Similarly, there is a sensation from Adenosine for chemical cardioversion that creates a hot flushing feeling inside your body as it spreads, and it's quite the sensation to feel it going from your chest down to your extremities in a few seconds.
2k years is a long time for gas dispersion in such a "small" volume as the earth's atmosphere. early weather behaviour probably affected the distribution unevenly, but by now it should be relatively evenly distributed across the globe. no more or less in rome or italy. this is, however, as we say in sweden, a "guy's guess".
Ex - we see consistent, long term, patterns in weather that make it unlikely that this dispersion is anything close to "ideal gas in a chamber" style dispersion.
Further - we have all sorts of compounding effects. Ex - atmospheric escape is a real thing, plants do nitrogen fixation, hydrogen and oxygen can be bound up in the oceans, etc...
Maybe 2000 years is enough time for real random dispersion, maybe it's not. But it's a huge assumption baked into this that doesn't feel especially reasonable to me.
All we have is this:
>If we assume that a breath diffuses evenly throughout the atmosphere and that these molecules are preserved over time (a reasonable assumption—nitrogen is relatively inert)
Which... I challenge is likely not a particularly reasonable assumption to base this on.
It's still an atmosphere mostly made of nitrogen, on a scale vastly exceeding 2000 years.
I don't have an intuition for how molecules actually disperse, but I do know that general climate trends certainly aren't "random dispersion".
Big volcano eruptions make for pretty sunsets across the world. Nuclear testing fallout is detectable in everything since atmospheric nuclear testing began. Everywhere we find the K-P boundary, we find iridium. The counter-assumption (which may well be true!) is the counter-intuitive one.
https://en.wikipedia.org/wiki/File:Aerial_Superhighway.ogv
The jetstream blows at around 110 mph, and Earth's circumference at mid-northern latitudes is around 12500 miles, so it takes 12500/110=114 hours or just under 5 days for the jets to complete a lap around the planet, assuming we choose a molecule that doesn't take a diverging path on that lap. That's 73 laps per year, so 2000 years is nearly 150,000 times that the faster parts of the atmosphere have circled the globe, twisting, breaking, and reconnecting paths the whole time.