Smash Company Splash Image

March 27th, 2017

In Philosophy

1 Comment

Complexity emerges when a system has transitions that demand a different kind of math

(written by lawrence krubner, however indented passages are often quotes). You can contact lawrence at: lawrence@krubner.com

Interesting:

When we observe the largest scale behaviors of a system, we simplify the mathematical description of the system because there are fewer distinguishable states, and only a limited set of possible behaviors. This also means that systems that look different on a microscopic scale may not look different at the macroscopic scale, and their mathematical descriptions become the same.

An important example of this arose in the study of phase transitions using the new mathematics of renormalization group. The transition when boiling a liquid to a gas has the same properties as the one that occurs when a heating a magnet up to the point where it becomes non-magnetic (ferromagnet to paramagnetic transition). Magnets have local magnetizations that fluctuate and interact at a critical point just like local changes of density at the water to vapor critical point. The result is that these two seemingly different types of systems map mathematically onto each other.

As renormalization group was more widely applied, other instances were found of systems that have the same behavior even though they differ in detail, a concept that became referred to as universality. Still, while many systems have the same behavior, there are multiple distinct behaviors. Together this means that systems fall into classes of behaviors, leading to the term ‘universality class.’ Since renormalization group focuses on how behaviors transform across scales leading to power laws, the value of the power law exponent became used as a signature of the universality class.

In a sense, the idea that many systems can be described by the same large scale behavior is used in traditional theory. Scientists use the normal distribution for many different biological and social systems. Any system having sufficiently independent components, satisfies the axioms of the central limit theorem, and therefore can be described by the normal distribution. When there are dependencies, the normal distribution no longer applies, but there are other behaviors that are characteristic of other kinds of dependencies. To study those behaviors, we have to determine the way different kinds of dependencies give rise to kinds of large scale behavior.

There are even more basic ways a common mathematical description of systems is used, e.g., point particle motion describes the motion of many distinct objects, and wave equations describe everything from music strings to water waves to light. Even though the specific systems are very different, the dependencies that give rise to their behaviors, and the behaviors themselves, are related mathematically.

How does universality work for complex systems? Unlike traditional renormalization group, we do not consider the limit of infinite size and power law exponents. Instead, the states of our representation must correspond to the states of the system at the scale of observation. Moreover, instead of describing the equilibrium energy, we describe dynamics and system response. The mathematical representation of one system at a particular scale may correspond to the behavior of other systems despite different underlying components.

Source



Check out my book:





RECENT COMMENTS

August 18, 2018 4:59 am

From Cat Mara on How to deal with a psychopath

"One of my favorite passages in the book. Like a lot of people in tech, I'm pretty introverted & would do a..."

August 17, 2018 9:37 am

From Richard Caldwell on Poland was shockingly liberal during the 1400s

"We may not have been taught this in school, but it wasn't forgotten by the Powers That Be. Whereas today’s mor..."

August 17, 2018 8:23 am

From Mikołaj Hołysz on Poland was shockingly liberal during the 1400s

"Most of this is not taught in polish schools either. I'm a pole myself and I haven't been taught most of those..."

August 17, 2018 7:01 am

From ignaloidas on Poland was shockingly liberal during the 1400s

"I'm glad to see that people start to learn this part of history. I'm not from Poland, but from Lithuania, and ..."

August 17, 2018 6:52 am

From Tom on Poland was shockingly liberal during the 1400s

"Thanks for this great article. All this would probably be clearer to the world if our magnificent neighbours ..."

August 11, 2018 8:10 pm

From lawrence on Why does anyone use Python?

"I should add, I feel that Python 1.x and 2.x added something to the world of technology. Early Python was both..."

August 11, 2018 10:44 am

From lawrence on Why does anyone use Python?

"Ryan, about this: " the default should be to not try to install packages system-wide". Agreed. I think that is..."

August 10, 2018 8:23 pm

From Ryan on Why does anyone use Python?

"I suspect your problem is that you missed the step in the instructions where it told you to run all those comm..."

August 5, 2018 4:41 pm

From Howard Ormsby on A developer asks for help, then gets angry at those who help them

"This one is very intense reading thank you..."

July 27, 2018 12:04 am

From Just An Observer on Why do women drop out of co-ed sports?

"Many coed leagues require a certain %age of women players, like volleyball 2 out of 5, in a lot of leagues. ..."

July 15, 2018 3:17 pm

From Anonymous on Friendly, intelligent flash cards?

"Good to know that I wasn't too late :)..."

July 15, 2018 8:32 am

From lawrence on Friendly, intelligent flash cards?

"Thank you for these comments regarding Anki. It sounds useful. I can imagine having the full context of the se..."

July 15, 2018 6:32 am

From Anonymous on Friendly, intelligent flash cards?

"Anki limits how much you review. I have 10000+ cards across my decks but the program arranges them in such a w..."

July 6, 2018 3:42 pm

From lawrence on Docker is the dangerous gamble which we will regret

"RK, your comment is irresponsible and pure FUD. These technologies are not being deprecated. Some of them have..."

July 5, 2018 10:21 am

From RK on Docker is the dangerous gamble which we will regret

"Deprecation warning: The Packer, Artifact Registry and Terraform Enterprise (Legacy) features of Atlas will no..."

1 COMMENT

November 19, 2017
2:14 pm

By lawrence

This is a branch of math that I hope to study a lot more. These transitions. Fractal math and dynamic systems.

Leave a Reply

Your email address will not be published. Required fields are marked *

*

You may use these HTML tags and attributes: <a href="" title=""> <abbr title=""> <acronym title=""> <b> <blockquote cite=""> <cite> <code> <del datetime=""> <em> <i> <q cite=""> <strike> <strong>