15 research outputs found
Statistical Mechanics Analysis of the Continuous Number Partitioning Problem
The number partitioning problem consists of partitioning a sequence of
positive numbers into two disjoint sets, and
, such that the absolute value of the difference of the sums of
over the two sets is minimized. We use statistical mechanics tools to study
analytically the Linear Programming relaxation of this NP-complete integer
programming. In particular, we calculate the probability distribution of the
difference between the cardinalities of and and show that
this difference is not self-averaging.Comment: 9 pages, 1 figur
On the structure pf genealogical trees in the presence of selection
We investigate through numerical simulations the effect of selection on two
summary statistics for nucleotide variation in a sample of two genes from a
population of N asexually reproducing haploid individuals. One is the mean time
since two individuals had their most recent common ancestor (), and
the other is the mean number of nucleotide differences between two genes in the
sample (). In the case of diminishing epistasis, in which the
deleterious effect of a new mutation is attenuated, we find that the scale of
with the population size depends on the mutation rate, leading then
to the onset of a sharp threshold phenomenon as N becomes large.Comment: 6 page
Phase transition between synchronous and asynchronous updating algorithms
We update a one-dimensional chain of Ising spins of length with
algorithms which are parameterized by the probability for a certain site to
get updated in one time step. The result of the update event itself is
determined by the energy change due to the local change in the configuration.
In this way we interpolate between the Metropolis algorithm at zero temperature
for of the order of 1/L and for large , and a synchronous deterministic
updating procedure for . As function of we observe a phase transition
between the stationary states to which the algorithm drives the system. These
are non-absorbing stationary states with antiferromagnetic domains for ,
and absorbing states with ferromagnetic domains for . This means
that above this transition the stationary states have lost any remnants to the
ferromagnetic Ising interaction. A measurement of the critical exponents shows
that this transition belongs to the universality class of parity conservation.Comment: 5 pages, 3 figure
Escaping from cycles through a glass transition
A random walk is performed over a disordered media composed of sites
random and uniformly distributed inside a -dimensional hypercube. The walker
cannot remain in the same site and hops to one of its neighboring sites
with a transition probability that depends on the distance between sites
according to a cost function . The stochasticity level is parametrized by
a formal temperature . In the case , the walk is deterministic and
ergodicity is broken: the phase space is divided in a number of
attractor basins of two-cycles that trap the walker. For , analytic
results indicate the existence of a glass transition at as . Below , the average trapping time in two-cycles diverges and
out-of-equilibrium behavior appears. Similar glass transitions occur in higher
dimensions choosing a proper cost function. We also present some results for
the statistics of distances for Poisson spatial point processes.Comment: 11 pages, 4 figure
Generalization in a Hopfield network
The performance of a Hopfield network in learning an extensive number of concepts having access only to a finite supply of typical data which exemplify the concepts is studied. The minimal number of examples which must be taught to the network in order it starts to create representations for the concepts is calculated analitically. It is shown that the mixture states play a crucial role in the creation of these representations
Landscape statistics of the binary perceptron
The landscape of the binary perceptron is studied by Simulated Annealing, exhaustive search and performing random walks on the landscape. We find that the number of local minima increases exponentially with the number of bonds, becoming deeper in the vicinity of a global minimum, but more and more shallow as we move away from it. The random walker detects a simple dependence on the size of the mapping, the architecture introducing a nontrivial dependence on the number of steps
Information processing in synchronous neural networks
The phase diagram of Little's model is determined when the number of stored patterns p grows as ρ = αN, where N is the number of neurons. We duplicate phase space in order to accomodate cycles of length two within the framework of equilibrium statistical mechanics. Using the replica symmetry scheme we determine the phase diagram including a parameter J0 able to control the occurrence of cycles. The second order transition between the paramagnetic and ferromagnetic phase becomes first order at a tricritical point. The retrieval region is some what larger than in Hopfield's model. We also find a low temperature paramagnetic phase with unphysical properties.Nous obtenons le diagramme de phase du modèle de Little quand le nombre p d'échantillons mémorisés croit comme ρ = αN, où N est le nombre de neurones. Nous dédoublons l'espace de phase de façon à accommoder des cycles de longueur deux dans le cadre de la mécanique statistique. Utilisant la méthode des répliques, nous déterminons le diagramme de phase incluant un paramètre J0 pour contrôler l'apparition des cycles. La transition de phase entre les phases para- et ferromagnétiques passe du second ordre au premier ordre au point tricritique. La région de recouvrement de l'information est un peu plus grande que dans le modèle de Hopfield. Nous trouvons également une phase paramagnétique à basse température qui a des propriétés physiquement inacceptables