The h Bar

More Anonymous

00:27

Some days I feel bad for Sabine

I was asked to keep this confidential

►

@ACuriousMind do you have any book recommendation for thermodyanmics?

Allie

03:09

hi

PM 2Ring

05:47

A ball bearing bouncing between a pair of "atomic trampolines", demonstrated by NileRed:

This metal is too bouncy

►

Nile made the amorphous metal trampolines himself from an alloy called Vitreloy 1 (41.2% Zr, 13.8% Ti, 12.5% Cu, 10% Ni, and 22.5% Be)

In retrospect, he decided that it would have been a lot easier if he'd used an alloy that didn't contain beryllium, since it is so toxic. You really don't want to get the dust or vapor from it or its oxides into your lungs.

Berylliosis

Berylliosis, or chronic beryllium disease (CBD), is a chronic allergic-type lung response and chronic lung disease caused by exposure to beryllium and its compounds, a form of beryllium poisoning. It is distinct from acute beryllium poisoning, which became rare following occupational exposure limits established around 1950. Berylliosis is an occupational lung disease. While there is no cure, symptoms can be treated. == Signs and symptoms == With single or prolonged exposure by inhalation the lungs may become sensitized to beryllium. Berylliosis has a slow onset and progression. Some people who...

> The onset of symptoms can range from weeks up to tens of years from the initial exposure. In some individuals, a single exposure to beryllium can cause berylliosis.

Notable physicists killed by beryllium include en.wikipedia.org/wiki/W._W._Hansen & en.wikipedia.org/wiki/Herbert_L._Anderson

Tobias Fünke

07:45

morning

Ryder Rude

08:35

hi

Samyak Marathe

08:48

Hi. Can someone help me with image magnification using fesnel diffraction and Fourier trasnformation?
I have the code but I am getting a distorted image (lots of diffraction). I need some help to figure out what's wrong

here's the complete question: stackoverflow.com/q/79441714/13845045

@RyderRude hello

@TobiasFünke morning

Signor Feynman

09:06

@ACuriousMind I'm a no-coordinate guy who uses coordinates to survive

imbAF

10:10

In different notes that I have read through I have seen that sometimes for the action functional the Hamilton's principle is written as $\delta S=0$ and sometimes $\frac{\delta S}{\delta \vec q(t)}$. But the former is known as the "first variation" while the latter as the functional derivative. So, which is the correct notation?

Signor Feynman

@imbAF the formal relation ia the same that you have for the differential of a function and the partial derivatives

Just that that now you have continuous variables, one for each $t\in\mathbb{R}$

imbAF

you mean $\delta S= \int \frac{\delta S}{\delta q}\delta q \vec x$ ?

Signor Feynman

Yes

imbAF

But this is one of the only things I couldn't understand while I was studying the topic yesterda

how is this derivated, or proved ?

I couldn't find it anywhere

Signor Feynman

$\delta S\sim dF, q(t)\sim x_i, \delta q(t)\sim dx_i,\delta S/\delta q(t)\sim \partial F/\partial x_i, \int dt\sim\sum_i$

imbAF

10:18

And in this way you can claim that?

Signor Feynman

What do you want me to claim?

imbAF

The idea that the functional derivative, is at the same time the directional derivative

of functional S at "point" $\vec q$, in direction $\delta \vec q$

Because, ultimately that is the claim done

that $\delta S$ is as such

while at the same time is defined as the derivative of the functional

Signor Feynman

Do you understand the intuitive analogy of going from discrete to continuous variables first?

imbAF

@SignorFeynman this?

Signor Feynman

Yes

imbAF

10:22

yes, it's quite clear.

Signor Feynman

Okay. I suppose you are familiar with the formal definition of the variation

imbAF

I might be, if you are more specific

variation as a phenomena in mathematics or what?

or you mean $f(x+\delta x)$ which can be written down via taylor expansion ?

Signor Feynman

$\delta S_\eta[f]:=\frac{d}{d\lambda}S[f+\lambda\eta]\bigg\lvert_{\lambda=0}$

This

imbAF

yes, it would be the what is known as the first variation. Which is the exact same as the definition of derivative for a function, but here for a functional

instead of f(x) you have F[f]

Signor Feynman

@imbAF good, now, you can see by explicit computation that the variation takes the form $\delta S_\eta=\int dt[...]\eta$

That bracket is what we call functional derivative

I'm assuming $\eta(t_1)=\eta(t_2)=0$

imbAF

10:28

I am not sure I get this part

Signor Feynman

Otherwise boundary terms will appear

@imbAF This is basically the usual derivation of the EL equations

imbAF

If what the box contains

is the EL eq

Signor Feynman

Yup, it is the LHS of the EL equations

imbAF

then, ok I know how to get it for the simple case where the functional is of one function one variable and up to one derivative

which would be the simplest of cases

Signor Feynman

In the general case you also get boundary terms

Typically we consider this case

imbAF

10:30

yes

and the EL is the directional derivative ?

Signor Feynman

The functional derivative

imbAF

ah yes

But if I am point out something

2 things actually

what the functional derivative here is : https://en.wikipedia.org/wiki/Functional_derivative#Determining_functional_derivatives:%7E:text=without%20a%20norm.-,Functional%20derivative,-%5Bedit%5D:~:text=then%20this%20function%20%CE%B4F/%CE%B4%CF%81%20is%20called%20the%20functional%20derivative%20of%20F%20at%20%CF%81
is different than what we are saying

Tobias Fünke

yes, that's the connection between the two concepts

it is like the directional derivative and the gradient

the first variation is the directional derivative, and the functional derivative is the gradient--so to speak

imbAF

@TobiasFünke But this is just a claim. Maybe it is evident when you look at it but I dont see it. And if I have to say how I understand the directional derivative, I'd say is the projection of the gradient to an arbitrary direction

Signor Feynman

I explained to you how you get it

imbAF

10:35

And to be precise $\delta S$ which is the functional differential, is also called directional derivative

@SignorFeynman yes, which is you getting the EL equation. What I fail to see is how that plays the role of directional derivative

Tobias Fünke

no, this is no claim, that's the analogy

Signor Feynman

@imbAF the differential and the directional derivative are basically the same thing, in the sense that the directional derivative is the differential evaluated at a direction

imbAF

that is what I fail to see, but I need to read it again

Tobias Fünke

@imbAF yes. now make the dot product explicit by summing. then replace indices by continuous variables and sum by integral

imbAF

Ok I will have to see it with this additional new info, because I feel like I am getting there. I only have one question for the moment, in order to summarize my understanding

Signor Feynman

10:39

Yes, nothing is better than a beer with pizza

Tobias Fünke

mbernste.github.io/posts/functionals

imbAF

Would it be accurate to write:
$S[f+\delta f]=S[f]+ S'[f]\delta f+...$

$S'[f]\equl \delta S=\int \frac{\delta S}{\delta q}\delta q d\vec x$ ?

Tobias Fünke

@SignorFeynman NOW? hehe

I'll have a coffee now :) enjoying the winter sun. it is cold and sunny==best weather

Signor Feynman

Oh, I thought that was the question imbAF wanted to ask

imbAF

I did ask the question. The above one. I am reducing alcohol , because it kills brain cells, and clearly I need what;s left cuz things are already hard

Tobias Fünke

10:42

lol

imbAF

Also to add to my question, lie @SignorFeynman wrote, I can write: $\delta S_\eta[f]:=\frac{d}{d\lambda}S[f+\lambda\eta]\bigg\lvert_\lambda=0$

Tobias Fünke

I think it is an urband legend that it literally kills brain cells (at least for "normal" amount of consumption)

imbAF

so are my 2 last statement correct ?

Signor Feynman

@imbAF $S[q+\eta]=S[q]+\int dt \frac{\delta S}{\delta q(t)} \eta(t)+\frac{1}{2}\int dt\int dt'\frac{\delta^2 S}{\delta q(t)\delta q(t')}\eta(t)\eta(t')+...$

Tobias Fünke

@imbAF what do you mean with "can I write"---that is literally a definition. But yes, sure, that is just the Gâteaux derivative/first variation

imbAF

10:44

@SignorFeynman so than I am correct

Tobias Fünke

it is completely analogous to the "normal" function case

imbAF

yes, what you get in the Perturbative expansion of a functional is the differential (total derivative), which in turns, if you have multiple variable dependency, summation of partial derivative for the function which translates to the integration for the functional

Signor Feynman

@imbAF I don't get that much alcohol, but even then I wouldn't have enough grey matter to do what one would expect by one protecting every single one of their brain cells :P

I will go now, farewell

imbAF

When we get the first chronic alcoholic nobel prize winner in physics, then the truth has settled

Tobias Fünke

lol

imbAF

10:47

I will have to read the discussion in order to make the interpretation of the functional derivative as a directional derivative, but I think I am getting there

Later on, I will be asking one more question, whose answer I couldn't figure out yesterday, but Ill try again with this additional knowledge

and @TobiasFünke if you remember our discussion about multiple arguments in a functional

this was very helpful

Euler–Lagrange equation

In the calculus of variations and classical mechanics, the Euler–Lagrange equations are a system of second-order ordinary differential equations whose solutions are stationary points of the given action functional. The equations were discovered in the 1750s by Swiss mathematician Leonhard Euler and Italian mathematician Joseph-Louis Lagrange. Because a differentiable functional is stationary at its local extrema, the Euler–Lagrange equation is useful for solving optimization problems in which, given some functional, one seeks the function minimizing or maximizing it. This is analogous to Fermat...

the action functional, falls into the category of a functional of several functions of several variables of first derivation (1:1:1)

Tobias Fünke

@imbAF yes, sure

@imbAF I've posted a link (a few messages above) where exactly this is explained in detail

imbAF

11:11

@TobiasFünke I realized how the analogy to the directional derivative is correct.In the directional derivative you have $\nabla f(\vec x)\vec v$ In the functional derivative you have $\delta S= \int \frac{\delta S}{\delta q}\delta q d\vec x$, where the integral part plus the differential play the role of $\nabla f$, but in functional the summation is replaced with the integration and $\vec v$ is $\delta q$, right?

Tobias Fünke

well, I do not like nor understand the $\delta q$ notation; but I also have written out explicitly the analogy, and you can compare to your terms/terminology

the analogy is $D_v(f)=\nabla f\cdot v=\sum\limits_i \partial_i f \,v_i$ vs $G_v(f)=\int\mathrm dx\, \delta f/\delta v \, v(x)$

$D_v$ denotes the directional derivative along $v$, and $G_v$ the Gâteaux derivative (the analogon)

so $\nabla f \leftrightarrow \delta f/\delta v$ in this sense.

imbAF

11:31

Yes, that is what I am saying

I mean $\delta q$ is the perturbative term in $S[\vecq + \delta \vec q]$ where S is the functional

and $\vec q(x^\mu)$ is the vector value function

But, at least I have the right understanding

Just one thing, to be sure

you say: $\nabla f \leftrightarrow \delta f/\delta v$
But shouldn't it be $\nabla f \leftrightarrow \int \delta f/\delta v$ ?

As said above in the discussion the summation is substituted for integration

Tobias Fünke

yes, the summation goes into integration

but in $\nabla f$ there is no summation

Please, just read my messages carefully or the link I've posted. In any case, it is just an analogy.

imbAF

@TobiasFünke how not? $\nabla f = \sum_{i}\frac{\partial f}{\partial q_i}$ ?

Tobias Fünke

and I bet if you just Google you will find a lot of nice texts discussing this

@imbAF no

imbAF

@TobiasFünke you are right

Tobias Fünke

that is simply not true. once again, we discuss "high level mathematics" but you are missing the basics...

imbAF

11:43

Yes I was mistaken

Tobias Fünke

good

the sum comes with the dot product

imbAF

I see

that would be $\vec v$ for functions and

$\delta f$ for the functional i presume

Tobias Fünke

?

please

read the link. (but take care: this is not rigorous there, just an analogy)

imbAF

funnily enough I have the same link, and I was going to read it xD

Tobias Fünke

it is probably the first link you get when using google

that's how I found it...

imbAF

11:49

Yeah, same

Ryder Rude

13:53

@SamyakMarathe hello

i feel i am not able to use philosophy as a way of life. i just keep it theoretical

doing philosophy and living philosophy r very different things

imbAF

16:13

If, in case of functionals, for the functional derivative: $\frac{\delta S}{\delta \rho}$ that can be interpreted as the gradient of the functional S at some "point" $\rho(\vec x)$, the value $\frac{\delta S}{\delta \rho}$ measures how much the functional S will change if the function $\rho$ changes in $ \vec x$. How can we describe the functional differential $\delta S$, which is the directional derivative of S at $\rho$ in direction of i.e $\eta$ $ (S[\rho + \eta])$?

Does it makes sense to say that $\delta S$ measures the change of the functional at the function $\rho(\vec x)$ in direction of $\eta$? Does it make sense to talk about the direction towards a function?

Tobias Fünke

16:43

@imbAF that's precisely the intuition about directional derivatives

I cannot understand what you mean with the first part of your message, though. The functional derivative is analogous to the gradient, as we've discussed now several times already

Wikipedia states: "In the calculus of variations, a field of mathematical analysis, the functional derivative (or variational derivative)[1] relates a change in a functional (a functional in this sense is a function that acts on functions) to a change in a function on which the functional depends."

More Anonymous

17:17

@ACuriousMind u probably missed my message. But do u have any thermodynamics book recommendation?

imbAF

18:05

@TobiasFünke Ok got it. Finally I am done with the math part of variation calculus and can jump to physics, with action S as a functional

DIRAC1930

Does anyone have a comprehensive list of various groups (mainly used in physics) and their covering groups?

imbAF

@DIRAC1930 I'd be interested in this as well

Allie

20:39

meow

Tobias Fünke

21:19

hey allie

Allie

hi tobias

Tobias Fünke

how are you doing? :)

Allie

im alright

im very excited to start phd

Tobias Fünke

did you move already to your new home in NYC? (or did I misunderstand?--IIRC you wanted to move, no?)

niiiiceee

Allie

just been doing studying, lots of it

Tobias Fünke

21:20

sure

Allie

I will be moving in the fall

been starting some basic stat mech stuff just to get a taste

but in this current moment i am studying DFT

Tobias Fünke

I see

yey, stat.mech and DFT hehe

Allie

stat mech seems mysterious.....

Tobias Fünke

I hope you can enjoy studying these topics

mhmh

Allie

i just got a basic introduction to partition functions

i have been

DIRAC1930

21:23

@imbAF Have you seen how in non-rel QM, a symmetry operator $\mathcal{O}$ is one that commutes with the Hamiltonian $H$ i.e. $[\mathcal{O},H]=0$?

Allie

its always cool to learn new fundamental aspects of natrue

and statistical mechanics is omething ive been curious about for a while :P

Signor Feynman

@Allie maybe you should keep it a mystery :P

Allie

why does everyone hate stat mechh

it seems kinda cool

imbAF

@DIRAC1930 I usually frame it as a conserved quantity, is when ..... but I guess saying a symmetry operator contains the same meaning

Signor Feynman

@Allie everyone has their own truth to hate Stat Mech

DIRAC1930

21:25

Okay, how are you getting along with group theory in relation to QM/QFT?

Tobias Fünke

stat. mech is cool ;)

imbAF

Until now I am doing ok. Through a differential geometry course I am tackling concepts which appear in the book, and via a linear algebra one I am refreshing old concepts

DIRAC1930

Stat mech is one of the most profound topics in the entirety of physics but I keep on forgetting it and having to relearn it

imbAF

@DIRAC1930 When calculating the action, we usually talk about paths of the system going from A to B, the paths are trajectories in phase space ?

Tobias Fünke

...

imbAF

21:27

@DIRAC1930 While stat. mechanics clearly does wonders, I refuse the idea of an indeterministic universe

Tobias Fünke

why would you think that?

@imbAF stat. mech does in no way arrive at an indeterministic universe

imbAF

@TobiasFünke "Action principles start with an energy function called a Lagrangian describing the physical system. The accumulated value of this energy function between two states of the system is called the action."

@TobiasFünke Then probability shouldn't be a thing.

Tobias Fünke

ok. how do you arrive at the phase space?

imbAF

In phase space, a state of a system is a point,

Tobias Fünke

@imbAF nope

DIRAC1930

21:29

The laws in stat mech can be deterministic

imbAF

@TobiasFünke How exactly dealing with a situation where probability is present, doesn't imply indeterminism ?

Tobias Fünke

I cannot reproduce a statistical mechanics textbook here

imbAF

ok

Tobias Fünke

if you throw a coin, is the universe indeterministic? if you throw a thousand coins, does it become indeterministic?

Allie

there's a 50% chance this coin will show up heads. therefore the coin will never be heads or tails

lol tobias

Tobias Fünke

21:31

hehe

imbAF

@TobiasFünke if you throw a coin it will land in some way, and that is deterministic. You just don't know which side. That's a humans inability I would say

Tobias Fünke

and just to be clear: Is the universe indeterministic if you describe the motion of one (classical) particle? Does it become indeterministic if you describe $10^{23}$?

great

Allie

thats the same issue you have in stat mech

our human inability to calculate the equations of motion for 10^23 particles is what stops us from knowing

Tobias Fünke

exactly

imbAF

@TobiasFünke The universe is deterministic

Tobias Fünke

21:33

I've not said that

imbAF

No, that is my belief

Tobias Fünke

You can believe what you want :)

but that was not the question

imbAF

The question is unclear

Tobias Fünke

The point I want to make is that whatever "ontology" you have for classical mechanics, you basically have the same for stat. mech. --the latter is basically an inference problem

as Allie has pointed out already

Allie, which books/papers do you follow for stat.mech?

Allie

well, just to start out im reading the chapters in mcquarries Physical Chemistry

i quite liked the first part of his book on quantum mechanics (And it is how I initially learned basic QM/quantum chem)

Tobias Fünke

21:37

I see

Allie

and the second part handles som basic thermo and stat mech

imbAF

@Allie Yes I fully agree. Is our incapability that gives rise to probabilistic approach to a situation. That means, that you are not 100% certain for the outcome, and that is indeterminism. But in reality, my belief is that the universe can't be indeterministic

Tobias Fünke

I don't know the phys chem literature; only Atkins (?) or so

Allie

nothing really intense. after that i might read his other book specifically on stat mech

i believe its jut called Statistical Mechanics

Tobias Fünke

@imbAF your last part of the sentence is your belief not a universally true statement

Allie

21:38

and also im reading some of Tuckerman stat mech which covers computaitonal aspects

Tobias Fünke

yeah

imbAF

@TobiasFünke fixed it

Allie

but i think i should ease into it so im just starting with the pchem book :3. my thermo is very weak

@imbAF regardless of what you believe, if you apply this exact argument to the coin flip scenario, you arrive at the conclusion that a coin flip is indeterministic

imbAF

Because of our incapability to accurately predict the outcome, see the future, not because of some indeterministic nature of the event

Allie

okay then apply that argument to the molecular case...

the reason we choose a statistical approach is because 10^23 particles is way too hard to do the exact calculations for

imbAF

21:41

That is a multi variable highly computational event that is beyond our capabilities

Allie

so is the coin flip.... >.>

imbAF

so you rely on statistics, to make an educated guess

Allie

and why does "our capabilities" have to do with anything

imbAF

Yeah, both are the same

Allie

yurp

so are you saying the coin is indeterministic?

because thats surely incorrect. lol

imbAF

21:42

can you guess with 100% accuracy a coin flip, 10 times in a row?

every time you do this

Allie

i think you dont understand what indeterministic means

imbAF

I don't ?

Allie

if you flip the coin with some initial trajectory (specify the initial conditions), there's no indeterminacy in what the outcome will be

it iwll be either heads or tails, no randomness is at play

imbAF

Yes, now do that for 10^23 coins

qwerty

yeah @imbAF that's not the definition of deterministic

Allie

21:45

the same case is true lol

if you know the initial conditions of those 10^23 coins they are all guaranteed to land a specific way based on those initial conditions

there's no indeterminacy there

imbAF

Because this is a carefully crafted scenario

Allie

yeah okay im done with this convo

good luck with your studies :3

qwerty

A deterministic model produces the same output from a given starting condition or initial state.

sensitivity to initial conditions does not imply lack of determinism

imbAF

when you have a system at some state, and you make a measurement , because of collapse , there's a certain probability that the state will be an eigenstate of some measurable quantity. The process itself is probabilistic in nature

Unless there's some new QM, where the probabilistic nature of it, doesn't exist anymore

@Allie xD bye

Tobias Fünke

wait

didn't you just state that your belief is that the universe is deterministic?

imbAF

21:50

yes

Allie

lul

Tobias Fünke

why do you talk about fundamental QM measurement indeterminism now?

imbAF

lmao

Tobias Fünke

do I misunderstand anything here?

imbAF

yes

Tobias Fünke

21:51

ok

Allie

meow

qwerty

@TobiasFünke I have the same questions as you xD

Allie

______

lol

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