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Concept# Microstate (statistical mechanics)

Summary

In statistical mechanics, a microstate is a specific configuration of a system that describes the precise positions and momenta of all the individual particles or components that make up the system. Each microstate has a certain probability of occurring during the course of the system's thermal fluctuations.
In contrast, the macrostate of a system refers to its macroscopic properties, such as its temperature, pressure, volume and density. Treatments on statistical mechanics define a macrostate as follows: a particular set of values of energy, the number of particles, and the volume of an isolated thermodynamic system is said to specify a particular macrostate of it. In this description, microstates appear as different possible ways the system can achieve a particular macrostate.
A macrostate is characterized by a probability distribution of possible states across a certain statistical ensemble of all microstates. This distribution describes the probability of finding the system i

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We are living in the era of "Big Data", an era characterized by a voluminous amount of available data. Such amount is mainly due to the continuing advances in the computational capabilities for capturing, storing, transmitting and processing data. However, it is not always the volume of data that matters, but rather the "relevant" information that resides in it.
Exactly 70 years ago, Claude Shannon, the father of information theory, was able to quantify the amount of information in a communication scenario based on a probabilistic model of the data. It turns out that Shannon's theory can be adapted to various probability-based information processing fields, ranging from coding theory to machine learning. The computation of some information theoretic quantities, such as the mutual information, can help in setting fundamental limits and devising more efficient algorithms for many inference problems.
This thesis deals with two different, yet intimately related, inference problems in the fields of coding theory and machine learning. We use Bayesian probabilistic formulations for both problems, and we analyse them in the asymptotic high-dimensional regime. The goal of our analysis is to assess the algorithmic performance on the first hand and to predict the Bayes-optimal performance on the second hand, using an information theoretic approach. To this end, we employ powerful analytical tools from statistical physics.
The first problem is a recent forward-error-correction code called sparse superposition code. We consider the extension of such code to a large class of noisy channels by exploiting the similarity with the compressed sensing paradigm. Moreover, we show the amenability of sparse superposition codes to perform
joint distribution matching and channel coding.
In the second problem, we study symmetric rank-one matrix factorization, a prominent model in machine learning and statistics with many applications ranging from community detection to sparse principal component analysis. We provide an explicit expression for the normalized mutual information and the minimum mean-square error of this model in the asymptotic limit. This allows us to prove the optimality of a certain iterative algorithm on a large set of parameters.
A common feature of the two problems stems from the fact that both of them are represented on dense graphical models. Hence, similar message-passing algorithms and analysis tools can be adopted. Furthermore, spatial coupling, a new technique introduced in the context of low-density parity-check (LDPC) codes, can be applied to both problems. Spatial coupling is used in this thesis as a "construction technique" to boost the algorithmic performance and as a "proof technique" to compute some information theoretic quantities.
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Kathryn Hess Bellwald, Gard Spreemann

One of the paramount challenges in neuroscience is to understand the dynamics of individual neurons and how they give rise to network dynamics when interconnected. Historically, researchers have resorted to graph theory, statistics, and statistical mechanics to describe the spatiotemporal structure of such network dynamics. Our novel approach employs tools from algebraic topology to characterize the global properties of network structure and dynamics.We propose a method based on persistent homology to automatically classify network dynamics using topological features of spaces built from various spike train distances. We investigate the efficacy of our method by simulating activity in three small artificial neural networks with different sets of parameters, giving rise to dynamics that can be classified into four regimes. We then compute three measures of spike train similarity and use persistent homology to extract topological features that are fundamentally different from those used in traditional methods. Our results show that a machine learning classifier trained on these features can accurately predict the regime of the network it was trained on and also generalize to other networks that were not presented during training. Moreover, we demonstrate that using features extracted from multiple spike train distances systematically improves the performance of our method.

Michael Christoph Gastpar, Fernando Rosas de Andraca

This paper introduces a model-agnostic approach to study statistical synergy, a form of emergence in which patterns at large scales are not traceable from lower scales. Our framework leverages various multivariate extensions of Shannon's mutual information, and introduces the O-information as a metric that is capable of characterizing synergy- and redundancy-dominated systems. The O-information is a symmetric quantity, and can assess intrinsic properties of a system without dividing its parts into “predictors” and “targets.” We develop key analytical properties of the O-information, and study how it relates to other metrics of high-order interactions from the statistical mechanics and neuroscience literature. Finally, as a proof of concept, we present an exploration on the relevance of statistical synergy in Baroque music scores.

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