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In this paper, we investigate the stability of the configurations of harmonic oscillator potential that are directly proportional to the square of the displacement. We derive expressions for fluctuations in partition function due to variations of the parameters, viz. the mass, temperature and the frequency of oscillators. Here, we introduce the Hessian matrix of the partition function as the model embedding function from the space of parameters to the set of real numbers. In this framework, we classify the regions in the parameter space of the harmonic oscillator fluctuations where they yield a stable statistical configuration. The mechanism of stability follows from the notion of the fluctuation theory. In sections 7 and 8, we provide the nature of local and global correlations and stability regions where the system yields a stable or unstable statistical basis, or it undergoes into geometric phase transitions. Finally, in section $9$, the comparison of results is provided with reference to other existing research., Comment: 15 pages, 1 figures

This paper presents the intrinsic geometric model for the solution of power system planning and its operation. This problem is large-scale and nonlinear, in general. Thus, we have developed the intrinsic geometric model for the network reliability and voltage stability, and examined it for the IEEE 5 bus system. The robustness of the proposed model is illustrated by introducing variations of the network parameters. Exact analytical results show the accuracy as well as the efficiency of the proposed solution technique., Comment: 8 pages, 4 figures, 2 tables, Index Terms -- Circuit modeling, geometric modeling, parameter space method, power system reliability, power system stability, transmission planning, nonlinear methods, geometric controls, components optimization

Under the fluctuation of the electric charge and atomic mass, this paper considers the theory of the thin film depletion layer formation of an ensemble of finitely excited, non-empty $d/f$-orbital heavy materials, from the thermodynamic geometric perspective. At each state of the local adiabatic evolutions, we examine the nature of the thermodynamic parameters, \textit{viz.}, electric charge and mass, changing at each respective embeddings. The definition of the intrinsic Riemannian geometry and differential topology offers the properties of (i) local heat capacities, (ii) global stability criterion and (iv) global correlation length. Under the Gaussian fluctuations, such an intrinsic geometric consideration is anticipated to be useful in the statistical coating of the thin film layer of a desired quality-fine high cost material on a low cost durable coatant. From the perspective of the daily-life applications, the thermodynamic geometry is thus intrinsically self-consistent with the theory of the local and global economic optimizations. Following the above procedure, the quality of the thin layer depletion could self-consistently be examined to produce an economic, quality products at a desired economic value., Comment: 22 pages, 5 figures, Keywords: Thermodynamic Geometry, Metal Depletion, Nano-science, Thin Film Technology, Quality Economic Characterization; added 1 figure and 1 section (n.10), and edited bibliography

This work explores the role of thermodynamic fluctuations in the two parameter Hawking radiating black hole configurations. The system is characterized by an ensemble of arbitrary mass and radiation frequency of the black holes. In the due course of the Hawking radiations, we find that the intrinsic geometric description exhibits an intriguing set of exact pair correction functions and global correlation lengths. We investigate the nature of the constant amplitude radiation and find that it's not stable under fluctuations of the mass and frequency. Subsequently, the consideration of the York model decreasing amplitude radiation demonstrates that thermodynamic fluctuations are globally stable in the small frequency region. In connection with quantum gravity refinements, we take an account of the logarithmic correction into the constant amplitude and York amplitude over the Hawking radiation. In both considerations, we notice that the nature of the possible parametric fluctuations may precisely be ascertained without any approximation. In the frequency domain $w \in (0, \infty)$, we observe that both the local and the global thermodynamic fluctuations of the radiation energy flux are stable in the s-channel. The intrinsic geometry exemplifies a definite stability character to the thermodynamic fluctuations, and up to finitely many topological defects on the parametric surface, the notion remains almost the same for both the constant amplitude and the York model. The Gaussian fluctuations over equilibrium radiation energy flux and fluctuating horizon configurations accomplish a well-defined, non-degenerate, curved and regular intrinsic Riemannian manifolds, for all the physically admissible domains of the radiation parameters. PACS numbers: 04.70.-s: Physics of black holes; 04.70.-Bw: Classical black holes; 04.70.Dy Quantum aspects of black holes, evaporation, thermodynamics; 04.60.Cf Gravitational aspects of string theory. Keywords: Quantum Gravity, Hawking Radiation; Horizon Perturbations; Vaidya Geometry; York Model; Statistical Fluctuations; Thermodynamic Configurations, Comment: 38 pages, 12 figures

A general investigation is made into the intrinsic Riemannian geometry for complex systems, from the perspective of statistical mechanics. The entropic formulation of statistical mechanics is the ingredient which enables a connection between statistical mechanics and the corresponding Riemannian geometry. The form of the entropy used commonly is the Shannon entropy. However, for modelling complex systems, it is often useful to make use of entropies such as the R\'{e}nyi and Tsallis entropies. We consider, here, Shannon, R\'{e}nyi, Tsallis, Abe and structural entropies, for our analysis. We focus on one, two and three particle thermally excited configurations. We find that statistical pair correlation functions, determined by the components of the covariant metric tensor of the underlying thermodynamic geometry, associated with the various entropies have well defined, definite expressions, which may be extended for arbitrary finite particle systems. In all cases, we find a non-degenerate intrinsic Riemannian manifold. In particular, any finite particle system described in terms of R\'{e}nyi, Tsallis, Abe and structural entropies, always corresponds to an interacting statistical system, thereby highlighting their importance in the study of complex systems. On the other hand, a statistical description by the Gibbs-Shannon entropy corresponds to a non-interacting system., Comment: 26 pages, 23 figures