Gramin (ACS) Mahavidyalaya, Vasantnagar Kotgyal.

DEPARTMENT OF PHYSICS

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B.Sc. Second Year

P-VI

Statistical Physics

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Dr. Kendre D.K.

Head,Department of Physics

Gramin (ACS) Mahavidyalaya, Vasantnagar Kotgyal.

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Statistical Basis and Thermodynamics

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Statistical basis, Probability, Probability and frequency,

Principle of equal a priori probability, Additive and multiplication rule of probability, Conditional probability, Permutation and combinations, Ensemble and average properties, Micro and macro states, Thermodynamic probability, Entropy and probability and relation connecting them

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Statistical basis

Statistics is the branch of science which deals with the collections, classification and tabulation of numerical data as the basis of explanation, description and comparison of various phenomenon. When these concepts are applied to the physics, called as Statistical Physics.

Statistical Physics deals with macroscopic systems .

Macroscopic system : The system which consists of large number of individual particles such as atoms, molecules etc. is called as Macroscopic system . ‘Or’ the system which has a collection of large number of particles of one type like atoms, molecules, protons, neutrons or electros.

It is impossible to apply to apply the ordinary laws of mechanics to a physical system containing large number of particles., particularly that of electrons which is successfully solved by Statistical mechanics.

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The larger number of particles in the physical system considered, the more nearly correct are the statistical predictions.

Smaller the number of particles in the mechanical system , the statistical mechanics goes on becoming meaningless.

The Classical statistics or Maxwell-Boltzmann Statistics successfully explained the phenomenon like temperature , pressure, energy etc. but failed to explain other observed phenomenon like black body radiation, specific heat at low temperature. For this, new approach was introduced by Bose –Einstein,Fermi-Dirac which is also called as Quantum Statistics.

There are three types of statistics:

1. M-B Statistics : This is applicable to identical & distinguishable Particles of any spin. For example : Gas molecules.
2. B-E Statistics: This is applicable to identical & indistinguishable Particles of zero or integral spin called as Bosons.
3. F-D Statistics : This is applicable to identical & indistinguishable Particles of half integral spin called as Fermions obeys Pauli exclusive principle.

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probability

• The probability of an event may be defined as the ratio of the number of cases in which the event occurs to the total number of cases.

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• This is explained as follows:

Throwing a coin: Suppose we toss a coin, either

Head or Tail come upward. i.e. an event can occur

in total two equally like ways. The number of

ways in which Head can come up is only one.

Therefore , the probability that the head may come

up is ½.

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principle of equal a priori probability

• It is supposed that, the coin will fall either its “Head” up or “Tail” up. Similarly if a 6 faced cubical dice is thrown , it is sure that the dice will fall with one of its six faces upward. In the same way, if we have a open box divide into two equal compartments X & Y and a small particle is thrown from a large distance in such a way that it must fall in either of the two compartments, then the probability of the particle to fall in the compartment marked X is equal to the probability that it may fall into the compartment Y . This principle of assuming equal probability for events which are equally likely to occur is known as the Principle of equal priori probability.
• This principle is the fundamental basis of Statistical Mechnics

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probability and frequency

Suppose the die is asymmetric by adding a little load or by wax to one of its face. Now all the outcomes are not equal. For such cases, Suppose , we toss a coin, say N times and we find that Head appears H times . Then frequency of event is given by the equation,

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Thus, if a coin is tossed 50 times and the Head appears 10 times , then the frequency of this event is

From the classical definitions of probability, the probability of occurrence of Head is 0.5. Hence we conclude that the frequency is not same as probability.. As the number of trials are increased then frequency of the event tends to stabilize and gradually reaches a constant value called as probability . We define Probability in terms of frequency as P =

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PROBABILITY AND FREQUENCY

additive and multiplication rule of probability

A) Additive Law of Probability : This law is applicable to mutually exclusive events. Two or more events are said to be mutually exclusive if the occurrence of any one of them prevents the occurrence of other. Such events never occurs simultaneously.

Let us consider two small non overlapping regions ∆V₁ & ∆V₂

in a box of volume V as shown in figure.A particle in ∆V₁ rules out the possibility of its being present , at the same instant in

∆V₂ and vice versa. This two events are

mutually exclusive. Suppose in N trials the

particle is found m₁ times in ∆V₁ and

m₂ times in ∆V_{2 .} The probabilities of finding

particles in two regions are

P_{1 = and} P_{2 =}

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∆V₁

∆V₂

The number of times that the particle will be found at least in one of the two regions in N trials is (m₁ +m₂ ). Hence , the probability will be, � P(∆V₁ or ∆V₂ ) = = P₁ + P₂

This is known as Additive Law of Probability.

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Example : A card is drawn from a well shuffled pack of 52 cards. Calculate the probability for this card to be either a king or a Queen.

Solution : The total number of ways in which the event can occur is N=52.

But we want draw the specific card king or Queen. There are 4 kings or queen in a pack of 52 cards . Therefore the number of ways favorable to first event is m1= 4

Therefore Probability of drawing a king is P_{1 = = =}

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Similarly , Probability of drawing a Queen is is P_{2 = = =}

_{Therefore the probability of the card drawn is either King or Queen is}

P₁ + P_{2 = + =}

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The probability of occurrence of one does not affect the probability of occurrence of other. The probability of A gets in into ∆V₁

at a particular instant is .The probability

of another molecule gets in into ∆V₂ at the same

instant is . We want to calculate the probability of joint occurance of these events at the same instant, i.e. the

probability of A gets into ∆V_{1 ,}B getting into ∆V₂ at

the same instant . Suppose in N trials, the molecule A is found m times in ∆V_{1 ,} If P₂ is the probability that B gets into ∆V₂ , irrespective of the presence of A in ∆V_{1 ,} the number of times the two events will occur simultaneously is mP_{2 .} Thus the joint probability of occurrence of these two events is,

since P1 = m/N Thus , the probability of joint occurrence of two independent events is equal to the product of the probabilities of each event.

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MULTIPLICATION RULE OF PROBABILITY

∆V₁

∆V₂

Conditional probability

The probability for event A to occur the condition that event B has occurred is called the Conditional Probability and is denoted by P ( )

This can be better understood by the following numerical.

Example: A bag contains 4 black and 3 white balls. What is the probability that on two successive draws , the ball drawn are both black?

Solution: Suppose the first ball drawn is black. Let the event be called as A. Its probability is P (A) = = . This event is followed by second event B in which the second ball drawn is also black ( Conditional). Since an event A is already realized , the ball remained are 3 black and 3 White, out of which the event B is taking place, i.e. second ball out of 6 balls. Hence the probability of the event B to occur under the condition that A has already is P ( ) = = 3/6= 1/2 . The first event thus has an effect on the probability of the second event. Therefore the probability of the composite event (A+B) is

P(A+B) = P(A) X P ( B/A) = X =

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Permutation and combinations

Permutation : Arrangement

Combination : Formation of Groups

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• Permutation : To understand the permutation , let us consider an example of four distinguishable objects marked as a,b,c,d. Taking any two objects at a time, the possible arrangements are,

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ab, ba, ac, ca, ad, da, bc, cb, bd, db, cd, dc.

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There are total 12 possible arrangements . In arranging these objects the order of their placing is also taken into consideration . Thus , ab and ba are two different permutations. Thus ,4 objects can be arranged in 12 ways by taking 2 objects at a time, i.e. No of permutations is 12.

Symbolically we can write as, ⁴P₂ = 12

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• If we take 3 objects at a time out of 4 objects a,b,c and d the various arrangements are as follows :

abc abd acd bcd

acb adb adc bdc

bca bad cda cbd

bac bda cad cdb

cab dab dac dbc

cba dba dca dcb

Thus, 4 objects can be arrange in 24 ways by taking 3 at a time out of 4 objects. Symbolically, ⁴P₃ = 24.

In general, the number of arrangements of n distinguishable objects by taking r at a time is given by,

ⁿP_r =

⁴P₃ = = = 24

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⁴P₂ = = = = 12

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• Combinations: The meaningful combinations. Thus the combinations of 4 objects a,b,c,and d taking two objects at a time,

ab, ac, ad, bc, bd, cd

i.e., only 6 combinations which are symbolically represented as ,

⁴C₂ = 6

similarly, combinations of 4 objects a,b,c,and d taking three objects at a time abc abd acd bcd

only 6 combinations which are symbolically represented as ,

⁴C₃ = 4

In general, the number of groups i.e. meaningful combinations of n distinguishable objects by taking r at a time is given by,

ⁿ C _r =

But ⁿP_r =

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ⁿ C _r =

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ⁿ C _{r =} ⁿP_r

=

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Ensemble and average properties

In statistical Mechanics an ensemble is a collection of identical systems with all accessible microstates represented in it with equal frequency.

Let us consider a system of 4 particles in two similar compartments . The system has 16 different microstates as shown in figure.

Let us consider one

system which at time

t = o has a particular

microstate , say (ab,cd)

Then due to interaction

among the particles ,

the system will pass

through all the

accessible microstates as time passes. In a short time, we may find that same microstates repeat frequently than the other.

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But if we observe for long time, then we should expect all the microstates are repeated with equal frequency. In present case of 16 accessible states, suppose we make 48 observations in all, then we expect that each microstate should appear just 3 times. But in practice , some microstates may occur even 7 times and some microstates may not occur at all these 48 observations.

On the other hand, if we make 48 X 10²⁰ observations in all, then the observed frequency of each of 16 microstates would not be different from 3 X 10²⁰ by more than 1 part in 10²⁰

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Micro and macro states

Macrostate : Each compartment wise distribution of a system of particles is known as Macrostate.

Let us consider 4 distinguishable particles and the distribution of these particles into two exactly similar compartments in an open box is as shown in table. Let the particles be a,b,c and d , when any particle is thrown into the box , it must fall into one of the two compartments. The possible ways in which 4 particles can be distributed in two compartments are shown in following table.

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Since the compartments are alike , the particles have the same priori probability of going into either of them and will be ½ . Thus, there are 5 different distributions (0,4),(1,3),(2,2),(3,1) and (4,0). In general, for a system of n particles , the macro states are (0,n),(1,n-1), (2,n-2), (n-1,1) and (n,0). Therefore , the total no. of macrostates for n particles is ( n+1).

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Microstate: Each distinct arrangement is known as Microstate of the System.

Since the particles are distinguishable ,the number of different possible arrangements in each compartment is as follows:

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The distribution (0,4) can have only one arrangement, distribution (1,3) can have 4 arrangement, distribution (2,2) can have 6 arrangement, distribution (3,1) can have 4 arrangement, distribution (4,0) can have only one arrangement.

Thus, a given macrostate may consists of a number of microstates.

In above example of 4 , the total number of microstates are 16= 2⁴

In general, for a system of n particles, the total number of microstates are 2ⁿ

Suppose we consider a thermodynamic system of N identical , non inter acting particles occupying a volume V. Let E₁,E₂,E₃,……E_i

be the energy of particles n₁,n₂,n₃,……n_i respectively , then total energy if the system is

E = n₁E₁ + n₂E₂ + n₃E₃ + n₄E₄ + ……….. +n_iE_i

= Σ n_iE_{i and} N = Σ n_i

The actual values of parameters N,V and E defines a particular macrostate (N,V,E) of a given system.

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Thermodynamic probability

The number of microstates corresponding to any given macrostate is called its Thermodynamic probability.

In other words , the Thermodynamic probability of a particular macrostate is defined as the number of microstates corresponding to that macrostate.

For a case of n particles and two compartment , if r is the number of particles in the compartment 1 and remaining (n-r) are in compartment 2, then

No. of meaningful arrangements = = ⁿC_r

Therefore , the number of microstates in a macrostate (r, n-r) or Thermodynamic probability is,

W _(r,n-r) = = ⁿC_r Applying this to a system of 4 distinguishable particles, for macrostate (1,3), r=1 n-r = 3 and n =4

W _(r,n-r) = = 4

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