Download 2. THERMODYNAMICS and ENSEMBLES (Part A) Introduction

2. THERMODYNAMICS and ENSEMBLES
(Part A)
R. Bhattacharya, Department of Physics, Jadavpur University, Kolkata – 32
Introduction
Statistical mechanics is concerned with the discussion of systems consisting of a very
large number of particles such as gases, liquids, solids, electromagnetic radiation
(photon) and most physical chemical or biological systems.
A discussion of these would naturally involve the interaction between and the nature of
the smallest units, i.e. atoms or molecules. This force arises from the well-understood
electromagnetic interactions. The laws of motions of the atoms are given by the familiar
laws of quantum mechanics. In many cases, classical Newtonian mechanics may be
applied. Thus, despite the great number of particles involved, one might be tempted to
assert that all the properties of a system may be derived from the basic laws of motion
and the known interactions, “in principle”. Let us pause to consider the complexity of the
equation of motion with approximate boundary conditions for 1023 particles! Apart from
the enormity of this aspect of the problem, we must appreciate that even simple
interactions between individual particles may give rise to unexpected qualitative features
for a large system. An example is the phase transition of a gas which abruptly condenses
to form a liquid with very different properties.
However, the adversity of large numbers can be turned to an advantage by considering
the gross, average or statistical properties of the system rather than trying even to set up
the hopeless task. Thus, we learn to distinguish a “microscopic” or a “small scale” system
from a “macroscopic” or a “large scale” one. Thus, we call a system to be microscopic if
it is roughly of the size of atoms. This system might consist of a single molecule. An
example of a macroscopic system would be something as large as the universe in its early
stages of evolution. For a macroscopic system, one is not normally interested in the
detailed motion of each particle of the system, but rather in certain macroscopic
parameters which characterize the system as a whole, i.e., quantities like volume,
pressure, magnetic moments and so on. If these “macro” parameters do not vary with
time, then the system is said to be in equilibrium. If they do vary with time for an isolated
system, then the system is not in equilibrium and will, after a sufficiently long time, reach
a state where the parameters will have constant values. Thus, equilibrium systems are
clearly easier to deal with than those not in equilibrium.
The study of the equilibrium properties of a macroscopic system from a
phenomenological point of view began in the nineteenth century. The laws thus
discovered formed the subject matter of “thermodynamics”. As the atomic constitution of
matter gained more and more general acceptance, macroscopic systems began to be
naturally understood from a fundamental microscopic point of view. We discuss briefly
the different disciplines that developed.
1) The oldest disciplines is thermodynamics. In classical thermodynamics, one tries to
make very general statements concerning relationships existing between the macroscopic
parameters of the system. The great generality of the method allows one to make valid
statements based on a minimum number of postulates without requiring any detailed
assumption the molecular or microscopic properties of the system. However only a
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relatively few statements can be made on such general grounds and many interesting
properties of the system remain outside the scope of the method.
2) For a system in equilibrium, one can again try to make statements consistently based
on the microscopic properties of the particles in the system and on the laws of mechanics
governing their behaviour. This is the subject matter of “statistical mechanics”. It gives
us all the results of thermodynamics in addition to a large number of general relations for
calculating the macroscopic parameters from a knowledge of its microscopic
constituents.
3) As has been mentioned before, for a system not in equilibrium one faces a much more
difficult task. The general method of making such statements belongs to the domain of
“statistical mechanics of irreversible process” or “irreversible thermodynamics”.
4) One may also study in detail the interaction of all the particles in the system and then
calculate parameters of macroscopic significance. This is the method of “kinetic theory”.
It can yield results even for system not in equilibrium. This is the most difficult method
to apply because it attempt to yield such detailed description.
In the present chapter, we shall confine ourselves to the development of the formal
structure of equilibrium statistical mechanics. In doing so, we shall, unfortunately, have a
limited scope of discussing the application of these methods, to real, interacting systems
of interest. However we shall study some systems of course. As we go along, we shall
discuss and identify the laws of thermodynamics. Having arrived at the thermodynamic
potentials starting from a microscopic picture, it will be easy for us to study the properties
of the system from its “equation of state”.
We should, strictly speaking, use only quantum mechanics, and the basic postulates of
statistical mechanics as our staring point. This should enable us to deduce all the results,
including the classical ones, in the appropriate limit. However, we shall also talk about
“classical statistical mechanics” using classical mechanics alone along with the postulates
of statistical mechanics.
But before getting into the heart of the subject matter, we shall first review, for the
sake of completeness, a few notions from the elementary theory of probability, such as a
probability distribution, means, and dispersion and so on, keeping in mind a particular
problem. These concepts will recur again and again in our subsequent discussion and
therefore, it may be useful to collect some of the results at the very beginning.
2.1 Probability
When we talk about probability, it is necessary to consider an ‘ensemble’ or a ‘collection’
consisting of a very large number of similarly prepared systems. The probability of
occurrence of a particular event is then given by the fraction of systems in this ensemble
characterised by the occurrence of this particular event. For example, in throwing a pair
of dice or in throwing a coin, one can give a statistical description by considering a very
large number of coins or dice thrown under similar circumstances. Alternatively, the
same coin may be thrown successively a large number of times under similar
circumstances. The probability of obtaining a ‘head’ is given by the fraction of the
experiments in which the outcome is a head. [The probability of a single toss producing a
head has no meaning]. We now discuss what is known as the random walk problem.
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2.1.1 Random Walk Problem (in one dimension)
l
x=0
Figure 2.1 Random Walk in One dimension
In a random walk in one dimension, a person starts out from the point x = 0 and takes
steps of length l. Whether he takes a step to the right or to the left is independent of the
previous step. If the probability of taking a step to the right is p, then that of taking a step
to the left is q = 1-p. What is the probability that the person has gone a length x = ml after
taking N steps
(m = integer, i.e., 0, +ve or –ve)?
We are really considering a very large number or an ensemble of persons undertaking
this random walk (or alternatively one can repeat the experiment N times). Some
examples where this problem has relevance to physics are:
i) An assembly of spin 1/2 particles having a magnetic moment μ. The spin can point ‘up’
or ‘down’. What is the magnetic moment of a collection of N such spins
(paramagnetism)?
ii) A molecule travels a mean distance l between collisions. How far is it likely to go after
N collisions (diffusion)?
Coming back to the problem of the random walk, after N steps the man is located at
x = ml, -N ≤ m ≤ N
(2.1)
Let n1= no. of steps to the right and n2 = no. of steps to the left. Then N = n1+n2.
The net displacement (measured to the right) is given by (in units of l )
m = n1 - n2
(2.2)
= n1 - (N-n1) = 2n1 - N
(2.3)
Now the steps are statistically independent of each other. Therefore, the probability of a
given sequence of steps (n1 to the right and n2 to the left) is
p. p p q. q q = p n1 q n2
 

n1
n2
(2.4)
But there are many ways of taking N steps so that n1 are to the right and n2 to the left.
N!
The member of such possibilities is
n1 ! n2 !
Therefore the probability of taking n1 steps to the right and n2 to steps to the left is given
by
N!
N!
WN ( n1 ) =
p n1 q n2 =
p n1 q N − n1
(2.5)
n1 ! n2 !
n1 !( N − n1 )!
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This probability function is a binomial distribution because it occurs in the binomial
expansion of (p + q)N.
N
N!
( p + q )N = ∑
[ p n q N −n ]
(2.6)
n =0 n!( N − n )!
Now if the person has taken n1 steps to the right and n2 steps to the left, then the
displacement m = n1 - n2 is determined. Therefore the probability PN (m) that the particle
is found at m after N steps is the same as WN (n1)
N+m
N −m
N!
p 2 (1 − p ) 2
 N + m  N − m

! 
!
 2   2 
N +m
N − m

Q n1 + n2 = N , n1 − n2 = m and so n1 = 2 , n2 = 2 
In the special case when p = q = 1/2,
N
N!
1
PN ( m ) =
 
 N +m  N −m 2
!
!

 

 2   2 
We also note that
∴ PN ( m ) = WN ( n1 ) =
N
∑W (n
n1 =0
n
1
(2.7)
(2.8)
N
) = ∑ p n1 q N − n1 /[ n1 !( N − n1 )!] = ( p + q )N = 1N = 1
(2.9)
n1
This is the normalisation condition for the probability. This says that the probability of
taking any number of steps n1 has to the be unity.
W(n1) or
P(m)
0 1 2 3 4 5 6 7 8 9
[p = q = ½]
Fig. 2.2. The binomial probability distribution.
2.1.2 Averages
If a variable U assumes the values U1, U2, U3,…..UN with probabilities, P(U1), P(U2),
P(U3), ….P(UN), then, the mean or average value of U is defined by
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U=
U 1 P(U 1 ) + U 2 P(U 2 ) +  + U N P(U N )
P(U 1 ) + P(U 2 ) +  + P(U N )
(2.10)
N
=
∑U
i =1
N
P(U i )
i
∑ P(U
i =1
i
N
∑U i P(U i ) because
=
i =1
)
N
∑ P(U
i =1
i
)=1
(2.11)
This is a normalisation condition which implies that the probability that U has any one of
its possible values is unity. Consider in the example of the random walk
N
N
n1 =0
ni =0
N
 ∂
N!
p
( p n1

n1 !( N − n1 )!  ∂ p
n1 = ∑ W( n1 )n1 = ∑
=∑
n1 =0
= p
N!
p n1 q N − n1 n1
n1 !( N − n1 )!

) q N − n1

∂ α
N!
p n1 q N − n1
∑
∂ p n1 =0 n1 !( N − n1 )!
∂
( p + q )N = pN( p + q )N −1 [ true for arbitrary p ]
∂ p
= Np
= p
(2.12)
Similarly,
n2 = Nq
Naturally, N (p + q) = N and
(2.13)
m = n1 − n 2 = N ( p − q ) = 0, if p = q
(2.14)
The dispersion
( ∆ n1 )2 ≡ ( n1 − n1 )2
= n12 − 2 n1 n1 + n12
(2.15)
= n12 − 2 n12 + n12 = n12 − n12
because ( ∆ n1 )2 ≥ 0 ]
[ thus n12 ≥ n12
N
Now, n1 = ∑ W ( n1 )n12
2
(2.16)
n1 =0
But,
2
 ∂  n1

∂  ∂
n1 p =  p
p
. p n1 
 p , i.e, p

∂ p ∂ p
 ∂ p

2
N
∴ n1 = ∑
2
n=0
n1
2
 ∂  n1 N − n1
N!
=
p
 p q
n1 !( N − n1 )!  ∂ p 
 ∂ 
p

 ∂ p
2
N
∑
n=0
N!
p n1 q N − n1
n1 !( N − n1 )
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2
=
=
=
=
 ∂ 
∂
N
 Np ( p + q )N −1 
p
 ( p+q) = p
∂ p
 ∂ p
p  N( p + q )N − 1 + pN( N − 1)( p + q )N − 2 
p [ N + pN( N − 1)] = Np [ 1 + Np − p ]
Np [ Np + q ] = ( Np )2 + Npq =
∴
( ∆ n1 )2
n12 + Npq
= n12 − n12 = Npq
The relative width of the distribution is
Npq
[ ( ∆ n1 )2 ] 1/ 2
=
=
n1
Np
Also,
q 1
.
p N
(2.17)
1
N
=
for q = p
∆m = ( m − m ) = ( 2n1 − N ) −( 2n1 − N ) = 2( n1 − n1 ) = 2 ∆n1
∴( ∆m )2 = 4 ( ∆n1 )2 = 4 Npq.
(2.18)
(2.19)
For large N, W (n1) has a sharp maximum at n1. We use this result to get an approximate
form for W (n1) large N. For large N, near the maximum, | W (n1+1) - W (n1) | 〈〈 W (n1)
dW
= 0 or
And n~1 is determined from
dn1
d
ln W = 0
(2.20)
dn1
[the ln W is more slowly varying function and is to be Taylor expanded]. Then, writing
n1 = %n1 + η
ln W ( n1 ) = ln W ( %n1 ) + B1η +
1
1
B2η 2 + B3η 3 + 
2!
3!
(2.21)
Where,
Bk =
dk
ln W
dn1k
(2.22)
Since we are expanding about a maximum, we have
B1 = 0 , B2 < 0 = − | B2 |
(2.23)
Since η is sufficiently small, we retain terms up to the second order in η .
% e− 12|B2 |η
ln W( n1 ) = ln W( %n1 ) − 21 | B2 |η 2 or W( n1 ) = W
2
[This is good approximation, for the range Npq << η << Npq i. e. Npq >> 1 ]
(2.24)
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N
~
W is determined from ∑ W (n1 ) = 1 . If we treat W and n1 as quasi-continuous variables,
n1 = 0
N
∑ W (n ) ≈ ∫ W (n ) dn
1
n1 = 0
1
1
α
= ∫ W (n1 ) dn1 = 1
(2.25)
−α
[Here the integrand is practically zero outside a small range so that the limits can be taken
from -∞ and +∞ .]
This gives
α
2π
−1 B η2
W ∫ e 2 2 dη =W
=1 , or
(2.26)
| B2 |
−α
| B2 | − 21 |B2 | (n1 −%n1 )2
e
2π
This is a Guassian distribution. By using the Stirling’s approximation, we have
W( n1 ) =
d
ln n ! = ln n
dn
and we can show that
(2.27)
(2.28)
~ = Np, B = 1 / Npq
n
1
2
(2.29)
 −( n1 − Np ) 
−1
W( n1 ) = ( 2 π Npq ) 2 exp 

 2 Npq 
−1
 −( n1 − n1 )2 
2 2

∴W ( n1 ) = 2 π ( ∆n1 )
exp 

2


 2( ∆n1 ) 
2
and thus get,
(2.30)
Also,
 − [m − N ( p − q )] 2
 N +m 
−1 / 2
P ( m) = W 
exp 
 = 2πNpq

8 Npq
 2 

[
]




(2.31)
Since n1 − Np = (1 2)( N + m − 2 Np ) = (1 / 2)[m − N ( p − q )]. But, x = ml, m = 2n1-N
and Δm changes in steps of 2. Therefore the probability that x lies between x and x + dx
is,
 ( x − µ) 2 
dx
1
dx
P ( x )dx = P (m)
=
exp −
(2.32)
2l
2σ 2 
2π σ

where
µ = ( p − q )Nl = x
(2.33)
[
σ = 2 Npql = ( ∆x) 2
]
1/ 2
(2.34)
2.2 Statistical Mechanics
In statistical mechanics, one tries to apply the ideas of statistics appropriate to a system
containing a large number of particles that are individually described by the laws of
mechanics (or quantum mechanics). To do so, one introduces the following ingredients:1) The state of the system: When one considers a system of 10 dice that are thrown from
a cup onto a table, a specification of the state of the system after the throw is the
statement as to which face is uppermost for each one of them.
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2) Statistical ensemble: In principle, if we know the positions and orientations of the
dice initially, as well as their velocities at the instant they are thrown, it is possible to
predict the outcome by solving the equation of motion. However, such detailed
knowledge is not available and then we start talking about the probability of outcome
of a particular event. The theoretical aim is to predict this probability on the basis of
some fundamental postulates.
3) Postulate about equal a priori probability: There is nothing in the laws of mechanics
which, for dice of regular shape and size, would lead us to expect that any given face
would appear on top preferentially. So we introduce the notion that the a priori (based
on our prior notion as yet unverified by experimental observation) probability of any
face appearing on top is the same. This does not contradict any of the laws of
mechanics. However, the conclusion following from this postulate may be checked
by experiment.
Example 2.1
Consider 3 spin - ½ particles in a magnetic field H along the z - axis. Corresponding to
the quantum number mi of each spin, the particles have an energy + µ H (energy of a


 
magnetic moment µ in a field H is −µ.H ).
Let us denote +1/2 by + and -1/2 by -. Then we can construct the following table.
Table 2.1 Magnetic moments and energies of three spins.
State Index r
1
2
3
4
5
6
7
8
Quantum Nos.
m1 m2 m3
+
+
+
+
+
+
+
+ +
+
+ +
-
Total. Magnetic Moment
3 µ
µ
µ
µ
-µ
-µ
-µ
-3µ
Tot Energy
- 3 µH
-µH
-µH
-µH
µH
µH
µH
3 µH
If this system is an isolated system in equilibrium, the probability of finding the system in
a given state is independent of time. All macroscopic parameters, including the total
energy are constants. Corresponding to given energy, there may be more than one state
accessible to the system.
The postulate about equal a priori probability then implies that an isolated system in
equilibrium is equally likely to in any one of its accessible states.
In our example of three spin -1/2 particles, suppose the energy is constant and equal to
- µH. then the possible states are
+ + +-+
-++
and at equilibrium, any one of these is equally likely. Let us express these things a little
more formally. Let Ω (E) be the total number of states accessible to the system having
29
energy between E and E + ζ E. Let Ω (E, yk) be the number of states with some
parameter y having the value yk.
Then the probability that y has the value yk is
P( yk ) =
and
yk =
Ω ( E, yk )
Ω(E )
∑ Ω ( E, y
k
k
(2.35)
) yk
(2.36)
Ω(E)
For the system with 3 spins and energy - µH. what is the probability that the 1st spin
points up? It is
P+ =
2
3
[ Note P+ ≠ 1 / 2 ]
(2.37)
Similarly,
P− =
1
3
(2.38)
and the average moment of the first spin is
µ1 = P+ . µ + P− ( − µ ) =
2
3
µ − 31 µ =
1
3
µ
(2.39)
2.2.1 The Quantum Description
We now turn from our model system of 3 spins to a realistic system composed of a great
number of ∼1023 particles. Now the particles in the system, however complicated, are
described by the laws of quantum mechanics. Thus a system can be described by a wave
function Ψ(q1, …..qs) which is a function of s variables required to specify the system. In
principle, Ψ at a later time can be calculated using the time dependent Schrödinger
equation.
Two comments are in order. When the number of degrees of freedom of a system is
very large, the density of energy levels for the system of becomes very large. This is easy
to understand qualitatively. If a given amount of energy is distributed between many
particles, there are many ways available for this to be distributed and the corresponding
level density becomes very large.
Secondly, because of the large density of levels and the corresponding closeness of
levels, the concept of a stationary state as a macroscopic state loses its meaning. The
energy of the system will always be “broadened” by an amount of the order of the
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interaction with the surroundings. The latter quantity is very large corresponding with the
separation between the levels.
Thus the microscopic state of the system may be specified by enumerating in some
convenient order all possible quantum states of the system.
2.2.2 The Classical Description
Sometimes a classical mechanical description instead of a quantum mechanical one is
very useful. To illustrate this, consider one particle with a single degree of freedom. The
specification is complete if q and p are known for this particle. Thus the state of the
particle can be represented by a point in the p-q plane as shown in Fig 2.2
p
q
Figure 2.3 Cells in p - q plane.
In order to make the number of states accessible to the system countable, one can divide
the range of (p, q ) into small intervals of the size δq and δp. Thus the phase space is
divided into small cells of area [‘two dimensional volume’] δp δq = ho. Thus if q lies
between q and q + δq and p lies between p and p + δp the representation point lies in a
particular cell. The value of ho can be taken to be arbitrarily small in the classical
description. But there is a limit to the smallness of h o in quantum mechanics if we recall
that
δ pδ q ≥ h
(2.40)
The extension to a system of N particles with 3N degree of freedom is straight forward.
Such a system is described by s coordinates q1…..qs and s momenta p1…..ps, where s =
3N. The set of 2s numbers q1…..qs , p1…..ps can be taken to correspond to a point in a 2s
- dimensional phase - space. This phase space can be subdivided into cells of volume δq1
δq2 ….δq δp1 , δp2 ….δps = hos. The state of the system corresponds to a particular cell in
phase space as before.
2.2.3 Approach to Equilibrium
Coming back to the quantum description of a system of particles, we note again that a
rigorous discussion in terms of the exact stationary states of the system is impossible. We
describe the system in terms of some complete set of approximate quantum states which
take into account most of the prominent dynamical features of the system without being
exact. Thus a system in one such state will make transitions to nearby states because of
small residual interaction between the particles. Now if a system is known to be in a
subset of all the states accessible to it, as the time progresses and the system tends to
equilibrium, it makes transitions to all the accessible states, so that, at equilibrium, it is
equally likely to be found in any one of all the states accessible to the system.
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Example 2.2 In our system of 3 spin in a magnetic field, suppose the state is known to
be in the state + + - . This is in equilibrium due to residual interaction between the spins
and after a sufficiently long time, the system will proceed to the equilibrium state so that
it is equally likely to be in any one of the states + + -, + - +, - + +, all of which correspond
to the same energy.
Example 2.3 Suppose a certain amount of gas is contained in the left half of a container
as shown. If the partition is removed, the right half now becomes accessible. As a result
of collisions between themselves and the wall, the molecules will redistribute themselves
so that at equilibrium they are equally likely to be at any point in the box, i.e., the density
is uniform.
V
V
Figure 2.4 A gas container separated by a partition.
2.2.4 The Density of States
A macroscopic state has many degree of freedom. If the energy of the system is E, we
can subdivide the energy scale into small equal ranges of magnitude δE. δE is the
measure of the precision within which one chooses to measure the energy of the system.
If the system is macroscopic, even a very small interval O(δE) [O(δE) means of the order
of δE] contains a very large number of levels of the system. Let Ω(E) be the number of
states which lie between E and E + δE. If δE is sufficiently small,
Ω( E ) = w( E ) δE
(2.41)
Where w(E) is the density of states and measures the number of states per unit energy
range. It can be shown that Ω(E) is an extremely rapidly rising function of E. An order of
magnitude relation is
Ω(E) α Eα
(2.42)
[thus, w has a similar dependence on E ].
2.2.5 Equilibrium Conditions and Constraints
For an isolated system whose energy lies in a narrow range, let Ω be the number of states
accessible to the system. Then, the fundamental postulate states that at equilibrium the
system is equally likely to be in any of these states.
Since a system satisfies certain constraints, the number of accessible states depends on
the these constraints, and are consistent with these constants. The constraints are
specified quantitatively by some parameters y1, y2,….yn, which characterise the system on
a macroscopic scale. Thus one can write
Ω = Ω (y1 ,y2…….yn )
(2.43)
where each of the yi lie between yi and yi + δyi .
Example 2.4
32
1) A gas is confined in the left half of a box. The partition acts as a constraint and the
number of states accessible to the system are those for which the co-ordinates of all
the molecules lie in the left half.
2) A0 consists of two subsystems A and A´ which are separated by an insulating wall.
The constraint is that no energy can be exchanges between A and A´
Only those states are accessible for which the energies of A and A´ are constant and equal
to E and E´ respectively.
..…….
…A…… A´
Figure 2.5 Compartments separated by a partition.
……….
A0 is the total system.
A0
insulation
If some of the constraints are now removed, then the states formerly accessible in general
change. Thus if Ωi and Ωf are the numbers of states accessible to the system initially and
finally.
Ωf ≥ Ωi
(2.44)
If Ωf ≥ Ωi , then at the time when the constraint are removed, the system cannot be in
an equilibrium state, because the system occupies only a fraction Ωi / Ωf of the states
accessible to it. At equilibrium, the system must be equally distributed over all the
accessible states.
Going back to our previous two examples, suppose that in (1) the partition is removed. It
is very improbable that molecules will remain confined to the left half of the box. Thus in
the equilibrium situation they are distributed uniformly throughout the box. In the
equilibrium state, what is the probability that all the molecules are in the left half? The
answer is:
N
1
−2 X 1023
(2.45)
  ≈ 10
2
For N = 6.02 X 10 23. In example (2), if the partitions are made conducting, the A and A´
can exchange energy with each other. The most probable final equilibrium situation
corresponds to the situation when an adjustment has taken place by virtue of heat transfer
between the subsystems.
In terms of the macroscopic variables y1 ,y2 ….yn, we can say that of one of the y’s i.e. yi
is allowed to vary, the probability P(y) of finding the system in the range y + dy is
P(y) α Ω(y)
(2.46)
where Ω(y) is the number of states accessible to the system when y lies between y , y
+ δy . If, before the removal of the constraints y = yi, then after the removal, y = yi is an
exceedingly improbable value. The situation changes in time until y changes to its most
y . Usually Ω(y) has a very sharp maximum at y = ~
y . Thus we can
probable value ~
summarise by saying that if the constraints of an isolated system are removed, the
parameters of the system change in such a way that
Ω ( y1 ,y2 ….yn ) → maximum
(2.47)
2.2.6 Reversible and Irreversible Processes
33
If some of the constraints for a system are removed, then, in general Ωf > Ωi. After the
system has redistributed itself, the constraint may be restored. But this does not
necessarily restore the system to its initial condition. In the two examples that we have
considered, restoring the partition after the gas has expanded or replacing the conducting
wall by as insulating one does not restore the system its initial state. Therefore if Ωf =
Ωi, the system in the representation ensemble is already distributed with equal probability
over all the accessible states. The system is always in equilibrium and the process is
reversible. If Ωf > Ωi , the system changes with time till at equilibrium a new state is
reached (which is the most probable final state). The process is irreversible.
2.2.7 Relaxation Time
This is the time taken by a system to reach an equilibrium state. Since we are always
making statements about equilibrium states, we cannot say any thing about the rate of a
process or its relaxation time τ. Such things can be known by considering the detailed
interaction which we have not considered.
Equilibrium statistical mechanics may be applied to the two situations.
i) τ << t exp : the system comes to equilibrium very quickly in this case.
ii) τ >> t exp : the system comes to equilibrium very slowly. Then the system can be
thought of as being always in equilibrium as the macroscopic parameters change so
slowly.
For τ ~ t exp, equilibrium statistical mechanics cannot be applied. Here, t exp is the time
over which the experiment is carried out.
2.2.8 System in Thermal Interaction
Consider now the thermal interaction between two systems A and A´ . The systems A
and A´ are free to exchange energy in the form of heat only (thus the external parameters
are supposed to remain fixed.). let A and A´ have energies E and E´ receptively. We
divide the energy scale into equal intervals of magnitude δE and δE´ . Let Ω ( E ) and Ω
´
(E´ ) be the number of states of A and A´ in the range between E, E + δE and E´, E´
+δE´, respectively.
A
A´
Figure 2.6 Interacting systems A and A´
Then the combined system Ao = A + A´ is isolated and its total energy E(o) is constant.
Assuming the interaction between A and A´ to be small, we can write E + E´ = E(o).
Consider the equilibrium between A and A´. The energy of A can have a large number of
possible values. If the energy of A is E (i.e. between E, E +δE ), that of A´ is
E´ = E(o) - E.
The number of states accessible to the entire system A(o) can thus be regarded as a
function of the single parameter E. Let Ω(o) (E) be the number of states accessible to
A(o) when A has an energy between E, E + δE . At equilibrium A(o) must be equally
likely to be found in one of its accessible states. Therefore the probability P(E) of finding
34
the combined system in a state for which A has an energy between E, E + δE is
proportional to Ω(o) (E) .then
P(E) α Ω(o) (E)
(2.48)
(o)
P(E) = CΩ (E)
(2.49)
But when A has an energy E, it can be in any one of Ω(E) possible states. Then A´ can
be in any of Ω´ ( E´ ) = Ω´( E(o) - E ) possible states. Since every possible state of A can
be combined with each one of the states of A´, we must have
Ω(o) (E) = Ω (E) Ω´( E(o) - E )
(2.50)
or
P(E)
= C Ω (E) Ω´( E(o) - E )
(2.51)
To find the dependence of P(E) on E we recall that Ω (E) is an extremely rapidly
increasing function of E . Ω´ ( E´ ) also has a similar dependence on E´. But when E
increases, E´ decreases and therefore Ω´( Eo - E ) decreases very rapidly. The product of
these two functions thus exhibits an extremely pronounced maximum for some value of
% . The maximum of P(E) is given by the condition,
E≈E
∂P
∂
= 0 or equivalantly by
ln P = 0
∂E
∂E
But, ln P = ln C + ln Ω ' (E ' ) + ln Ω(E)
(2.52)
'
∂
∂
∂
∂E
∂
∂
∴
ln P =
ln Ω(E) +
ln Ω ' (E ' )
=
ln Ω(E) −
ln Ω' (E ' )
'
'
∂E
∂E
∂E
∂E ∂E
∂E
P(E)
~
E
E
~
Figure 2.7 The function P(E) which is sharply peaked at E
Define
∂
ln Ω
∂E
~
~
Then the maximum of P is given by β(E ) - β′(E ′) = 0, or
β (E) =
(2.53)
% = β' (E% ' )
(2.54)
β (E)
~
~
´
'
where E and E are the energies of A and A at the maximum. β has the dimensions
(energy )-1. Introduce a parameter T by
35
1
1
, kT =
kT
β
Then, k has the dimension of energy divided by T. Also introduce the function
S ≡ k ln Ω
β ≡
(2.55)
(2.56)
1 ∂S
=
(2.57)
T ∂E
S is, of course, the entropy. The condition of maximum probability is expressible as
ln P(E) = maximum
(2.58)
or
S + S´ = maximum
(2.59)
when this maximum occurs, we have
β = β´
or
T = T´
(2.60)
Comment :
Ω(E) can be written as Ω(E) = w(E) δE, Where, w(E) = density of states. Therefore,
Then,
β( E ) =
∂
∂
ln Ω( E ) =
( ln w( E ) + ln δE ) = ∂ ln w( E )
∂E
∂E
∂E
(2.61)
As δE is independent of E.
If we have a different scale of energy subdivision interval δ*E instead of δE , and a new
Ω*(E) from the number of states between E and E + δ*E, then
Ω( E ) Ω*( E )
= *
δE
δ E
Ω*( E ) = Ω( E )
Then,
δ*E
δE
δ * E 
ln Ω * ( E ) = ln Ω ( E ) + ln 

δ E 
But, Ω α E s
δ * E 
ln Ω * ( E ) ≅ s ln E + ln 

δ E 
But, even if we choose δ * E = 10 23 δ E,
ln Ω * ( E ) = ln Ω ( E ) + ln( 10 23 )
= O( s ) + Oln s = 10 24 + 55
(2.62)
∴
ln Ω * ( E ) = ln Ω ( E )
*
and S = S to a very good approximate and the entropy does not depend on the size of
subdivision of the energy interval.
2.2.9 Approach To Thermal Equilibrium
Let us consider the situation where A and A´ are initially separately in equilibrium and
isolated from each other, their respective energies being very close to Ei and Ei´. Their
respective mean energies are: Ei = E i , E i ′ = Ei ′ . When they are placed in thermal
∴
36
contact, they are free to exchange energy with each other. The situation therefore changes
with time until the systems attain final mean energies E f and E f′ such that
~
~
E f = E , E f′ = E ′
(2.63)
so that the probability P(E) becomes maximum. Then,
β f = β ( E f ) = β f′ = β ( E f′ )
(2.64)
Since the final probability is a maximum and never less than the original one, one can
write
S ( E f ) + S ′( E ′f ) ≥ S ( Ei ) + S ′( Ei′)
(2.65)
We have also
E f + E ′f = Ei + Ei′
S ( E f ) − S ( E i ) + S ( E ′f ) − S ′( E i′ ) ≥ 0
∆S + ∆ S ′ ≥ 0
E f − Ei
i.e., Q + Q´ = 0,
+ E f ′ − Ei ′ = 0
Q =Ef
−Ei ,
Q´ = E ′f − Ei′
(2.66)
(2.67)
(2.68)
(2.69)
(2.70)
Thus there can be two situations
~
1) The initial energies are such that βi = βi′ . Then E i = E and system remain in
equilibrium.
2) If βi ≠ βi ′ , the situation changes towards one of maximum probability or entropy
~
when, finally, E f = E and β f = β f′ .
2.3 Thermodynamics
Thus we see that
1) If two system have the same value of β , they will remain in equilibrium with each
other when put in thermal contact, and
2) If they have different values of β , they will not remain in equilibrium when put in
thermal contact.
Thus if we have three systems A, B, C and if A and B are in equilibrium βA = βB.
Similarly if A and C are in equilibrium when brought into thermal contact, βA = βC .
Therefore βB = βC, i.e., B and C will remain in equilibrium when brought in thermal
contact with each other. This is the zeroth law of thermodynamics.
2.3.1 Some Properties Of The Absolute Temperature
The definition of the absolute temperature is
1
∂
=β=
(ln Ω )
(2.71)
kT
∂E
Since Ω is a very rapidly increasing function of E, β > 0 and hence T > 0.
[ Remark : There are exceptional cases when as E increases, Ω increases upto a certain
point and then decreases. This is true when one does not take into account the
translational degree of freedom of a system but focuses attention on the spin degree of
freedom only. In this case the system has both an upper as well as lower bound to its
37
energy (corresponding to the cases when all spins are lined up anti-parallel or parallel to
the applied field ). Then Ω spin(E) first increases with E and then decreases and it is
possible to get temperature which are positive as well as negative.
ln Ω
E
2.3.2 Equipartition of Energy
Now,
Ω( E ) = E s , ln Ω( E ) = s ln E + C
(2.72)
where C is a constant.
β=
∂
s
ln Ω( E ) =
∂E
E
(2.73)
where, E = E = E%
Thus the energy per degree of freedom is of the order of kT
2.3.3 Direction of Heat Flow
Take two system A, A´ at slightly different initial temperatures βi , βi´ . Then the condition
that the probability must increase as they are brought into thermal contact reads
ΔS + ΔS´≥0, or
∂S
∂S ′
∆E +
∆E ′ ≥ 0 ,or
∂E
∂E ′
E f − Ei E ′f − Ei′
+
≥0
Ti
Ti′
But, E f − Ei + E ′f − Ei′ = 0
Letting Q =
E f −Ei ,
(2.74)
(2.75)
(2.76)
(2.77)
we get
1
1
Q −  ≥ 0
 Ti Ti′ 
(2.79)
and if Q > 0, then
1
1


−
T
 ≥ 0 and Ti′ >Ti
 i Ti′ 
(2.80)
Therefore since A gains energy (Q > 0) , we conclude that heat has flown from A´ to A.
Thus heat flows from a body to a higher temperature to one at a lower temperature.
38
2.3.4 Heat Reservoirs
Consider two systems; one of which is very large compared to the other, in thermal
contact. In terms of the notation used already let A´ be the system much larger than A.
A´ then acts as heat reservoir or heat bath. This means the temperature of A´ remains
unchanged even if it absorbs any amount of heat (Q ´) from the smaller system A. Thus,
∂β′
Q ′ << β′
∂E ′
(2.81)
∂β ′
β′
E
Since
is of the order of
and Q´ is ~ E we have
<< 1
∂E ′
E′
E′
Therefore we have,
∂
1 ∂2
2
′
′
ln Ω ′ ( E ′ + Q′ ) − ln Ω ′ ( E ′ ) = ln Ω ′ ( E ′ ) Q′ +
ln
Ω
Q
+
2
∂ E′
2 ∂ E′
(2.82)
= β ′ Q′ +
1 ∂β ′ 2
Q ′ + ........
2 ∂ E′
= β ′ Q′ =
Q′
kT ′
(2.83)
(2.84)
Q′
(2.86)
T′
This relation is true for any system which absorbs an infinitesimal amount of heat from
another at a slightly higher temperature.
∴∆S ′ = k [ ln Ω′( E ′ + Q ′) − ln Ω′( E ′) ] =
dS =
dQ
T
(2.87)
2.3.5 Additivity of S
We saw that for two systems A and A' which interact thermally, one has for the number
of states accessible to the total system A( o ) = A + A' ,
Ω(o) (E) = Ω (E) Ω ' ( E(o) - E )
(2.88)
~
We also argued that this function has a very sharp maximum at E = E .
The total number of states accessible to the entire system A(o) is
∴ Ω (toto) = ∑ Ω ( o ) ( E )
(2.89)
E
where the sum runs over all possible values of the energies E of A.
∴ Ω (toto) = ∑ Ω( E ) Ω ′ ( E ( o ) − E )
E
(2.90)
But this can be written as
~
~ ∆* E
( 0)
Ωtot
= Ω( E )Ω′( E ( 0 ) − E )
δE
(2.91)
39
where ∆*E is the region where Ω(o) is appreciably different from zero and approximately
equal to its maximum value. Then,
∆* E
~
( 0)
S ( 0 ) = k ln Ωtot
= k ln Ω( 0 ) ( E ) + k ln
δE
(2.92)
The second term is utterly small compared to the first as it is of the order if ln(s) at most
~
~
(0) ~
S ( 0 ) = k ln Ωtot
( E ) = k ln Ω( E ) + k ln Ω′( E ′) = S ( E ) + S ( E ′)
(2.93)
Thus the entropy has the desired additivity property.
2.3.6 Equilibrium Between Interacting Systems
If two systems can interact with each other thermally and mechanically, then, the system
A can be characterised by its total energy E and external parameters x1….xn. A´ is
specified by E´ and x1´ ….xn´. The total system A(o) = A + A' is isolated and its energy
E(o) is E(o) = E + E' is constant.
E' is determined if E is known. Similarly x' depends x-s . Thus in this case,
Ω(o) = Ω(o) (E , x1….xn )
(2.94)
~
~
(o)
x
=
x
and Ω has a sharp maximum for E = E and α
. The equilibrium situation
α
~
xα .
corresponds to one of maximum probability when E = E and xα = ~
2.3.7 Infinitesimal Quasi-static Process
A quasi-static process for a system is one for which the system is practically always in
equilibrium. Consider a quasi-static process in which A interacts with A´ and changes
from an equilibrium state described by E and xα to an infinitesimally different
equilibrium state described by E + dE , xα + dxα . Then,
∂ ln Ω
∂
d ln Ω =
ln Ω dE + ∑
dxα
(2.95)
∂E
∂ xα
α
Consider the case where there is one external parameter, the volume of the system V.
∂ ln Ω
d ln Ω = β dE +
dV
(2.96)
∂V
Define the mean pressure p as
∂ ln Ω
βp=
(2.97)
∂V
Then, d ln Ω=βdE +β p dV =βdQ
(2.98)
Or, T dS =dE + p dV
(2.99)
Recognising pdV as the work done by the system, we write (In a lot of current
chemistry literature, a different convention is used, where, dE =Q +dW and this
dW = −pdV which is the work done on the system)
dQ =T dS =dE +dW
(2.100)
Here dQ is the heat absorbed by A,
and dS = dQ / T
(2.101)
40
for a quasi-static change when the external parameters vary. In the special case when the
system is thermally isolated dQ = 0,
∴ dS = 0.
(2.102)
Therefore if the external parameters of a system that is thermally isolated are changed
quasi-statically to even a finite amount,
∆S = 0.
(2.103)
2.3.8 Equilibrium
Considering the case where the external parameter is the volume V, we have for two
interacting systems A and A' , and the combined system A(o),
Ω ( o ) ( E ,V ) = Ω( E ,V ) Ω ′ ( E ′,V ′)
E (o) = E + E ′
(2.104)
V (o) = V + V ′
The maximum value for Ω(o) corresponds to equilibrium.
∴
d ln Ω ( o ) = 0
or
d ( ln Ω + ln Ω ′ ) = 0
(2.105)
∂
∂
ln Ω dE +
ln Ω dV
∂E
∂V
= β dE + β p dV
and d ln Ω ′ = β ′ dE ′ + β ′ p′ dV ′ = − β ′ dE − β ′ p′ dV
∴
( β − β ′ )dE + ( β p − β ′ p′ )dV = 0
Since this is true for arbitrary changes dE, dV,
β = β ′ , β p = β ′ p′
∴
p = p′
But
d ln Ω =
(2.106)
2.3.9 Limiting Behaviour of The Entropy
As one goes to lower energy, every system approaches the lowest possible energy Eo of
its ground state. Corresponding to the ground state energy there is one or at most a
relatively small number of states for the system. At relatively high energies Ω α Es and
S = k ln Ω ~ kS. However for the lowest energy itself, S ~ k ln s or less. Since this is
negligible compared to ks , one may say
as E → Eo , S → 0
(2.107)
Now E → Eo corresponds to T → 0.
∴ S → 0, as T → 0.
(2.108)
For two system in interaction, ∆E = - ∆E´, and
dS =
∂S
1 ∂2 S
∆E +
( ∆E ) 2 + ....
2
∂E
2 ∂E
∂ S′
1 ∂ 2S′
2
′
dS ′ = ∆ E ′ +
(
∆
E
)
+ ....
2
∂ E′
2 ∂ E′
41
′ ∂ S ∂ S′  1 2 ∂ S ∂ S′ 
dS = dS + dS =  −  ∆ E + (∆ E)  2 2 + 2 
 ∂ E ∂ E′  2  ∂ E ∂ E′ 
2 2
( )0
At or near equilibrium,
∂S
∂S ′
=
∂E ∂E ′
∂ S
∂ 1
1 ∂T
=
=− 2
,
2
∂E T
∂E
T ∂E
2
dS
Hence,
(0)
∂ 2S′ 1 ∂ T ′
=− 2
2
∂ E′ T ′ ∂ E
1 (∆E ) 2  ∂T ∂T ′ 
=−
+

>0
2 T 2  ∂E ∂E ′ 
∂T
∂E
>0;
>0
∂E
∂T
(2.109)
Now we can summarize the basic thermodynamic laws and the corresponding statistical
mechanical relations.
2.3.10 The Laws of Thermodynamics
Zeroth law : Already described earlier.
First law : Conservation of energy : If an equilibrium macrostate of a system goes
from one state to another
∆E = − W + Q
where, W is work done by the system and Q, the heat absorbed by the system.
(We have already pointed that a different convention also exists where, ∆E =Q +W
where, this W is the work done on the system)
A system in isolation can be characterised by E (internal energy ) such that E is
constant.
Second law : An equilibrium marcostate of a system can be characterised by a quantity
S ( the entropy ) which has properties:
a) in any process in which a thermally isolated system goes from one state to another,
∆S ≥ 0
(2.110)
42
b) if the system is not isolated and undergoes a quasi-static infinitesimal process in
which it absorbs heat dQ , then
dQ
(2.111)
ds =
T
where, T is the absolute temperature of the system.
Third law : The entropy of a system has the property that
as T → 0, S → So
(2.112)
The statistical relations that we have used are
1) S = k ln Ω
2) Probability P α Ω α eS/k
These allow us to relate the microscopic knowledge of the system to macroscopic
thermodynamic laws. Using statistical mechanics, one can calculate the thermodynamic
quantities from first principles as well as the fluctuations of quantities around their mean
values.
Assumptions of statistical mechanics :1) All accessible states of a system are equally likely to occur.
2) Averages of physical quantities over an ensemble are equal to their equilibrium
values. For a large system, the equilibrium values correspond to the most probable
values.
3) An equilibrium state corresponds to a more probable state (law of increase of
entropy).