Quantum Techniques In Stochastic Mechanics

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 interaction of molecules in a stochastic rather than quantum way. Computer scientists use a different but equivalent formalism called ‘stochastic Petri nets’ to describe collections of randomly interacting entities. These two equivalent formalisms also underlie many models in population biology and epidemiology. Surprisingly, the mathematics underlying these formalisms is very much like that used in the quantum theory of interacting particles—but modified to use probabilities instead of complex amplitudes. In this text, we explain this fact as part of a detailed analogy between quantum mechanics and the theory of random processes. To heighten the analogy, we call the latter ‘stochastic mechanics’. We use this analogy to explain two major results in the theory of chemical reaction networks. First, the ‘deficiency zero theorem’ gives conditions for the existence of equilibria in the approximation where the number of molecules of each kind is treated as varying continuously in a deterministic way. Our proof uses tools borrowed from quantum mechanics, including a stochastic analogue of Noether’s theorem relating symmetries and conservation laws. Second, the ‘Anderson–Craciun–Kurtz theorem’ gives conditions under which these equilibria continue to exist when we treat the number of molecules of each kind as discrete, varying in a random way. We prove this using another tool borrowed from quantum mechanics: coherent states. We also investigate the overlap of quantum and stochastic mechanics. Some Hamiltonians can describe either quantum-mechanical or stochastic processes. These are called ‘Dirichlet operators’, and they have an intriguing connection to the theory of electrical circuits. In a section on further directions for research, we describe how the stochastic

Noether theorem simplifies for Hamiltonians that are Dirichlet operators, and explain some connections between stochastic Petri nets and computation.