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WORLD SCIENTIFIC eBooks

Quantum Techniques in Stochastic Mechanics

John C. Baez, Jacob Biamonte

Published May 29, 2017• 44 citations
Publisher

Abstract

We introduce the theory of chemical reaction networks and their relation to stochastic Petri nets — important ways of modeling population biology and many other fields. We explain how techniques from quantum mechanics can be used to study these models. This relies on a profound and still mysterious analogy between quantum theory and probability theory, which we explore in detail. We also give a tour of key results concerning chemical reaction networks and Petri nets.

DOI
10.1142/10623

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We introduce the theory of chemical reaction networks and their relation to stochastic Petri nets — important ways of modeling population biology and many other fields. We explain how techniques from quantum mechanics can be used to study these models. This relies on a profound and still mysterious analogy between quantum theory and probability theory, which we explore in detail. We also give a tour of key results concerning chemical reaction networks and Petri nets.
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.
Reaction networks are a general formalism for describing collections of classical entities interacting in a random way. While reaction networks are mainly studied by chemists, they are equivalent to Petri nets, which are used for similar purposes in computer science and biology. As noted by Doi and others, techniques from quantum physics, such as second quantization, can be adapted to apply to such systems. Here we use these techniques to study how the “master equation” describing stochastic time evolution for a reaction network is related to the “rate equation” describing the deterministic evolution of the expected number of particles of each species in the large-number limit. We show that the relation is especially strong when a solution of master equation is a “coherent state”, meaning that the numbers of entities of each kind are described by independent Poisson distributions. Remarkably, in this case the rate equation and master equation give the exact same formula for the time derivative of the expected number of particles of each species.
Anderson, Craciun, and Kurtz have proved that a stochastically modelled chemical reaction system with mass-action kinetics admits a stationary distribution when the deterministic model of the same system with mass-action kinetics admits an equilibrium solution obeying a certain complex balance condition. Here we present a proof of their theorem using tools from the theory of second quantization: Fock space, annihilation and creation operators, and coherent states. This is an example of stochastic mechanics, where we take techniques from quantum mechanics and replace amplitudes by probabilities.