4,367 research outputs found
The Entropy Influence Conjecture Revisited
In this paper, we prove that most of the boolean functions, satisfy the Fourier Entropy Influence (FEI) Conjecture
due to Friedgut and Kalai (Proc. AMS'96). The conjecture says that the Entropy
of a boolean function is at most a constant times the Influence of the
function. The conjecture has been proven for families of functions of smaller
sizes. O'donnell, Wright and Zhou (ICALP'11) verified the conjecture for the
family of symmetric functions, whose size is . They are in fact able
to prove the conjecture for the family of -part symmetric functions for
constant , the size of whose is . Also it is known that the
conjecture is true for a large fraction of polynomial sized DNFs (COLT'10).
Using elementary methods we prove that a random function with high probability
satisfies the conjecture with the constant as , for any constant
.Comment: We thank Kunal Dutta and Justin Salez for pointing out that our
result can be extended to a high probability statemen
Phase transitions and configuration space topology
Equilibrium phase transitions may be defined as nonanalytic points of
thermodynamic functions, e.g., of the canonical free energy. Given a certain
physical system, it is of interest to understand which properties of the system
account for the presence of a phase transition, and an understanding of these
properties may lead to a deeper understanding of the physical phenomenon. One
possible approach of this issue, reviewed and discussed in the present paper,
is the study of topology changes in configuration space which, remarkably, are
found to be related to equilibrium phase transitions in classical statistical
mechanical systems. For the study of configuration space topology, one
considers the subsets M_v, consisting of all points from configuration space
with a potential energy per particle equal to or less than a given v. For
finite systems, topology changes of M_v are intimately related to nonanalytic
points of the microcanonical entropy (which, as a surprise to many, do exist).
In the thermodynamic limit, a more complex relation between nonanalytic points
of thermodynamic functions (i.e., phase transitions) and topology changes is
observed. For some class of short-range systems, a topology change of the M_v
at v=v_t was proved to be necessary for a phase transition to take place at a
potential energy v_t. In contrast, phase transitions in systems with long-range
interactions or in systems with non-confining potentials need not be
accompanied by such a topology change. Instead, for such systems the
nonanalytic point in a thermodynamic function is found to have some
maximization procedure at its origin. These results may foster insight into the
mechanisms which lead to the occurrence of a phase transition, and thus may
help to explore the origin of this physical phenomenon.Comment: 22 pages, 6 figure
Quark matter revisited with non extensive MIT bag model
In this work we revisit the MIT bag model to describe quark matter within
both the usual Fermi-Dirac and the Tsallis statistics. We verify the effects of
the non-additivity of the latter by analysing two different pictures: the first
order phase transition of the QCD phase diagram and stellar matter properties.
While, the QCD phase diagram is visually affected by the Tsallis statistics,
the resulting effects on quark star macroscopic properties are barely noticed.Comment: 10 pagens, 5 figure
Metric Features of a Dipolar Model
The lattice spin model, with nearest neighbor ferromagnetic exchange and long
range dipolar interaction, is studied by the method of time series for
observables based on cluster configurations and associated partitions, such as
Shannon entropy, Hamming and Rohlin distances. Previous results based on the
two peaks shape of the specific heat, suggested the existence of two possible
transitions. By the analysis of the Shannon entropy we are able to prove that
the first one is a true phase transition corresponding to a particular melting
process of oriented domains, where colored noise is present almost
independently of true fractality. The second one is not a real transition and
it may be ascribed to a smooth balancing between two geometrical effects: a
progressive fragmentation of the big clusters (possibly creating fractals), and
the slow onset of a small clusters chaotic phase. Comparison with the nearest
neighbor Ising ferromagnetic system points out a substantial difference in the
cluster geometrical properties of the two models and in their critical
behavior.Comment: 20 pages, 15 figures, submitted to JPhys
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