
Prakash Gupta Statistical Physics
1 Classical Statistical Mechanics:
Statistical Mechanics deal with systems containing of many particles and methods employed are to get
a collective or macroscopic property of the system without taking into account the individual motion
of the particles.
It is not practically possible to determine the property of each particle individually, a statistical approach
is made using the concept of probability of distribution. This statistical approach helps in determining
the bulk or macroscopic property of the system as a whole.
The idea of probability does not imply that the particles move in a random way without obeying the
laws.
Phase Space
From the statistical point of view, a mono-atomic gas constitutes the simplest system. The state of the
gas is completely known if the position and momentum of each molecule of the gas is specified, i.e., we
must specify six quantities x, y, z, px, py, pzfor each of the molecules.
In a purely mathematical concept, we may imagine six dimensional space in which the state of a point
will be described a set of six co-ordinates x, y, z, px, py, pz. This six-dimensional space is called phase
space (µ-space).
The 6N dimensional space where we specify co-ordinates and momenta of N molecules is called τ-space.
Volume of Phase Space
In three dimensional space, we consider an element of volume dxdydz. Similarly, we consider three
dimesional volume element dpxdpydpzin this momentum space.
thus, dxdydzdpxdpydpz= elementary volume of phase space.
The element of volume in this space is termed as a cell. According to Hisenberg’s uncertainty principle,
dxdpx≥h, dydpy≥h&dzdpz≥h
so, volume of each state is 'h3
Thus, in phase space, the co-ordinates of a particle can be specified only to the extent that the particle
under consideration has the position and momentum lying within an element of phase space of volume
h3.
Microstates and Macrostates
The specification of a molecule in a cell is known as microstate. If the individual molecules remains
always in the same region of cell, there is no change in microstate.
The specification of number of molecules in a cell is known as macrostate. if the number of molecules
always remains same in the confined region of space (cell) whichever may be the molecule, the macrostate
remains same. If the initial no of molecules in the cell is not same with final number of molecules, there
will be change in macrostate.
cell C
cell B
cell A
a, b, c
p,q
f
The specification of molecule a in cell A, p in cell B and f
in cell C are their respective microstates.The total number
of molecules in cell A is 3,in cell B is 2 and in cell C is 1
refers to macrostate of cell B and C respectively.