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

- 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).
- 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!