# Second Law of Thermodynamics - Postulate or Provable?

1. Dec 17, 2009

### boderam

Second Law of Thermodynamics - Postulate or Provable??

I know that the 2nd law of thermodynamics is considered an experimental fact, but I have yet to find any sort of structure when I see a discussion of the law via proving it as a theorem. I know there are mathematical formulas that express the law, but are any of them derivable in the way that the conservation of momentum is derived from Newton's laws, i.e. from first principles?

2. Dec 18, 2009

### atyy

Re: Second Law of Thermodynamics - Postulate or Provable??

The second law as a statement about entropy is derived from the first law of thermodynamics, and the Kelvin and Clausius statements of the second law.

3. Dec 18, 2009

### boderam

Re: Second Law of Thermodynamics - Postulate or Provable??

so it is provable, assuming the kelvin/clausius statements are derivable from first principles as well...do you have a reference for this, every book and website i have come across so far doesn't have a real mathematical sort of proof for it, a lot of handwaving i find very frustrating.

4. Dec 18, 2009

### atyy

Re: Second Law of Thermodynamics - Postulate or Provable??

The Kelvin and Clausius statements are "common sense" postulates that seem to summarise lots of experiments - you cannot transfer heat from cold to hot without doing work. From those statements one can get to the mathematical statement that entropy stays the same or increases. Try Mehran Kardar's notes http://ocw.mit.edu/OcwWeb/Physics/8-333Fall-2007/LectureNotes/index.htm [Broken].

Last edited by a moderator: May 4, 2017
5. Dec 18, 2009

### ideasrule

Re: Second Law of Thermodynamics - Postulate or Provable??

The second law of thermodynamics can be derived from the fact that heat flows from objects with higher temperature to objects with lower temperature. That's much more of a "first principle" than either Newton's laws or the conservation of momentum.

6. Dec 18, 2009

### Gerenuk

Re: Second Law of Thermodynamics - Postulate or Provable??

I find the proposals here old-fashioned. Nowadays, general entropy can be applied just about to any abstract system like a deck of cards. And then it is not clear how to define "heat", "reversible processes" and "temperature" at all! Also I wouldn't call it a postulate as many people think the second law at macroscopic scales at least most likely won't be violated, but for small systems it may well be. So the derivation that I find most natural is saying that
"assuming all states are equally probable, a system will most likely go from a small set of states to the larger set of states". This is intuitive and can yield the equation for entropy:

7. Dec 19, 2009

### heldervelez

Re: Second Law of Thermodynamics - Postulate or Provable??

The second law is not 'general'.
It has to be considered in systems that are 'closed systems'. This is an useful aproximation, as there exist no real closed systems, afaik.
There exist a range of applicability.
Choose a system, considerer it as aprox closed, define temp/entropy then yes.

The gravitation provides 'free energy' (is it ? I think not!) that changes everything.

From what we know the evolution of the universe at large is contrary to the 2nd law:
From a well mixed begining, the gravity starts to build more and more complex structures as time goes by: superclusters, clusters, galaxies,stars, planets, life, humans, brains.
Some radiation is lost, but stuctures are building up and 2nd law does not apply.

Then, to prove the 2nd law we have to clarify terms, limits of application, etc...

Last edited: Dec 19, 2009
8. Dec 20, 2009

### xxChrisxx

Re: Second Law of Thermodynamics - Postulate or Provable??

All law's of thermodynamics are considered always because there has never been a situation where they have been wrong.

By the strictest definition (a mathematical proof) I don't believe there is one (although i'm not sure as my maths isn't as stong as it should be). For puroposes of use, it's proved correct by NEVER having been proven wrong.

9. Dec 20, 2009

### Gerenuk

Re: Second Law of Thermodynamics - Postulate or Provable??

I definitely agree that laws of thermodynamics almost surely will never fail, because thermodynamics is about system with a huge number of particles.
Entropy in general however could be applied to non-thermodynamic system which have far less particles. Then it could fail more often.

10. Dec 20, 2009

### xxChrisxx

Re: Second Law of Thermodynamics - Postulate or Provable??

The OP was talking about the classical laws of thermodynamics, specifically mentioning the 2nd law.

I don't think he really wanted to consider statistical thermodynamics.

11. Dec 20, 2009

### boderam

Re: Second Law of Thermodynamics - Postulate or Provable??

i posted this in the classical section because i was under the impression the 2nd law of thermodynamics IS classical, if there is a non classical version of it i would be interested in knowing about it though, especially a "mathematical" proof of it. i watched some of susskind's statistical mechanics lectures on youtube, made it to the boltzmann distribution and the mathematical definition of entropy, but the lecture on the 2nd law was very unclear to me. i am looking for a very clear/detailed mathematical proof, if there is one.

12. Dec 20, 2009

### xxChrisxx

Re: Second Law of Thermodynamics - Postulate or Provable??

It is, Gerenuk was giving a statistical approach to entropy, which isn't quite the same thing.

There is no mathematical proof of the laws of thermodynamics.

You either have empirically derived formulas, that have never been proven wrong. OR A statisticial method

13. Dec 20, 2009

### boderam

Re: Second Law of Thermodynamics - Postulate or Provable??

so how are the two related then? mathematically the statements of the 2nd law look very different from statistical mechanics to classical thermodynamics, is there a way to "convert" one to the other via relating abstract states of the system to temperature? i'm getting a bit confused because they seem to be two different subjects yet they are both claiming a 2nd law about entropy. and saying there is a statistical method for the 2nd law, this does mean there is a derivation from first principles? this is what i am looking for. thank you for the discussion so far.

14. Dec 20, 2009

### Gerenuk

Re: Second Law of Thermodynamics - Postulate or Provable??

He was looking for a proof and the only proof I find reasonable is the statmech one. Of course one could derive the 2nd law from statements about heat flow, but that is really a shift of definition only. If one wishes to derive the law while ignoring the microscopic particles of thermodynamics, then this isn't possible from logic alone I believe. Then the dissatisfying answer is "no, you cannot prove the second law from logic".

I suppose classical mean "non-quantum-mechanical". The second law isn't even really physics. It is a general probabilistic statement about general systems. So it is classical.

Ask questions if you wish. However, I believe most lectures just define entropy and don't prove why it is increasing.

Have you looked at my earlier post I referenced?
To understand the increase of entropy I suggest the following computer experiment:
Your data is an array of numbers from 1 to N. For example "7,1,5,9,8,1,2,3,5,1,8,9,2" or so. For a quantity called entropy you calculate $S=-\sum p_i\ln p_i$ where $p_i$ is the fraction of the numbers which is equal to number i. The "physical time evolution" will correspond to changing one number at a random position to another randomly chosen number. Now you can observe what happens to the quantity S and you will find that is almost always increases. This law is more precise the more and larger numbers you deal with.
If you understand this computer program, then you understand why entropy increases. Stating this in mathematical probabilitic terms this is a proof from first principles.
One more point: you shouldn't start with a randomly chosen sequence of numbers, but a self-made structured one. Otherwise you most likely start the already a maximum entropy state.

All of the thermodynamics laws can be derived from statmech, but not vice versa.
In the same forum link that I posted above you can find how to derive thermodynamics from the statmech definition.
(it might useful to have heard a lecture though)

15. Dec 20, 2009

### xxChrisxx

Re: Second Law of Thermodynamics - Postulate or Provable??

I'll say before I carry on, I am not too hot at maths, and utterly dreadful at statistics.

*snip - probably wrong*

Statiscal methods are a wider method of calculating (or estimating as it's statistics) entropy in a system. They are different becuase they look at different things.

Classical method is a system (macroscopic view). So if we had a box of gas, we could tell you thermodynamic information about it. Such as entropy, etc.

Statistical methos is a microspoic approach that can be used to find a system value: this metod will use probability to find a distribution and average entropy for every single molecule. These are then summed to get a system value.

The classical method had no way to predict the value for a single molecule, but statistical thermodynamics does.

As far as I can remember anyway. I'm kind of relying on someone better to correct any mistakes, which im sure they will jump at :P

EDIT: You beat me to it :P

16. Dec 20, 2009

### hkyriazi

Re: Second Law of Thermodynamics - Postulate or Provable??

Boderam, I'm no expert myself, but I've researched this question thoroughly. There is no formal proof of the law of increasing entropy, at least as it applies to an ideal gas. Boltzmann himself, after it was brought to his attention by one of his students, stated that it, and all its derivatives, are based on the assumption of molecular chaos. Maxwell made the assumption implicitly while deriving his distribution of molecular speeds in an ideal gas.

It was a reasonable assumption, and seems to hold true for all the gases we've ever dealt with.

17. Dec 20, 2009

### Andy Resnick

Re: Second Law of Thermodynamics - Postulate or Provable??

Setting aside the fact that the conservation of momentum not really "derived" from Newton's Law F = dp/dt, the second Law of Thermodynamics is based purely on observation- heat never flows from a cold body to a hot one. Microscopically, the entropy (as do all physical properties) can and does fluctuate (the fluctuation-dissipation theorem).

To be clear- macroscopically, the Second law has never been observed to be false, while microscopically it has been observed to be violated (for short times):

http://www.aip.org/enews/physnews/2002/598.html [Broken]
http://prola.aps.org/abstract/PRL/v89/i5/e050601

Last edited by a moderator: May 4, 2017
18. Dec 21, 2009

### Gerenuk

Re: Second Law of Thermodynamics - Postulate or Provable??

That is an interesting reference
Is there a theory that calculates how much the entropy will fluctuate once it reaches its maximum?
I think usual parameter fluctuation theory assumes that entropy is at a constant maximum.

19. Dec 21, 2009

### Gerenuk

Re: Second Law of Thermodynamics - Postulate or Provable??

For demonstration I wrote this python program. Here is the program
Code (Text):

from __future__ import division
from random import randrange
from math import log
import sys,os
P=[100,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0] # all 100 particles in the first state

f=file("entropy.dat","w")
tot=sum(P)
for a in range(1000):
for b in range(10):
while 1:
j=randrange(len(P))
if P[j]>0: break
while 1:
i=randrange(len(P))
if i!=j: break
P[i]+=1
P[j]-=1
S=-sum((p/tot*log(p/tot) if p>0 else 0) for p in P)/log(len(P))
print >>f,a,S
f.close()
os.system("gnuplot -persist - <<< 'plot [0:*] [0:1] \"entropy.dat\" t \"\"'")

and the output is attached.

So basically entropy has nothing to do with physics or dynamical processes. It is a purely probabilistic statement equivalent to saying "for coin tosses the most probable ratio of head to tails is 1:1".

#### Attached Files:

• ###### entropy.jpg
File size:
18.1 KB
Views:
158
20. Dec 21, 2009

### Andy Resnick

Re: Second Law of Thermodynamics - Postulate or Provable??

I'm not sure... it's somewhat outside my expertise.