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 =

If a is the number of cases in which an event occurs and b 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.

casesofnumberTotal

occurseventthewhichincasesofNumber

ba

a

ba

b

Some Probability Considerations

Tossing of a coin : If we toss a coin. Either the head can

come upward or the tail i.e PH =PT =

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,

PHH = = PH. PH

2

1

4

1

Independent eventsTwo 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.

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

2i

1ii 4n

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. Probability of a microstate = Probability of a macrostate = (number of

microstates in that microstate)

(Probability of one miscrostate)

=

=Thermodynamic probability

probability of each microstate

= W.p.

smicrostateofnumberTotal

1

esmiscrostatofnumberTotal

itinesmiscrostatofNumber

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, n2),….. (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 waysnPn = 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)

W(n1, n2) =

The total no. of microstates=2n

2nn

1nn

1121CC

)!n–n(!n

!n

!n!n

!n

!n!n

!n

2

1)n,n(

21n21

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

P(r, n r) = The probability of the most probable macrostate

Pmax =

2

n,

2

n

n2

1

!2

n!

2

n!n

n2

1

)!rn(!r

!n

Probability of macrostate x < < n

slightly different from most probable state is

Px =

Px = Pmax ….(1)

Using stirling’s formula ln n = nln nn and using Taylor’s theorem

ln (1+ y) = y provided | y | < 1

(1) on simplification gives

Where f= = fractional deviation from most

probable no. of particles in a cell.

x

2

n,x

2

n

n2

1

!x2

x!x

2

n

!n

!x2

n!x

2

n

!2

n 2

........................3

y

2

y 32

2

xfexpPP

2

maxx

2/n

x

Discussion

For f = 103

Then

The ratios for different values of n are given in the table

It is apparent that as n increases the probability for 0.1% deviation from the most probable state goes in decreasing very rapidly..

710n5

max

x eP

P

n

103 0.999

106 0.607

108

1010

max

x

P

P

50e

1

5000e

1

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 > n3

(2x / n)

0.2 0.1 0 0.1 0.2

n3

n2

n1

max

x

P

P

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 systemOr 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 macrostateor t W

= 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

N

W

t

t

0

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 the thermodynamic probability is

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

k

1ii

k21k21

!n

!n

!n!.....n!n

!n)n.......n,n(W

knk

2n2

1n1

k21k21 )g....()g()g(

!n!.....n!n

!n)n.......n,n(W

k

1i i

ini

!n

)g(!nW

)n.......n,n( k21

Thanks