chua thermodynamics

8/2/2019 Chua Thermodynamics

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Thermodynamics andthe Gibbs Paradox

Presented by: Chua Hui Ying Grace

Goh Ying Ying

Ng Gek Puey Yvonne



Overview

The three laws of thermodynamics

The Gibbs Paradox

The Resolution of the Paradox

Gibbs / Jaynes

Von Neumann

Shu Kun Lin’s revolutionary idea

Conclusion



The Three Laws of Thermodynamics

1st Law Energy is always conserved

2nd Law Entropy of the Universe always increase

3rd Law

Entropy of a perfect crystalline substance istaken as zero at the absolute temperatureof 0K.



Unravel the mysteryof The Gibbs Paradox



The mixing of

non-identical gases



Shows obvious increase in entropy (disorder)



The mixing of identical gases



Shows zero increase in entropy as action is reversible



Compare the two scenarios of

mixing and we realize that……



To resolve the Contradiction

Look at how people do this

1. Gibbs /Jaynes

2. Von Neumann

3. Lin Shu Kun



Gibbs’ opinion

When 2 non-identical gases mix and entropyincrease, we imply that the gases can be

separated and returned to their original state When 2 identical gases mix, it is impossible to

separate the two gases into their originalstate as there is no recognizable difference

between the gases



Gibbs’ opinion (2)

Thus, these two cases stand ondifferent footing and should not be

compared with each other The mixing of gases of different kinds

that resulted in the entropy change wasindependent of the nature of the gases

Hence independent of the degree of similarity between them



Entropy

Smax

Similarity

S=0

Z=0 Z = 1



Jaynes’ explanation

The entropy of a macrostate is given as

)(log)( C W k X S

Where S(X) is the entropy associated with a chosen

set of macroscopic quantities

W(C) is the phase volume occupied by all the

microstates in a chosen reference class C



Jaynes’ explanation (2)

This thermodynamic entropy S(X) is not aproperty of a microstate, but of a certain

reference class C(X) of microstates For entropy to always increase, we need to

specify the variables we want to control andthose we want to change.

Any manipulation of variables outside thischosen set may cause us to see a violation of the second law.



Von Neumann’s Resolution

Makes use of the quantum mechanicalapproach to the problem

He derives the equation

2log21log11log12

Nk S

Where measures the degree of orthogonality, which

is the degree of similarity between the gases.



Von Neumann’s Resolution (2)

Hence when = 0 entropy is at its highestand when = 1 entropy is at its lowest

Therefore entropy decreases continuouslywith increasing similarity



Entropy

Smax

SimilarityS=0

Z=0 Z = 1



Resolving the Gibbs Paradox - Using Entropy and its

revised relation with Similarity proposed by Lin Shu Kun.

• Draws a connection between information theory and entropy

• proposed that entropy increases continuously with similarityof the gases



Analyse 3 concepts!

(1) high symmetry = high similarity,

(2) entropy = information loss and

(3) similarity = information loss.

Why “entropy increases with similarity” ?

Due to Lin’s proposition that

• entropy is the degree of symmetry and

• information is the degree of non-symmetry



(1) high symmetry = high similarity

• symmetry is a measure of indistinguishability

• high symmetry contributes to high indistinguishability

similarity can be described as a continuous measure ofimperfect symmetry

High Symmetry Indistinguishability High

similarity



(2) entropy = information loss

an increase in entropy means an increase in

disorder.

a decrease in entropy reflects an increase in order.

A more ordered system is more highly organized

thus possesses greater information content.



Do you have any

idea what the picture is all about?



From the previous example,

• Greater entropy would result in least information registered

Higher entropy , higher information loss

Thus if the system is more ordered,

• This means lower entropy and thus less information loss.



(3) similarity = information loss.

1 Particle (n-1) particles

For a system with distinguishable particles,

Information on N particles

= different information of each particle

= N pieces of information

High similarity (high symmetry)

there is greater information loss .

For a system withindistinguishable particles,

Information of N particles

= Information of 1 particle

= 1 piece of information



Concepts explained:

(1) high symmetry = high similarity

(2) entropy = information loss and

(3) similarity = information loss

After establishing the links between the various concepts,

If a system is

highly symmetrical high similarity

Greater information lossHigher entropy



The mixing of identicalgases (revisited)



Lin’s Resolution of the Gibbs Paradox

Compared to the non-identical gases, we have lessinformation about the identical gases

According to his theory,

less information=higher entropy

Therefore, the mixing of gases should result in anincrease with entropy.



Comparing the 3 graphs

Entropy

Smax

Similarity

S=0

Z=0 Z = 1

Entropy

Smax

Similarity

S=0

Z=0 Z = 1 Z=0

Entropy

Smax

Similarity

S=0

Z = 1

Gibbs Von Neumann Lin



Why are there different ways inresolving the paradox?

Different ways of considering Entropy

Lin —Static Entropy: consideration of configurations of fixed particles in a system

Gibbs & von Neumann —Dynamic Entropy:dependent of the changes in the dispersal of energy in the microstates of atoms andmolecules



We cannot compare the twoways of resolving the paradox!

Since Lin’s definition of entropy isessentially different from that of Gibbsand von Neumann, it is unjustified tocompare the two ways of resolving the

paradox.



Conclusion

The Gibbs Paradox poses problem tothe second law due to an inadequate

understanding of the system involved. Lin’s novel idea sheds new light on

entropy and information theory, but

which also leaves conflicting grey areasfor further exploration.



Acknowledgements

We would like to thank

Dr. Chin Wee Shong for her support and

guidance throughout the semesterDr Kuldip Singh for his kind support

And all who have helped in one way or another

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