Virial Extension for Discrete Data Series

The Virial theorem has been applied with considerable success in various fields of natural sciences. This work proposes an extension of the theorem applied to discrete data series. This application will be called the Virial theorem extension and can be applied to the numerical solution of nonlinear dynamic systems represented by difference equations, such as logistic, discubic and random number generators, the numerical solution of differential equations like the nonlinear double pendulum and a series of pseudorandom numbers and its reciprocals. For this purpose, a coefficient was derived from the discrete Virial formalism. This coefficient can be used to detect when a time series is obtained as the solution of a differential equation, in which case the coefficient is close to 1, and when the data come from other sources, in which case it takes different values. With reference to chaotic dynamic systems, the discrete Virial coefficient shows the feasibility in the detection of a change in behavior, as an alternative to the traditional calculation of Lyapunov exponents, and it is a thousand times faster. The convergence speed of the final value of the discrete Virial coefficient of a dynamic system in a non-chaotic regime is between one and five orders of magnitude greater than in the chaotic regime, thus extending results in non-Hamiltonian systems, previously found by another author in Hamiltonian systems. The results obtained show that the proposal characterizes and distinguishes different types of behavior from the series under study. It also shows great sensitivity to the evolution of the series, even anticipating critical points. The proposed method to construct the discrete Virial extension does not require the existence of a Hamiltonian, which allows its application to a series obtained experimentally or from any differential equation. From a general point of view, this research shows a series of properties that can be reinterpreted in light of the discrete Virial coefficient, providing a novel and versatile tool, given its minimal applicability requirements. For pseudorandom number series, the extension reveals a consistent, quasi-mirror behavior between its kinetic and potential factors, suggesting an underlying structural property.