arXiv subjects

Allen Lobo

Publications and source records attributed to Allen Lobo.

7 recordsLinked to original sources

arXiv2025

Feedback Driven Convergence, Competition, and Entanglement in Classical Stochastic Processes

We present a dynamical theory of statistical convergence in which the law of large numbers arises from outcome-outcome feedback rather than assumed independence. Defining the convergence field and its derivative, we show that empirical frequencies evolve through coupling, producing competition, finite-m fluctuations, and classical entanglement. Using the Kramers-Moyal expansion, we derive an Ito-Langevin and Fokker-Planck description, reducing in the symmetric regime to a time-dependent Ornstein-Uhlenbeck process. We propose variance-based witnesses that detect outcome-space entanglement in both binary sequences and coupled Brownian trajectories, and confirm entanglement through numerical experiments. Extending the formalism yields multi-outcome feedback dynamics and finite-time cross-diffusion between Brownian particles. The results unify convergence, fluctuation, and entanglement as consequences of a single feedback-driven stochastic principle.

cond-mat.stat-mech↗

arXiv2025

On the dynamical evolution of randomness Part B: Geometrisation and the origin of convergence in LLN

In classical probability theory, the convergence of empirical frequencies to theoretical probabilities: as captured by the Law of Large Numbers (LLN): is treated as axiomatic and emergent from statistical assumptions such as independence and identical distribution. In this work, a novel dynamical framework is constructed in which convergence arises as a consequence of structured evolution in outcome space, rather than a statistical postulate. Through this formalism, statistical convergence is derived dynamically, revealing an internal structure to randomness and exposing entanglement between successive trials. The system recovers classical LLN behaviour in the large-number limit, while predicting deviations, transient fluctuations, and geometric asymmetries in the early regime. This work inaugurates a new paradigm: dynamical probability mechanics: in which randomness is modelled not as a sequence of disconnected stochastic events, but as a physically structured, feedback-driven process. The theory provides a novel explanatory layer beneath statistical laws and opens pathways toward a mechanistic foundation of probability itself.

physics.data-an↗

arXiv2025

On the dynamical evolution of randomness Part A: Random experiments as dynamical systems

In this work, Bernoulli's Law of Large Numbers, also known as the Golden theorem, has been extended to study the relations between empirical probability and empirical randomness of an otherwise random experiment. Using the example of a coin toss and a dice role, some interesting results are drawn. Analytically and using numerical computations, empirical randomness of each outcome has been shown to increase by \textit{ chance}, which itself depends on the growth rate of empirical probabilities. The analyses presented in this work, apart form depicting the nature of flow of random experiments in repetitions, also present dynamical behaviours of the random experiment, and experimental and simulation-based verifications of the mathematical analyses. It also presents an appreciation of the beauty of Bernoulli's Golden theorem and its applications by extension.

physics.data-an↗

arXiv2024

Vlasov-Poisson simulation study of phase-space hole coalescence in a cylindrically wave-guided plasma

In this work, coalescence of phase-space holes of collision-less, one-dimensional plasmas is studied using kinetic simulation techniques. Phase-space holes are well-known Bernstein-Greene-Kruskal waves known for exhibiting coalescence, are numerically simulated and their coalescence is observed. Relations between the hole speed, potential, phase-space vorticity and phase-space depth are then obtained using the simulation data. This study involves the study of electron phase-space hole coalescence in a cylindrically wave-guided plasma. Using the recently developed phase-space hydrodynamic analogy, it is shown that the coalescence phenomena can be explored in terms of the fluid-analogous vortical nature of the phase-space holes. Coalescence occurs due to the interaction of the phase-space velocity fields associated with these phase-space vortices. Results obtained from the study describes various parametric relations between the coalesced hole characteristics and the characteristics of the colliding holes.

physics.plasm-ph↗

arXiv2024

Diffusion and turbulence in phase-space and formation of phase-space vortices

In this work, the recently introduced fluid-like treatment of the phase-space has been further extended and some interesting outcomes have been presented. A modified form of the Vlasov equation has been presented which describes the diffusion of the phase-space density. This anisotropic diffusion is analysed and the flow of the phase-space probability field is shown. Growth of phase-space vortices is then shown due to increased turbulent-like flow, which is marked by the dominating inertial flow above the diffusive flow. The nature of this flow is judged by using a parameter for the phase-space. It is then shown that the formation of phase-space vortices is due to growth of turbulent-like flow in the phase-space. On the bases of the diffusion parameters, the vorticity field transport of the hydrodynamic phase-space is studied and a Schamel-KdV form of the vorticity transport equation is derived, suggesting solitary modes of the phase-space vorticity waves.

physics.plasm-ph↗

arXiv2024

A numerical integration scheme for vectorised phase-space of one-dimensional collision-free, electrostatic systems

The kinetic analyses of many-particle soft matter often employ many simulation studies of various physical phenomena which supplement the experimental limitations or compliment the theoretical findings of the study. Such simulations are generally conducted by the numerical integration techniques of the governing equations. In the typical case of collisionless electrostatic systems such as electrostatic plasmas, the Vlasov-Poisson (VP) equation system governs the dynamical evolution of the particle phase-space. The one-dimensional position-velocity (1D-1V) particle phase-space, on the other hand, is known to exhibit direct analogy with ordinary two-dimensional fluids, wherein the Vlasov equation resembles the fluid continuity equation of an in-compressible fluid. On the basis of this fluid-analogy, we present, in this work, a new numerical integration scheme which treats the 1D-1V phase-space as a two-dimensional fluid vector space. We then perform and present analyses of numerical accuracy of this scheme and compare its speed and accuracy with the well-known finite splitting scheme, which is a standardised technique for the numerical Vlasov-Poisson integration. Finally, we show some simulation results of the 1D collisionless electrostatic plasma which highlight the higher speed and accuracy of the new scheme. This work presents a fast and sufficiently accurate numerical integration technique of the VP system which can be directly employed in various simulation studies of many particle systems, including plasmas.

physics.plasm-ph↗

arXiv2023

Theory of electron and ion holes as vortices in the phase-space of collision-less plasmas

This article studies the vortical nature and structure of phase-space holes -- nonlinear B.G.K. trapping modes found in the phase-space collision-free plasmas. A fluid-like outlook of the particles' phase-space is explored, which makes it convenient to analytically identify electron and ion holes as vortices -- similar to that of ordinary two-dimensional fluids. A fluid velocity is defined for the phase-space of the electrons and ions, continuity and momentum equations describing the flow of the phase-space fluid representing the particle system are then developed. Pressure formation and associated diffusion in phase-space of such systems is introduced and a vorticity field of the phase-space is then defined. Using these equations, electron holes and ion holes are analytically identified as vortices in the phase-space of the plasma. A relation between Schamel's trapping parameter ($\beta$), hole speed ($M$), hole phase-space depth ($-\Gamma$) and hole potential amplitude ($\chi_0$) is derived. The approach introduces a new technique to study the phase-space holes of collision-less plasmas, allowing fluid-vortex-like treatment to these kinetic structures. Phase-space distribution functions for electron hole regions can then be analytically derived from this model, reproducing the schamel-df equations and thus acting as a precursor to the pseudo-potential approach, avoiding the need to assume a solution to the phase-space density.

physics.plasm-ph↗