1. Maxwell - boltzmann statistics
Harvinder Kaur,
Associate Professor,
Head of Department, Physics
GCG 11, Chandigarh

2. INTRODUCTION
The Maxwell-Boltzmann distribution is an important relationship that finds many applications in Physics and Chemistry. It forms the basis of the Kinetic Theory of Gases, which accurately explains many fundamental gas properties, including pressure and diffusion. The Maxwell-Boltzmann distribution also finds important applications in Electron Transport and other phenomena.
The Maxwell-Boltzmann distribution can be derived using Statistical Mechanics. It corresponds to the most probable energy distribution, in a collisionally-dominated system consisting of a large number of non- interacting particles. Since interactions between the molecules in a gas are generally quite small, the Maxwell-Boltzmann distribution provides a very good approximation of the conditions in a gas. This probability distribution is named after James Clerk Maxwell and Ludwig Boltzmann.

3. PHASE SPACE
In Mathematics and Physics, phase space, introduced by Willard Gibbs in 1901, is a space in which all possible states of a system are represented, with each possible state of the system corresponding to one unique point in the phase space. For mechanical systems, the phase space usually consists of all possible values of position and momentum variables
A combination of position and momentum space is known as phase space which is a six dimensional space. So, if a system consists of n particles then we need 6n co-ordinates to describe the behaviour of the system in the phase space.
At any instant of time, suppose one of the particles have its position co-ordinates lying between x and x+x,y and y+y, z and z+z and let its momentum coordinates lie between px and px+ px, , py and py+ py ,pz and pz+ pz. Then, the particle is said to lie in a phase space compartment having a volume,
 = xyzpxpypz.

4. PHASE SPACE DIVISION
Let the compartments in phase space be further divided into a very large number of elementary cells of equal volume, d.
d = dx dy dz dpx dpy dpz = h3o (say)
where ho has the dimensions of length * momentum.
In Classical Physics the choice of phase space cell size (ho) is entirely in our hands. But in Quantum Mechanics we are not free to make ho as small as we like. The minimum volume of an elementary cell in phase space in Quantum Mechanical system is h3, where h is Planck’s constant
Total number of elementary cells in phase space = Total volume in phase space
Volume of one elementary cell
= dxdydz    dpx dpy dpz d
where d = h3o (which may approach zero) in Classical system
h3 in Quantum Mechanical system

5. FERMI-DIRAC STATISTICS
TYPES OF STATISTICS
STATISTICS
CLASSICAL STATISTICS
(MAXWELL – BOLTZMANN)
QUANTUM STATISTICS
BOSE - EINSTEIN
STATISTICS
FERMI-DIRAC STATISTICS

6. MAIN DIFFERENCES IN THREE STATISTICS
PARAMETER
CLASSICAL OR M-B STATISTICS
BOSE-EINSTEIN STATISTICS
FERMI-DIRAC STATISTICS
Particles
Particles are distinguishable
Particles are indistinguishable
Size of phase space cell
The size of phase space cell can be as small as we require
The size of the phase space cell cannot be less than h3
The size cannot be less than h3
Number of cells
If ni be the number of particles and gi the number of cells then gi>>ni so ni/gi<<1. Thus,number of cells can be made as large as possible
The number of cells is less than or comparable to the number of particles gi < ni
ni/gi  1
The number of cells has to be greater or equal to the number of particles so ni /gi  1
Restriction on particles
No restriction
Restriction due to Pauli Exclusion Principle
Two particle distribution in two cells
The particles are distinguishable and can be arranged in four ways
The particles are indistinguishable and can be arranged in three ways
No two particles can occupy the same cell. Hence, only one arrangement
A B
B A AB AB

7. BASIC APPROACH IN THE THREE STATISTICS
The basic approach in three statistics is essentially the same but we get the different results due to different assumptions. The common approach is as follows :
In any dynamic isolated system, the total number of particles (n) and the total energy (U) has to remain constant
n =  ni = constant
dn =  dni = (1)
dU =  ui dni = (2)
When the system is in equilibrium then it exists only in the most probable state. From the assumptions of a given kind of statistics, we calculate thermodynamical probability (W) for any given macrostate. In the most probable state of the system W must be maximum.
For all natural systems W is a very large number. We, therefore, deal with ln W. For most probable state W is maximum 
d(ln W) = (3)
The three equations (1), (2) and (3) must be simultaneously satisfied by the system irrespective of the kind of statistics to be applied. These three conditions can be incorporated into a single eq. by the method of Lagrange’s undetermined multipliers. We multiply eq. (1) with - , eq. (2) by - and add to eq. (3)
d(ln W) -   dni -   ui dni = 0
d(ln W) -  (  +  ui ) dni = (4)

8. Maxwell – boltzmann statistics applied to an ideal gas in Equilibrium
Consider n molecules of an ideal gas enclosed in a vessel of volume V, moving about in all directions, colliding against one another and against walls of the enclosure with energies ranging from 0 to uk . Let U represents the total energy of all the molecules in the gas which remains constant for an isolated system . In the equilibrium position , the system is in the most probable state. u1 u2 uk
Suppose n1,n2,---,nk be the number of molecules, g1,g2---,gk be the number cells in the phase space corresponding to the energy intervals 1,2 ,---,k respectively. Then, k
Thermodynamic Probability , W(n1 , n nk) = n!  (gi)ni/ni!
i=1
Taking Natural Logarithms
k k
ln W = ln (n !)+  ni ln gi -  ln (ni!)
i= i=1
Apply Stirling formula, ln n! = n ln n –n
k k
ln W = n ln n – n +  ni ln gi -  (ni ln ni - ni)
i= i=1

9. Maxwell – boltzmann statistics Continuedk
d ln W =  ( ln gi –ln ni) dni
i=1
In the equilibrium state, W is maximum, so that
(d ln W) = 0 k
 ( ln gi –ln ni) dni = ………(5)
i=1
i.e.,
 dni = 0 …..(6)
 ui dni = 0 …….(7)
Multiplying eq. 6 with - and eq 7 with - and adding to eq. 5 we get, k
 ( ln gi –ln ni -  - ui) dni = ………(8)
i=1
i.e.,
ln gi –ln ni -  - ui =0
Hence,
ni = gi/(e eui)
ni  e-ui
This law is known as Maxwell-Boltzmann law of energy distribution and e-ui is Boltzmann constant

10. Number of phase space cells
Number of molecules in a unit interval around u are
n(u) = g(u) e- e-u
u = p2/2m
So, in momentum coordinates,
n(p) = g(p) e- e -p2 /2m
Number of cells in phase space corresponding to the momentum interval p to p+dp
g(p) dp = dxdydz    dpx dpy dpz
ho3
Considering the spherical volume between the momentum value p and p+dp
g(p)dp = ( V4p2 dp) / ho3
n(p) dp = 4Vp2 dp e- e -p2 /2m
ho3
Total number of available phase space cells, 
 g(p)dp = (4V / ho3) 0 p2 dp or
= (V/ ho3)(4/3) p3 max= V / ho3
where,  = Total volume in momentum space
& V = Total volume in position space

11. Values of  and  As stated earlier, total number of molecules, n,
Using standard integral,
 x2 e-ax2 dx = /(4 a3/2 )
n = n(p) dp = 4V e- ( / (/2m) 3/2 )
4ho3
Therefore,
e- = (n ho3 /V) (/2m) 3/2
u = p2/2m
Using
n(u) du = (2n/  ) () 3/2 ue -u du ……..(9)
n = n(p) dp = 4V e- p2 e -p2 /2m dp
ho3
n(p) dp = 4 n (/2m) 3/2 p2 dp e -p2 /2m

12. Values of  and  continued
Total energy for the system of n molecules is given by,
U =  u n(u) du = 3/2 nkT
i.e.,
3/2 nkT = (2n/  ) () 3/2  u3/2e -u du
Using standard integral,
 x3/2 e-ax dx = (3/4 a2) (/a)
we get,
 = 1/kT
THIS VALUE OF  IS APPLICABLE TO ALL THREE STATISTICS
Substituting the value of  in eq. 9,
n(u) du = (2n/ (kT)3/2 ) u. e –u/kt du ………..(10)
n(p) dp = (2n/ (mkT)3/2 ) p2e –p2/2mkt dp
Using
p = m v
n(v) dv= (2nm3/2/ (kT)3/2 ) v2e –mv2/2kt dv ………...(11)
Eq. 10 and 11 are known as Maxwell-Boltzmann law of distribution of energy and law of distribution of velocity respectively

13. GRAPHICAL DISTRIBUTION OF MAXWELL-BOLTZMANN SPEED DISTRIBUTION
Let f = n(v)/n denote the fraction of molecules which have speeds lying in a unit velocity interval around the value v.
If we plot f(v) against v, using equation (11), we get this graph for a particular value of temperature, T. The value corresponding to the most probable speed (vmp), df/dv = 0
f = Cv2 e –mv2/2kT
C = 2 (m/kT)3/2
vmp = f/v = (2kT/m)
The average speed of molecules = v is defined as :
v n(v) dv
v =
 n(v) dv
v = (8kT/m)
The root mean square speed (vrms) is given by the expression :
vrms =  ( v 2) = (3kT/m)

14. MAXWELL –BOLTZMANN DISTRIBUTION : AN EXPERIMENTAL VERIFICATION
I.F Zartman and C.C Ko performed an experiment to study the distribution of velocities in a molecular beam and hence to verify the theoretical results obtained from Maxwell Boltzmann velocity distribution.
To Vacuum
Pump
Apparatus
An oven (in which bismuth was heated at about 800oC
Slits S1, S2 and S3 along one side of the drum
A drum (D) rotating about Z axis (~ 100 rps)
A glass plate P, fixed at the face opposite to S3.
The whole system is enclosed in a vessel which is highly evacuated to ensure that Bi molecules can go into the drum from S1 to drum without suffering any collision with air molecules

15. WORKING OF ZARTMAN & C. C. KO EXPERIMENT
When the drum is rotating the slit S3 comes in line with the S1 and S2 once and a bit of Bi vapours enter into the drum. These vapours will be deposited on the different points on the glass plate depending upon their velocities and speed of the rotation of the drum.
The oven is maintained at fixed temperature and the speed of rotation of drum is also constant. The slower the molecule, more is the time taken by it to reach the glass plate. So, slower molecules get deposited at larger distance from the point O.
The distance from point O is thus a measure of molecular speed. The density of Bi atom at various points is determined. This gives the relative number of Bi molecules at various speeds and this distribution is found to agree with the theoretical predictions of Maxwell-Boltzmann Statistical Law Of Molecular Speeds

16. SUMMARY
A knowledge of 6n coordinates in the phase space for all the n particles of the system completely specifies the position and momentum of the particles.
Maxwell – Boltzmann distribution treats the particle distinguishable since number of cells is much greater than the number of particles.
The M–B distribution describes particle speeds in gases, where the particles do not constantly interact with each other but move freely between short collisions. It describes the probability of a particle's speed as a function of the temperature of the system, the mass of the particle.
The most probable, average and root mean square speeds of particles are (2kT/m) , (8kT/m) and (3kT/m) respectively.

17. Thanks