The h Bar

user228700

12:34 AM

Hello :) @JohnRennie You were telling me about the fact that entropy doesn't really have a physical significance in that it's not a thing like mass/pressure, no?

user228700

Well, I found a definition that doesn't quite agree with this and was wondering what you think about it-

user228700

"Entropy is a measure of the unavailable energy that a system has, with which it can't do work".

WillO

Okay....this is in response to a five-hour old post, but: what does it even mean to "disagree with" the axiom of choice? Some models of ZF satisfy AC and some don't. When you're studying group theory, do you have any problem accepting that some groups are commutative and others aren't? Or do you feel like you have to "agree" or "disagree" with commutativity?

0celo7

12:50 AM

@WillO What?

I think AoC is unreasonable.

WillO

What does that mean? Can the color red be unreasonable?

0celo7

There are certain set theory axioms I accept and some I do not

I do not accept AoC

WillO

Some things are red and some things are not red. It can be unreasonable for some particular thing to be red. But red, by itself, cannot be reasonable or unreasonable.

If I tell you that there are some colors I accept and some I do not, does that make sense to you?

0celo7

Yes.

Completely different things, but yes.

WillO

Oh.

0celo7

12:54 AM

I see what you're trying to get at.

But I disagree with your analogy, and I think it's a poor analogy.

WillO

I think it's exactly the right analogy.

But if you don't like it, here's another:

Does it make sense to "refuse to accept", say, the Hausdorff property for topological spaces?

0celo7

@WillO I cannot tell you exactly why I dislike AoC. I have had nightmares about it and whenever it is used I get a guilty feeling in my gut.

WillO

What would that mean? Would it mean denying that Hausdorff spaces exist? if so, it would just be wrong.

0celo7

@WillO This isn't a very good analogy either.

WillO

Nevertheless, AoC is sometimes true. Surely your gut feelings should be trumped by truth!

0celo7

12:56 AM

What? When is AoC true?

When is any axiom true?

WillO

It's true in any model of ZFC.

You can construct such models by forcing.

0celo7

I don't think ZFC should be a model of mathematics.

WillO

what does "a model of mathematics" mean?

0celo7

*standard

@WillO Standard rules of set theory.

I don't think AoC should be among them.

WillO

I have no idea what "standard" means in this context.

0celo7

12:57 AM

Ah. You're just being difficult.

WillO

No, I do not intend to be.

I honestly haven't the foggiest idea what you're talking about.

Most sincerely.

0celo7

That's too bad.

WillO

Let's go back to the Hausdorff spaces.

Are we agreed that some spaces are Hausdorff and others aren't?

And that no gut feeling can change that?

0celo7

@WillO I see what analogy you're trying to make, and I reject it.

WillO

Why?

0celo7

1:04 AM

@WillO Hausdorff spaces are not set theory axioms

WillO

No, the Hausdorff property is a topology axiom.

0celo7

No

It's a property some topological spaces have

WillO

Right!!!

And AoC is a property some set theoretic universes have.

0celo7

Did I deny that?

I do not wish to do mathematics within such a universe.

WillO

One does not do mathematics within such a universe. One does mathematics within one's brain.

0celo7

1:06 AM

ugh

WillO

And one goes where the mathematics leads one.

0celo7

Now you're being difficult.

WillO

Certainly not intentionally.

Just to make sure----are you aware that there's a theorem that says that if ZF has a model, then so does ZFC?

0celo7

1:42 AM

@WillO I don't know what a model is

And I don't particularly care

David Z

4:05 AM

@KaumudiHarikumar that strikes me as an analogy, not a definition - or at least, what it's defining may not be entropy. It might be related somehow, but I don't see how it's the same.

user228700

4:21 AM

@DavidZ Oh, but many websites have defined it like this :/

user228700

Also, can you please answer this question if you know:

user228700

1

Q: The Carnot engine and entropy?

The Clausius statement of the second law of thermodynamics is as given below: "Heat can never pass from a colder to a warmer body without some other change, connected therewith, occurring at the same time." Why is this statement true and exactly what does it have to do with entropy? In te...

John Rennie

I must admit I had never seen that analogy, though on googling I find this nice explanation.

David Z

@KaumudiHarikumar when I have time, perhaps

user228700

@DavidZ OK :)

user228700

4:35 AM

@JohnRennie Riight. OK, understood. Thanks sir :) Also, good morning!

0celo7

You know it's late when JR is up.

@JohnRennie Apple replaced my charger under warranty, thankfully.

John Rennie

@0celo7 Result! :-)

0celo7

I was able to bootstrap a charge off a MacBook Air charger and survived.

@JohnRennie Why would an author remove sections of a book between editions?

John Rennie

A maths book?

0celo7

Yeah.

John Rennie

4:42 AM

Maybe they thought the deleted sections didn't add anything.

0celo7

@JohnRennie What?

@JohnRennie My advisor was confused too

The removed sections were quite topical

0celo7

5:37 AM

$\partial_r:=D\exp_p\partial_r$. What kind of crappy notation is this

user116211

6:28 AM

@KaumudiHarikumar Check my post; oh man; it took over an hour to complete the writings ;P

user116211

1

A: The Carnot engine and entropy?

Second Law of Thermodynamics: Postulate of Kelvin: A transformation whose only final result is to transform into work, heat extracted from a source which is at the same temperature throughout the process is impossible. Postulate of Clausius: A transformation whose only final result is ...

user116211

If there is any query, ping me.

user228700

6:47 AM

I read your answer! Over an hour?! Wow, thanks so much! :D

user228700

I didn't read a small portion of the math carefully enough and so, I haven't understood it yet but I'll give the whole thing a proper read again and I will let you know if I need help understanding anything in particular.

user228700

Again, thanks a ton! :D

user116211

I have left the discussion of Clausius' Inequality and its derivation based on the second law as I had no time; if you want then I can add that later.

user228700

Oh, no no, don't bother with all that!

user116211

sure.

user228700

6:53 AM

Something else that I didn't ask there...

user116211

@KaumudiHarikumar Leave here; unfortunately I have to go back to my university; anyone interested would give a reply.

user228700

Okay, sure. Thank you :)

user228700

Also, your dog(if that is your dog) is very cute :D

user116211

@KaumudiHarikumar :D yes, it's mine.

user228700

Since you have to go and all, I'll ask about his/her name and all later. Thanks again!

user116211

6:56 AM

@KaumudiHarikumar oh, that's MAF. Nice Day.

user228700

7:07 AM

For anyone who'd like to clear my query; the equation $S=Q/T$ (This is about as much MathJax as I know :P) is valid only for reversible processes, yes?

John Rennie

7:18 AM

@KaumudiHarikumar Yes.

yuggib

7:47 AM

@0celo7 you should learn about weaker forms of the axiom of choice, and how todo proofs in a theory with an aforementioned weaker choice (like dependable or countable choice)

If else, you can choose to not care about it, and to not complain ;-P

Secret

8:48 AM

1

Q: A question on an assumption of space-time

"A four-dimensional differentiable (Hausdorff and paracompact) manifold $M$ will be called a space time if it possesses a pseudo-Riemannian metric of hyperbolic normal signature $(+,-,-,-)$ and a time orientation. There will be no real loss of generality in physical applications if we assume t...

general-relativity

Exotic $\mathbb{R}^4$ will surely make things a lot more interesting

user228700

@JohnRennie Okay, so what about irreversible processes? Since they happen too, clearly their entropy increased but how to calculate that increase?

user228700

My textbook has given one way, which is to consider a reversible process that takes the system through the same initial and final states and then, calculate the change in entropy for that process. Since entropy is a state variable, it will remain the same for both processes.

user228700

But I was wondering if there was any other way to do this...

Secret

9:03 AM

Entropy production, which explicitly calculate the entropy being produced in a process, is widely used for open systems (a subject of nonequlibrium thermodynamics, which is way beyond high school level because high school thermodyanmics only deal with equlibiurm, and closed or isolated systems). However under the context of high school thermodynamics, they are basically the same thing as the textbook formulations

We exploit the fact that entropy is a state variable in order to compute it via a reversible path that joined the same points, as the process that happens in an irreversible path does not necessary have nice formula or even analytic formula on relating work and heat

and then of course, there's always the statistical mechanical way to calculate it by counting microstates and using partition functions (whcih is something you don't want to write in a high school exam paper)

typo: entropy being produced in an irreversible process

Free expansion is an example of an irreversible process. Since during that process, the system is far from equlibrium, there are no curves that corresponds it in a PV and TS diagram (where points on them are equilibiurm states)

John Rennie

9:21 AM

@KaumudiHarikumar bear in mind that thermodynamics isn't my favourite subject, so what I say may not be completely reliable. OK, with this warning: the thermodynamics properties like entropy, Gibbs free energy and even temperature are only defined for systems at equilibrium.

In an irreversible process the system is not in equilibrium so the entropy isn't defined. The only way to calculate how much it changes is to let the process complete then let the system come to equilbrium again. Then you can calculate the change because entropy is, as you say, an equation of state.

There is an area called non-equilibrium thermodynamics that deals with systems that aren't in equilibrium. However I know next to nothing about this.

Danu

@DanielSank ο/

So I spent some time camping on an island in Greece---it was so, so great! :D

Secret

@JohnRennie , I also commonly heard in physical chemistry circles that they like to say entropy is a measure of the extent energy is spread. Is this an accurate description. If it is not, what ingredient or subtlety is it missing when compared with the microstate description?

John Rennie

@Secret Honestly, I don't know. I find it hard to get excited about thermodynamics and I've never taken the trouble to learn it really thoroughly.

Secret

ok nvm

2physics

hello everybody

user228700

9:28 AM

@JohnRennie I see.

user228700

@Secret OK, then I won't dig any further into all this.

2physics

new day, new questions.. :D

user228700

@2physics Hi! Yep! :D

Secret

It has been a long time since I last revisited about my quest about entropy and temperature that took place 3 years ago, Kaumudi's questions brought me back to those times

peterh

And new close votes... it is very funny to see, that the number of closed question per day is around 20-24... we have a VtC circle about 5 members, and not more. The probable reason, why they can't grow, that it is the required minimum to close questions, and any bigger VtC circle would start to expel their own members.

user228700

9:35 AM

@Secret Do you enjoy doing thermo.? If not, I'm sorry :P

Secret

I am kinda neutral towards it, but I do spent a lot of time understanding entropy

this has something to do with my preference of making things that rapidly cool down to very low temperatures, and entropy play an important role on that

user228700

@Secret Oh, I see...

Secret

However, compared to entropy, I am even more confused about the notion of temperature, expecially when the concept of negative temperature is learnt, it is no longer easy for me to visualise what temperature actually is

user228700

@Secret You've scared me just a little :P Cut it out. Brr.

Secret

Before your learn stat mech, the books often said that temperature in a gas is the average kinetic energy, but once you get down to stat mech, temperature is something very different, but important

user228700

9:39 AM

@Secret Oh, I see. Sounds interesting :D Scary but interesting.

John Rennie

@KaumudiHarikumar: when you first realise that you understand something it's a great buzz, and that feeling is probably why most of us put so much energy into physics. But as you get to know the subject better you'll sometimes realise that, well, actually it isn't that interesting.

Secret

I still yet to fully grasp temperature, but perhaps when you get there you might be better

John Rennie

For me thermodynamics is definitely in the not interesting category, while e.g. general relativity just gets more and more interesting the more I learn about it.

user228700

@JohnRennie Oh. That's a bummer :/ Have you felt this way a lot?

Secret

I like QM and GR. But GR is kinda more like an affair or a means for me to do scifi stuff. So far my only field that I showed near 100% genuine interest in physics is QM

user228700

9:42 AM

@JohnRennie I see :) Well, I hope it keeps getting better.

John Rennie

@KaumudiHarikumar I wouldn't say it's a bummer. We can't be interested in everything and physics is an enormous subject. Inevitably we all find there are some bits of physics we really like and other bits that we grudgingly learn just to pass exams.

2physics

I want to know the precise definitions and the exact differences between these expressions : model, theory, Over-arching theories, hypothesis, postulate, law, axiom, corollary, principle, fact, assumption, supposition, rule, theorem, lemma, thesis, proposition, proposal, term, statement and sentence... anybody knows a reference which proposes a comprehensive discussion about such definitions and terms?

Secret

> we grudgingly learn just to pass exams

Electromagnetism strikes me as that, and I did the worst on this subject. However I have been trying to understand better so I can change my attitude to it

John Rennie

@2physics that's probably something you should ask in the maths chat. Outside of maths I'm not sure the terms have strictly defined meanings.

user228700

@JohnRennie Yes, I guess that's true. As long as there are bits that get us jumping-up-ans-down-can't-control-it excited, it's worth all the effort, no? Would you say so?

John Rennie

9:44 AM

@Secret YES!! I found electrodynamics a huge pain at university. It seemed to be just an excuse for examiners to set absurdly complicated problems with weird arrangements of charges and currents.

2physics

@JohnRennie physics uses a mathematical logic to interpret the phenomena, doesn't it?

John Rennie

@KaumudiHarikumar It's always worth the effort for anything you enjoy doing! :-)

user228700

@Secret Oh, really? This might be 'cause I'm just in high school and haven't learned about the vast majority of physics left to learn but I quite enjoy electromagnetism.

John Rennie

@2physics yes, but physicists are famously careless about using maths - it drives mathematicians to distraction! :-)

Secret

The problem is that, gravity and electromagnetism rule our worlds in basically all scales, and it is just too important to left it out. The nature of light is needed to understood GR

@KaumudiHarikumar I won't say I hate it, but I realyl REALLY suck at it both maths and intuition wise

which is an irony because I love matrices and vector stuff

user228700

9:47 AM

@Secret Oh, I see. This is the first time I'm talking to professionals IRL and I gotta tell you-it's great to know that you're not gonna be great at all of physics :)

John Rennie

@KaumudiHarikumar: you won't find out what areas of physics you really like until you study them, so the best course is to study everything with as much energy as you can. I ended up working for 12 years in an area that I didn't even know existed when I was an undergraduate!

Secret

In fact the very fact we humans mastered electromagnetism first of the 4 fundamental forces means our everyday life is surrounded by it, whcih is why a uni simply cannot cut away this course

user228700

@JohnRennie :D I see. It's so great to hear from you guys on this. If you don't mind me asking...when did you develop a passion for science?

2physics

but I used to think the only difference between mathematicians and physicists is that physicists are "physicist" and "mathematician" at the same time while mathematicians are always just "mathematician".. :D

user228700

@2physics Here we go again :P 'Bout how many times a week do you guys discuss this?

Secret

9:50 AM

Mine dates all the way back to childhood. The trigger can be traced back to chemsitry experiments, which like most, I am fascinated by the colors explosions etc., but deep down, you might say I am actually born to like science because since childhood, it has been my guiding light throughout my life

I know from very early what I am passionate on and what I don;t thus plannign for the future is very straightforward for me, with the support of my parents and friends

John Rennie

@KaumudiHarikumar I can't remember a time when I wasn't fascinated by science. Even as a small boy I wanted to be a scientist. And I got to be one - who says dreams can't come true? :-)

Secret

My physics passion remains dormant because of not understanding the maths, until my grade 11 physics teacher who is a very good teacher, cause my interest on particle physics and quanutm mechanics to explode

2physics

@KaumudiHarikumar I haven't discussed it and didn't know about such debates ever..

user228700

@JohnRennie That is so wonderful to hear :D

Secret

and since then, my chemsitry and physics interest are almost the same level, with only chemsitry being slightly higher because of wet labs countering the dryness of analysing databases and not able to touch a e.g. electron with bare hands

user228700

9:53 AM

@Secret I see :)

Secret

This is one reason I plan to do a chemistry phD and not physics, because I need to get my hands dirty in order to not be bored out by excel spreadsheets of data

and there is nothing like wet labs in physics

user228700

@Secret Haha, I see :)

John Rennie

@KaumudiHarikumar I saw your question yesterday about what career to pursue. I can't offer any advice because my knowledge of careers is so badly out of date. What I can say is that if you do what you love it will be a great life, even if you don't make that much money :-)

I speak from experience!

Secret

@JohnRennie This is the path I have always been taking since childhood

user228700

@JohnRennie :) Thanks sir.

user228700

9:56 AM

One more...as a scientist, was it ever possible to devote time for anything else?

John Rennie

@Secret the three years I spent doing my PhD were the happiest of my life. I know that's a cliché but it's true. To be studying what i loved without having to worry about anything else was just fantastic.

2

Secret

what makes what I said about dry excel spreadsheets an irony is that my chemsitry professors, basd on the personality tratis they observe about, me, all agree that I actually performed better at computational chemsitry. However my interest on it is not as high as wet labs

and in fact, I am clumsier in synthesis

user228700

@JohnRennie Wow, I definitely need to tell that to my mother! She needs to hear about happy PhD stories too :P

Secret

Right now, my future phD project will be a combination of computational and synthetic chemsitry, thus I can satisfy my demand of deep understanding on how things happened, while haviing a wet lab component to counteract databse dryness

John Rennie

@KaumudiHarikumar we hear stories about unhappy PhD students, but I suspect the majority, maybe the vast majority, of PhD students really enjoy their time doing a PhD.

user228700

9:59 AM

@JohnRennie Oh, wow, I see...

2physics

@JohnRennie you know sometimes the conditions and constrains of peoples life , prevent them from pursuing and doing what they love..

Secret

I can asure that, I have at least 10 phD friends randomyl met in other unis that they all say phD is very fruitful

John Rennie

@2physics yes, and I consider myself fortunate that didn't happen to me.

2physics

great

user228700

Just one more; is it possible to devote time to any other small passions alongside science? I suspect that the answer is "Not really" but still...

John Rennie

10:00 AM

Depends what you mean by small passions.

user228700

@Secret Oh, that's great :)

Secret

same. In fact, when counting the numebr of coincidence I have throughout my life, there are at least 3 instance I would have died had it not happened otherwise

John Rennie

I used to spend a lot of time canoeing and reading science fiction while I was doing my PhD.

user228700

@JohnRennie I don't really know. I mean, the word "scientist" brings up the mental picture of forgetful, insanely intelligent people stuck in a lab all their lives to many people's minds.

Secret

@KaumudiHarikumar Well ,if you look at my AMA page, I have a lot of small passions coming and going every now and then. For example, I started to become interested in nonscience on 2014, and art start to grow on 2015. Spirituality started a bit later

@KaumudiHarikumar This is just outdated, your mother should really do some google search on that. Scientists who worked in fields (such as geologists, ecologists etc.) are very dynamic people and like the share about their passions

John Rennie

10:04 AM

@KaumudiHarikumar I bet if you asked the site members here you'd find that all of them enjoy extracurricular activities. You can't work at one thing all the time - you'd burn out.

user228700

@JohnRennie Oh, that's nice!

user228700

@Secret Yes, I realize this :(

user228700

@JohnRennie That's definitely relieving.

user228700

My mom read an article online. Well, not a real article. Just this answer on quota:

user228700

quora.com/…

user228700

10:07 AM

Since then, she's been really skeptical.

2physics

@JohnRennie why do you learn and pursue physics? can you answer it shortly in one sentence?

John Rennie

@KaumudiHarikumar If you want to be a scientist in a university working on physics then you'll be paid a lot less than your engineering friends who work as consultants. I'm afraid that's true.

But you'll be having a great time.

user228700

@JohnRennie But certainly, I will have the time to start my own family and all..?

John Rennie

Honestly I wouldn't worry about it.

user228700

My parents are also worried that perhaps I won't and that all other aspects of my life might just fall away.

John Rennie

10:13 AM

You may end up deciding you don't want to remain in academia.

user228700

@JohnRennie But I'm not! I've got to find ways to convince my parents! And I sure will too but I don't quite know how yet.

John Rennie

After my PhD I realised I was more interested in applied science i.e. solving real problems. So I went to work for a large company.

user228700

@JohnRennie Oh, you can do stuff like that? That's awesome!

John Rennie

This one in fact though I worked at the UK research lab not in India.

In the UK, and I'd guess it's similar in India, companies are desperate for clever motivated people.

user228700

@JohnRennie That's so cool!

John Rennie

10:16 AM

If you complete a PhD you've proved you can work hard and succeed at something hard and that makes you very attractive to any technology company.

user228700

@JohnRennie Yes, I guess that's true :)

user228700

Thanks for talking to me about this!

John Rennie

So if you want to stay at university and become a professor you can do that, or if you prefer a more applied career you can work in industry.

The thing is that at your age you simply don't know things are going to turn out.

So I wouldn't worry about it. If you find something fascinating then study it - even if it is thermodynamics :-)

Secret

For me, I am a wavefunction, and I don't like to stick in one place or another for too long. Thus my future path might be a professor along with some ties with research companies

I especially see potentials in small sized private sectors, which complements well with the large government companies

user228700

@JohnRennie Yep. I'm sometimes worried that it's disastrous that I haven't got all this stuff figured out yet. The whole career situation is entirely different in India and we're forced into deciding exactly what we want to do for the rest of our lives. It's spectacularly screwed up.

user228700

10:20 AM

'Course, we don't all end up with what we chose when we were 17 but still, it's tortuous.

Secret

Not being able to change where to go is like the most boring thing to me. I even once said to my chrsitian friend that I like neither heaven nor hell, thus I will not stay forever in either place

user228700

@Secret Haha, that's an interesting stand!

Secret

Eternally unchanging of something is the most boring thing. My meaning of life is to NOT get bored, and my purpose of life is to unlock the mysteries of nature

user228700

Anyway, I will speak with you all later! Thanks for sharing and speaking to me about all this. Bubye!

2physics

I think choosing to be a physicist or engineer(to choose future career or future choices which sb is faced with in life), so much depends on sb's values and character actually. because it depends on what you find enjoyable and fascinating, what you find worthy to spend your time on... family and school have a great impact on shaping this in people.

it highly depends and how and who we are

our values have a great and important role here. It's the values of our lives which leads us to find and pursue our way through future..

Secret

10:32 AM

0

Q: Can entropy be regarded as energy dispersal?

In several answers here the claim has been made that thermodynamic entropy can be regarded as energy dispersion. See, in particular here, and here and here. This is apparently the pet theory of a chemistry professor, Frank Lambert. Apparently (at least according to his Wikipedia page) this defini...

thermodynamics statistical-mechanics entropy

FINALLY after 2.5 years, a counterxample to "entropy is a measure of energy dispersal"!

2physics

ladies and gentlemen I'm done with my speech :D:D

@Secret your messages here adhere Second law of thermodynamics :D

Secret

Yeah, and now I am wondering how to construct a system so that the probability of accessing the microstate is the thing that changes, but energy remains constant. I am interested on seeing what happens in such isoenergetic systems

That is, unless I a mistaken by what I read, if there is a process that can increase the number of microstates without any change in energy, then we might be able to find a way to bypass the inefficiency probelm, while keeping the second law happy

Ultimately I am not very interested in the energy inefficiency problem, I am just trying to find more ways to make heat flow from cold to hot

Xasel

Hello everybody here

2physics

@Secret does increasing the number of microstates increase the amount of entropy necessarily?

@Xasel hi and welcome

Xasel

I have one confusion and I want your guys opinion

2physics

10:43 AM

what's that

Xasel

How would you guys go about proving that U(x) of potential energy system is always differentiable

Secret

well, if you increase the number of microsate that can be accessed by the system, not just increase the microstate (that won't really do anything if the system cannot access it). The issue is then how to prevent energy from flowing into those microstates as the system access them

Xasel

BACKSTory:Recently I was skimming thorugh physics text and encountered a defination of force as

Secret

However I might have made a conceptual error someehwre, I need to sit down and think carefullly about the stat mech of this

Xasel

F = -dU(x)/dx

In calculus we were taught about class of functions that are continuous everywhere but differentialble no where

it implies thta U(x) should alsway be differentiable

Secret

10:46 AM

The point is, if we can find a process that increase the system's access to microstates without a heat tranfer, then we can break the situation where the second law must imply unavaliable energy from irreversible prcoesses

Xasel

but how should I go about proving so @2phys

@2physics

2physics

@Secret I dunno .. what's your idea about @xasel question?

Secret

@Xasel I don't think you can really prove it, it's just because nearly all physical systems have forces that are bound, and does not suddenyl disappear or suddenly change for some values of x, thus the potential has to be differentiable

2physics

@Xasel your question is why and how we consider the U(x) function differentiable , right?

Xasel

yeah

2physics

10:49 AM

that's a good question

Xasel

intuitionally it seems right but how should I rigorously prove it mathematically

2physics

well , what is potential?

Xasel

hmm..energy possesd by an object by the virtue of it's postion or configuration

2physics

well let me convert your question, why the distance is differentiable?

Xasel

hmm...interesting...well enlighten me on this

@2physics:As of know I don't know the answer,,,please enlighten me more on this

2physics

10:57 AM

a differentiable function of one real variable is a function whose derivative exists at each point in its domain.

I also don't know and and I'm trying to figure it out.

When a function is differentiable it is also continuous. But a function can be continuous but not differentiable. For example the absolute value function is actually continuous (though not differentiable) at x=0.

(googled it)

Xasel

yeah I know that but how will I go about proving that derivatives exist

les ask this on STkSE

2physics

I think we need to consider it to be partially differentiable to be able to differentiate it

yup you also can ask it on SE.

Secret

@Xasel Actually I have a counterexample of this: In classical electromagnetism, the electric potential at the electric charge is finite, continous but not differentiable (U(x) forms a cusp there )

Xasel

@JohnRennie:Sir can enlighten us on this

2physics

his question is about why we consider the potential energy function differentiable

Secret

11:03 AM

because most physical systems the forces don't exhibit jumps and are bounded

Xasel

how will you prove it mathematically with rigour

Secret

You cannot, as i said above, we have counterexample to "U(x) must be differentiable". But in all real world systems, singularities tend to get smooth out one way or another, thus ensuring continuity

it's a physical observation, not a mathematical reasult

2physics

I don't think we can mathematically prove that distance is discrete or continues.

Xasel

Well then can you contruct the mathematical model which approximated to the observation

Secret

Our mathematical model that fit such observation thus require the potential to be differentiable, otherwise we would have seen jumps everywhere and other weird things such as objects suddenl turnign around with no reason

@2physics That boils down to whether spacetime is discrete

That is, U(x) must have the property that it must be differentiable in order to account for the behaviour of the forces we observed (at least for classical mechanics)

Xasel

11:10 AM

0

Q: Why we consider potential energy function U(x) differentiable?

Recently when skimming through my physics-text I encountered an interesting defination of Force $$F(x) = -\frac{dU(x)}{dx}$$ We were taught that some function are continuous but not differentibale. So for the force to exist U(x0 to be differentiable .So how can we prove that U(x) is differentiab...

quantum-mechanics classical-mechanics mathematical-physics potential

2physics

suppose $[U = mgh]\$ which results in.. $[F = \frac{{dU}}{{dx}} = \frac{{dmgh}}{{dx}} = mg\frac{{dh}}{{dx}} = mg]\$ therefore maybe your question can be applied to why we consider the distance diffrentiable..

Xasel

hmmm...lets see if anybody can enlighten us on this

2physics

it's considered to be locally differentiable

$[F = \frac{{dU}}{{dx}} = \frac{{dmgh}}{{dx}} = mg\frac{{dh}}{{dx}} = mg]$

@Xasel anyway, good question. it applies to many other physics equations. applies to differentiablity of physical quantities.

Xasel

11:28 AM

@2physics:lead to this by @Qmechanic: physics.stackexchange.com/questions/1324/…

@2physics:physics.stackexchange.com/questions/1324/…

2physics

@Xasel I don't understand the question and its anwers lol :D

Xasel

C^infty means a smooth function

hmm..no answer till now

2physics

@Xasel I think such questions should be answered in such a way that be understandable for a student which is just has started the university.

Xasel

@2physics:better if answered for high-school students like me

2physics

yup

Xasel

11:38 AM

are you univ grad?

2physics

I'm an engineer actually

Xasel

in which field?(mech/EE/comp)

2physics

Electrical eng

Xasel

that the wonderful field...well wht is the most interersting thing about EE for you

2physics

the most interesting thing for me is to understand the physical laws behind phenomena and then using them; I think a better engineer is who knows more about physics more. EE is nothing but of physics

Xasel

11:44 AM

a better engineer knew how to manage hte tradeoff of physicals phenomencal most optimally and effectiviely

2physics

yup, it needs to know those phenomena as well.

Xasel

what's the most interesting phenomena for you

Time traveler

Hi! I have been told to ask one question here since it cover a ground too broad. My question is:

2physics

there are so many things that I find interesting, such as fields and waves, electricity , etc

Time traveler

what is the general procedure to obtain a theory from two theories? How do you unificate the math of two theories?

2physics

11:49 AM

@Timetraveler I have no idea actually :D but you can ask other people around here, probably they can help

@Xasel what do you wanna study at the university

Xasel

@2physics:Can't decide sometime my interest dangles to maths other to physics or to EE or music

2physics

cool

Xasel

you experienced same?

2physics

main theories and basic ideas which E engineers develop are most always proposed by physicists..

no, actually when I was at school I realized that I like it to learn more about electricity and then I decided to choose Electronics to study

while pursuing my BSc, at the last years I realized that I'm more interested in communications engineering

therefore I took some courses such as fields and waves, filters, and more stuff on communications..

when I took those courses I realized that yes! it was more like what I always loved to pursue and learn . so I decided to continue as a communications eng in EE

and for my MSc I changed the fields of study from electronics to communications

after a while I thought maybe it was learning physics, what I've always been looking for

after getting my MSc degree I've just decided to study physics .. or maybe sth between EE and physics which yet I don't know what it is. lol

ACuriousMind

@Timetraveler That's not a well-defined question. There's no process by which you could take any two theories and produce a third one from them.

Sometimes you can build a new theory that contains your original ones, sometimes you can't, but in neither case is there a recipe for it

2physics

12:04 PM

maybe Electrical Engineering (with a minor in Physics) is what I'm looking for

@ACuriousMind Hi what's your idea about this question and can you propose an answer which may seem understandable to a high-school student?

0

Q: Why do we consider potential energy function $U(x)$ differentiable?

Recently when skimming through my physics-text I encountered an interesting definition of Force $$F(x) = -\frac{\mathrm dU(x)}{\mathrm dx}$$ We were taught that some functions are continuous but not differentiable. So for the force to exist $U(x)$ has to be differentiable. So how can we prove th...

classical-mechanics mathematical-physics potential-energy differentiation

user116211

@2physics Did you read yuggib's post Qmech linked to?

ACuriousMind

@Xasel The point is that we define a potential as the function associated to a conservative force (which we may define without recourse to differentiability e.g. by saying its line integrals only depend on the start- and endpoints) such that $F = -\nabla V$. A potential is therefore differentiable by definition, and there is nothing to prove (except that that function exists for every conservative force, for which e.g. see Wikipedia)

user116211

@Secret Not a counter-example, I would say; see the whole comment section.

ACuriousMind

@2physics I just answered it in my last message. And now I guess I'm going to put it there as an answer

2physics

@MAFIA36790 yup. I think those are two different questions in a student(like me)'s view. lol

user116211

12:09 PM

Nevertheless, entropy survives without such interpretations also. However, the dispersal is much more lucid than the term disorder which seems to be vague quite some times.

2physics

@ACuriousMind he is a high-school student and maybe doesn't know about gradient of a function and line integrals..

user116211

@2physics Every introductory under-grad book does cover that; it's not a difficult topic for a high-schooler, IMO.

Secret

@MAFIA36790 I am trying to find a system that can increase microstates it can explore in an interaction, but without any energy change. That will mean the microstate in question cannot stem from a spatial degrees of freedoom (thus the object before and after the change in the number of accessible microstates, will have the same energy in basically the same spatial distribution), just more degenerate states it can hop into). If

ACuriousMind

@2physics Well, but the answer to why potential energies are differentiable involves these things. Am I supposed to not mention them because someone might not yet know what they are?

2physics

@MAFIA36790 I personally remember that the first time I met those strange vector calculus operators was in my first year of university

Secret

12:15 PM

that is not a counterexample, it means a change in number of accessible microstates must acompanying with a change in the energy distribution in the system, thus there is no way to have $\frac{\partial E}{\partial S}\neq 0$ without both the change in E and S being nonzero, which means the second law must imply the loss of efficiency

...uh wait a second, if E is independent of S, then the partial derivive wil be zero... Ok I better re read what exactly is temperature, I think what I just said does not really make much sense

2physics

@ACuriousMind well we'd be grateful if you first explain it in a rather convincable way to a high-school student, and then develop the answer using those useful tools. I dunno if it's possible or not, but I believe it's a question which may be asked by any high-school student which has studied differentiating and physics problems associated to it

user116211

0

Q: I How to turn model of Schrödinger's Equation 2D?

I am a student in the Netherlands, currently 17 years old and at the end of my 'middelbare school', meaning that next year I'll be a bachelor student at a university. I am doing an extended essay/research thing that is custom you do in your last year, with a friend of mine. We picked the topic o...

wavefunction schroedinger-equation probability models

user116211

Is it on-topic? At the first sight, it seems to be concerned with code and not any concept; but I've not read that.....

peterh

12:31 PM

@MAFIA36790 Most of the physicists code a lot as their daily work...

@MAFIA36790 The most awesome job I can imagine would be to work on such software which solves physical problems numerically.

user116211

@peterh Well, that doesn't make it an on-topic, is it? Whatever it is, I have not commented as I'm not sure.

peterh

@MAFIA36790 Only this doesn't, but the current reality is that programming is an important part of the physics.

user116211

@peterh This is not relevant in discussion; the criteria would be whether there is any conceptual query; if there is any dearth of conceptual query, then it has to be closed; I'm waiting for what others think.

peterh

@MAFIA36790 I wrote a comment to ACM below the question: "I don't think he would need help in a programming problem, he needs help in some deeper physical problem. Solving real physical problems numerically, we can experience that there are a lot of problems which looked somehow unimportant thinking purely in theory, and these are not trivial bugtracking/bugfixing tasks, but about the algorithmization of the physical problem. I suspect some similar could be also here."

user116211

@peterh Such response should come not from you, but from the OP himself. Currently, IMO, it dearths conceptual query; however, OP can edit it; I've no problem with that.

peterh

12:43 PM

@MAFIA36790 Actually, what I've found: if you want some complex thing, which is purely programming and not physical (for example, using the data SE to get some non-trivially reachable data from it), one can experience, after some cycles of code-try-track-fix, it will work. Doing the same for any physical thing, I've found that it doesn't work, because only making good-working code is not enough, often there are also problems from the modelling side.

ACuriousMind

@2physics If I explain every technical term I want to use before using it I wouldn't ever get anything done. All those terms are standard enough that googling them will lead to plenty of resources explaining them.

Balarka Sen

@2physics The point is conservative, which means "on my vector field, integrating along paths only depends on the two endpoints and not on the path itself", is equivalent to having a differentiable function such that the vector field is the gradient of the function. This is by definition the potential function.

peterh

@MAFIA36790 Last time I've made an aerodynamical simulator for airplane drones. From the programming side, it is trivial: there is a main loop, there are the inputs from the fin controls, there is the state of the plane, and calculate its state after the next delta-t. The result was a plane, which flied as it had to, it was even stable, but somehow it has flown very clearly not as a normal plane had to fly. For example, in a weightlessness situation, its angular velocity was unstable.

Balarka Sen

If you want to understand why those two notions of conservative fields are equivalent, it's essentially this. Integrate your vector field along a path from origin (or any fixed point, really) to some point $p$. This is independent of the choice of the path due to the nature of the field, hence is actually a function of $p$.

You should be able to convince yourself this is really the potential function aka antiderivative.

peterh

@MAFIA36790 But I've played also a little bit with leightweight Schroedinger calculations with Matlab. What I experienced: 1) with base matlab (without any extensions), even the most trivial calculations require a lot of computing capacity 2) the wavefunction wanted somehow nearly never to look as it should have to. Although the formulas and the code was correct. Why?

Balarka Sen

12:50 PM

The proof is slightly technical but all you need to do is to choose appropriate polygonal paths (which, again, you can do by conservatism of the field), and then writing down the definition of derivative.

peterh

1:01 PM

@MAFIA36790 ...because I didn't have the needed understand of the basic QM (at the time) to understand from my code, what it does from a physical viewpoint. I could make it bugfree as program easily, but I couldn't make it bugfree from physical view. I.e. the worst bugs solving numerical problems are not the programming bugs, they are the physical bugs.

John Duffield

1:34 PM

@3075 : mathematics doesn't follow the scientific method. As such, it isn't a science.

Space Otter

science
noun
the intellectual and practical activity encompassing the systematic study of the structure and behaviour of the physical and natural world through observation and experiment.

Yup that's no maths

experiment being the key word

John Duffield

@BalarkaSen : see above. This is not to say mathematics lacks rigor or value. Mathematics is a vital tool for physics, we can't do physics without it. But it isn't what physics is.

Balarka Sen

I never said I disagree with all what you said there.

Space Otter

::secretly whispers to JohnD:: Psssst... the definition for maths has the word "science" in it... o.o

user116211

@SpaceOtter Mathematics is beyond science.

Space Otter

1:45 PM

In some ways

John Duffield

@SpaceOtter : what definition?

Space Otter

The beauty in mathematics is that it a mathematical proof can be given, beyond a doubt. Such is the nature of maths.

However, it is scientifically accepted that nothing can ever be proven by science and statistics, such is the nature of scientific method.

@JohnDuffield Google

So with maths, we can work out the workings of a hypothetical universe and all it's laws beyond a doubt, however to test if our universe behaves the way we have predicted we must experiment, and at that point we are doing physics.

John Duffield

@SpaceOtter : I googled. The first thing I found is this: "Update 2015: After much thought, I no longer think math is a science. Science must be empirical, meaning it must be based on observations of nature, and it must be potentially falsifiable by new observations of nature..."

@0celo7 : it's because I'm the expert now.

Space Otter

English is an ever evolving language and so the terms used in defining something can barely be used as grounds for a debate such as this. I totally agree with you, Maths is not a science as defined by the nature of the scientific method.

Balarka Sen

Most of the discussions here are either metaphysics/metamathematics, philosophical, about video games, or 0celo7-conversations.

Space Otter

1:59 PM

We have moved on from Aristotelian logic, and so we cannot assume our mathematical findings to be meaningful to our study of the universe.

Without mathematics we are crippled, and without physics it would all be quite meaningless.

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