Finite TYCZ expansions and cscK metrics

Let $(M, g)$ be a Kaehler manifold whose associated Kaehler form $\omega$ is integral and let $(L, h)\rightarrow (M, \omega)$ be a quantization hermitian line bundle. In this paper we study those Kaehler manifolds $(M, g)$ admitting a finite TYCZ expansion. We show that if the TYCZ expansion is finite then $T_{mg}$ is indeed a polynomial in $m$ of degree $n$, $n=dim M$, and the log-term of the Szeg\"{o} kernel of the disc bundle $D\subset L^*$ vanishes (where $L^*$ is the dual bundle of $L$). Moreover, we provide a complete classification of the Kaehler manifolds admitting finite TYCZ expansion either when $M$ is a complex curve or when $M$ is a complex surface with a cscK metric which admits a radial Kaehler potential

Symplectic geometry of Cartan–Hartogs domains

This paper studies the geometry of Cartan–Hartogs domains from the symplectic point of view. Inspired by duality between compact and noncompact Hermitian symmetric spaces, we construct a dual counterpart of Cartan–Hartogs domains and give explicit expression of global Darboux coordinates for both Cartan–Hartogs domains and their dual. Further, we compute their symplectic capacity and show that a Cartan–Hartogs domain admits a symplectic duality if and only if it reduces to be a complex hyperbolic space

Immersions of Sasaki–Ricci solitons into homogeneous Sasakian manifolds

We discuss local Sasakian immersion of Sasaki-Ricci solitons (SRS) into fiber products of homogeneous Sasakian manifolds. In particular, we prove that SRS locally induced by alarge class of fiber products of homogeneous Sasakian manifolds are, in fact, eta-Einstein. The results are stronger for immersions into Sasakian space forms. Moreover, we show an example of a Kähler-Ricci soliton on C^n which admits no local holomorphic isometry into products of homogeneous bounded domains with flat Kähler manifolds and generalized flag manifolds

Minimal symplectic atlases of Hermitian symmetric spaces

In this paper we estimate the minimal number of Darboux charts needed to cover a Hermitian symmetric space of compact type M in terms of the degree of their embeddings in CPN. The proof is based on the recent work of Rudyak and Schlenk (Commun Contemp Math 9(6):811–855, 2007) and on the symplectic geometry tool developed by the first author in collaboration with Loi et al. (J Sympl Geom, 2014). As application we compute this number for a large class of Hermitian symmetric spaces of compact type

Vibrational origin of the fast relaxation processes in molecular glass-formers

We study the interaction of the relaxation processes with the density fluctuations by molecular dynamics simulation of a flexible molecule model for o-terphenyl (oTP) in the liquid and supercooled phases. We find evidence, besides the structural relaxation, of a secondary vibrational relaxation whose characteristic time, few ps, is slightly temperature dependent. This i) confirms the result by Monaco et al. [Phys. Rev, E 62, 7595 (2000)] of the vibrational nature of the fast relaxation observed in Brillouin Light Scattering (BLS) experiments in oTP; and ii) poses a caveat on the interpretation of the BLS spectra of molecular systems in terms of a purely center of mass dynamics.Comment: RevTeX, 5 pages, 4 eps figure

Equilibrium cluster phases and low-density arrested disordered states: The role of short-range attraction and long-range repulsion

We study a model in which particles interact with short-ranged attractive and long-ranged repulsive interactions, in an attempt to model the equilibrium cluster phase recently discovered in sterically stabilized colloidal systems in the presence of depletion interactions. At low packing fraction particles form stable equilibrium clusters which act as building blocks of a cluster fluid. We study the possibility that cluster fluids generate a low-density disordered arrested phase, a gel, via a glass transition driven by the repulsive interaction. In this model the gel formation is formally described with the same physics of the glass formation.Comment: RevTeX4, 4 pages, 4 eps figure

Equilibrium and out of equilibrium thermodynamics in supercooled liquids and glasses

We review the inherent structure thermodynamical formalism and the formulation of an equation of state for liquids in equilibrium based on the (volume) derivatives of the statistical properties of the potential energy surface. We also show that, under the hypothesis that during aging the system explores states associated to equilibrium configurations, it is possible to generalize the proposed equation of state to out-of-equilibrium conditions. The proposed formulation is based on the introduction of one additional parameter which, in the chosen thermodynamic formalism, can be chosen as the local minima where the slowly relaxing out-of-equilibrium liquid is trapped.Comment: 7 pages, 4 eps figure

Kovacs effects in an aging molecular liquid

We study by means of molecular dynamics simulations the aging behavior of a molecular model of ortho-terphenyl. We find evidence of a a non-monotonic evolution of the volume during an isothermal-isobaric equilibration process, a phenomenon known in polymeric systems as Kovacs effect. We characterize this phenomenology in terms of landscape properties, providing evidence that, far from equilibrium, the system explores region of the potential energy landscape distinct from the one explored in thermal equilibrium. We discuss the relevance of our findings for the present understanding of the thermodynamics of the glass state.Comment: RevTeX 4, 4 pages, 5 eps figure

A Simple Theory of Condensation

A simple assumption of an emergence in gas of small atomic clusters consisting of $c$ particles each, leads to a phase separation (first order transition). It reveals itself by an emergence of ``forbidden'' density range starting at a certain temperature. Defining this latter value as the critical temperature predicts existence of an interval with anomalous heat capacity behaviour $c_p\propto\Delta T^{-1/c}$. The value $c=13$ suggested in literature yields the heat capacity exponent $\alpha=0.077$.Comment: 9 pages, 1 figur