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Precursor phenomena in frustrated systems
To understand the origin of the dynamical transition, between high
temperature exponential relaxation and low temperature nonexponential
relaxation, that occurs well above the static transition in glassy systems, a
frustrated spin model, with and without disorder, is considered. The model has
two phase transitions, the lower being a standard spin glass transition (in
presence of disorder) or fully frustrated Ising (in absence of disorder), and
the higher being a Potts transition. Monte Carlo results clarify that in the
model with (or without) disorder the precursor phenomena are related to the
Griffiths (or Potts) transition. The Griffiths transition is a vanishing
transition which occurs above the Potts transition and is present only when
disorder is present, while the Potts transition which signals the effect due to
frustration is always present. These results suggest that precursor phenomena
in frustrated systems are due either to disorder and/or to frustration, giving
a consistent interpretation also for the limiting cases of Ising spin glass and
of Ising fully frustrated model, where also the Potts transition is vanishing.
This interpretation could play a relevant role in glassy systems beyond the
spin systems case.Comment: Completely rewritten. New data. New result
Phase transitions in simplified models with long-range interactions
We study the origin of phase transitions in some simplified models with long
range interactions. For the ring model, we show that a possible new phase
transition predicted in a recent paper by Nardini and Casetti from an energy
landscape analysis does not occur. Instead of such phase transitions we observe
a sharp, although without any non-analiticity, change from a core-halo to an
only core configuration in the spatial distribution functions for low energies.
By introducing a new class of solvable simplified models without any critical
points in the potential energy, we show that a similar behaviour to the ring
model is obtained, with a first order phase transition from an almost
homogeneous high energy phase to a clustered phase, and the same core-halo to
core configuration transition at lower energies. We discuss the origin of these
features of the simplified models, and show that the first order phase
transition comes from the maximization of the entropy of the system as a
function of energy an an order parameter, as previously discussed by Kastner,
which seems to be the main mechanism causing phase transitions in long-range
interacting systems
Graphical representation of covariant-contravariant modal formulae
Covariant-contravariant simulation is a combination of standard (covariant)
simulation, its contravariant counterpart and bisimulation. We have previously
studied its logical characterization by means of the covariant-contravariant
modal logic. Moreover, we have investigated the relationships between this
model and that of modal transition systems, where two kinds of transitions (the
so-called may and must transitions) were combined in order to obtain a simple
framework to express a notion of refinement over state-transition models. In a
classic paper, Boudol and Larsen established a precise connection between the
graphical approach, by means of modal transition systems, and the logical
approach, based on Hennessy-Milner logic without negation, to system
specification. They obtained a (graphical) representation theorem proving that
a formula can be represented by a term if, and only if, it is consistent and
prime. We show in this paper that the formulae from the covariant-contravariant
modal logic that admit a "graphical" representation by means of processes,
modulo the covariant-contravariant simulation preorder, are also the consistent
and prime ones. In order to obtain the desired graphical representation result,
we first restrict ourselves to the case of covariant-contravariant systems
without bivariant actions. Bivariant actions can be incorporated later by means
of an encoding that splits each bivariant action into its covariant and its
contravariant parts.Comment: In Proceedings EXPRESS 2011, arXiv:1108.407
What can nuclear collisions teach us about the boiling of water or the formation of multi-star systems ?
Phase transitions in nuclei, small atomic clusters and self-gravitating
systems demand the extension of thermo-statistics to ``Small'' systems. The
main obstacle is the thermodynamic limit. It is shown how the original
definition of the entropy by Boltzmann as the volume of the energy-manifold of
the N-body phase space allows a {\em geometrical} definition of the entropy as
function of the conserved quantities. Without invoking the thermodynamic limit
the whole ``zoo'' of phase transitions and critical points/lines can be
unambiguously defined. The relation to the Yang--Lee singularities of the
grand-canonical partition sum is pointed out. It is shown that just phase
transitions in non-extensive systems give the complete set of characteristic
parameters of the transition {\em including the surface tension.} Nuclear
heavy-ion collisions are an experimental playground to explore this extension
of thermo-statisticsComment: Invited talk for Bologna 2000, 8 pages, 3 figures.ep
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