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Thermodynamics Y Y Shan

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Part II Statistical thermodynamics—The Microscopic theory

Introduction

The classical theory of thermodynamics leaves certain questions unanswered:

The statistical theory can answer all these.

The M a crostates and M i crostates of a thermodynamic system:

Classical thermodynamics describes macroscopic systems in terms of a few

variables (state functions). A macrostate of a system refers to its macroscopic properties

such as temperature and pressure, etc.

But such a system is really made of millions of atoms, so a much richer

description must be possible in principle: we could specify the quantum states of all

individual atoms. This suggests we can calculate the macro-state of a system by

averaging over the corresponding huge amount of microstates of the system. We can

derive thermodynamics from the quantum behavior (quantum states) of atoms and

molecules.

In Statistical thermodynamics, a microstate of a system describes a specific

detailed microscopic configuration of a system. The crucial link from microscopic to

macroscopic properties is as follows. If the value of some quantity in the th

microstate is i X , and the probability that the system is in that microstate is

i p , then the

value of in the macrostate is the average :

!="#i

ii X p X

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Chapter 6 Background knowledge and concepts for statistical thermodynamics

1. A (quantum) State of a particle: For micro-particles, they have quantum states,

described by their wave functions,n

ψ .

If a “particle” were a die, for instance, obviously it has six states:

For the throw of a single die, all the six states are equally probable. Given one throw of a

die, what is the probability of getting a state, such as “2-dots” state?

If a “particle” is a token (blue on one side and green on the other), it has two

states: state “blue” and state “green”. Given one toss of a token, the probability of getting

a “blue” state (or a “green” state) is ½.

2. A Microstate of a system (consists of many particles): defined in terms of the current

behavior of all the constituent particles(atoms). If a system contains many micro-

particles, the system will have many microstates, depending on the combination of the

states of each individual particle.

Example (i): A system has two “particles”—two dice (distinguishable)

The combination of the states from each “particle” forms microstates of the system of

two-dice, such as: are five different

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microstates of the “2-particle” system. Its total number of the microstates of the system

is:

36661

6

1

6 =×=⋅=Ω PP

where the distribution showing total-2-dots (T2) of the system has one “microstate”, thedistribution showing total-6-dots (T6) has five “microstates”

Example (ii): A system having 36 chips (indistinguishable—identical chips): one

side is blue and the other is green.

Now we toss them and place them on a 6 by 6 checkerboard. Many different

patterns are possible to be generated. Each pattern (formation) stands for a microstate of

the system, the following shows six possible microstates of the system.

The total number of microstates of the system:

( ) 1026261

2

1

2

1

2

1

21087.62 ×===⋅⋅⋅=Ω PPPP ! .

3. The principle of equal a priori probabilities

The fundamental postulate (assumption) in statistical mechanics is the principle of

equal a priori probabilities.

It states: An isolated system in equilibrium, each microstate of the system has the

same probability to be observed, i.e., the system is equally likely (equal probability) to be

found in any of its accessible microstates. Or a system in equilibrium does not have any

preference for any of its available microstates.

We useΩ

for the number of such microstates, so the probability of the systembeing in any one microstate is

11

,1

=

Ω=

Ω= !!

ΩΩ

iiii

p p

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Examples: (i) For the two-dice system,

What is the probability of throwing to get a microstate of “2-dots + 3-dots” ?

The probability of the system at any microstates:

is same and equal to:3611 =

Ω

Example (ii): For the 36-pieces counter system: the following shows six

microstates of the system.

Each microstate of the system is equally likely to occur. The probability of the systemaccessing to any microstate is same and equal to:

11

361046.1

2

11 −×==

Ω=

i p

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4. Distribution, and Distributions in statistical physics

4.1 Distribution, the number of microstates for a distribution ,and most probable

distribution

The concept of distribution: A distribution (of a system, having N particles) is agroup of microstates which all satisfy a particular condition (or constrain), i.e. each

microstate of the distribution has something in common. These conditions can be: fixed

total energy of all microstates of a distribution, particular particle characters

(distinguishable, indistinguishable, etc.)

Example: For the system of two-dice

If a condition set for the microstates is: the total “dots” must be five (T5),

therefore the number of microstates of the distribution will be 4. If the condition set for

the microstates is: the total “dots” must be seven (T7), then the total number of

microstates of the distribution is 6.

There are 6 ways (6 states) to get a total of seven-dots (T7), but only 1 way (1 state) to

get two-dots (T2). From the principle of equal a priori probabilities, the probability of

showing each microstate is same and equals to 1/36 . The probability of getting the

distribution consisting of T4 states is: 111.0)36 / 1(4 =× . The probability of getting

the distribution consisting of T7 states is the highest: 167.0)36 / 1(6 =× .

It is concluded that the more microstates a distribution has, the higher probability

the distribution will appear. A distribution has maximum number of microstates is called

a most probable distribution.

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4.2 System with Distinguishable or indistinguishable particles

Example (i) : the 36-counters system: Looking into a particular distribution (of

the system), of which all its microstates are under the constraint of having “15-Green”

(the rest “21-Blue”), 15 ,36 == n N .

(a) Distinguishable particles: If those “Green”s and those “Blue”s can be

distinguished, what is the total number of microstates for this particular

distribution, of witch all the microstates have 15-Green (and 21-Bluie)?

There are ways of putting distinguishable counters

411071993.312...3536!36! ×=⋅⋅⋅⋅===Ω N dis

(b) Indistinguishable particles: If “Green”s and “Blue”s can not be distinguished,

what is the total number of microstates for this particular distribution, where all its

microstates have 15-Green?

This is the common problem of splitting a group of into two smaller groups, of

and n N − , without caring about the ordering in each group, and the number of ways

of doing it is

)!(!

!

n N n

N C n N Ind

−==Ω

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There are ways of putting distinguishable counters. However if of the

counters are green, there are ways of arranging the green counters among themselves

without changing the pattern, and )!( n N − ways of arranging the blues. Here,

For , so the total is

915

36 10568.5)!12!15 /(!36 ×=

⋅==

Ω C .

For there are only810

3610542.2)!26!10 /(!36 ×=⋅==Ω C ,

It is found that:

For ,918

3610075.9)!18!18 /(!36 ×=⋅==Ω C , is the the maximum. We

call a distribution which has maximum number of microstates the “most probable

distribution”. Hence the above distribution with 18-green among 36 in total is the most

probable distribution. The numbers )!(!

!

n N n

N

C

n

N −=

are called the binomial coefficients,

since they enter the binomial expansion.

Example (ii) two-dice system.

(a) Distinguishable particles:

If , for e.g. , then the two

“particles” (dice) are said to be distinguishable.

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As for the particular distribution: “total-dots-is-seven”, its microstates is 6, which is

the most probable distribution. The total microstates of the system from all

distributions (11) are 36.

(b) Indistinguishable particles:

If, for e.g., , then

the two “particles” (dice) are said to be indistinguishable. Now the number of

microstates of the “total-dots-is-seven” distribution reduces 6 to 3, while the total

number of microstates of the system decreases from 36 down to 21.

(c) If two indistinguishable particles can not have the same individual states (particles

are so called Fermions), Now the total number of microstates of the system further

reduced from 21 to 15. While, for e.g., the number of microstates for the distribution of“total-dots-is-six” will be reduced from 3 to 2.

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4.3 Distributions in particle statistics:

For those examples in 4.2, the energies of each particle at its different individual

states (for real particles, these will be quantum states, such as electrons, atoms etc.) arenot assigned (or taken into account).

However, in statistical thermodynamics, the total energy of a system containing large

amount of particles is very important. Therefore, energies of (quantum) states of

individual particles must be under consideration when discussing the energies of

microstates of different distributions from systems consisting of real particles.

These distributions, concerning energies of individual states, are namely Maxwell–

Boltzmann distribution (particles are distinguishable), Fermi–Dirac distribution, and

Bose–Einstein distribution (particles are in distinguishable),

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5. Some mathematics:

(i) 12...)2()1(! ⋅⋅⋅−⋅−⋅= nnnn , 1!0 ≡

(ii)i

i

iaaaa ⋅⋅⋅=∏ ...

21

(iii) Permutations and Combinations

If you have a collection of N distinguishable objects, then the number of ways

you can pick a number r of them (r < N) is given by the permutation relationship:

)!(

!

r N

nP r

N −

=

For example if you have six persons for tennis, then the number of pairings for

singles tennis is

3056)!26(

!62

6 =⋅=

−=P

But this really double counts, because it treats the match of A vs B as distinct from

the match of B vs A for players A and B. If you don't want to take into account the

different permutations of the elements (i.e. r), then you must divide the above expression

by the number of permutations of r (here r=2). This result is called a "combination". The

combination relationship is

!)!(

!

r r N

N C r

N −

=

The number of tennis matches is then the combination

152

56

!2)!26(

!62

6 =

⋅=

−=C

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