thermal physics lecture note 10

8/10/2019 Thermal Physics Lecture Note 10

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10. Introduction to statistical mechanics

For a system consisting of large number of particles moving freely (gas system), there are two

ways to describe its behaviour.

(1) Consider individual particle motion. This is not possible for large number of particles.

(2) Consider the energy distribution of the particles.

If there areN number of particles in the system, where

n1particles each with energyE1

n2particles each with energyE2

and so on.

so that, N=

nii

and E=

niEii

(total energy)

Here E= niEii

is the kinetic energy only if there is no interaction between the particles;


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E= niEii

+ potential energy

if there is interaction between particles

potential energy = V12+ V13+....... + V23+ V24....

If this system of multi particles is left closed and isolated for a sufficiently long time,

an equilibrium state will be achieved, and the values for n1,n2 , ........ will not be changed again

The distribution of the values of ni will be expressed by a certain distribution function following

certain suitable distribution law. Three type of distribution laws are commonly used:

Maxwell-Boltzmann

Bose-Einstein

Fermi-Dirac

Studies on these distribution functions and how they vary with time is called

Statistical Mechanics


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In statistical mechanics, the behaviour of the particles in a system is described by usingdistribution function

f(r, v, t)which represents the number of particles at positionrat timethaving velocity between

vxdan vx+dvx, vydan vy+dvy , vzdan vz+dvz

r : (x , y , z ) is referred to as spatial space

v: ( vx, vy, vz) is referred to as velocity space

(r , v ) is referred to as phase space


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The number of particles at positionr at timet: n(r,t)=

f(r,v,t)dv

For any property of the system Q(r,v, t), its average value is :

=

1

Nf(r,v,t)Q(r,v,t)dv

, N : total number of particles

Example : For a gas at equilibrium, the overall distribution function is

f(v) function of v

Average velocity of particles: v=1

Nvf(v)dv

rms velocity v2=

1v2f(v)dv

Average kinetic energy E=1

2mv2


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For this course, we introduce three types of distribution functions,

Maxwell-Boltzmann distribution

This is often referred to as the classical distribution function

(where the kinetic energy of the electrons are assumed to be continuous)

It is suitable for system with high temperature (room temperature and above)

All particles are identical and distinguishable

The particles are considered to be distributed at energy stateE1,E2......

Every energy state has intrinsic probability of g

(statistical weight)


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Fermi-Dirac distribution

For quantum system each energy level is splitted into sub-levels

the energy state is definied by three quantum numbers : principle, orbit dan spin

The number of splitted levels is expressed by degeneracy gifor energy levelEi

Each energy state is filled up by one particle (Exclusive Pauli Principle)

Bose-Einstein Distribution

Similar to F-D distribution, but the Exclusive principle is not obeyed

The general form of the three types of distribution function for energy level ican be written as

i

ii

Egn

exp


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For M-B distribution:ZN

=

)exp(kT

1=

= 0

where

iinN and

i

ii

kT

EgZ exp

Z = electronic partition function

For F-D distribution:kT

F

kT

1= = 1

For B-E distribution:kT

kT

1= = -1


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M-B B-E F-D


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Consider ideal gas using M-B distribution,

0

)()exp( dEEgkT

EZ

here dEEh

mVdEEg 2

1

3

3 21

)2(4)(

= from quantum physics

Therefore,

0

2

1

3

3

exp)2(4 2

1

dEkT

EE

h

mVZ

3

0

2

1

2

1exp kTdE

kT

EE

3

23

)2(

h

mkTVZ

=


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Number of particles with energy EandE + dE :

dEkT

EE

h

mV

Z

NdEEg

kT

E

Z

Ndn

exp)2(4

)(exp 212

1

3

3

ReplaceZ:

kTEE

kTN

dEdn exp2 2

1

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This can also be expressed in terms of v:

TakedE

dnmv

dv

dE

dE

dn

dv

dn=

vfkT

mvv

kT

mN

dv

dn=

=

2exp

24

22

2/3


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Most probable value of v[peak of the functionf(v)] is given by

m

kTvm

2= which is when kTmvm=

2

2

1

By using f(v), we can obtain

Average velocity

=

02

23

3exp

4)(

1dv

v

vv

vdvvvf

Nv

mm

which ismkTvv m

82=

and the root mean square (rms) velocity

21

2

1

02

24

3

2 exp4)(1

=

=

dvv

vvv

dvvfvN

v

mm

rms


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Which is,mkTvv mrms 3

23

=

dv

dn


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We can also express Maxwell-Boltzmann distribution as

the number of particles having velocity between vand v+dvper unit volume in the velocity space,

or particle density in the velocity space

=

kT

mv

kT

m

Ndvv

dnv 2exp24

22/3

2

v is referred to asMaxwell-Boltzmann velocity distribution function

v has a maximum at v= 0

thermal physics lecture note 10

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