1. Basic ideas of statistical physics

2. Basic ideas of statistical physics
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.

3. Trial → experiment→ tossing of coin
Event → outcome of experiment
Sample Space A set of all possible distinct outcomes of a experiment.
S = {H, T } for tossing of a coin
Exhaustive events The total number of possible outcomes in any trial
For tossing of coin exhaustive events = 2

4. Favorable events number of possible outcomes (events) in any trial
Number of cases favorable 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
E.g: Tossing of two coin
Simultaneous events two or more events occurring at the same time
E.g: Tossing of two coin together
Probability
The probability of an event =

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 =
Probability of failing 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.

7. 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.

8. For mutually exclusive events:
For two mutually exclusive events A and B, the probability of occurrence of either event A or B is
= P(A)+P(B)
For independent events:
For two independent events A and B, the probability that both the events occur is
= P(A)×P(B)
For n-events

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.

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

11. Macrostate
The specification of the number of particles in each compartment is called macrostate of the system. Or
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)
= 24
If n particles are to be distributed in 2 compartments. The no. of microstates is
= 2n
=(Compts)particles

13. Thermodynamic probability or frequency (W)
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)
(2,2)
(1,3)
(0,4)

15. Probability of a microstate =
All the microstates of a system have equal a priori
probability.
Probability of a microstate =
Probability of a macrostate =
(no. of microstates in that microstate) 
(Probability of one miscrostate)
= thermodynamic probability× prob. Of one
microstate

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. Accessible and inaccessible states
The macrostates / microstates which are allowed under
given constraints are called accessible 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
Consider n distinguishable particles in two compartments of equal size of a box.
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
The total no. of microstates = 2n

19. n particles can be arranged among themselves in
nPn = n! ways
These arrangements include meaningful as well as meaningless arrangements.
Total number of ways = (no. of meaningful ways)  (no. of meaningless ways)

20. 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.

21. Now, the number of meaningful arrangements in a given macrostate is equal to the number of microstates in that macrostate.
The number of microstates in a given macrostate is called Thermodynamic probability (W).
therefore thermodynamic probability of macrostate (n1, n2) is

22. Prob. of distribution (r, n-r)
Now, the probability of a macrostate is equal to the ratio of thermodynamic probability to total number of microstates in the system
Therefore probability of macrostate (n1, n2)is given by
The total no. of microstates = 2n
Prob. of distribution (r, n-r)

23. Maximum Probability:
When r=n/2 (n=even)
Therefore
Minimum Probability:
When r=0 or r=n
Therefore

24. Stirling’s formula

25. Deviation from the state of Maximum probability
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

26. Let us deviate probability of macrostate slightly from most probable state by x ( x << n )
Then new macrostate will be

27. stirling’s formula
Taylor’s theorem

29. Discussion: Consider deviation of the order of 0.1 i.e. 10-3 n 103 106
0.999
106
0.607
108
1010

30. n1 > n2 > n3
(2x / n)
0.2
0.1
0.1
0.2 n3 n2 n1

31. Thus we conclude that as n increases, the prob
Thus we conclude that as n increases, the prob. of a macrostate decreases more rapidly even for small deviations w.r.t. the most probable state.
If a graph is plotted between and f , then the probability distribution curve becomes narrower and narrower as n increases.
Thus if n is very large then the system stays very near to most probable state.

32. 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.

33. 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 microstate
Let macrostate has frequency

34. Time spend in macrostate
That is the fraction of the time spent by a dynamic system in the macrostate is equal to the probability of that state

35. 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.

36. Distribution of n distinguishable particles in k compartments of unequal sizes
The thermodynamic prob. for macrostate
Let the comp. 1 is divided into no. of cells
Particle 1st can be placed in comp.1 in = no. of ways
Particle 2nd can be placed in comp.1 in = no. of ways
Particle can be placed in comp.1 in = no. of ways

37. particles in comp. 1 can be placed in =
particles in comp. k can be placed in =
total no. ways in which n particles in k comparmrnts can be arranged in the cells in these compartments is given by

38. Thermodynamic probability for macrostate is