# Thermodynamics and the Gibbs Paradox - MDPI

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27 Οκτ 2013 (πριν από 4 χρόνια και 8 μήνες)

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Thermodynamics and

Presented by: Chua Hui Ying Grace

Goh Ying Ying

Ng Gek Puey Yvonne

Overview

The three laws of thermodynamics

Gibbs / Jaynes

Von Neumann

Shu Kun Lin’s revolutionary idea

Conclusion

The Three Laws of
Thermodynamics

1
st

Law

Energy is always conserved

2
nd

Law

Entropy of the Universe always increase

3
rd

Law

Entropy of a perfect crystalline substance is
taken as zero at the absolute temperature
of 0K.

Unravel the mystery

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……

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 entropy
increase, 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 original
state as there is no recognizable difference
between the gases

Gibbs’ opinion (2)

Thus, these two cases stand on
different footing and should not be
compared with each other

The mixing of gases of different kinds
that resulted in the entropy change was
independent of the nature of the gases

Hence independent of the degree of
similarity between them

Entropy

S
max

Similarity

S=0

Z=0

Z = 1

Jaynes’ explanation

The entropy of a macrostate is given as

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 a
property 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 and
those we want to change.

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

Von Neumann’s Resolution

Makes use of the quantum mechanical
approach to the problem

He derives the equation

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 highest
and when

= 1 entropy is at its lowest

Therefore entropy decreases continuously
with increasing similarity

Entropy

S
max

Similarity

S=0

Z=0

Z = 1

-

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 similarity
of 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

s
imilarity can be described as a continuous measure of
imperfect 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

p
ossesses greater information content.

Do you have any
idea what the

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)

gr敡瑥r

information

loss
.

For a system with
indistinguishable 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

loss

Higher
entropy

The mixing of identical

gases (revisited)

Lin’s Resolution of the Gibbs Paradox

Compared to the non
-
identical gases, we have less

According to his theory,

less information=higher entropy

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

Comparing the 3 graphs

Entropy

S
max

Similarity

S=0

Z=0

Z = 1

Entropy

S
max

Similarity

S=0

Z=0

Z = 1

Z=0

Entropy

S
max

Similarity

S=0

Z = 1

Gibbs

Von Neumann

Lin

Why are there
different

ways in

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 and
molecules

We cannot compare the two

Since Lin’s definition of entropy is
essentially different from that of Gibbs
and von Neumann, it is unjustified to
compare the two ways of resolving the

Conclusion

The Gibbs Paradox poses problem to
the 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 areas
for further exploration.

Acknowledgements

We would like to thank

Dr. Chin Wee Shong for her support and
guidance throughout the semester

Dr Kuldip Singh for his kind support

And all who have helped in one way or another