Maxwell Boltzmann statistics Statistical theory of a system consisting of a large number of particles, that obey classical laws of motion.

Consider for example a system of monatomic gas, consisting of N molecules. Each molecule has six coordinates given by,

x, y, z, v_x, v_y, v_z

which are to be specified so as to describe the system. In a six dimensional hyperspace , called the phase space, each molecule will be represented by a point. The phase space is sub divided into a large number of cells, each with volume , H = dx.dy.dz..dv_x.dv_y.dv_z . H is small compared to the total volume of the system in the phase space, but large enough to contain a large number of molecules. In statistical mechanics we are interested in the distribution of phase points in the cells.

A specification of the number of phase points in each cell defines a macrostate of the system. A macrostate can be realized in several ways. Fig. m , shows all possible ways of arrangements of a hypothetical system with two phase points named a and b in two cells I and II. Each one of such arrangement is called a microstate. Thus a microstate specifies where each phase point lies within the limits of dimension of cells.

One of the fundamental hypothesis of statistical mechanics is that all microstates are equally probable; that is over a long period of time any one microstate occurs as often as other. The number of microstates corresponding to any given macrostate is called the thermodynamic probability, W of the macrostate.

In a general case of N phase points and n cells, the thermodynamic probability of a macrostate is given by,

W= N / (N₁ N₂ N₃ …) = N / P N_i

where N is the number of phase points in cell 1, N in cell 2, and so on. The symbol P is used to denote a product.

The thermodynamic equilibrium is the state of maximum thermodynamic probability. This implies that,

d W = 0

or d ln W = 0

Þ S ln N_i d N_i = 0 (1)

There are also two constraints on the system. (i) The total number of particles is a constant. Any increase in the population in some cells must be balanced by decrease in others.

d N = S d N_i = 0 (2)

(ii) The change in the internal energy, when the number of phase points in the ith cell changes by d N_i , is w _id N_i . Since the total energy remains constant, the sum of energy changes must be zero.

d U = S w _id N_i = 0 (3)

From the three equations (1),(2) and (3) it can be shown that,

N_i = (N/z) exp ( -b w _i ) (4)

where b is a constant. Using the Boltzmann theorem* , the constant b is found to be,

b =1/kT

The constant z = S exp(-b w _i )

is called the partition function. The eq.(4) provides an expression for the number phase points N_i in the ith cell for the macrostate of maximum probability.

The equation (4) can be applied on varied systems, as in the case of monatomic ideal gas consisting of N molecules, each of mass m, in an enclosure of volume V. The energy of a molecule in the ith cell is w _i =(1/2) mv_i² . The partition function is given by,

z = S exp (-mv_i/2kT)

The summation can be replaced by an integration and the partition function z can be evaluated.

z = V/H [ 2p kT/m]^3/2

Then N_i is given by,

N_i = NH / V [ m/ (2p k T) ] ^3/2 exp [ - m v_i² /(2kT) ]

N_i = N / V [ m/ (2p k T) ] ^3/2 exp [ - m v ² /(2kT) ] dx dy dz dv_x dv_y dv_z

Integrating over x, y, z

d³N = N [ m/ (2p k T) ] ^3/2 exp [ - m v ² /(2kT) ] dv_x dv_y dv_z

This gives the velocity distribution of the molecules. This equation is called the Maxwell Boltzmann distribution law.