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Chemical Kinetics, Markov Chains, and the Imaginary Itô Interpretation

Álvaro Correales, Carlos Escudero, Mariya Ptashnyk

Abstract

The abstract chemical reaction $$ A+A \to \emptyset, $$ understood as a Markov chain in continuous time, has been studied in the physical literature for several years. It has been claimed that this reaction can be described by means of the stochastic differential equation $$ d φ= - φ^2 dt + i \, φ\, dW_t, $$ where $i$ is the imaginary unit. This affirmation is, at least, intriguing, and has led to controversy and criticisms in the literature. The goal of this work is to give partial evidence that such a description may be possible.

Chemical Kinetics, Markov Chains, and the Imaginary Itô Interpretation

Abstract

The abstract chemical reaction understood as a Markov chain in continuous time, has been studied in the physical literature for several years. It has been claimed that this reaction can be described by means of the stochastic differential equation where is the imaginary unit. This affirmation is, at least, intriguing, and has led to controversy and criticisms in the literature. The goal of this work is to give partial evidence that such a description may be possible.

Paper Structure

This paper contains 17 sections, 23 theorems, 190 equations.

Table of Contents

  1. Introduction
  2. Quantum Mechanics of Chemical Kinetics: The Formal Approach
  3. The Abstract Vector Space Representation
  4. Coherent States
  5. Generating Functions: A Classical Approach
  6. Imagine the Noise were Real
  7. No Noise
  8. Real Noise
  9. Imaginary Noise is not Unreal
  10. Compactly Supported Initial Conditions
  11. Stochastic Dynamics
  12. Conclusions
  13. Van Kampen System Size Expansion
  14. Equations for the General Reaction
  15. An Explicit Distributional Solution
  16. ...and 2 more sections

Key Result

Lemma 1

The commutator of the annihilation and creator operators is the identity operator

Theorems & Definitions (65)

  • Definition 1: Annihilation and creation operators
  • Lemma 1
  • proof
  • Remark : Combinatorics and operators
  • Lemma 2
  • proof
  • Remark
  • Theorem 1
  • proof
  • Remark
  • ...and 55 more