On the Local Equivalence Between the Canonical and the Microcanonical Ensembles for Quantum Spin Systems

We study a quantum spin system on the d-dimensional hypercubic lattice \(\Lambda \) with \(N=L^d\) sites with periodic boundary conditions. We take an arbitrary translation invariant short-ranged Hamiltonian. For this system, we consider both the canonical ensemble with inverse temperature \(\beta _0\) and the microcanonical ensemble with the corresponding energy \(U_N(\beta _0)\). For an arbitrary self-adjoint operator \(\hat{A}\) whose support is contained in a hypercubic block B inside \(\Lambda \), we prove that the expectation values of \(\hat{A}\) with respect to these two ensembles are close to each other for large N provided that \(\beta _0\) is sufficiently small and the number of sites in B is \(o(N^{1/2})\). This establishes the equivalence of ensembles on the level of local states in a large but finite system. The result is essentially that of Brandao and Cramer (here restricted to the case of the canonical and the microcanonical ensembles), but we prove improved estimates in an elementary manner. We also review and prove standard results on the thermodynamic limits of thermodynamic functions and the equivalence of ensembles in terms of thermodynamic functions. The present paper assumes only elementary knowledge on quantum statistical mechanics and quantum spin systems.     

Radical polymerization of allylbiguanide

Monoallyl compounds are not readily homopolymerized by a conventional free‐radical mechanism. However, we successfully performed the radical polymerization of allylbiguanide hydrochloride in a concentrated acid solution (hydrochloric acid or phosphoric acid) in the presence of a radical initiator at 50 °C. The polymer product was precipitated from the reaction solution through the addition of an excess amount of acetone. The precipitated crude polymer [polyallylbiguanide (PAB)] was then purified by dialysis. PAB was confirmed by elemental analysis, infrared spectroscopy, and ¹H NMR. The molecular weight range of PAB was 10,340–113,200, and PAB exhibited a low polydispersity (weight‐average molecular weight/number‐average molecular weight = 1.04–1.68) by multi‐angle laser light scattering. The polymerization of allylbiguanide was quite sensitive to the protonic concentration of the inorganic acid. © 2004 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 42: 1707–1711, 2004     

Gaussian bounds for correlations in lattice spin systems

In quite generalN-component ferromagnetic spin systems, it is proved that an arbitrary correlation function is bounded by the corresponding correlation function of a Gaussian model. The bound is useful for the analysis of high-temperature behavior of the system. Similar bounds for truncated correlation functions are also obtained for a class of single-component spin systems.     

The Coefficient of Restitution Does Not Exceed Unity

We study a classical mechanical problem in which a macroscopic ball is reflected by a non-deformable wall. The ball is modeled as a collection of classical particles bound together by an arbitrary potential, and its internal degrees of freedom are initially set to be in thermal equilibrium. The wall is represented by an arbitrary potential which is translation invariant in two directions. We then prove that the final normal momentum can exceed the initial normal momentum at most by O(■mkT), where m is the total mass of the ball, k the Boltzmann constant, and T the temperature. This implies the well-known statement in the title in the macroscopic limit where O(■mkT) is negligible. Our result may be interpreted as a rigorous demonstration of the second law of thermodynamics in a system where a macroscopic dynamics and microscopic degrees of freedom are intrinsically coupled with each other.     

Generalized spectral decompositions of mixing dynamical systems

We introduce a method for the explicit computation of the eigenvalue problem of the evolution operator of mixing dynamical systems. The method is based on the subdynamics decomposition of the Brussels–Austin groups directed by Professor I. Prigogine. We apply the method to three different representatives of mixing systems, namely, the Renyi maps, baker's transformations, and the Friedrichs model. The obtained spectral decompositions acquire meaning in suitable rigged Hilbert spaces that we construct explicitly for the three models. The resulting spectral decompositions show explicitly the intrinsic irreversibility of baker's transformations and Friedrichs model and the intrinsically probabilistic characters of the Renyi maps and baker's transformations. The dynamical properties are reflected in the spectrum because the eigenvalues are the powers of the Lyapunov times for the Renyi and baker systems and include the lifetimes for the Friedrichs model. © 1993 John Wiley & Sons, Inc.     

Precise retinal shape measurement by alignment error and eye model calibration

Optical Review - To evaluate the repeatability of the optical coherence tomography (OCT) retinal shape measurement with and without alignment correction in children and adults. 62 eyes of the 31...     

Metallic ferromagnetism in the Hubbard model: a rigorous example

We present the first rigorous example of the Hubbard model in any dimension which exhibits metallic ferromagnetism. The model is a genuine Hubbard model with short-range hopping and on-site Coulomb repulsion, and has many single-electron bands. In the limit where the band gap and the Coulomb repulsion become infinite, we prove that the ground states are completely ferromagnetic and at the same time conducting.