The h Bar

Yashas

12:05

does "particles in the same state" mean that the particles are having the same position and velocity?

In a system consisting of independent molecules (PE = 0), if two molecules are having different position but same velocity(hence, the same energy), they are still in different states, right?

and the degeneracy of that energy state is 2 (in case only those two states were allowed)?

user400188

same state usually means same energy

degeneracy would be two because there are 2 particles in with the same energy (in the same state)

Yashas

what if an energy level has a degeneracy of n? The different states which belong to the same energy level are truly different, right?

if they were perfectly identical, then we could have grouped them into single state and claim that the degeneracy is 1

user400188

degeneracy is a count of the number of particles in the same state.

Mithrandir24601

@Yashas If they're degenerate, yes; if the states are coalescing (which is a niche bit of a niche bit of physics), no

Yashas

Why does my textbook use this to represent different energy levels:

(3,3)

(2, 1) (2, 1) <--- same energy

(2, 2)

(2, 1) and (1, 2) are different states, right?

Anonymous

12:12

Well, the context is very important here. From what I can see, they mean (1,2) and (2,1) are equivalent macrostates

Yashas

I think so because in the context where I am reading it from, the numbers are numbers representing different translational states in one dimension. So 2, 1 is like 2 units in X, and 1 unit in Y and 1, 2 is 1 unit in X and 2 units in Y. That would mean velocity of the two states of the particle are different.

0celo7

@Phase you said something about SR 3 and 4

Phase

Oh right

Saints Row

Anonymous

@Yashas I'm totally confused what you're talking about now. Upload a picture of your textbook's page :P

Anonymous

Where does translation even come from...

Yashas

12:13

I am deal with a particle in a plane

Phase

In a 3 dimensional box?

If so then you have degeneracy

Yashas

2D plane

Phase

Yeah but is there a third dimension for potential

Yashas

they are independent particles

so V is taken as 0

Phase

So they're free particles...?

Yashas

12:15

yea

Anonymous

Doesn't make any sense even then. I don't know how you jumped from stat mech to translation of particles in a plane all of a sudden

Yashas

I need to write down the partition function for it :/

so I had to first make a list of states and ....

and then I got a doubt about the definition of state

Anonymous

Write down the full question. Only then can someone help you...

Anonymous

If you're talking about the canonical partition function you need to sum over all the energy states and take into account all degeneracy.

Anonymous

$Z=\sum_{\text{states i}} e^{-\beta E_i}$

Yashas

12:20

I have N free particles in a plane of size LxL. I need to find the partition function for it.

I took N as 2 (even though it is a bad thing to do as it breaks approximations but idc)

I just realized that taking N as 2 doesn't affect my answer.

nvm screw it

Anonymous

@Yashas Sure, that's a property of the partition function being an exponential function

Yashas

Say I am dealing with a particle in a box

I can compute the wavefunction and get a list of energy levels

the expression for energy turns out to be E = (n^2 + m^2 + l^2) * (some constant)

where n, m, l are quantum numbers for each direction

Anonymous

Dude. You're doing statistical mechanics. Not quantum mechanics.

Anonymous

No need of wavefunctions here

Yashas

but I need the QM stuff to write partition function --,-

Anonymous

12:25

No.

Yashas

some basics was taught in one class

how do I even figure out the energy levels without some QM stuff?

Anonymous

Partition function (statistical mechanics)

In physics, a partition function describes the statistical properties of a system in thermodynamic equilibrium. Partition functions are functions of the thermodynamic state variables, such as the temperature and volume. Most of the aggregate thermodynamic variables of the system, such as the total energy, free energy, entropy, and pressure, can be expressed in terms of the partition function or its derivatives. Each partition function is constructed to represent a particular statistical ensemble (which, in turn, corresponds to a particular free energy). The most common statistical ensembles have...

Yashas

anyway if I have two states (2, 2, 1) and (2, 1, 2), they are different right?

Anonymous

You don't need any QM stuff here

Anonymous

I really recommend learning SM properly first

Anonymous

12:27

You are confusing lot of stuff now

Yashas

Don't I need the energies of the states to write the partition function?

If I was given the energies, it would be easy work.

Anonymous

@Yashas They are the same macrostate

Anonymous

Different microstates

Yashas

ok

so two different velocities = two different microstates?

if the velocity differs only in direction, even then they are two different microstates? but the same macrostate?

Anonymous

$v_x=1$ and $v_y=2$, and $v_x=2$ and $v_y=1$ are equivalent microstates

Anonymous

12:30

Same macrostate

Yashas

what if the particles are free particles and their positions are different but their velocities are the same, even then in my context, they are taken to be the same microstate, right?

if the particles were interacting, then different positions with same velocity would be different microstates, right?

Anonymous

@Yashas Right. Here, position doesn't matter

Anonymous

Only momenta matters

Anonymous

Use $E=p^2/2m$

Yashas

?

Anonymous

12:32

and integrate over all the states

Yashas

??

Anonymous

@Yashas No. You haven't understood statistical mechanics

Anonymous

Please watch Balakrishnan's lectures (first 6-7)

Anonymous

For stat mech

Anonymous

You are saying strange things. It's impossible to address so many misconceptions over chat

Yashas

12:35

if the particle's position matters for its energy (becaz it is interacting), then different positions can give different potential energy and say if they had the same velocity, the energy of that whatever state would be different (becaz T is same but V changed and hence E changed)?

Anonymous

Suppose in a system consisting of a gas, it doesn't matter where individual gas molecules are for finding the partition function

Anonymous

@Yashas Noooooo.....potential interations are ignored in stat mech!!!!

Yashas

@Blue I was just asking in general

Anonymous

....

Anonymous

The problem is a stat mech problem

Yashas

12:36

if the energies are different, then they have to be in different microstates?

@Blue but I'm too curious :(

well, I know what to write on paper to get full marks for that question but I need to understand what actually is happening :/

Anonymous

It's impossible to take into account potential interactions for so many particle systems. N-body problems are still research problems.

Anonymous

@Yashas If you want to learn about N-particle systems, first you need to learn QM properly

Anonymous

We can learn it together after your exams are over :)

Anonymous

In fact my next year's summer project will probably be based on that

Yashas

13:12

My textbook says "there is a configuration with so great a weight that it overwhelms all the rest in importance to such an extent that the system will almost always be found in it"

is the most probable configuration really has way more arrangements than the 2nd most probable configuration?

to such an extent that the 2nd probable configuration is extremely rare

Anonymous

Yes, greater the number number of particles, lesser will be the standard deviation (variance) from the most probable distribution of energy.

Yashas

do u have a nice pictorial representation of that fact?

Anonymous

@Yashas Yuck. "there is a configuration with so great a weight"!!!

Yashas

I am too curious to see how much the two configurations differ for different values of N

Anonymous

What book are you using...

Anonymous

13:15

@Yashas It's given in any BSc physics book. You can check Reif

Yashas

Oxford: Atkins' Physical Chemistry, Peter Atkins | Julio de Paula

Anonymous

Duh. You're reading from a Chemistry book. :'D

Anonymous

It's given here: books.google.co.in/…

Anonymous

But you need to search for it

Balarka Sen

It's a much more general statistical property that the standard deviation from the expected value decreases if the data is sufficiently large.

Also known as "law of large numbers"

There are also statements about how well the standard deviation estimates the general fluctuations of the data from the expected value - if you're normally distributed, with probability close to 70% a random data point will lie inside $[\mu - \sigma, \mu + \sigma]$

Anonymous

13:32

@BalarkaSen Nice! I didn't know that exact value

Anonymous

I need to learn some statistics...

Anonymous

Never had a formal statistics course in school

Balarka Sen

In fact if you look at a slightly larger interval $[\mu - 3\sigma, \mu + 3\sigma]$ the probability jumps to 0.997

That's pretty damn close to 1

Anonymous

Cool

Anonymous

I heard something similar about the birthday problem too

Balarka Sen

13:34

Oh yeah that's a different phenomenon, but surprising nonetheless

Anonymous

It was something like even if you have a 200 students in a class, the probability that two of them have the same birthday is like >90% or something (don't remember the exact value)

Anonymous

@BalarkaSen Yup

Balarka Sen

The experiment of determining whether 23 persons in a room have at least 2 persons with equal birthdays is probabilistically the same experiment as tossing a coin

i.e., the answer "yes" to the first question has probability 1/2

@Blue 200 is not a surprising number, because there are already 365 days in a year

If you have 365 students the probability is... 1

Anonymous

@BalarkaSen I think it was less...I don't remember

Balarka Sen

That it's already 1/2 when you have 23 students is damn surprising

Anonymous

13:37

@BalarkaSen Uh, really? 365 students could have 365 different birthdays. It's not exactly 1 for sure

Sid

@Blue make that 366. Happy? :P

Anonymous

(Assuming 365 days in a year)

Balarka Sen

I meant 366

Anonymous

@Sid Yup :D

Balarka Sen

Thanks @Sid

Yashas

13:38

@BalarkaSen Pigeon hole principle :D

Balarka Sen

Quite

user685252

stop fighting over the leap year :P

Anonymous

We didn't even start...

user685252

/joke

Balarka Sen

@user685252 We are not fighting over the leap year... Your joke is moot bro

@Blue But yeah you should definitely learn some statistics at some point. I really like it.

It's quite surprising to see how small-scale randomness collectively cancel out to show large-scale regularity

Anonymous

13:42

@BalarkaSen Keep it in your teaching bucket list then. :) Lots of cool things up in the planner. How are your PDE/ODE reading sessions going with the ISI prof?

Balarka Sen

I'd hardly be qualified to teach you; what's in my school syllabus is relatively basic things. I myself want to learn it thoroughly at one point.

Yashas

@blue can you give the definitions for microstate, macrostate, arrangement, distribution of energy?

Balarka Sen

We meet next month

user685252

@BalarkaSen one on one?

Anonymous

Sure. I wish these damned exams get over fast. I need to learn the random variables transformations and the Gaussian, Poisson stuff. Also measure theory (slowed down with it a bit)

Anonymous

13:45

Oh, btw I got the Shakarchi Stein books today morning

Balarka Sen

I should learn measure theory someday

Nice!

Anonymous

They seem to be really nice small books (and have good typography)

Anonymous

:)

Balarka Sen

Oh you bought all three?

Anonymous

@BalarkaSen Yeah, except the functional analysis one. I'm using Kreysig for that

Anonymous

13:46

It was on sale

Balarka Sen

Got it

The Fourier analysis book looks real nice

Anonymous

And I got a few bucks as gift from one of my uncles, sooo...that was the best way to spend it

Anonymous

:D

Anonymous

@BalarkaSen Strangely it comes before real analysis

Anonymous

In that series

Balarka Sen

13:47

Indeed.

Anonymous

And the real analysis book has measure theory and hilbert spaces stuff

Balarka Sen

S-S's approach to analysis is interesting

I think complex analysis also comes before real analysis

Anonymous

Yup

Anonymous

that too

Anonymous

hehe

Balarka Sen

13:48

They approach analysis through a Fourier analytic point of view

Anonymous

I think that's the author's area of research: harmonic functions and Fourier analysis...that's why

Anonymous

(From what I read somewhere)

Balarka Sen

I think it's a refreshing POV

Anonymous

mhm

Anonymous

@Yashas It will get too long, man. I'm a bit busy today

Balarka Sen

13:50

Real analysis is way harder than complex analysis

Anonymous

@BalarkaSen Most people say the opposite :P

Balarka Sen

Really? I have never seen anyone disagree.

Complex analysis is super rigid.

I guess what you could say is that multivariable complex analysis is harder, because it is.

But 1 variable complex analysis is really really nice and easy

Anonymous

Yeah, I guess they meant multivariable complex analysis

Balarka Sen

There are not many complex analytic functions. Everything's either a polynomial, or grows exponentially or more

Anonymous

I actually don't know much of either. So I shouldn't comment based on hearsay :P

Anonymous

13:54

@BalarkaSen I see

Sid

Today is a sad day...

Prathyush Poduval

good evening to you all

Balarka Sen

14:12

toppata morning

Phase

14:28

@BalarkaSen oh hey there jack spedicy

Qmechanic

14:51

0

Q: Quantum Field Theory - Representation Theory (of the Lorentz Group)

I am now in the middle of a one-year-course in QFT. I have one major problem, which somebody else also had and already asked in the forums (Lorentz group representations for QFT with mathematical language), but sadly it was left unanswered. I am also familiar with Representation theory for finite...

Close as duplicate?

0celo7

@BalarkaSen I disagree.

Understanding the logarithm involves more conceptual hurdles than anything in real analysis.

Balarka Sen

15:12

@0celo7 ? The logarithm is very easy to understand.

One only needs to realize $e^z$ is a periodic function in the complex land

Cows

I am going to try to understand two quantum mechanics topics on here at about 1:30 pm pacific time. I encourage most of you to be here then so I can bounce thoughts back and forth. For the level of rigor please look thought the chat conversations I had with @Secret on the math chat last night. I am hoping to continue the conversation in that room.

Balarka Sen

15:39

Damn Squarepusher is pretty good

some trippy shit

alan2here

16:14

Hey

John Rennie

I knew coffee was good for you

alan2here

I've been thinking about entropy being about energy/mass/disorder/complexity/density/heat.

Anonymous

@JohnRennie Excuses to drink more coffee ;)

Anonymous

Actually that looks like an interesting read

John Rennie

@Blue I'll live to be 200 years old with the amount of coffee I drink :-)

Anonymous

16:20

Nice. The world really needs you (for javascript advice, dirty jokes and physics). Keep drinking! :P

Phase

@JohnRennie do Mocha's count or does that tip the scale towards diabeetus

alan2here

And that many lifelike and generations cellular automata, for example "Conways Life" or even more apparently in the "Star Wars" rule, starting in a bound universe with a random distribution of cells, have an entropy like behaviour where they tend towards cyclic behaviour, or in the infinite case any region tends towards purely cyclic behaviour, and complexity has to come in from outside but gradually dies away.

Also, yes coffee rocks :)

John Rennie

@Phase no idea - drink them anyway! :-)

Phase

@JohnRennie tryna get me killed smh

Mithrandir24601

@JohnRennie What about tea?

John Rennie

16:31

That causes premature death and genital stunting in males.

Mithrandir24601

@JohnRennie No, no, I'm not talking about your eating and drinking habits - I mean a good cup of hot water infused with tea leaves :)

John Rennie

:-)

Phase

genital stunting

Is that when your wedding tackle does a sick wheelie over a ramp

alan2here

also to the above, blobs of matter preserve "entropy" better than disperse stuff, again like real entropy

enumaris

uhhhh

John Rennie

16:40

@Phase please don't post the video on YouTube :-)

Balarka Sen

it has already been done before

Phase

@JohnRennie what else am I supposed to do with these hot wheels?

CooperCape

16:55

ohai

what's poppin

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