I'll ask this from the point of view of physics more than of theoretical mathematics. I'm searching for a mathematical discussion of stochastic resonance interpreted in a PDE sense. This is a good mathematical discussion of the one dimensional case: http://www.univ-orleans.fr/mapmo/membres/berglund/noisres.html. I'll gladly discuss what I think is important about this from the point of view of theoretical physics, but in the comments. I'm not even sure if a discussion about the SPDE case corresponding to that even exists, so it may well be worth it to open up one in this medium.

Here's an example of what might interest a physics audience: what's the relationship between stochastic resonance and the Boltzmann equation?

I would also love to be linked to protocols for doing "experiments" in SPDE, for example powerful simulations of noise.

(To be one hundred percent clear, the trajectory of the particle shown in the videos in that website's exposition should be extended here to a trajectory in an infinite dimensional Banach space)

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    $\begingroup$ One place to start for the interest in stochastic resonance that's been, at least for me, local here in Cambridge is a video on the progress of Jeremy England, which was posted by the MIT Physics Department. Here it is: <youtube.com/watch?v=ka8573QQKW4&feature=youtu.be> $\endgroup$ – Ravi Andrew Bajaj Jan 10 '15 at 17:53
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    $\begingroup$ Consider G. da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions (1992). If I remember correctly they also discuss asymptotic properties of solutions, besides the usual existence, uniqueness and continuous dependence properties. $\endgroup$ – Uwe Stroinski Jan 10 '15 at 18:43

What is the relationship between stochastic resonance and the Boltzmann equation? Let's break it down.

  1. The Boltzmann equation defines entropy as a function of the total number of microstates that comprise an observable macrostate (e.g., how many different specific configurations of particles in space and momentum could account for an observed pressure, temperature, volume, etc).
  2. Stochastic resonance is a phenomenon where random noise produces coherent activity. This occurs, for instance, in neuronal populations that start firing consistently once they receive sufficient random input. In the case of neurons, this can occur when inhibitory input is periodic, such that random excitatory input will cause action potentials when the inhibition levels are low. In the case of physical systems, stochastic resonance occurs with systems that have a resonant frequency. For instance, random noise will cause a taut string to vibrate at many different frequencies, but only those frequencies (wavelengths) that have a whole number integer relationship with the string length will be amplified; others will dissipate as heat. Therefore, noise causes strings to resonate at particular frequencies.

Now, let's bring the concepts together. In a closed thermodynamic system, coherence/resonance/synchrony is improbable. That is, there are far fewer microstates where particles are coherent (e.g., moving back and forth, in synchrony, at a particular frequency) than microstates where particles are incoherent (moving in any direction). Therefore, coherent states are lower entropy states. But, with stochastic resonance, random noise causes coherence. So, does stochastic resonance transform high-entropy noise into lower entropy coherence? Not without producing more entropy! But, where is it? I've mentioned that the string dissipates heat from non-resonant frequencies. However, Jeremy England's 2017 (Self-Organized Resonance during Search of a Diverse Chemical Space_ work suggests another source of dissipation: coherently moving molecules (i.e., in resonance) appear to dissipate heat more effectively than randomly moving molecules.

Note that closed systems can be expected to produce stochastic resonance as a function of the size of the container (e.g., the resonant frequency of the container)-- however, this effect will diminish as the system tends towards equilibrium. The Casimir effect, on the other hand, is an equilibrium resonance phenomenon that occurs due to virtual particles popping out of the vacuum!

I hope this answers your initial question. Unfortunately, I've been unable to find work that specifically quantifies the relationship between coherence and entropy, at least in a thermodynamic setting. Let me know if you find something!

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