377 research outputs found
Free-energy bounds for hierarchical spin models
In this paper we study two non-mean-field spin models built on a hierarchical
lattice: The hierarchical Edward-Anderson model (HEA) of a spin glass, and
Dyson's hierarchical model (DHM) of a ferromagnet. For the HEA, we prove the
existence of the thermodynamic limit of the free energy and the
replica-symmetry-breaking (RSB) free-energy bounds previously derived for the
Sherrington-Kirkpatrick model of a spin glass. These RSB mean-field bounds are
exact only if the order-parameter fluctuations (OPF) vanish: Given that such
fluctuations are not negligible in non-mean-field models, we develop a novel
strategy to tackle part of OPF in hierarchical models. The method is based on
absorbing part of OPF of a block of spins into an effective Hamiltonian of the
underlying spin blocks. We illustrate this method for DHM and show that,
compared to the mean-field bound for the free energy, it provides a tighter
non-mean-field bound, with a critical temperature closer to the exact one. To
extend this method to the HEA model, a suitable generalization of Griffith's
correlation inequalities for Ising ferromagnets is needed: Since correlation
inequalities for spin glasses are still an open topic, we leave the extension
of this method to hierarchical spin glasses as a future perspective
Using X-ray variability to estimate the nature of the compact objects powering ULXs
It is not very clear if the spectral transitions in the Ultra-luminous X-ray sources (ULXs) are due to stochastic variability in the wind or orbital modulation in the accretion rate or in the source geometry. In this talk I will compare the results obtained on two different variable ULXs: NGC 55 ULX-1 and HOLMBERG II X-1. The XMM-Newton satellite collected data that were modelled with a double ther- mal component, adding a power-law component especially for the latter source that presents an harder spectrum. The Luminosity-Temperature relation for both ther- mal components broadly agrees with the results expected from theoretical models of thin discs. However, at higher luminosities significant deviations are present. If such deviations are due to the accretion rate exceeding the Eddington limit or the supercritical rate, a stellar-mass black hole is forecasted in both accreting sources
On quantum and relativistic mechanical analogues in mean field spin models
Conceptual analogies among statistical mechanics and classical (or quantum)
mechanics often appeared in the literature. For classical two-body mean field
models, an analogy develops into a proper identification between the free
energy of Curie-Weiss type magnetic models and the Hamilton-Jacobi action for a
one dimensional mechanical system. Similarly, the partition function plays the
role of the wave function in quantum mechanics and satisfies the heat equation
that plays, in this context, the role of the Schrodinger equation in quantum
mechanics. We show that this identification can be remarkably extended to
include a wide family of magnetic models classified by normal forms of suitable
real algebraic dispersion curves. In all these cases, the model turns out to be
completely solvable as the free energy as well as the order parameter are
obtained as solutions of an integrable nonlinear PDE of Hamilton-Jacobi type.
We observe that the mechanical analog of these models can be viewed as the
relativistic analog of the Curie-Weiss model and this helps to clarify the
connection between generalised self-averaging and in statistical thermodynamics
and the semi-classical dynamics of viscous conservation laws.Comment: Dedicated to Sandro Graffi in honor of his seventieth birthda
Anergy in self-directed B lymphocytes from a statistical mechanics perspective
The ability of the adaptive immune system to discriminate between self and
non-self mainly stems from the ontogenic clonal-deletion of lymphocytes
expressing strong binding affinity with self-peptides. However, some
self-directed lymphocytes may evade selection and still be harmless due to a
mechanism called clonal anergy. As for B lymphocytes, two major explanations
for anergy developed over three decades: according to "Varela theory", it stems
from a proper orchestration of the whole B-repertoire, in such a way that
self-reactive clones, due to intensive interactions and feed-back from other
clones, display more inertia to mount a response. On the other hand, according
to the `two-signal model", which has prevailed nowadays, self-reacting cells
are not stimulated by helper lymphocytes and the absence of such signaling
yields anergy. The first result we present, achieved through disordered
statistical mechanics, shows that helper cells do not prompt the activation and
proliferation of a certain sub-group of B cells, which turn out to be just
those broadly interacting, hence it merges the two approaches as a whole (in
particular, Varela theory is then contained into the two-signal model). As a
second result, we outline a minimal topological architecture for the B-world,
where highly connected clones are self-directed as a natural consequence of an
ontogenetic learning; this provides a mathematical framework to Varela
perspective. As a consequence of these two achievements, clonal deletion and
clonal anergy can be seen as two inter-playing aspects of the same phenomenon
too
Topological properties of hierarchical networks
Hierarchical networks are attracting a renewal interest for modelling the
organization of a number of biological systems and for tackling the complexity
of statistical mechanical models beyond mean-field limitations. Here we
consider the Dyson hierarchical construction for ferromagnets, neural networks
and spin-glasses, recently analyzed from a statistical-mechanics perspective,
and we focus on the topological properties of the underlying structures. In
particular, we find that such structures are weighted graphs that exhibit high
degree of clustering and of modularity, with small spectral gap; the robustness
of such features with respect to link removal is also studied. These outcomes
are then discussed and related to the statistical mechanics scenario in full
consistency. Lastly, we look at these weighted graphs as Markov chains and we
show that in the limit of infinite size, the emergence of ergodicity breakdown
for the stochastic process mirrors the emergence of meta-stabilities in the
corresponding statistical mechanical analysis
Hierarchical neural networks perform both serial and parallel processing
In this work we study a Hebbian neural network, where neurons are arranged
according to a hierarchical architecture such that their couplings scale with
their reciprocal distance. As a full statistical mechanics solution is not yet
available, after a streamlined introduction to the state of the art via that
route, the problem is consistently approached through signal- to-noise
technique and extensive numerical simulations. Focusing on the low-storage
regime, where the amount of stored patterns grows at most logarithmical with
the system size, we prove that these non-mean-field Hopfield-like networks
display a richer phase diagram than their classical counterparts. In
particular, these networks are able to perform serial processing (i.e. retrieve
one pattern at a time through a complete rearrangement of the whole ensemble of
neurons) as well as parallel processing (i.e. retrieve several patterns
simultaneously, delegating the management of diff erent patterns to diverse
communities that build network). The tune between the two regimes is given by
the rate of the coupling decay and by the level of noise affecting the system.
The price to pay for those remarkable capabilities lies in a network's capacity
smaller than the mean field counterpart, thus yielding a new budget principle:
the wider the multitasking capabilities, the lower the network load and
viceversa. This may have important implications in our understanding of
biological complexity
Meta-stable states in the hierarchical Dyson model drive parallel processing in the hierarchical Hopfield network
In this paper we introduce and investigate the statistical mechanics of
hierarchical neural networks: First, we approach these systems \`a la Mattis,
by thinking at the Dyson model as a single-pattern hierarchical neural network
and we discuss the stability of different retrievable states as predicted by
the related self-consistencies obtained from a mean-field bound and from a
bound that bypasses the mean-field limitation. The latter is worked out by
properly reabsorbing fluctuations of the magnetization related to higher levels
of the hierarchy into effective fields for the lower levels. Remarkably, mixing
Amit's ansatz technique (to select candidate retrievable states) with the
interpolation procedure (to solve for the free energy of these states) we prove
that (due to gauge symmetry) the Dyson model accomplishes both serial and
parallel processing. One step forward, we extend this scenario toward multiple
stored patterns by implementing the Hebb prescription for learning within the
couplings. This results in an Hopfield-like networks constrained on a
hierarchical topology, for which, restricting to the low storage regime (where
the number of patterns grows at most logarithmical with the amount of neurons),
we prove the existence of the thermodynamic limit for the free energy and we
give an explicit expression of its mean field bound and of the related improved
boun
From Dyson to Hopfield: Processing on hierarchical networks
We consider statistical-mechanical models for spin systems built on
hierarchical structures, which provide a simple example of non-mean-field
framework. We show that the coupling decay with spin distance can give rise to
peculiar features and phase diagrams much richer that their mean-field
counterpart. In particular, we consider the Dyson model, mimicking
ferromagnetism in lattices, and we prove the existence of a number of
meta-stabilities, beyond the ordered state, which get stable in the
thermodynamic limit. Such a feature is retained when the hierarchical structure
is coupled with the Hebb rule for learning, hence mimicking the modular
architecture of neurons, and gives rise to an associative network able to
perform both as a serial processor as well as a parallel processor, depending
crucially on the external stimuli and on the rate of interaction decay with
distance; however, those emergent multitasking features reduce the network
capacity with respect to the mean-field counterpart. The analysis is
accomplished through statistical mechanics, graph theory, signal-to-noise
technique and numerical simulations in full consistency. Our results shed light
on the biological complexity shown by real networks, and suggest future
directions for understanding more realistic models
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