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Stochastic Determinism in Physical Systems: A Unified Perspective on Quantum and Classical Stability

Nikos Demopoulos

Published Mar 17, 2025• 1 citation
Publisher

Abstract

The role of randomness in physical systems remains a pivotal topic, influencing the transition from quantum mechanics to classical stability and shaping complex dynamic behaviors. This paper explores the concept of stochastic determinism, illustrating how stochastic processes contribute to order in classical and quantum systems. By integrating stochastic differential equations (SDEs), entropy-driven self-organization, and fluctuation-dissipation principles, we demonstrate that randomness is not merely an obstacle to predictability but a key mechanism for structure formation. The study further examines applications in thermodynamics, turbulence, and quantum decoherence, bridging stochasticity and determinism as complementary aspects of physical reality.

DOI
10.20944/preprints202503.1177.v1

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This book studies the mathematical aspect of path integrals and Hamiltonians – which emerge from the formulation of quantum mechanics. The theoretical framework of quantum mechanics provides the mathematical tools for studying both quantum indeterminacy and classical randomness . Many problems arising in quantum mechanics as well as in vastly different fields such as finance and economics can be addressed by the mathematics of quantum mechanics, or quantum mathematics in short. All the topics and subjects in the various chapters have been specifically chosen to illustrate the structure of quantum mathematics, and are not tied to any specific discipline, be it quantum mechanics or stochastic systems. The book is divided into the following six parts, in accordance with the Chapter dependency flowchart given below. Part one addresses the Fundamental principles of path integrals and (Hamiltonian) operators and consists of five chapters. Chapter 2 is on the Mathematical structure of quantum mechanics and introduces the mathematical framework that emerges from the quantum principle. Chapters 3 to 6 discuss the mathematical pillars of quantum mathematics, starting from the Feynman path integral, summarizing Hamiltonian mechanics and introducing path integral quantization. Part two is on Stochastic processes . Stochastic systems are dissipative and are shown to be effectively modeled by the path integral. Chapter 7 is focused on the application of quantum mathematics to classical random systems and to stochastic processes.
Quantum-Classical convergence describes the inevitable merging of quantum and classical systems, often observed in complex systems exhibiting chaotic behavior. This paper explores a novel mathematical framework for understanding this convergence, proposing a hybrid approach that integrates concepts from quantum mechanics (superposition, entanglement) and classical chaos theory. We introduce a model where classical elements are intrinsically influenced by quantum fluctuations, aiming to illuminate the dynamic interplay between these two fundamental scales. The paper details the core mechanism, outlining the integration of quantum and chaotic elements into a unified system. The resulting model offers a new perspective on the emergence of patterns and stability within complex dynamical systems, particularly in the context of chaotic systems. This work provides a foundation for advanced theoretical investigations into the fundamental principles underlying the convergence of these seemingly disparate realms.