Statistical mechanics

Summary

In physics, statistical mechanics is a mathematical framework that applies statistical methods and probability theory to large assemblies of microscopic entities. It does not assume or postulate any natural laws, but explains the macroscopic behavior of nature from the behavior of such ensembles. Sometimes called statistical physics or statistical thermodynamics, its applications include many problems in the fields of physics, biology, chemistry, and neuroscience. Its main purpose is to clarify the properties of matter in aggregate, in terms of physical laws governing atomic motion. Statistical mechanics arose out of the development of classical thermodynamics, a field for which it was successful in explaining macroscopic physical properties—such as temperature, pressure, and heat capacity—in terms of microscopic parameters that fluctuate about average values and are characterized by probability distributions. The founding of the field of statistical mechanics is generally credite

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Parallel retrieval of correlated patterns: From Hopfield networks to Boltzmann machines

Andrea De Antoni

In this work, we first revise some extensions of the standard Hopfield model in the low storage limit, namely the correlated attractor case and the multitasking case recently introduced by the authors. The former case is based on a modification of the Hebbian prescription, which induces a coupling between consecutive patterns and this effect is tuned by a parameter a. In the latter case, dilution is introduced in pattern entries, in such a way that a fraction d of them is blank. Then, we merge these two extensions to obtain a system able to retrieve several patterns in parallel and the quality of retrieval, encoded by the set of Mattis magnetizations {m(mu)}, is reminiscent of the correlation among patterns. By tuning the parameters d and a, qualitatively different outputs emerge, ranging from highly hierarchical to symmetric. The investigations are accomplished by means of both numerical simulations and statistical mechanics analysis, properly adapting a novel technique originally developed for spin glasses, i.e. the Hamilton-Jacobi interpolation, with excellent agreement. Finally, we show the thermodynamical equivalence of this associative network with a (restricted) Boltzmann machine and study its stochastic dynamics to obtain even a dynamical picture, perfectly consistent with the static scenario earlier discussed. (c) 2012 Elsevier Ltd. All rights reserved.

Elsevier2013

Copper wires each one containing a pseudo-spin valve of Co/Cu/Co were produced by electrodeposition inside nanoporous polycarbonate membranes. These wires have a diameter of 40 nm and a length of 6000 nm of which only 50 nm relates to the spin valve. In order to be able to measure the magnetic behavior of a single wire, the entire deposition was carried out in a Co/Cu multibath and then electrical contact of a single wire was done with a special equipment using a pure Copper bath. Knowing that such wire has a resistance of the order of 500 Ω, with a Giant Magneto-Resistance ratio of about 0.2 %, one needs a sufficiently sensitive measuring apparatus be able to detect such variations. After having carried out a magnetoresistive characterization, we used a set-up built with a Wheatstone bridge in which the sample is integrated. The resistance variation associated with the magnetization reversal being weak, we were very concerned with the sample quality during these three years of research. As high current densities are injected, a behavior of "Two Level Fluctuation" appeared between parallel and antiparallel states. Such phenomena can be treated only by statistics. Therefore, several thousands of measurements were taken in order to be able to understand the statistical behaviors of the current induced magnetization switching. To better understand the mechanisms of the reversal, we studied its dynamics with a model of double potential well by measuring the average time of residence in parallel (respectively in antiparallel) state as a function of current and applied field for different amplitudes and signs. From our analyses, it arises clearly that the dynamics of the magnetization reversal induced by a high current density, in pseudo-spin valves, could be formulated in terms of Spin-Transfer-Torque. We show that the agreement with this model is remarkable.

EPFL2004

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We consider the phase retrieval problem of reconstructing a n -dimensional real or complex signal X ⋆ from m (possibly noisy) observations Y μ = | ∑ n i = 1 Φ μ i X ⋆ i / √ n | , for a large class of correlated real and complex random sensing matrices Φ , in a high-dimensional setting where m , n → ∞ while α = m / n = Θ ( 1 ) . First, we derive sharp asymptotics for the lowest possible estimation error achievable statistically and we unveil the existence of sharp phase transitions for the weak- and full-recovery thresholds as a function of the singular values of the matrix Φ . This is achieved by providing a rigorous proof of a result first obtained by the replica method from statistical mechanics. In particular, the information-theoretic transition to perfect recovery for full-rank matrices appears at α = 1 (real case) and α = 2 (complex case). Secondly, we analyze the performance of the best-known polynomial time algorithm for this problem --- approximate message-passing--- establishing the existence of statistical-to-algorithmic gap depending, again, on the spectral properties of Φ . Our work provides an extensive classification of the statistical and algorithmic thresholds in high-dimensional phase retrieval for a broad class of random matrices.

Curran Associates, Inc.2020