold masses, while a sector (or part of it) is warmed up and opened to atmosphere. A reasonable scenario of sectorisation, namely with four 650-750 m long sectors per arc, and each consisting of 3 or 4 insulation vacuum sub-sectors with two to four half-cells, would represent an additional total cost of about 6.6 MCHF for the machine. It is estimated that this capital investment would be paid off by time savings in less than three long unscheduled interventions such as the change of a cryomagnet

On the Mixing of Diffusing Particles

We study how the order of N independent random walks in one dimension evolves with time. Our focus is statistical properties of the inversion number m, defined as the number of pairs that are out of sort with respect to the initial configuration. In the steady-state, the distribution of the inversion number is Gaussian with the average ~N^2/4 and the standard deviation sigma N^{3/2}/6. The survival probability, S_m(t), which measures the likelihood that the inversion number remains below m until time t, decays algebraically in the long-time limit, S_m t^{-beta_m}. Interestingly, there is a spectrum of N(N-1)/2 distinct exponents beta_m(N). We also find that the kinetics of first-passage in a circular cone provides a good approximation for these exponents. When N is large, the first-passage exponents are a universal function of a single scaling variable, beta_m(N)--> beta(z) with z=(m-)/sigma. In the cone approximation, the scaling function is a root of a transcendental equation involving the parabolic cylinder equation, D_{2 beta}(-z)=0, and surprisingly, numerical simulations show this prediction to be exact.Comment: 9 pages, 6 figures, 2 table

The closest elastic tensor of arbitrary symmetry to an elasticity tensor of lower symmetry

The closest tensors of higher symmetry classes are derived in explicit form for a given elasticity tensor of arbitrary symmetry. The mathematical problem is to minimize the elastic length or distance between the given tensor and the closest elasticity tensor of the specified symmetry. Solutions are presented for three distance functions, with particular attention to the Riemannian and log-Euclidean distances. These yield solutions that are invariant under inversion, i.e., the same whether elastic stiffness or compliance are considered. The Frobenius distance function, which corresponds to common notions of Euclidean length, is not invariant although it is simple to apply using projection operators. A complete description of the Euclidean projection method is presented. The three metrics are considered at a level of detail far greater than heretofore, as we develop the general framework to best fit a given set of moduli onto higher elastic symmetries. The procedures for finding the closest elasticity tensor are illustrated by application to a set of 21 moduli with no underlying symmetry.Comment: 48 pages, 1 figur

Pattern Avoidance in Poset Permutations

We extend the concept of pattern avoidance in permutations on a totally ordered set to pattern avoidance in permutations on partially ordered sets. The number of permutations on $P$ that avoid the pattern $\pi$ is denoted $Av_P(\pi)$. We extend a proof of Simion and Schmidt to show that $Av_P(132) \leq Av_P(123)$ for any poset $P$, and we exactly classify the posets for which equality holds.Comment: 13 pages, 1 figure; v2: corrected typos; v3: corrected typos and improved formatting; v4: to appear in Order; v5: corrected typos; v6: updated author email addresse

Intrinsic and extrinsic properties of quantum systems

The paper attempts to convince that the orthodox interpretation of quantum mechanics does not contradict philosophical realism by throwing light onto certain properties of quantum systems that seem to have escaped attention as yet. The exposition starts with the philosophical notions of realism. Then, the quantum mechanics as it is usually taught is demoted to a mere part of the theory called phenomenology of observations, and the common impression about its contradiction to realism is explained. The main idea of the paper, the physical notion of intrinsic properties, is introduced and many examples thereof are given. It replaces the irritating dichotomy of quantum and classical worlds by a much softer difference between intrinsic and extrinsic properties, which concern equally microscopic and macroscopic systems. Finally, the classicality and the quantum measurement are analyzed and found to present some still unsolved problems. A possible way of dealing with the Schr\"{o}dinger cat is suggested that is based on the intrinsic properties. A simple quantum model of one classical property illustrates how our philosophy may work.Comment: 20 pages, no figure. Comments are wellcom

Manufacturing features and performances of long models and first prototype for the LHC project

This paper reports about the 10-m-long models and one 15-m-long prototype. Their main design features are a 5-block coil cross section, an intra-beam distance of 194 mm at room temperature and a 15-mm-wide superconducting cable. The collared coil of the 10-m-long models were built in Industry and the assembly of the magnetic circuit and cold mass was done at CERN while the 15-m-long prototype was entirely made in Industry. Manufacturing features, assembly steps and quench performances of each magnet are presented. Results of magnetic measurements taken in the course of magnet assembly, during and after the cold test campaigns are also given

Abelian Sandpile Model on the Honeycomb Lattice

We check the universality properties of the two-dimensional Abelian sandpile model by computing some of its properties on the honeycomb lattice. Exact expressions for unit height correlation functions in presence of boundaries and for different boundary conditions are derived. Also, we study the statistics of the boundaries of avalanche waves by using the theory of SLE and suggest that these curves are conformally invariant and described by SLE2.Comment: 24 pages, 5 figure