The Strongly Intensive Cumulants Made Simple: A Simplifjed - PowerPoint PPT Presentation

The Strongly Intensive Cumulants Made Simple: A Simplifjed Explanation Geared T owards Intuitive Understanding Evan Sangaline

I was going to reuse a talk, but... Wojtek bets on a bottle of a good wine that it is impossible to get strongly intensive measures of the order higher then three. He could not get through your formulas. Can you show that he is wrong (or right) in a simpler way? ~Viktor Begun So instead I’m going to try to make this as simple as possible… at the expense of some mathematical rigor

Some Basic Notation

Moment Generating Function We can construct an arbitrary function... Such that the coeffjcients of the T aylor series are the moments of X 0 and X 1 ...

Cumulant Generating Function We can make a new function that depends on the last The coeffjcients of this new T aylor series are the cumulants of X 0 and X 1 ... ...by defjnition!

The Generating Function is Key Anytime you’ve seen... ● Closed form equations for cumulants ● Recursion relations for cumulants ● Properties of cumulants ...they have been derived from an arbitrarily defjned generating function.

Why was it defjned like that? It’s basically a mathematical trick to make cumulants additive under convolution. convolved variables The Taylor series coeffjcients/cumulants simply add.

From Convolution to Volume T ake a tiny little box with a tiny little cumulant generating function ~ dV And add them up to make a big box with a big cumulant generating function Ψ is extensive Ψ’ is intensive , it has ZERO volume dependence ~ V

Going Back to the Moments For any volume, are moments are still determined by the T aylor series coeffjcients of ϕ. We can now express this with an explicit volume dependence Ψ’ is still intensive , it has ZERO volume dependence

The Problem We are measuring the moments of a mixed distribution over systems with difgerent volumes. moments are mixed difgerent volumes

We’re Measuring The Moments Averaged Over Difgerent Volumes There is an explicit dependency on the distribution of volume over the ensemble. Reality Check: It’s OK if you don’t understand what this function “means.” It’s just a mathematical tool that makes this volume dependence explicit in a simple way. What you must understand is that: - The T aylor series coeffjcients of each grouped ξ term correspond to the moments that you measure in an experiment. - You can tell that these terms will, in general, depend on fmuctuations in the volume.

The Solution So, we know we can measure the coeffjcients of And we know that these depend on bad volume fmuctuations. Let’s come up with a new function that doesn’t. Explicitly, has ZERO Only depends on things dependence on volume or we can measure in an volume fmuctuations experiment. (e.g. “Strongly Intensive”)

Strongly Intensive Cumulants We defjne a new generating function Ψ* using this quantity which we just showed was explicitly “strongly intensive” And the strongly intensive cumulants, κ*, are then defjned as the T aylor series coeffjcients of Ψ*. This is in direct analogy to how the cumulants are quite literally defjned as the coeffjcients of their generating function Ψ.

But what am I supposed to do with the generating function? Use it to come up with recursion relationships that you can use to calculate arbitrary strongly intensive cumulants in terms of moments. I’ve already done this for you in the strongly intensive cumulants paper (https://arxiv.org/pdf/1505.00261.pdf). This is the same thing that you have to do for cumulants and, no, it’s not particularly easy. Can you calculate κ 4,1 without looking up the equation? If the answer is no then you shouldn’t expect to be able to do it for κ* 4,1 either. They’re equally diffjcult, and intimately related, problems.

Precalculated equations The recursion relation simplifjes dramatically for κ* n,0 which leads to simple, ready to use expressions Use these if you want to measure physics instead of experimental artifacts.

Conclusion ● The strongly intensive cumulants are fmuctuation measures that are completely independent of the volume distributions. ● They exists for 1 st , 2 nd , 3 rd , and 4 th order (all the way up to infjnity). ● They exists for an arbitrary number of dimensions (any > 1). ● The proof that they’re strongly intensive is relatively simple. ● The generalized recurrence relations are not simple, but have been calculated for you already. Use them. ● They have a lot of really interesting properties that I can’t get into in a 15 minute talk...

Bonus Slides! I indirectly misquoted Wojtek once, this time it’s straight from the source! Challenge: Can you get the Mrowczynski measure, which in this case is ********************************************************************* Q_1 kappa_3 (A/<A> − A/<A>) = kappa_3 (a/<a> − b/<b>) from your technique? ******************** ~Wojtek Broniowski

Relating The Quantities From https://arxiv.org/abs/1704.01532 Symmetric combinations of 2 nd order strongly intensive cumulants 3 rd order

Which should you use? From an email to Wojtek... I mentioned that I had started out doing something similar to what you're doing, and I also was attracted to these equations that have symmetry and a number of terms canceling. After exploring it more though, I worry that this sort of attraction is based on aesthetics rather than physics. The strongly intensive cumulants have a much more direct connection to physical susceptibilities and, in fact, reduce to them directly under certain conditions. It's in some sense analogous to <X^4> being a nice simple equation, but one that, beneath the surface, mixes the susceptibilities of order <= 4 in a complicated way. It's better to measure k_4, which has a more complicated equation, but simultaneously provides more physical clarity.

More Bonus! Mathematica implementation of the 2-dimensional recursion relations for deriving SICs of any order. a[r1_, r2_] := Sum[Sum[If[r1 == 0 && r2 == 0, 1/Moment[{0, 1}], If[i1 == r1 && i2 == r2, 0, -Binomial[r1, i1]*Binomial[r2, i2]*a[i1, i2]* Moment[{r1 - i1, r2 + 1 - i2}]/Moment[{0, 1}] ]], {i1, 0, r1}], {i2, 0, r2}] k[r1_, r2_] := Sum[Sum[ Binomial[r1 - 1, i1]*Binomial[r2, i2]*a[i1, i2]* Moment[{r1 - i1, r2 - i2}] , {i1, 0, r1 - 1}], {i2, 0, r2}]

For The Particularly Stubborn These equations are trivial to verify explicitly if you work out the volume fmuctuation dependence of the moments.

Second Order This is literally just plugging and chugging. Wow! All of the volume distribution dependent terms canceled!

Third Order This is literally just plugging and chugging. Wow! All of the volume distribution dependent terms canceled!

Fourth Order This is literally just plugging and chugging. Wow! All of the volume distribution dependent terms canceled!