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# Notes - Quantum techniques for stochastic mechanics, Baez and Biamonte

edited March 2020

Let's work through some of this one:

• John C. Baez and Jacob Biamonte, Quantum techniques for stochastic mechanics, arXiv:1209.3632 [quant-ph]. Text includes treatment from the ground up of Petri nets, both stochastic and deterministic. Example includes SI, SIR and SIRS models.

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1.
edited March 2020

Simplified predator-prey reaction network.

Note: I'll use reaction networks rather than Petri nets. They are equivalent.

• $$birth(\beta): rabbit \rightarrow rabbit + rabbit$$

• $$predation(\gamma): wolf + rabbit \rightarrow wolf + wolf$$

• $$death1(\rho): rabbit \rightarrow$$

• $$death2(\delta): wolf \rightarrow$$

The values in parentheses are the transition rates.

The rate equations for these reactions are a system of nonlinear ODEs.

Let x(t) be the number of rabbits at time t, and y(t) be the number of wolves. Then:

• $$x' = \beta x - \gamma x y - \rho x$$

• $$y' = \gamma x y - \delta y$$

Comment Source:Simplified predator-prey reaction network. Note: I'll use reaction networks rather than Petri nets. They are equivalent. * \$$birth(\beta): rabbit \rightarrow rabbit + rabbit\$$ * \$$predation(\gamma): wolf + rabbit \rightarrow wolf + wolf\$$ * \$$death1(\rho): rabbit \rightarrow \$$ * \$$death2(\delta): wolf \rightarrow \$$ The values in parentheses are the transition rates. The rate equations for these reactions are a system of nonlinear ODEs. Let x(t) be the number of rabbits at time t, and y(t) be the number of wolves. Then: * \$$x' = \beta x - \gamma x y - \rho x\$$ * \$$y' = \gamma x y - \delta y\$$
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2.

This is a special case of the Lotka-Volterra predator-prey model, which has the following rate equations:

• $$x' = \beta x - \gamma x y$$

• $$y' = \epsilon x y - \delta y$$

Comment Source:This is a special case of the Lotka-Volterra predator-prey model, which has the following rate equations: * \$$x' = \beta x - \gamma x y\$$ * \$$y' = \epsilon x y - \delta y\$$
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3.
edited March 2020

The SI model is an extremely simple model of a disease process:

• S - susceptible population
• I - infected population

• $$infection(\beta): S + I \rightarrow I + I$$

Comment Source:The SI model is an extremely simple model of a disease process: * S - susceptible population * I - infected population * \$$infection(\beta): S + I \rightarrow I + I\$$
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4.
edited March 2020

Here is the rate equation:

• $$S' = - \beta S I$$
• $$I' = \beta S I$$

Assuming a constant population size $$S + I = P$$, we can substitute to get a differential equation in one variable:

• $$I' = \beta I (P - I)$$

This an instance of the logistic equation.

Comment Source:Here is the rate equation: * \$$S' = - \beta S I\$$ * \$$I' = \beta S I\$$ Assuming a constant population size \$$S + I = P\$$, we can substitute to get a differential equation in one variable: * \$$I' = \beta I (P - I)\$$ This an instance of the **logistic equation**.
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5.
edited March 2020

Replace rabbit with lemming and wolf with arctic fox, and look at this analysis over in the logistic thread

The Lotka-Volterra in comment #1 is missing the death rate of the prey species

• $$death(\rho): lemming\rightarrow$$

The survival of the lemming appears to be more dependent on climate cycles since the cyclic lemming population period is always ~3.8 years, which happens to be the lunisolar tidal forcing period that I use in the ENSO model. So $$\rho$$ is actually $$\rho(t)$$ and this follows a cyclic pattern as described in that thread. What this means is that the lemming population never crashes due to excess predation by the arctic fox (or snowy owl) but by the harshness of the climate, and so the predatory feedback factor in the Lotka-Volterra equation is likely not controlling. Those nonlinear limit cycles likely never occur for lemming/fox but may for rabbit/wolf.

Interesting and perfectly plausible how this missing factor completely changes the character of the solution.

Comment Source:Replace **rabbit** with **lemming** and **wolf** with **arctic fox**, and look at this analysis over in the [logistic thread](https://forum.azimuthproject.org/discussion/comment/21969/#Comment_21969) The Lotka-Volterra in comment #1 is missing the death rate of the prey species * \$$death(\rho): lemming\rightarrow \$$ The survival of the lemming appears to be more dependent on climate cycles since the cyclic lemming population period is always ~3.8 years, which happens to be the lunisolar tidal forcing period that I use in the [ENSO model](https://forum.azimuthproject.org/discussion/comment/21894/#Comment_21894). So \$$\rho \$$ is actually \$$\rho(t) \$$ and this follows a cyclic pattern as described in that thread. What this means is that the lemming population never crashes due to excess predation by the arctic fox (or snowy owl) but by the harshness of the climate, and so the predatory feedback factor in the Lotka-Volterra equation is likely not controlling. Those nonlinear limit cycles likely never occur for lemming/fox but may for rabbit/wolf. Interesting and perfectly plausible how this missing factor completely changes the character of the solution.
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6.
edited March 2020

The Lotka-Volterra in comment #1 is missing the death rate of the prey species

Right, I just edited it in.

Comment Source:> The Lotka-Volterra in comment #1 is missing the death rate of the prey species Right, I just edited it in.
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7.
edited March 2020

If that text is about stochastic mechanics and QM, shouldn't it have an introduction to the ideas behind a Wiener process, i.e. a fluctuating driving force being added to the fundamental equations at the quantum level?

It would be interesting to get some clarification here. IMO the concept of stochastic mechanics is essentially that of a stochastic process or Markov process where the future is independent of the present.

Moreover, there's also some indication that Stochastic Mechanics defines an alternative approach to understanding QM

Stochastic mechanics This interpretation, or perhaps explanation, of quantum mechanics leaves logic intact but adds a new physical process. The modern and enduring branch of stochastic mechanics began with a 1966 paper by Edward Nelson that begins boldly:

“We shall attempt to show in this paper that the radical departure from classical physics produced by the introduction of quantum mechanics forty years ago was unnecessary.”

The main result of the paper is impressive: the author derives the Schrödinger equation, the central equation of quantum mechanics, by assuming that particles are subject to a rapidly fluctuating random force. Microscopic particles such as electrons are therefore described as executing something similar to Brownian motion, and Nelson uses much of the related mathematics from statistical physics in his derivation.

Since Nelson’s paper, the field has grown steadily and attracted a large community of researchers. Some of its intriguing successes include an explanation for quantized angular momentum (“spin”), quantum statistics, and the famous two-slit experiment. However, stochastic mechanics is still far from replacing the Copenhagen Interpretation or conventional quantum mechanics. It includes what appears to be an unphysical instantaneous action at a distance and gives incorrect predictions for certain types of measurements. But its proponents have not given up. As Nelson asks in a review of the subject, “How can a theory to be so right and yet so wrong?”

from this article, https://arstechnica.com/science/2017/07/a-brief-history-of-quantum-alternatives/, the author of which I just had a twitter discussion this morning. That may explain why I was confused by the use of the phrase Stochastic Mechanics, which apparently may not be part of the consensus vocabulary.

Comment Source:If that text is about stochastic mechanics and QM, shouldn't it have an introduction to the ideas behind a Wiener process, i.e. a fluctuating driving force being added to the fundamental equations at the quantum level? ![](https://imagizer.imageshack.com/img921/3765/Wyaf9R.png) It would be interesting to get some clarification here. IMO the concept of stochastic mechanics is essentially that of a stochastic process or Markov process where the future is independent of the present. Moreover, there's also some indication that Stochastic Mechanics defines an alternative approach to understanding QM > **Stochastic mechanics** >This interpretation, or perhaps explanation, of quantum mechanics leaves logic intact but adds a new physical process. The modern and enduring branch of stochastic mechanics began with a 1966 paper by Edward Nelson that begins boldly: > “We shall attempt to show in this paper that the radical departure from classical physics produced by the introduction of quantum mechanics forty years ago was unnecessary.” > The main result of the paper is impressive: the author derives the Schrödinger equation, the central equation of quantum mechanics, by assuming that particles are subject to a rapidly fluctuating random force. Microscopic particles such as electrons are therefore described as executing something similar to Brownian motion, and Nelson uses much of the related mathematics from statistical physics in his derivation. > Since Nelson’s paper, the field has grown steadily and attracted a large community of researchers. Some of its intriguing successes include an explanation for quantized angular momentum (“spin”), quantum statistics, and the famous two-slit experiment. However, stochastic mechanics is still far from replacing the Copenhagen Interpretation or conventional quantum mechanics. It includes what appears to be an unphysical instantaneous action at a distance and gives incorrect predictions for certain types of measurements. But its proponents have not given up. As Nelson asks in a review of the subject, “How can a theory to be so right and yet so wrong?” from this article, https://arstechnica.com/science/2017/07/a-brief-history-of-quantum-alternatives/, the author of which I just had a twitter discussion this morning. That may explain why I was confused by the use of the phrase Stochastic Mechanics, which apparently may not be part of the consensus vocabulary.
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8.
edited March 2020

Caveat: I'm only starting to work through this, so I can't speak will full confidence about it.

It's not about quantum mechanics per se, but rather the application of certain mathematical techniques from QM to at least a certain class of stochastic systems - namely reaction networks / Petri nets, so it at least applies to these Markov processes.

They call a probability distribution a stochastic state, and pursue the analogy with quantum states. Both of these types of systems are driven by a Hamiltonian operator which gives the law of motion.

Ideas like annihilator and creation operators are taken from the quantum context and applied in this stochastic context. For this to work the stochastic state for a N-species reaction network is represented by a power series in n-variables, where the coefficients are the probabilities - which they describe as a "trick" taken from quantum mechanics. So then the annihilator is given by a derivative operator on power series.

They show how the Hamiltonian for a stochastic reaction network can be formulaically constructed by transforming its graph and coefficient structure into a formula involving annihilator and creation operators. The usages of the annihilation operators coincide with inputs to a transition, and the usages of the creation operators coincide with outputs to a transition.

There's more that this, as the book goes on, but this gives some sense of it.

I would describe it as exploratory foundational work.

Comment Source:Caveat: I'm only starting to work through this, so I can't speak will full confidence about it. It's not about quantum mechanics per se, but rather the application of certain mathematical techniques from QM to at least a certain class of stochastic systems - namely reaction networks / Petri nets, so it at least applies to these Markov processes. They call a probability distribution a stochastic state, and pursue the analogy with quantum states. Both of these types of systems are driven by a Hamiltonian operator which gives the law of motion. Ideas like annihilator and creation operators are taken from the quantum context and applied in _this_ stochastic context. For this to work the stochastic state for a N-species reaction network is represented by a power series in n-variables, where the coefficients are the probabilities - which they describe as a "trick" taken from quantum mechanics. So then the annihilator is given by a derivative operator on power series. They show how the Hamiltonian for a stochastic reaction network can be formulaically constructed by transforming its graph and coefficient structure into a formula involving annihilator and creation operators. The usages of the annihilation operators coincide with inputs to a transition, and the usages of the creation operators coincide with outputs to a transition. There's more that this, as the book goes on, but this gives some sense of it. I would describe it as exploratory foundational work.
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9.

Abstract:

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.

Comment Source:Abstract: >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.
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10.

This Wikipedia page at least tries to explain the name as the stochastic interpretation of QM

https://en.wikipedia.org/wiki/Stochastic_quantum_mechanics "Perhaps the most widely known theory where quantum mechanics is assumed to describe an inherently stochastic process was put forward by Edward Nelson [6] and is called stochastic mechanics. This was also developed by Davidson, Guerra, Ruggiero and others."

Nelson developed his "dynamical theory of Brownian motion" in 1966.

Comment Source:This Wikipedia page at least tries to explain the name as the stochastic interpretation of QM > https://en.wikipedia.org/wiki/Stochastic_quantum_mechanics > "Perhaps the most widely known theory where quantum mechanics is assumed to describe an inherently stochastic process was put forward by Edward Nelson [[6]](https://www.worldcat.org/title/dynamical-theories-of-brownian-motion-part-1/oclc/25799122) and is called stochastic mechanics. This was also developed by Davidson, Guerra, Ruggiero and others." Nelson developed his "dynamical theory of Brownian motion" in 1966. --- And then there's this : https://archive.is/20170607120820/https://golem.ph.utexas.edu/category/2011/09/the_inconsistency_of_arithmeti.html