A system in which the distribution generated by one particle (or person) over time is identical to the distribution of all particles (or persons) at one point in time.
I am not sure that I understand the argument here. As a generalization, averages across two groups if large enough can expose a latent quantity. The latent quantity is not confidently measurable at the individual level because it is obscured i.e. buried in noise. This becomes increasingly important as the noise expands to be many times larger than the putative signal. On the other hand, if we can assume that that a quantity coexists with random noise, then summed over a large number of samples, the random noise effectively reduces to zero. Once the "fog" of noise is removed, the latent quantity i.e. signal, becomes clearly visible. The issue of ergodicity applies when ensemble averages are compared to time averages. This is not the same as being able to apply the group averages to individuals within the group.
I think the generalization is mistaken. For instance, the non-ergodicity scenario would be one where the 'noise' is not of Markovian nature, that is, not random. What was highlighted in the presentation was the lack of correlation between individual and group levels. While it might be difficult to tell apart signal from noise when there isn't enough bulk of the sample, this does not completely thwart all attempts at correlation, at least not systematically. Non-ergodicity, on the other hand, systematically upsets such attempts. Imagine a population comprised of X many individuals of colour blue, and Y many of colour red. Averaging over the entire population would give you a nonsensical result - let us say we calculate the average wavelength or something. This is because the (phase) space of individuals of the first kind is cut off from that of the other kind. This would be a non-ergodic system, or a system with broken ergodicity since the phase space is effectively reducible to non overlapping sub-spaces, such that averages within each of the sub-spaces would be meaningful. Ergodicity, broadly speaking, is about the irreducibility of the phase space - this is of course consistent with the ergodic hypothesis that equates ensemble and time averages. There are other ways to get at this ergodic/non-ergodic distinction.
Great examples! Thanks for sharing.
4:00 What is an ergodic system?
A system in which the distribution generated by one particle (or person) over time is identical to the distribution of all particles (or persons) at one point in time.
@@complexsystemsinbehavioura5723 Thank you. That simple statement of a deep concept ought to be the first line on the relevant Wikipedia page. Lovely
Lovely. Where is the rest
This is an excerpt from a longer talk. You can find it here: th-cam.com/video/BXJN_KhGtrs/w-d-xo.html
I am not sure that I understand the argument here. As a generalization, averages across two groups if large enough can expose a latent quantity. The latent quantity is not confidently measurable at the individual level because it is obscured i.e. buried in noise. This becomes increasingly important as the noise expands to be many times larger than the putative signal.
On the other hand, if we can assume that that a quantity coexists with random noise, then summed over a large number of samples, the random noise effectively reduces to zero. Once the "fog" of noise is removed, the latent quantity i.e. signal, becomes clearly visible.
The issue of ergodicity applies when ensemble averages are compared to time averages.
This is not the same as being able to apply the group averages to individuals within the group.
I think the generalization is mistaken. For instance, the non-ergodicity scenario would be one where the 'noise' is not of Markovian nature, that is, not random. What was highlighted in the presentation was the lack of correlation between individual and group levels. While it might be difficult to tell apart signal from noise when there isn't enough bulk of the sample, this does not completely thwart all attempts at correlation, at least not systematically. Non-ergodicity, on the other hand, systematically upsets such attempts.
Imagine a population comprised of X many individuals of colour blue, and Y many of colour red. Averaging over the entire population would give you a nonsensical result - let us say we calculate the average wavelength or something. This is because the (phase) space of individuals of the first kind is cut off from that of the other kind. This would be a non-ergodic system, or a system with broken ergodicity since the phase space is effectively reducible to non overlapping sub-spaces, such that averages within each of the sub-spaces would be meaningful. Ergodicity, broadly speaking, is about the irreducibility of the phase space - this is of course consistent with the ergodic hypothesis that equates ensemble and time averages. There are other ways to get at this ergodic/non-ergodic distinction.