Method of Ensembles: Partition Functions, Lecture notes of Statistical Physics

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Programme: M.Sc. Physics Semester: 2nd

Method of Ensembles: Partition Functions

Lecture Notes Part-1 (Unit - IV)

PHYS4006: Thermal and Statistical Physics

In earlier lectures, we have discussed the concept of

Ensembles viz. Microcanonical, Canonical and Grand-

canonical.

Herein, we will use that concept in deriving the

thermodynamical functions of a thermodynamic system.

Before we proceed, let us recall the concept of partition

function.

The quantity Z represents the sum of the Boltzmann factor over all the accessible states and is called the partition function (derived from German term

Zustandssummae). The quantity Z indicates how the gas molecules of an assembly are

distributed or partitioned among the various energy levels.

The energy term in the expression for partition function does not

mean only the translational component but also may contain the

components corresponding to other degrees of freedom too e.g.

rotational, vibrational and electronic too.





i

i

i

Z g e



i i kT Z e e

    /  

  











 

  

i

i

i i i

i

i

i i

i

i

i

i

g e

g e

N

n obability P

g e

Ng e n Z

N A









Pr , 

• This partition function can be used for calculating the

various thermodynamical properties of ensembles having

independent systems (obeying classical laws) irrespective

of whether the ensembles have distinguishable or

indistinguishable independent systems.

• Consider an assembly of classical gas where the

distribution of energy states is considered to be

continuous. So, the number of energy levels in the

momentum range p and (p+dp) is given by -

dp h

Vp g p dp 3

2 4 ( )

 

After solving, we get

This gives the translational partition function for a

gas molecule.

 mkT 

kT

m

h

V

Z

h

V

Z



 

3 / 2

3

3 / 2

3

3 / 2

Partition Function and its relation

Thermodynamic Quantities

1. with Entropy (S):

Consider an assembly of ideal gas molecules obeying

M-B distribution law and according to Boltzmann’s

entropy relation –

The maximum thermodynamic probability is given by -

S  k ln W  k ln .........( i )

! ii n

g W N i (^) i

n i

i  

but for an ideal gas,

from (iv) and (v),

ln ............( )

ln ln ln

iv T

E

S Nk Z

kT

kE S k W k N Z E Nk Z

E  NkT v

S  Nk ln Z  Nk vi

**2. with Helmholtz Free Energy (F):

3. with Total Energy (E):**

Average energy of a system of N particles is given by,

11

ln ............( )

ln

F NkT Z vii

T

E F E TS E T Nk Z

 

 

  

      

..........( viii ) Z

g e

E

g Ae

g A e

n

n

N

E

E

i

i i

i

i i

i

i i

i

i

i

i i

i

i

i























4. with Enthalpy (H):

Enthalpy is given by,

H = E+PV = E+RT (for an ideal gas, PV=RT)

5. with Gibb’s Potential (G):

putting the value of S from eqn. (iv) in above, we get

log  .........( )

2 H NkT Z RT x T (^) V

   



 







G H TS NkT  Z  RT TS T (^) V

      



 





 log

2 

putting the value of E from eqn. (ix), we get

 

G NkT   RT NkT Z E

T

E G NkT RT T Nk Z

Z T

Z T

V

V

   

 

  

    

 



 







 



 







ln

ln

log

log

2

2

     

  







 

  





     Z T

Z T (^) V V

G NkT^2 log RT NkT ln Z NkT^2 log

G  RT  NkT ln Z ..........( xi )

References: Further Readings

1. Statistical Mechanics by R.K. Pathria

2. Statistical Mechanics by K. Huang

3. Statistical Mechanics by B.K. Agrawal and M. Eisner

4. Statistical Mechanics by Satya Prakash