Phase Transitions and Quantum Effects in Anharmonic Crystals

Abstract

<jats:p>The most important recent results in the theory of phase transitions and quantum effects in quantum anharmonic crystals are presented and discussed. In particular, necessary and sufficient conditions for a phase transition to occur at some temperature are given in the form of simple inequalities involving the interaction strength and the parameters describing a single oscillator. The main characteristic feature of the theory is that both mentioned phenomena are described in one and the same setting, in which thermodynamic phases of the model appear as probability measures on path spaces. Then the possibility of a phase transition to occur is related to the existence of multiple phases at the same values of the relevant parameters. Other definitions of phase transitions, based on the nondifferentiability of the free energy density and on the appearance of ordering, are also discussed.</jats:p>

Equilibrium States, Phase Transitions and Dynamics in Quantum Anharmonic Crystals

“…According to Theorem 2.2 quantum stabilization occurs if the interaction intensity is smaller than the effective rigidity, see [2,4] and Part 2 of [3] for a physical interpretation of this effect. Note that R m can be made arbitrarily big either by making m small or ∆ big (e.g., by making the wells closer to each other).…”

“…By analogy, we call R m quantum effective rigidity, which, however, depends on m as just discussed. The sufficient condition mentioned above is, see [8,Theorem 4.6] According to Theorem 2.2 quantum stabilization occurs if the interaction intensity is smaller than the effective rigidity, see [2,4] and Part 2 of [3] for a physical interpretation of this effect. Note that R m can be made arbitrarily big either by making m small or ∆ big (e.g., by making the wells closer to each other).…”

“…Essential results in this direction were obtained by means of methods developed in stochastic analysis, mostly in the approach in which states of such systems are constructed as probability measures on infinite dimensional path spaces. A substantial part of these results appeared due to Michael Röckner's research activity, see, e.g., [1,2,3,4,5,12]. The aim of this work is to outline the main aspects of the theory of equilibrium states of quantum anharmonic crystals obtained in [3] in the approach based on path measures.…”

A Phase Transition in a Quenched Amorphous Ferromagnet

“…However, for unbounded spins, not all extreme Gibbs measures may have physical meaning. It is believed that the measures corresponding to observed thermodynamic states should be supported on spin configurations with 'tempered growth' see [6,29,37,38] or a more recent development in [2,28] and [1,Chap. 3].…”

“…3]. In this approach, only tempered Gibbs measures are taken into account, and hence a phase transition is related to the existence of multiple tempered Gibbs measures 2 . We take this approach and study quenched Gibbs measures introduced in Definition 2 with a priori prescribed support properties.…”