Transcript Statistics_Probability

SOME BASIC IDEAS OF
STATISTICAL PHYSICS
Mr. Anil Kumar
Associate Professor
Physics Department
Govt. College for Girls,
Sector -11, Chandigarh.
Introduction


Statistics is a branch of science which deals with
the collection, classification and interpretation of
numerical facts or data. The aim of this science
is to bring out a certain order in the data by the
use of laws of probability. The application of
statistical concepts of Physics has given rise to a
new branch of physics known as Statistical
Physics.
Statistical Physics establishes the relation
between macroscopic behaviours (bulk
properties) of the system in terms of its
microscopic behaviour (individual properties). It
is not concerned with the actual motions or
interactions of the individual particles but
explores the most probable behaviour of
assembly of particles.
Probability
 The probability of an event =
Number of cases in which the event occurs
Total number of cases
 If a is the number of cases in which an event
occurs and b the number of cases in which an
event fails, then
a
Probability of occurrence of the event = a  b
Probability of failing of the event =
b
ab
The sum of these two probabilities i.e. the total
probability is always one since the event may
either occur or fail.
Some Probability Considerations
 Tossing of a coin :
If we toss a coin. Either the head can come
1
upward or the tail i.e PH =PT = 2
 Tossing of two coins :
The following combinations of Heads (H)
and Tails (T) facing upwards are possible :
HH. HT, TH, TT,HT. , the chance of one of
them taking place (say that of HH) is,
1
PHH = 4 = PH. PH
Independent events
Two or more events are said to be independent if
the occurrence of one is not influenced by the
occurrence of others.
Consider two independent events which occur
simultaneously or in succession.
Let P1 and P2 be the probabilities of the individual
events,
The probability of occurrence of the composite
event
P = P1  P2
Similarly, for n independent events to take place
together the probability
P = P1 . P2 ……. Pn
This is known as multiplicative law of probability.
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.
Distribution of 4 different Particles in two
Compartments of equal sizes



Both the compartments are exactly alike.
The particles are distinguishable from one
another. Let the four particles be called as a, b,
c and d.
The total number of particles in two
compartments is 4 i.e.
i2
 ni  4
i 1
The meaningful ways in which these four
particles can be distributed among the two
compartments is shown in table.
Contd..
Microstate and Macrostate
 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 sixteen
microstates are possible. If n particles are to be
distributed in 2 compartments. The no. of
microstates is 2n.
 Macrostate
The arrangement of the particles of a system
without distinguishing them from one another is
called macrostate of the system.
For example, if four particles are to be distributed
in two compartments without distinguishing among
the particles, then there are five possible
macrostates. If n particles are to be distributed in 2
compartments. The no. of macrostates is (n+1).
 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(3,2) = 6
W(1,3) = 4
W(3,1) = 4
W(0,4) = 1
W depends upon the distinguishable or
indistinguishable nature of the particles.
For indistinguishable particles,W=1
 It must be noted that
 All the microstates of a system have
equal a priori probability.
1
 Probability of a microstate = Total number of microstates
 Probability of a macrostate = (number of
microstates in that microstate) 
(Probability of one miscrostate)
Number of miscrostates in it
= Total number of miscrostates
=Thermodynamic probability 
probability of each microstate
= W.p.
Constraints
The set of condition that must be obeyed by a
system are called constraints.
 All those macrostates / microstates which are
allowed by the constraints on the system are
known as accessible macrostates/microstates and
the macrostates/ microstates forbidden on account
of constrains are known as inaccessible
macrostates/microstates.
The constraints on the system play an important
role in determining the number of accessible
macrostates / microstates. Greater the number of
constraints, smaller the number of accessible
microstates.
Distribution of n Particles in 2
Compartments
The various macrostates (distributions) of the
system are : (o, n) (1, n, 1) (2, n2),….. (n 0), i.e.,
(n + 1) in number. 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 a
total number of ways
nP = n!
n
These arrangements include meaningful as well as
meaningless arrangements.
Total number of ways = (Number of meaningful
arrangements)  (Number of meaningless
arrangements)
n1 particles in compartment 1 and n2 particles in
compartment 2 can be arranged in heir respective
compartments in n1 !  n2 ! meaningless ways.
The number of meaningful arrangements (i.e, the
number of microstates) in the macrostate (n1, n2)
n!
n!
n
n


C

C n2
W(n1, n2) =
n1
n1! n 2 ! n1! (n – n1 )!
The total no. of microstates=2n
1
n!
( n 1 , n 2 ) 

n n !n !
2
1 2
Deviation from the state of Maximum
probability
When n particles are distributed in two
compartments, the number of macrostates comes
out to be (n+1). The macrostate
(r, n r) is of maximum probability if r = n/2,
provided n is even. The probability of the
macrostate (r, n r) is
n!
1

P(r, n  r) =
r! ( n  r )! 2 n
n n
The probability of the most probable macrostate  , 
2 2
n!
1

Pmax = n n
n
2
! !
2 2
n
n

 x,
x

2
2

Probability of macrostate
x<<n
slightly different from most probable state is
Px =
Px =
n!
1

n
 x

2n
  x !   x  !
2
 2

2
n 
 !
2 
Pmax
n
 n

 x  !
 x !

2
 2

….(1)
Using stirling’s formula
ln n = nln nn
and using Taylor’s theorem
y2
y3
ln (1+ y) = y 
provided | y | < 1

 .......... .......... ....
2
3
(1)
on simplification gives

f 2x

Px  Pmax exp 

2

x




Where f= n / 2 = fractional deviation from most
probable no. of particles in a cell.
Discussion
For f = 103
Px
5n 107
e
Then
Pmax
The ratios for different values of n are given in the table
n
Px
Pmax
103
0.999
106
0.607
108
1
1010
e 50
1
e
5000
It is apparent that as n increases the probability for 0.1%
deviation from the most probable state goes in decreasing very
rapidly..
Thus, as n increases, the probability of a macrostase falls off more
and more rapidly even for a slight deviation with respect to the
most probable macrostate. The probability distribution curve
(drawn between Px/Pmax versus f) becomes narrower and
narrower as n increases (figure). When n is very large the
macrostates, deviating even by very-very slight amount w.r.t. the
most probable macrostate, become extremely improbable and
the system may be expected to exist practically in the most
probable macrostate. Therefore, the properties of the system will
be the same as those deduced from the most probable state.
n1 > n2 > n 3
Px
Pmax
n3
n2
n1
0.2
0.1
(2x / n)
0
0.1
0.2
Static and Dynamic systems
Static systems:
If the constituent particles of a system
remain at rest in a particular microstate,
it is called static system.
Dynamic systems:
If the constituent particle of a system
can move so that the system goes from
one microstate to another, it is called
dynamic system.
Time spent by a dynamic system in a
Particular Macrostate
A dynamic system continuously changes from one microstate to another. 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 to be the time of observation.
So, on the average, we can assume that :
t0  total number of microstates of the system
Or
t0  N (say)
Let ‘t’ be the time spent by the system in a particular macrostate.
Then
t  number of microstates in the macrostate.
But number of microstates = thermodynamic probability (W) of the macrostate.

t  thermodynamic probability of the macrostate
or
tW
t
t0

W
N
= P, Probability of the macrostate
That is the fraction of the time spent by a dynamic system in the
macrostate is equal to the probability of that state
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.
Distribution of n Distinguishable Particles
in k compartments of unequal sizes
 For the macrostate (n1 , n 2 ....... n k ) the thermodynamic
probability is
W ( n 1 , n 2 ....... n k ) 
n!

n 1! n 2 !..... n k !
n!
k
 ni!
i 1
 If the space is divided into cells. Let g1, g2…… gk be
the number of cells in compartments 1, 2,……k
respectively. Thermodynamic probability for
macrostate is
n!
W(n1, n 2 ....... n k ) 
 (g 1 ) n1 (g 2 ) n 2 ....( g k ) n k
n1!n 2!..... n k !
k
W  n!

i 1
(g i ) n i
ni!
Thanks