Violating the second law of thermodynamics in a dynamical system through equivalence closure via mutual information carriers of a 5-tuple measure space

Time and space average of an ergodic systems following the 5-tuple relations (A,~,J,Σ,μ) through the initial increment from a+bθ to a+c+bθ indicates the entropy to be reserved in the deterministic yet dynamical and conservative systems to hold for the set S_p= S_1 ∑_(i=2)^∞_S_i keeping S as the entropy ∃(S_∞=⋯S_3=S_2 )>S_1 obeying the Poincare ́ recurrence theorem throughout the constant attractor A. This in turn states the facts of the equivalence closure as the property of the induced systems to resemblance an entropy conserving scenarios.

Violating the second law of thermodynamics in a dynamical system through equivalence closure via mutual information carriers of a 5-tuple measure space

Time and space average of an ergodic systems following the 5-tuple relations (A,~,J,Σ,μ) through the initial increment from a+bθ to a+c+bθ indicates the entropy to be reserved in the deterministic yet dynamical and conservative systems to hold for the set S_p= S_1 ∑_(i=2)^∞_S_i keeping S as the entropy ∃(S_∞=⋯S_3=S_2 )>S_1 obeying the Poincare ́ recurrence theorem throughout the constant attractor A. This in turn states the facts of the equivalence closure as the property of the induced systems to resemblance an entropy conserving scenarios.

Data-Driven Reaction Coordinate Discovery in Overdamped and non-Conservative Systems: Application to Optical Matter Structural Isomerization

Powerful technique as it is to study the collective fluctuations and structural transitions in under-damped conservative systems, normal mode analysis cannot be applied to the optical matter (OM) system, an overdamped system subject to non-conservative forces, which consists of (nano-)particle constituents in solution that can self-organize into ordered arrays bound by electrodynamic interactions. We propose a data-driven approach based on principal component analysis (PCA) of deviations from a reference structure to determine the soft collective modes of non-conservative overdamped systems, such as OM structures, and harmonic linear discriminant analysis (HLDA) of time trajectories to estimate the reaction coordinate for structural transitions, which is demonstrated for structural isomerization of a six-particle OM system. The reaction coordinate we discover is in accord with committor analysis and the identified mechanism is in agreement with experiment. The PCA-HLDA approach to data-driven discovery of reaction coordinates aids in the understanding of non-conservative and overdamped systems.