Thermodynamic properties of Bardeen black holes in dRGT massive gravity

The Bardeen black hole solution is the first spherically symmetric regular black hole based on the Sakharov and Gliner proposal which is the modification of the Schwarzschild black hole. We present the Bardeen black hole solution in presence of the dRGT massive gravity, which is regular everywhere in the presence of a nonlinear source. The obtained solution interpolates with the Bardeen black hole in the absence of massive gravity parameter and the Schwarzschild black hole in the limit of magnetic charge g=0. We investigate the thermodynamical quantities viz. mass (M), temperature (T), entropy (S) and free energy (F) in terms of horizon radius for both canonical and grand canonical ensembles. We check the local and global stability of the obtained solution by studying the heat capacity and free energy. The heat capacity flips the sign at r = r_c. The black hole is thermodynamically stable with positive heat capacity C>0 for i.e., globally preferred with negative free energy F < 0. In addition, we also study the phase structure of the obtained solution in both ensembles.

Erratum: Wang, J., et al. Microcanonical and Canonical Ensembles for fMRI Brain Networks in Alzheimer’s Disease. Entropy 2021, 23, 216

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Homotopy Particle Filter and Data Assimilation

&lt;p&gt;A homotopy schedule is proposed, wherein from a known probability distribution the normalization constant for an improper probability density function can be found. An improper distribution is one for which the normalization is not known, but its functional form is. In the statistical mechanics constant this amounts to finding the canonical ensemble for the improper distribution. Along the way, the method will generate samples from the target distribution.&lt;/p&gt;&lt;p&gt;This homotopy schedule can be adopted to particle filters used for Bayesian estimation with the aim of improving estimates of the mean path and the uncertainty of a noisy dynamical system, for which noisy observations are available. The method is useful when the dynamics are highly nonlinear, especially if the observations that inform the likelihood have low uncertainty. In the context of data assimilation we require that the stochastic dynamics of the system have an asymptotic stationary distribution, which we use as a the known distribution in the homotopy procedure.&lt;/p&gt;&lt;p&gt;In this talk we present the methodology, apply it to the estimation of canonical ensembles and present numerical comparisons of the standard particle filter estimates with those of the homotopy data assimilation.&amp;#160;&lt;/p&gt;&lt;p&gt;&amp;#160;&lt;/p&gt;

Microcanonical and Canonical Ensembles for fMRI Brain Networks in Alzheimer’s Disease

This paper seeks to advance the state-of-the-art in analysing fMRI data to detect onset of Alzheimer’s disease and identify stages in the disease progression. We employ methods of network neuroscience to represent correlation across fMRI data arrays, and introduce novel techniques for network construction and analysis. In network construction, we vary thresholds in establishing BOLD time series correlation between nodes, yielding variations in topological and other network characteristics. For network analysis, we employ methods developed for modelling statistical ensembles of virtual particles in thermal systems. The microcanonical ensemble and the canonical ensemble are analogous to two different fMRI network representations. In the former case, there is zero variance in the number of edges in each network, while in the latter case the set of networks have a variance in the number of edges. Ensemble methods describe the macroscopic properties of a network by considering the underlying microscopic characterisations which are in turn closely related to the degree configuration and network entropy. When applied to fMRI data in populations of Alzheimer’s patients and controls, our methods demonstrated levels of sensitivity adequate for clinical purposes in both identifying brain regions undergoing pathological changes and in revealing the dynamics of such changes.

Infinite Ergodic Walks in Finite Connected Undirected Graphs

The micro-canonical, canonical, and grand canonical ensembles of walks defined in finite connected undirected graphs are considered in the thermodynamic limit of infinite walk length. As infinitely long paths are extremely sensitive to structural irregularities and defects, their properties are used to describe the degree of structural imbalance, anisotropy, and navigability in finite graphs. For the first time, we introduce entropic force and pressure describing the effect of graph defects on mobility patterns associated with the very long walks in finite graphs; navigation in graphs and navigability to the nodes by the different types of ergodic walks; as well as node’s fugacity in the course of prospective network expansion or shrinking.