301 research outputs found
Techniques of replica symmetry breaking and the storage problem of the McCulloch-Pitts neuron
In this article the framework for Parisi's spontaneous replica symmetry
breaking is reviewed, and subsequently applied to the example of the
statistical mechanical description of the storage properties of a
McCulloch-Pitts neuron. The technical details are reviewed extensively, with
regard to the wide range of systems where the method may be applied. Parisi's
partial differential equation and related differential equations are discussed,
and a Green function technique introduced for the calculation of replica
averages, the key to determining the averages of physical quantities. The
ensuing graph rules involve only tree graphs, as appropriate for a
mean-field-like model. The lowest order Ward-Takahashi identity is recovered
analytically and is shown to lead to the Goldstone modes in continuous replica
symmetry breaking phases. The need for a replica symmetry breaking theory in
the storage problem of the neuron has arisen due to the thermodynamical
instability of formerly given solutions. Variational forms for the neuron's
free energy are derived in terms of the order parameter function x(q), for
different prior distribution of synapses. Analytically in the high temperature
limit and numerically in generic cases various phases are identified, among
them one similar to the Parisi phase in the Sherrington-Kirkpatrick model.
Extensive quantities like the error per pattern change slightly with respect to
the known unstable solutions, but there is a significant difference in the
distribution of non-extensive quantities like the synaptic overlaps and the
pattern storage stability parameter. A simulation result is also reviewed and
compared to the prediction of the theory.Comment: 103 Latex pages (with REVTeX 3.0), including 15 figures (ps, epsi,
eepic), accepted for Physics Report
DNA waves and water
Some bacterial and viral DNA sequences have been found to induce low
frequency electromagnetic waves in high aqueous dilutions. This phenomenon
appears to be triggered by the ambient electromagnetic background of very low
frequency. We discuss this phenomenon in the framework of quantum field theory.
A scheme able to account for the observations is proposed. The reported
phenomenon could allow to develop highly sensitive detection systems for
chronic bacterial and viral infections.Comment: Invited talk at the DICE2010 Conference, Castiglioncello, Italy
September 201
Hierarchy of random deterministic chaotic maps with an invariant measure
Hierarchy of one and many-parameter families of random trigonometric chaotic
maps and one-parameter random elliptic chaotic maps of type with an
invariant measure have been introduced. Using the invariant measure
(Sinai-Ruelle-Bowen measure), the Kolmogrov-Sinai entropy of the random chaotic
maps have been calculated analytically, where the numerical simulations support
the resultsComment: 11 pages, Late
Diagonalization of replicated transfer matrices for disordered Ising spin systems
We present an alternative procedure for solving the eigenvalue problem of
replicated transfer matrices describing disordered spin systems with (random)
1D nearest neighbor bonds and/or random fields, possibly in combination with
(random) long range bonds. Our method is based on transforming the original
eigenvalue problem for a matrix (where ) into an
eigenvalue problem for integral operators. We first develop our formalism for
the Ising chain with random bonds and fields, where we recover known results.
We then apply our methods to models of spins which interact simultaneously via
a one-dimensional ring and via more complex long-range connectivity structures,
e.g. dimensional neural networks and `small world' magnets.
Numerical simulations confirm our predictions satisfactorily.Comment: 24 pages, LaTex, IOP macro
Exact Minimum Eigenvalue Distribution of an Entangled Random Pure State
A recent conjecture regarding the average of the minimum eigenvalue of the
reduced density matrix of a random complex state is proved. In fact, the full
distribution of the minimum eigenvalue is derived exactly for both the cases of
a random real and a random complex state. Our results are relevant to the
entanglement properties of eigenvectors of the orthogonal and unitary ensembles
of random matrix theory and quantum chaotic systems. They also provide a rare
exactly solvable case for the distribution of the minimum of a set of N {\em
strongly correlated} random variables for all values of N (and not just for
large N).Comment: 13 pages, 2 figures included; typos corrected; to appear in J. Stat.
Phy
Airy Distribution Function: From the Area Under a Brownian Excursion to the Maximal Height of Fluctuating Interfaces
The Airy distribution function describes the probability distribution of the
area under a Brownian excursion over a unit interval. Surprisingly, this
function has appeared in a number of seemingly unrelated problems, mostly in
computer science and graph theory. In this paper, we show that this
distribution also appears in a rather well studied physical system, namely the
fluctuating interfaces. We present an exact solution for the distribution
P(h_m,L) of the maximal height h_m (measured with respect to the average
spatial height) in the steady state of a fluctuating interface in a one
dimensional system of size L with both periodic and free boundary conditions.
For the periodic case, we show that P(h_m,L)=L^{-1/2}f(h_m L^{-1/2}) for all L
where the function f(x) is the Airy distribution function. This result is valid
for both the Edwards-Wilkinson and the Kardar-Parisi-Zhang interfaces. For the
free boundary case, the same scaling holds P(h_m,L)=L^{-1/2}F(h_m L^{-1/2}),
but the scaling function F(x) is different from that of the periodic case. We
compute this scaling function explicitly for the Edwards-Wilkinson interface
and call it the F-Airy distribution function. Numerical simulations are in
excellent agreement with our analytical results. Our results provide a rather
rare exactly solvable case for the distribution of extremum of a set of
strongly correlated random variables. Some of these results were announced in a
recent Letter [ S.N. Majumdar and A. Comtet, Phys. Rev. Lett., 92, 225501
(2004)].Comment: 27 pages, 10 .eps figures included. Two figures improved, new
discussion and references adde
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