# Second Law from Statistics

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1. Nov 3, 2014

### Quentin_C

Hi all I hope you can help me with the statistical origins of the Second Law. I cannot find anything that mathematically proves that order from disorder is impossible only improbable.

Leading me to think that a system (Kelvin engine) that allows order to be created from disorder (work from ambient temp) is possible if it is probable.
To be more specific what prevents fundamentally prevents unidirectionality.
Can someone assist to shoot this down? Help me move on !!!
Working down from
• Entropy a statistical law which says that it is highly improbable that an equilibrium state will occupy anything but the most probable (disordered states)
Supports
• Second law of thermodynamics states that the arrow of time will only ever allow disorder to evolve travel from the current (ordered) state to a more probable (disordered) state
Supports
• Claussis statement heat will never travel from cool to hot (without work) as the resultant equilibrium microstate (of the system) is not the most probable therefore entropy would decrease.
Supports
• Kelvin Statement – heat can never be extracted and made to do work from a single reservoir heat source as this is allows and is equivalent to/and would allow the Claussis statement.
So really all the Kelvin statement says is that a single reservoir source of work is not allowed as it is improbable that you can create order from disorder

Or framed another way
Isn’t the equivalence argument cyclical as if the imagined Kelvin Engine is generated from an original statistical bias (a way to create order from disorder) then all the assumptions proving this are baseless as we are just using a the absence of statistical bias to prove we can’t have a statistical bias ?
This is the assumption that I ask you to suspend – Assume that a fixed number of microstates all are equally probable and all microstates encourage order.
The scenario I propose (however far-fetched) is one where order can be created by movement between fixed equally probable microstates at static** temperature–
1. Fixed Microstates - Entropy does not increase nor decrease.
2. Equally Probable – the equilibrium microstate is the most probable.
3. Static Temp - as work (order) is extracted that TH constantly resupplies TC so they can be assumed fixed and are so identical over time
What I am chasing is a mathematical proof that prevents such a statistical bias existing creating work (order) from random movement (disorder) and nothing changing over time as Q enters the system to balance W removed.
Can anyone help??
Before we bring them up the to me Smoluchowski trapdoor , Brownian Ratchet all only prove that if all microstates are equally probable then no net work can be extracted as states that extract work are equally probable as states that require work. Not that a bias can assure only positive states are probable.

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2. Nov 3, 2014

### Simon Bridge

That's because there isn't anything that does that ... which is because it is not impossible to get order from disorder.

3. Nov 4, 2014

### Quentin_C

Hi Simon, Thanks for that could you explain further as that's my problem, the second law seems built on the assumption that order from disorder is improbable and will only occur if work is put into the system. That's doesn't seem absolute but yet agrees with all known observation.
If order from chaos without work (which I have seen referred to as unidirectionality) is not fundamentally prohibited by something else than statistics then the existence of a 'Kelvin' system that creates work from heat is not precluded by anything else in the second law ?

4. Nov 4, 2014

### Staff: Mentor

There is a nice video by Sixty Symbols that discusses "entropy is not disorder."

5. Nov 4, 2014

### Staff: Mentor

The laws of thermodynamics, like Newton's laws and the law of gravity, were originally based on observation. With gravity, we observed that all masses are drawn together by a force given by $F=Gm_1m_2/r^2$, and as it became more and more clear that that really was all masses all the time, with no exceptions ever, we started describing this as a "law of nature". The laws of thermodynamics came to be "laws" in the same way; they are never ever violated in nature.

It was only after the discovery of statistical mechanics that we understood that there was a probabilistic mechanism underlying classical thermodynamics. And yes, in principle it is possible that we might observe a spontaneous violation of one of these laws of thermodynamics - if by random chance all the molecules in a volume of gas all just happened to be moving in the same direction. In practice that never happens, as the difference between zero and something like one chance in $2^{(10^{30})}$ is irrelevant for all practical purposes.

6. Nov 4, 2014

### dextercioby

The famous Poincare recurrence theorem, one of the results that haunted Boltzman's last part of life.

7. Nov 4, 2014

### Simon Bridge

In addition to what the others have posted:

The Laws of thermodynamics are currently understood in terms of emergent properties from more fundamental laws in relation to complex systems - like thermodynamic systems. They are taken as axiomatic, initially, because nobody had ever observed a violation (so it was like a really strong rule of thumb), but nowadays because there are good reasons to suspect that they are extremely rarely violated for complex systems. As explained quite well in the above posts.

The 2nd law is not built on the assumption that order from disorder is improbable - because that is not an assumption: it is an empirically established fact. The usual statement of the law does not say that order from disorder is improbable - it says that it is impossible. But see below.

In the field of thermodynamics, it is taken as axiomatic that decreases in entropy ("disorder" etc are imprecise terms here) cannot occur for closed thermodynamic systems, however it is not a general principle of physics in the same sense as the law of conservation of energy (which is behind the 1st law). This should be understood by the careful student that they need to be careful when applying laws to make sure they stay inside the bounds that the model holds. In this case: to thermodynamic systems. Even then, it is more correct to say that the probability for a spontaneous decrease in entropy is vanishingly small for very large numbers of particles... though not exactly impossible, it is very difficult to overstate how unlikely it is. You can think of "impossible" in this case as "functionally impossible" - don't hold your breath waiting for one.

The main thing to realize is that the statement you are trying to prove in absolute terms cannot be proven because, in absolute terms, it is just not true. It is not intended as an absolute statement. No physicist believes that it is true, only that it is an extremely good rule to follow.

You will get further with your original investigation by working out exactly how improbable a particular spontaneous entropy change actually is, for some system you are interested in... then you'll get the idea.

Careful though: most people trying to look for violations to the laws of thermodynamics are interested in perpetual motion.
This is a fruitless task - if that's what you are exploring, give up now: it won't work.
The creative effort wasted in this search drives grown physicists to tears.