/sci/ - Science & Math

>>10085369
i thought gas particles had gaussian velocity distributions whereas wealth follows a power law

>>10085425

Sorry I misspoke there it's not the velocity distribution it's the energy distribution (to the extent these things are distinct?)

Please don't confuse me with someone scientifically literate - I'm trying to understand this as a layman so I can explain it to laymen because I feel as if it's important, speaking of which:

>pic related

Is this a correct description of the equation (Eq. (1)) in the picture? It's from page 4:

The fundamental law of equilibrium statistical mechanics
is the Boltzmann-Gibbs distribution. It states
that the probability P(ε) of finding a physical system or
subsystem in a state with the energy ε is given by the
exponential function where T is the temperature, and c
is a normalizing constant (Wannier, 1987). Here we set
the Boltzmann constant kB to unity by choosing the
energy units for measuring the physical temperature T .

Eq. (1) can be derived in different ways (Wannier,
1987). All derivations involve the two main ingredients:
statistical character of the system and conservation of
energy ε. One of the shortest derivations can be summarized
as follows. Let us divide the system into two
(generally unequal) parts. Then, the total energy is the
sum of the parts: ε = ε1 + ε2, whereas the probability
is the product of probabilities: P(ε) = P(ε1) P(ε2). The
only solution of these two equations is the exponential
function (1).

>>10085369
>I want to be able to explain this to a layman and given I am one I thought I'd do it here.
Perhaps you should try /diy/. They're mostly highschool educated or have backgrounds at polytechnics or in engineering.

Sorry I don't have time to help you. But it's nice that Marx is finally being proved right, after all.

>>10085459

>Thus far I haven't demonstrated I can answer your questions - however I'll point you in the direction of people ostensibly less versed in science and mathematics than I am.

>That way I might imply can but it's beneath me.

So it's you and Bill Maher doing the contemptuous intellectual insecurity shtick - anyone else...

>>10085479
There is no question in the OP.

>>10085459

>Thus far I haven't demonstrated I can answer your questions - however I'll point you in the direction of people ostensibly less versed in science and mathematics than I am.

>That way I might imply I can but it's beneath me.

So it's you and Bill Maher doing the contemptuous intellectual insecurity shtick - anyone else.

>>10085465

>But it's nice that Marx is finally being proved right, after all.

It's got nothing to do with that.

>>10085482
There is no question in the OP.

>>10085448
How can you even begin to talk about probability without formally defining your sigma algebra? Have you never heard of Bertrand's paradox?
https://en.wikipedia.org/wiki/Bertrand_paradox_(probability)
This type of thing might be okay with laymen but we mathfags deal with absolute truth.

>>10085481

>>10085448
First of all is the picture here an accurate description of the equation? Sorry I'm a visual thinker I find this sort of notation clumsy.

>>10085482
>from the initial derivations of the boltzmann gibbs distribution on page 4

1. Why would you create an equation to acquire an "expectation value" of "any physical variable"?

Is this so you have a placeholder for anything you can measure empirically in terms of joules or kelvin in equation 2?

2. All states of the system, does that mean:

A. All aspects of a stirling engine in a cycle it goes through and its energy sources?

B. The full range of states of something like H20 - gas, liquid, solid, etc.

3. Numerical coefficient of the order of 1.

Does this just mean it isn't to the power of anything other than 1?

>>10085496
https://en.wikipedia.org/wiki/Expected_value

>>10085448
yeah that's correct for classical thermodynamic systems -- it's basically the definition of temperature for classical thermodynamic systems

anyhow the analogy to wealth distributions needs to be fleshed out -- you need to explain how that works to me for me to comment on it. because i claim that the analogy is not true. let me give an example

if you look at the distribution of wealth among individuals or countries or cities or etc. then you find that typically you have something like the top wealth holder has something like say 10% of the wealth, the next has something like 5% of the wealth, the next has like 2.5%, and so forth, that's a power law (the actual numbers are contrived, but a distribution like that, using a power law, is what tends to show up when you look at wealth distributions)

you do not see that with thermodynamic systems; with thermodynamic systems you see something like the top most energetic particles have something like 10% of the total energy, then 8%, then 7%, then 6.5%, then 6.3%, then 6.2%, etc, not a power law (again these numbers are kind of contrived but i hope you see my drift)

>>10085489
Questions:
>>10085496

Let's take the formal definition of the applicable "sigma algebra" here to be whatever you would use studying the energy content and distribution of a gas in a testable system in a laboratory instead of going on a tangent into the philosophical wilderness of less parochial and reality-bound aspects of mathematics shall we?

>>10085511
Depending on your sigma algebra you get different probability spaces with different probabilities and thus different expected values. Bertrand's paradox is a simple example of a scenario where there are multiple conflicting "right answers". I only do math and have no interest in the physical world. I don't know what physics people would consider appropriate or if they even understand the fine technical details (physics people are notoriously reckless and bad at math). Fortunately, mathematics has nothing to do with reality and if you're clear about the technical details then mathematicians have no problem discussing the topic, whatever it is.

>>10085508

>the analogy to wealth distributions needs to be fleshed out

Data analysis of the empirical distributions of wealth and income reveals a
two-class distribution. The majority of the population belongs to the lower class, characterized
by the exponential (“thermal”) distribution, whereas a small fraction of the population in the
upper class is characterized by the power-law (“superthermal”) distribution. The lower part is
very stable, stationary in time, whereas the upper part is highly dynamical and out of equilibrium.

>you need to explain how that works to me for me to comment on it

That's what I'm trying to do - I think economics is one of many bullshit fields wherein all such "experts":

1. Rely on laymen who don't know what they don't know by any other referent not seeing how intellectually shallow their field is.

2. Knowing just enough to appear to know more than the person they're selling that "expertise" to.

I trust physics as a field - when someone makes an analogy between a physical system and an economic one I take it as more trustworthy because it's much harder to sneak subjective presumptions into physics in the same way as it is economics.

I'm just illiterate in physics - that doesn't mean I can't understand it - I want to explain it without requiring quite such a degree of literacy.

>the analogy

Perhaps in a given moment but consider the truism that self-perpetuating wealth generally lasts 3 generations - that's similar to the highly dynamical super-thermal distribution.

>>10085501

Okeedoke so the expectation value of a coin toss is 50/50 heads / tails I get it - however why when he's saying "any physical variable" I presume he means anything he can empirically measure in a set unit?

>>10085524

Alright have fun in Infinite Fun Space I'm dealing with base reality here.

>>10085508

Thanks for answering my questions so far - again - I just don't know what I don't know so I'm confirming anything I could interpret ambiguously or make an improper interpretation of without knowing otherwise.

>1. Why would you create an equation to acquire an "expectation value" of "any physical variable"?

>Is this so you have a placeholder for anything you can measure empirically in terms of joules or kelvin in equation 2?

>2. All states of the system, does that mean:

>A. All aspects of a stirling engine in a cycle it goes through and its energy sources?

>B. The full range of states of something like H20 - gas, liquid, solid, etc.

>3. Numerical coefficient of the order of 1.

>Does this just mean it isn't to the power of anything other than 1?

>>10085534
The other guy is trolling you. But in essence a sigma-algebra is the set of all possible events. But you would not really call it a sigma-algebra in this context.

>>10085529
>The majority of the population belongs to the lower class, characterized by the exponential (“thermal”) distribution, whereas a small fraction of the population in theupper class is characterized by the power-law (“superthermal”) distribution.
ah okay so in the paper they agree with me -- that wealth distribution is described by a power law, at the top. and the thermal part just describes the relatively poor people. that makes sense to me.

well, the fact that the top IS a power law, as they admit, means that that's where all the money is. and this is what people usually talk about when they worry about income inequality -- that the wealth is sharply peaked at the top (according to the power law) and therefore the wealthiest tiny fraction of people control a hefty majority of the total wealth. i don't think many people really care about the distribution of the wealth among poor people as much, maybe that's a topic you could study in economics, but it's not very notable to the greater audience, who care more about the wealth inequality at the top, where it is described by a power law.

so basically if you're just writing a paper for the general audience, and you focus on where the wealth distribution is boltzmann-distributed, then it's pretty dry because you'd be ignoring the real income inequality and instead focusing on just one chunk of the population who are all poor basically. the interesting phenomenon is the super-concentration near the top where it is power-law

as an example i made a little graph for you to compare a thermal distribution with a power law distribution (convoluted with a sigmoid function to make it more realistic). the blue line is a power law and the purple is a boltzmann distribution. what those guys say is that wealth at the top is the blue line, and that's what people focus on when they discuss income inequality. the purple line is what they focus on, which they say is how money is distributed among the poor people

>>10085551

I gathered - someone trying to disorient via field-specific jargon gives off a similar vibe no matter who's jargon it is.

So questions 1, 2, 3 (?) they're not complicated I just want to confirm so I can move onto the next parts.
>>10085544

>>10085555

>I don't think many people really care about the distribution of the wealth among poor people as much

I don't have an axe to grind so much as I think there is an interesting parallel between a physical system in this instance and a physical system in terms of the flow of capital and income and society - if so, I want to understand why the former works the way it does and what conditions it because arguably that's what's "natural" (whether or not it's desirable is up for grabs - however you have to know what that is first).

I think wealth inequality is inevitable and the "thermal" and "super-thermal" phenomena attest to that - the question is to what extent (and what variables affect) the system in such a fashion inequality becomes so dire it creates a "phase transition" or crosses some other critical threshold which makes it implode or cycle back.

If you think about it obviously there are differing phenomena:

Lots of people work incredibly hard but if they don't "catch a break" they either slide back to square 1 or they are propelled forwards on the crest of a pareto distribution which is a feature of lots of systems - and once in that "thermal-distribution" it is to some extent self-perpetuating but at the same time in keeping with the "rise & fall" over a few generations that wealth is lost - as so for the lack of equillibrium in the super-thermal distribution.

So there clearly are two dynamics at work in society as shown - an exponential dynamic and a power-law dynamic - the question is what amplifies inequality in the interaction between these two dynamics and at what point it actually alters the parameters which keep those two dynamics intact.

>>10085555

In the paper it says "all states of the system", does that mean:

>A. All aspects of a stirling engine in a cycle it goes through and its energy sources?

>B. The full range of states of something like H20 - gas, liquid, solid, etc.

I'll give up on asking these questions here and try something else - maybe I'm expecting to be spoonfed too much or perhaps this is just the wrong place to ask.

>>10085496
Thoiiic