Quantum Techniques in Stochastic Mechanics

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.

Contents:

• Stochastic Petri Nets
• The Rate Equation
• The Master Equation
• Probabilities vs Amplitudes
• Annihilation and Creation Operators
• An Example from Population Biology
• Feynman Diagrams
• The Anderson–Craciun–Kurtz Theorem
• An Example of the Anderson–Craciun–Kurtz Theorem
• A Stochastic Version of Noether's Theorem
• Quantum Mechanics vs Stochastic Mechanics
• Noether's Theorem: Quantum vs Stochastic
• Chemistry and the Desargues Graph
• Graph Laplacians
• Dirichlet Operators and Electrical Circuits
• Perron–Frobenius Theory
• The Deficiency Zero Theorem
• Example of the Deficiency Zero Theorem
• Example of the Anderson–Craciun–Kurtz Theorem
• The Deficiency of a Reaction Network
• Rewriting the Rate Equation
• The Rate Equation and Markov Processes
• Proof of the Deficiency Zero Theorem
• Noether's Theorem for Dirichlet Operators
• Computation and Petri Nets
• Summary Table

Readership: Graduate students and researchers in the field of quantum and mathematical physics.
Key Features:

• It's a light-hearted introduction to a deep analogy between probability theory and quantum theory
• It explains how stochastic Petri nets can be used in modeling in biology, chemistry, and many other fields
• It gives new proofs of some fundamental theorems about chemical reaction networks