Mathematics

1. Basic ideas of statistical physics
Dr. J P SINGH
Associate Professor in Physics
PGGC-11, Chandigarh

2. Statistics is a branch of science which deals with
the collection, classification and interpretation
of numerical facts. When statistical concept are
applied to physics then a new branch of science
is called Physics Statistical Physics.
Basic ideas of statistical physics

3. Trial → experiment→ tossing of coin
Event → outcome of experiment
Exhaustive events The total number of
possible outcomes in any trial
For tossing of coin exhaustive events = 2

4. Favourable events number if possible
outcomes (events) in any trial
Number of cases favourable in drawing a king from a
pack of cards is 4.
Mutually exclusive events no two of them
can occur simultaneously.
Either head up or tail up in tossing of coin.
Equally likely events every event is equally
preferred.
Head up or tail up

5. Independent events if occurrence of one
event is independent of other
Tossing of two coin
Probability
The probability of an event =
ways
of
number
Total
occurs
event
the
which
in
cases
of
Number

6. If m is the number of cases in which an event occurs and
n the number of cases in which an event fails, then
Probability of occurrence of the event =
The sum of these two probabilities i.e. the total
probability is always one since the event may either
occur or fail.
Probability of failing of the event =
n
m
m

n
m
n

n
m
n
n
m
m
or




1

7. Tossing of two coins :
The following combinations of Heads up (H) and
Tails up(T) are possible :
2
1
,
2
1
2
1 
 H
H
2
1
,
2
1
2
1 
 T
T
4
1
2
1
2
1
2
1 


H
H
4
1
2
1
2
1
2
1 


T
T
4
1
2
1
2
1
2
1 


H
H
4
1
2
1
2
1
2
1 


H
H
n
P
P
P
P .
..........
2
1 



8. Principle of equal a priori probability
The principle of assuming equal probability for
events which are equally likely is known as the
principle of equal a priori probability.
A priori really means something which exists in
our mind prior to and independently of the
observation we are going to make.

9. Distribution of 4 different Particles in
two Compartments of equal sizes
Particles must go in one of the compartments.
Both the compartments are exactly alike.
The particles are distinguishable. Let the four particles
be called as a, b, c and d.
The total number of particles in two compartments is 4
i.e.



2
1
4
i
i
n

10. The meaningful ways in which these four particles can
be distributed among the two compartments is shown
in table.

11. Macrostate
The arrangement of the particles of a system without
distinguishing them from one another is called
macrostate of the system.
In this example if 4 particles are distributed
in 2 compts, then the possible macrostates (4+1) =5
If n particles are to be distributed in 2 compts.
Then the no. of macrostates is = n+1

12. Microstate
The distinct arrangement of the particles of a system is
called its microstate.
For example, if four distinguishable particles
are distributed in two compartments, then
the no. of possible microstates (16)
If n particles are to be distributed in 2
compartments. The no. of microstates is
= 24
= 2n
=(Compts)particles

13. Thermodynamic probability or frequency
The numbers of microstates in a given macrostate is
called thermodynamics probability or frequency of that
macrostate.
For distribution of 4 particles in 2 identical
compartments
W(4,0) =1
W(3,1) =4
W(2,2) = 6
W(1,3) = 4
W(0,4) =1

14. W depends on the distinguishable or indistinguishable
nature of the particles. For indistinguishable particles,
W=1
Micro-
Comp 1
States
Comp 2
macrostate Frequency
W
probability
(4,0) 1
(3,1) 1
(2,2) 1
(1,3) 1
(0,4) 1
5
1
5
1
5
1
5
1
5
1

15. All the microstates of a system have equal a priori
probability.
Probability of a microstate =
microstate
of
no
Total .
1
Probability of a macrostate =
(no. of microstates in that microstate) 
(Probability of one miscrostate)
n
W
W
W
2
1
2
1
16
1
4






= thermodynamic probability× prob. Of one
microstate
n
2
1
2
1
16
1
4




16. Constraints
Restrictions imposed on a system are called constraints.
Example
total no. particles in two compartments = 4
Only 5 macrostates (4.0), (3,1), (2,2),(1,3),(0,4) possible
The macrostates (1,2), (4,2), (0,1), (0,0) etc not possible

17. The macrostates / microstates which are allowed under
given constraints are called accessible states.
Accessible and inaccessible states
The macrostates/ microstates which are not allowed
under given constraints are called inaccessible states
Greater the number of constraints, smaller the number
of accessible microstates.

18. Distribution of n Particles in 2 Compartments
The (n+1) macrostates are
(0, n) (1, n, 1)… (n1, n2)…… (2, n2),….. (n 0),
Out of these macrostates, let us consider a particular
macrostate (n1, n2) such that
n1 + n2 = n
n particles can be arranged among themselves in
nPn = n! ways

19. These arrangements include meaningful as well as
meaningless arrangements.
Total number of ways = (no. of meaningful ways)
 (no.of meaningless ways)
n1 particles in comp. 1 can be arranged in
= n1 ! meaningless ways.
n2 particles in comp. 2 can be arranged in
= n2 ! meaningless ways.
n1 particles in comp. 1 and n2 particles in comp. 2 can be
arranged in
= n1 !  n2 ! meaningless ways.

20. ways
s
meaningles
of
no.
ways
of
number
Total
ways
meaningful
of
no. 
)!
–
(
!
!
!
!
!
)
,
(
1
1
2
1
2
1
n
n
n
n
n
n
n
n
n
W 

The total no. of microstates = 2n
n
n
n
n
n
n
2
1
.
!
!
!
)
,
(
2
1
2
1 

Prob. of distribution (r,n-r)
n
r
n
r
n
r
n
r
2
1
.
)!
(
!
!
)
,
(





21. Deviation from the state of Maximum
probability
n
r
n
r
n
r
n
r
2
1
.
)!
(
!
!
)
,
(




The probability of the macrostate (r, n r) is
When n particles are distributed in two comp., the
number of macrostates = (n+1)
The macrostate (r, n r) is of maximum probability if r =
n/2, provided n is even.
The prob. of the most probable macrostate 






2
,
2
n
n

22. n
n
n
n
P
2
1
!
2
!
2
!
max 

Probability of macrostate is slightly deviate from most
probable state by x.(x<<n)
Then new macrostate will be 






 x
n
x
n
2
,
2
n
x
x
x
x
n
n
P
2
1
!
2
!
2
!

















23. !
2
!
2
!
2
2
max





















x
n
x
n
n
P
Px
stirling’s formula n
n
n
n 
 ln
!
ln
Taylor’s theorem
1
|
y
|
provided
......
3
2
)
1
ln(
3
2






y
y
y
y
Do at home to get

24. 









2
exp
2
max
n
f
P
Px
cell
a
in
particles
of
no.
probable
most
from
deviation
fractional
the
is
2
/
f
Where
n
x


25. Discussion
n
103 0.999
106 0.607
108
1010
50
e
1
5000
1
e
max
P
Px

26. Thus we conclude that ass n increases the prob. of a
macrostate decreases more rapidly even for small
deviations w.r.t. the most probable state.
n1 > n2 > n3
(2x / n)
0.2 0.1 0 0.1 0.2
n3
n2
n1

27. Static and Dynamic systems
Static systems: If the particles of a system remain
at rest in a particular microstate, it is called static
system.
Dynamic systems: If the particles of a system
are in motion and can move from one microstate to
another, it is called dynamic system.

28. Equilibrium state of a dynamic system
A dynamic system continuously changes from one
microstate to another. Since all microstates of a
system have equal a priori probability, therefore, the
system should spend same amount of time in each of
the microstate.
If tobs be the time of observation in N microstates
The time spent by the system in a particular macrostate
N
t
t obs
m 
Let microstate has frequency
)
,
( 2
1 n
n )
,
( 2
1 n
n
W

29. Time spend in macrostate )
,
( 2
1 n
n
microstate
of
No.
microstate
each
in
spend
time
Average 

)
,
( 2
1 n
n
t
)
,
(
)
,
( 2
1
2
1 n
n
W
N
t
n
n
t obs


)
,
(
)
,
( 2
1
2
1 n
n
P
t
n
n
t obs 

obs
t
n
n
t
n
n
P
)
,
(
)
,
( 2
1
2
1 
That is the fraction of the time spent by a dynamic
system in the macrostate is equal to the probability
of that state

30. Equilibrium state of dynamic system
The macrostate having maximum probability is termed
as most probable state. For a dynamic system consisting
of large number of particles, the probability of deviation
from the most probable state decrease very rapidly.
So majority of time the system stays in the most
probable state. If the system is disturbed, it again tends
to go towards the most probable state because the
probability of staying in the disturbed state is very
small. Thus, the most probable state behaves as the
equilibrium state to which the system returns again and
again.

31. Distribution of n distinguishable
particles in k compartments of unequal
sizes
The thermodynamic prob. for macrostate )
....
,
,
( 3
2
1 k
n
n
n
n
!
!.....
!
!
)
.......
,
(
2
1
2
1
k
k
n
n
n
n
n
n
n
W 


 k
i
i
n
n
1
!
!
Let the comp. 1 is divided into no. of cells
Particle 1st can be placed in comp.1 in = no. of ways
1
g
1
g
Particle 2nd can be placed in comp.1 in = no. of ways
1
g
1
g
Particle can be placed in comp.1 in = no. of ways
th
n1

32. particles in comp. 1 can be placed in =
1
n 1
1
n
g
particles in comp. 2 can be placed in =
2
n 2
2
n
g
particles in comp. k can be placed in =
k
n k
n
k
g
total no. ways in which n particles in k comparmrnts
can be arranged in the cells in these compartments is
given by
k
n
k
n
n
n
g
g
g
g ......
.
. 3
2
1
3
2
1

i
n
i
k
i
g
1




33. Thermodynamic probability for macrostate is
k
n
k
n
n
k
k g
g
g
n
n
n
n
n
n
n
W )
....(
)
(
)
(
!
!.....
!
!
)
.......
,
( 2
1
1 2
2
1
2
1 




k
i i
n
i
n
g
n
W
i
1 !
)
(
!