*The papers in question get at one of the fundamental difficulties of dealing with real systems in physics, namely that there are generally only two problems we know how to solve exactly: two particles, and an infinite number of particles.* …

*If you let the number of particles in a system become infinite, lots of parameters that start out discrete and discontinuous become smooth continuous functions, and you can use calculus to get nice answers in the form of equations you can actually write down and solve.* …

*The game is to assert that you’re in the “thermodynamic limit,” where you have so many particles that it might as well be an infinite number. And for most measurable quantities of stuff, that’s a pretty good approximation– you’re talking about 100,000,000,000,000,000,000 particles on up– and the equations you get by assuming an infinite number work very well.*

*It’s interesting to ask, though, where that kicks in. That is, how many particles do you need to have before you can start using the infinite-limit answers to describe the properties. This is a hard problem to address, because the really nice mathematical tools don’t work for intermediate numbers, and then you need to do things like simulating huge numbers of particles on a supercomputer to try to figure out the answer.*

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One, Two, Many, Lots: Investigating the Start of Many-Body Physics

Two papers with a similar theme crossed my social media feeds in the last couple of days. You might think this is just a weird coincidence, but I’m choosing to take it as a sign to write about them for the blog. So, what are these papers, and what’s the theme? One is the final…

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Just starting off reading and I'm reminded of a great calculus book I was lent after first year, Calculus Made Easy (and it's in the public domain, http://www.gutenberg.org/ebooks/33283). It took the infinitesimals approach instead of the delta epsilon limit approach. It made way more sense when still learning the early stages. Later, re-examining the limit approach, limits made far more sense as a rigorous definition when the idea of infinitesimals was cemented in my mind. The idea was, like here, that for any given scenario there comes a degree of resolution beyond which, for the scenario in question, differences are meaningless. If I have $103,433,902,362.54, honestly, it's not really different from having $103,433,902,362.55 and I can ignore errors of pennies in a bank statement without it really affecting me. If I have $0.17, then I cannot ignore pennies and would complain to the bank they made an error. So, we are just asked that in what we are doing, what fractions are so small that it just doesn't matter if we're wrong with them. Whatever that is, an infinitesimal is much smaller, but still non-zero in size. So our derivatives can easily work even though we don't assume we have a tangent point, we assume we have a secant but the error is so small we just round off the infinitesimals and act like it's the tangent.