# Second Law from Statistics

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 fundamentally prevents a system creating unidirectionality. If you can achieve this somehow then the second 'law' falls down. doesn't sound like a law at all?

Can someone assist to shoot this down?

Further explanation

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

upload_2014-11-4_12-7-30.png 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 initial statistical bias existing which is creating work (order) from random movement (disorder) and nothing changing over time as Q enters the system to balance W removed.

Before we bring them up, To me the 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.

• improbable means unlikely, under all circumstances! – Wolphram jonny Nov 4 '14 at 7:30

## 2 Answers

I think you have a misunderstanding of the second law. The second law is statistical, not deterministic. For instance, once you reach thermodynamic equilibrium, that is, the largest entropy state, fluctuations will start to occur in any direction. It is possible that a broken egg goes back and reconstitutes itself, but the likelihood of that even has been calculated. A fluctuation to a state of entropy smaller than the equilibrium entropy by an amount $\Delta S$ requires a time with a likelihood proportional to $e^{t\Delta S}$. See http://en.wikipedia.org/wiki/Poincar%C3%A9_recurrence_theorem

• So, perhaps an interesting questions is, what could have happened in 'reverse' direction since the beginning of the universe? Would $\Delta S \sim 1/13.7 Gyr$ yield some interesting events? – AtmosphericPrisonEscape Nov 4 '14 at 15:01

It seems to me you are conflating micro and macrostates. Entropy is not a property defined for a specific microstate but for an ensemble. When we describe a system in terms of pressure, entropy, etc instead of the momentum and position of each particle we are giving up the possibility of discussing specific microstates. The 'states of high order' you are thinking of are specific instances of microstates included in the initial ensemble. It is possible that such a specific microstate is also included in the ensemble of other macrostates but that overlap is fortuitous and not something that can be exploited.