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Quantum diffusion and periodic reactions

Mareš, Jiří J., P. (Pavel) Hubík, J. (Jaroslav) Šesták

Published Jan 1, 2013• 0 citations
OpenAlex

Abstract

Periodic chemical reactions typical for biological objects controlled by diffusion in which the diffusion action attains the value of Planck’s quantum are explained in terms of stochastic theory where the random force is identified with temperature independent molecular quantum jiggling.

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It is a remarkable empirical fact, known for a long time, that in certain self-organized periodic chemical reactions, such as Liesegang's or Belousov-Zhabotinsky's reactions, the product of molecular weight of precipitate, precipitation length period, and speed of precipitation is of the order of universal Planck's quantum of action h. Based on the fact that the classical and quantum diffusions are processes, which are indistinguishable in the configuration space, a quantum criterion in terms of diffusion constants has been established. This criterion enables one to find out conditions under which the quantum behavior of self-organized periodic reactions can be observed.
Some ideas from quantum theory are just beginning to percolate back to classical probability theory. For example, there is a widely used and successful theory of ‘chemical reaction networks’, which describes the interactions of molecules in a stochastic rather than quantum way. Computer science and population biology use the same ideas under a different name: ‘stochastic Petri nets’. But if we look at these theories from the perspective of quantum theory, they turn out to involve creation and annihilation operators, coherent states and other well-known ideas—but in a context where probabilities replace amplitudes. We explain this connection as part of a detailed analogy between quantum mechanics and stochastic mechanics. We use this analogy to present new proofs of two major results in the theory of chemical reaction networks: the deficiency zero theorem and the Anderson–Craciun–Kurtz theorem. We also study the overlap of quantum mechanics and stochastic mechanics, which involves Hamiltonians that can generate either unitary or stochastic time evolution. These Hamiltonians
Some ideas from quantum theory are just beginning to percolate back to classical probability theory. For example, there is a widely used and successful theory of "chemical reaction networks", which describes the interactions of molecules in a stochastic rather than quantum way. Computer science and population biology use the same ideas under a different name: "stochastic Petri nets". But if we look at these theories from the perspective of quantum theory, they turn out to involve creation and annihilation operators, coherent states and other well-known ideas - but in a context where probabilities replace amplitudes. We explain this connection as part of a detailed analogy between quantum mechanics and stochastic mechanics. We use this analogy to present new proofs of two major results in the theory of chemical reaction networks: the deficiency zero theorem and the Anderson-Craciun-Kurtz theorem. We also study the overlap of quantum mechanics and stochastic mechanics, which involves Hamiltonians that can generate either unitary or stochastic time evolution. These Hamiltonians are called "Dirichlet forms", and they arise naturally from electrical circuits made only of resistors.
Understanding of processes such as carrier mobility, electron transfer reactions, chemical reactions in fluids, electron solvation in fluids and electron attachment and localization in clusters relies crucially on the understanding of electron dynamics in fluids at the molecular level. Because of its very small mass, an electron is a quantum object and some of its properties such as diffusion coefficient of a solvated electron in water can be explained only by resorting to quantum mechanical formulation. In this study, we have solved the Bloch NMR flow equations to describe the evolution of the wavelike properties and find the wave functions which can be useful to solve a particular fluid flow problem quantum mechanically. Based on the uncertainty principle, a wave packet is assumed to initially describe the fluid particle (electron) under study. Then, when the particle encounters a force (so its potential energy is no longer zero), the force modifies the wave packet. Finding such propagation techniques, and applying them appropriately can provide useful techniques to find solutions to biological, medical and physical problems which otherwise could not be easily solved.