On stability of thermodynamic systems: a fluctuation theory perspective

• Published in: European physical journal plus 2023-06, Vol.138 (6), p.525, Article 525
• Main Authors: Tiwari, Bhupendra Nath, Thakur, Rahul Kumar
• Format: Article
• Language: English
• Subjects: Applied and Technical Physics; Atomic; Bifurcations; Complex Systems; Compressibility; Condensed Matter Physics; Configurations; Critical phenomena; Fixed points (mathematics); Fluctuation theory; Fluctuations; Fourier series; Fourier transforms; Gravity; Integrals; Mathematical and Computational Physics; Mathematical models; Molecular; Optical and Plasma Physics; Parameters; Phase transitions; Physics; Physics and Astronomy; Quantum field theory; Regular Article; Series expansion; Specific heat; Stability analysis; String theory; Structural stability; Theoretical; Thermal expansion; Thermodynamics
• ISSN: 2190-5444; 2190-5444
• DOI: 10.1140/epjp/s13360-023-04000-6

In this paper, we study the stability analysis of thermodynamic configurations in the space of matrices, arising from an embedding of real thermodynamic spaces under fluctuations of the model parameters. The model parameters under the present consideration are the thermal expansion parameter, and the isothermal and/or adiabatic compressibilities. Under parametric fluctuations, we examine the matrix valued Fourier series and Fourier transforms by invoking the limiting local and global stability structures. The resulting embeddings are taken as the specific heat capacity at a constant pressure and that of at a constant volume, their ratio, and Mayer’s relation. In conclusion, by examining the Fourier series and Fourier transforms in the space of regular matrices, we have explicated the stability properties of such configurations under fluctuations of experimentally measurable quantities. To be specific, our method determines positions of the underling critical phenomena such as the phase transition points/ curves, bifurcations, cross-overs and others. Indeed, we provide highlights for possible geometric extensions by discussing the standing notions, such as the Onsager region, local stability structures and global parameter space invariants, and others. Moreover, an extension of our approach for the thermodynamic systems away from attractor fixed points may be examined for moduli space configurations through certain geometric and algebraic techniques. Finally, we have discussed prospective research directions toward the existence of the series representations of matrix valued functions, series expansions with discontinuities, integral kernels, and their multi-matrix extensions in the light of thermodynamic (in)stabilities under fluctuations of the system parameters.