44 research outputs found

### Perturbative large deviation analysis of non-equilibrium dynamics

Macroscopic fluctuation theory has shown that a wide class of non-equilibrium
stochastic dynamical systems obey a large deviation principle, but except for a
few one-dimensional examples these large deviation principles are in general
not known in closed form. We consider the problem of constructing successive
approximations to an (unknown) large deviation functional and show that the
non-equilibrium probability distribution the takes a Gibbs-Boltzmann form with
a set of auxiliary (non-physical) energy functions. The expectation values of
these auxiliary energy functions and their conjugate quantities satisfy a
closed system of equations which can imply a considerable reduction of
dimensionality of the dynamics. We show that the accuracy of the approximations
can be tested self-consistently without solving the full non- equilibrium
equations. We test the general procedure on the simple model problem of a
relaxing 1D Ising chain.Comment: 21 pages, 10 figure

### Dynamic message-passing approach for kinetic spin models with reversible dynamics

A method to approximately close the dynamic cavity equations for synchronous
reversible dynamics on a locally tree-like topology is presented. The method
builds on $(a)$ a graph expansion to eliminate loops from the normalizations of
each step in the dynamics, and $(b)$ an assumption that a set of auxilary
probability distributions on histories of pairs of spins mainly have
dependencies that are local in time. The closure is then effectuated by
projecting these probability distributions on $n$-step Markov processes. The
method is shown in detail on the level of ordinary Markov processes ($n=1$),
and outlined for higher-order approximations ($n>1$). Numerical validations of
the technique are provided for the reconstruction of the transient and
equilibrium dynamics of the kinetic Ising model on a random graph with
arbitrary connectivity symmetry.Comment: 6 pages, 4 figure

### Mean field spin glasses treated with PDE techniques

Following an original idea of F. Guerra, in this notes we analyze the
Sherrington-Kirkpatrick model from different perspectives, all sharing the
underlying approach which consists in linking the resolution of the statistical
mechanics of the model (e.g. solving for the free energy) to well-known partial
differential equation (PDE) problems (in suitable spaces). The plan is then to
solve the related PDE using techniques involved in their native field and
lastly bringing back the solution in the proper statistical mechanics
framework. Within this strand, after a streamlined test-case on the Curie-Weiss
model to highlight the methods more than the physics behind, we solve the SK
both at the replica symmetric and at the 1-RSB level, obtaining the correct
expression for the free energy via an analogy to a Fourier equation and for the
self-consistencies with an analogy to a Burger equation, whose shock wave
develops exactly at critical noise level (triggering the phase transition). Our
approach, beyond acting as a new alternative method (with respect to the
standard routes) for tackling the complexity of spin glasses, links symmetries
in PDE theory with constraints in statistical mechanics and, as a novel result
from the theoretical physics perspective, we obtain a new class of polynomial
identities (namely of Aizenman-Contucci type but merged within the Guerra's
broken replica measures), whose interest lies in understanding, via the recent
Panchenko breakthroughs, how to force the overlap organization to the
ultrametric tree predicted by Parisi

### A simple analytical description of the non-stationary dynamics in Ising spin systems

The analytical description of the dynamics in models with discrete variables (e.g. Isingspins) is a notoriously difficult problem, that can be tackled only undersome approximation.Recently a novel variational approach to solve the stationary dynamical regime has beenintroduced by Pelizzola [Eur. Phys. J. B, 86 (2013) 120], where simpleclosed equations arederived under mean-field approximations based on the cluster variational method. Here wepropose to use the same approximation based on the cluster variational method also for thenon-stationary regime, which has not been considered up to now within this framework. Wecheck the validity of this approximation in describing the non-stationary dynamical regime ofseveral Ising models defined on Erdos-R Ìenyi random graphs: westudy ferromagnetic modelswith symmetric and partially asymmetric couplings, models with randomfields and also spinglass models. A comparison with the actual Glauber dynamics, solvednumerically, showsthat one of the two studied approximations (the so-called âdiamondâapproximation) providesvery accurate results in all the systems studied. Only for the spin glass models we find somesmall discrepancies in the very low temperature phase, probably due to the existence of alarge number of metastable states. Given the simplicity of the equations to be solved, webelieve the diamond approximation should be considered as the âminimalstandardâ in thedescription of the non-stationary regime of Ising-like models: any new method pretending toprovide a better approximate description to the dynamics of Ising-like models should performat least as good as the diamond approximation

### 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