Transition vitreuse dans les polymères amorphes. Etude phénoménologique

New systematic approximants are proposed for exponential functions, operators and inner derivation δ
 H
 . Remainders of systematic approximants are evaluated explicitly, which give degrees of convergence of approximants. The first approximant corresponds to Trotter's formula [1]: exp(A+B)=
 
$$\mathop {\lim }\limits_{n \to \infty } $$

 [exp(A/n) exp(B/n)]
 n
 . Some applications to physics are also discussed.

/pdf/generalized-trotter-s-formula-and-systematic-approximants-of-2uvmgk3hes.pdf

Generalized Trotter's formula and systematic approximants of exponential operators and inner derivations with applications to many-body problems

We investigate the general monomer-dimer partition function, P(x), which is a polynomial in the monomer activity, x, with coefficients depending on the dimer activities. Our main result is that P(x) has its zeros on the imaginary axis when the dimer activities are nonnegative. Therefore, no monomer-dimer system can have a phase transition as a function of monomer density except, possibly, when the monomer density is minimal (i.e. x = 0). Elaborating on this theme we prove the existence and analyticity of correlation functions (away from x = 0) in the thermodynamic limit. Among other things we obtain bounds on the compressibility and derive a new variable in which to make an expansion of the free energy that converges down to the minimal monomer density. We also relate the monomer-dimer problem to the Heisenberg and Ising models of a magnet and derive Christoffell-Darboux formulas for the monomer-dimer and Ising model partition functions. This casts the Ising model in a new light and provides an alternative proof of the Lee-Yang circle theorem. We also derive joint complex analyticity domains in the monomer and dimer activities. Our considerations are independent of geometry and hence are valid for any dimensionality.

https://projecteuclid.org/download/pdf_1/euclid.cmp/1103857921

Theory of monomer-dimer systems

We prove the existence of spontaneous magnetization at sufficiently low temperature, and hence of a phase transition, in a variety of quantum spin systems in three or more dimensions. The isotropic spin 1/2 x−y model and the Heisenberg antiferromagnet with spin 1, 3/2,...and with nearest neighbor interactions on a simple cubic lattice are included.

/pdf/phase-transitions-in-quantum-spin-systems-with-isotropic-and-21rbktlx62.pdf

Phase Transitions in Quantum Spin Systems with Isotropic and Nonisotropic Interactions

This motivating textbook gives a friendly, rigorous introduction to fundamental concepts in equilibrium statistical mechanics, covering a selection of specific models, including the Curie–Weiss and Ising models, the Gaussian free field, O(n) models, and models with Kac interactions. Using classical concepts such as Gibbs measures, pressure, free energy, and entropy, the book exposes the main features of the classical description of large systems in equilibrium, in particular the central problem of phase transitions. It treats such important topics as the Peierls argument, the Dobrushin uniqueness, Mermin–Wagner and Lee–Yang theorems, and develops from scratch such workhorses as correlation inequalities, the cluster expansion, Pirogov–Sinai Theory, and reflection positivity. Written as a self-contained course for advanced undergraduate or beginning graduate students, the detailed explanations, large collection of exercises (with solutions), and appendix of mathematical results and concepts also make it a handy reference for researchers in related areas.

Statistical Mechanics of Lattice Systems: a Concrete Mathematical Introduction

It is shown that all the zeros of the partition function lie on the unit circle in the complex `fugacity' plane and the first inequality of Griffiths, Kelly and Sherman for the spin correlation functions also holds for the anisotropic Heisenberg ferromagnets as in the Ising model. The isotropic Heisenberg ferromagnets and the Ising ferromagnets are involved as special cases and thus the methods of this article provide alternative proof of the corresponding theorems in the Ising model.

Theorems on the Partition Functions of the Heisenberg Ferromagnets

It is shown that the zeros of the partition function of the anisotropic Heisenberg ferromagnet locate on the unit circle in the complex "fugacity" plane, and all the spin correlation functions are non-negative when no external magnetic field is present.

Lee-Yang Theorem and the Griffiths Inequality for the Anisotropic Heisenberg Ferromagnet

Generalization of the Lee-Yang Theorem

Complex Young's modulus, rigidity and bulk modulus are measured at 33 kc/s, 66 kc/s and 1 Mc/s for polystyrene, polymethyl methacrylate, polyvinyl acetate and phenol resin. In each material, the storage modulus-temperature curve has a discontinuity in the slope, irrespective of frequency and type of modulus, at the glass transition temperature T g determined from thermal expansion. This discontinuity takes place through a strong dependence of the modulus on the specific volume. Besides, the first three materials exhibit secondary discontinuities at T g ', about thirty degrees below T g , both in thermal expansion and modulus-temperature curve. Some possible interpretations are given to this secondary discontinuity, resulting in the conclusion that the glass transition occurs over a rather wide range of temperature and is completed at T g with increasing temperature. The ratio of the temperature coefficients of the modulus, taken below and above T g , agrees well with that of volume expansion curve. This i...

On the Dynamic Mechanical Properties of Polymers at Ultrasonic Frequencies in Relation to Their Glass Transition Phenomena

Lee-Yang's theorem which states that all the roots of the partition function of the ferromagnetic Ising model of spin 1/2 lie on the unit circle in the complex `fugacity' plane is generalized to the case of S =1 and 3/2, where S is the magnitude of the spin.

Taro Asano

Papers

Theorems on the Partition Functions of the Heisenberg Ferromagnets

Lee-Yang Theorem and the Griffiths Inequality for the Anisotropic Heisenberg Ferromagnet

Generalization of the Lee-Yang Theorem

On the Dynamic Mechanical Properties of Polymers at Ultrasonic Frequencies in Relation to Their Glass Transition Phenomena

Generalized Lee-Yang's Theorem