** STATISTICAL MECHANICS**

**Phase space**

In classical mechanics , the instantaneous dynamical state of a particle is completely specified by its three position coordinates (x, y, z) and three corresponding momentum coordinates (Px, Py, Pz). Thus six, co-ordinates are needed to specify a “one particle system” completely. These six coordinates (x, y, z, Px, Py, Pz) are marked along six mutually perpendicular axes in six-dimensional space. This space is called “phase space” ( **Γ **space).

**Phase point**

The point in the phase space represents the instantaneous state of the particle is known as phase point.

**Phase density**

The number of phase points per unit volume of phase space is called phase density.

**Phase cells**

When a phase space (or position-momentum space) is divided into tiny six dimensional cells whose sides are

then such tiny cells are called “phase cells”.

**Micro-state of a system**

A system of atomic dimension or smaller size is called microscopic system.

If the state of a system of particles are specified by quoting the position and momentum of individual particles then it is called micro-state or microscopic state of the system.

Example : a molecule , an atom, an ion.

### **Macro-stale of a system**

A system which is large enough to be observable in the ordinary sense, is called macroscopic system.

The macroscopic state or macro-state of a system specified by quoting the macroscopic parameters as like pressure, volume, temperature, energy, Chemical potential.

Example : Solid, liquid , gas.

### Illustration of micro states and macro-state of a system

Let us consider a system of three molecules a, b, c which are distributed in two halves of a box ; left half-L and right half-R:

The distribution of molecules are shown as

Hence there are four possible distributions —

a) 3 molecules in L and O molecule in R i.e. (3,0).

b) 0 molecule in L and 3 molecules in R i.e. (0,3).

c) 2 molecules in L and 1 molecule in R i.e. (2,1).

d) 1 molecule in L and 2 molecules in R i.e. (1,2).

Total number of ways in which three molecules can occupy two halves of the box = 1+1+3+3 = 8 = 2^{3} corresponding to four different distributions.

Each way of arrangement of molecules is a micro-state of the system while each different distribution of molecules is a macro-state.

Thus there are 8 micro-states and a macro-states in the above example.

### Accessible state

The micro-states which are permitted, under the constraints imposed upon the system are called accessible states.

If there are n1 molecules in a cell 1; each having energy **ε**1,

n2 molecules in call 2 ; each having energy **ε**2

and so on, then constraints are –

#### Details

The micro-states which are allowed under given restrictions are called accessible micro-states.

For example : In case of 3 molecules, a,b,c to be restricted between two halves of a box. If none of can be outside the box, then (a,b,c), (a,bc), (ac, b) are accessible micro-states while (a,b), (b,c), (a,c) etc. are in accessible micro-states.

### Postulate of **equal** **a priory probability**

Accessible micro-states corresponding to possible macro-states are equally probable. In other words this states that the probability of finding the phase point in any one region is identical with that for any other regions which are equally well with the given conditions.

Thus, this postulate is known as postulate of equal a priory probability.

The probability of finding the phase point for a given system in any one region of phase space is identical with that for any other region of equal extension of volume. This postulate is known as the postulate of equal a priory probability.

**Density of quantum state**

OR Show that for a particle in an enclosure of volume V, the number of states in which the particle has a momentum between p and p+dp

ans- For a single particle free to move in a three dimensional space. We have six dimensional phase space. Three position co-ordinate

and three momentum co-ordinates

specify the micro-state of the particle in the phase space ,

So, volume of phase cell

————-1

Total volume of the phase space

—————–2

Volume of momentum space containing momentum between p and p+dp will be given by the volume of a spherical shell of radius p and thickness dp i.e.

————3

So, Number of micro-states in the momentum range p and p+dp

—————–4

Switching over the energy E from momentum p via the relation

We get, number of quantum state (or micro-state) in the energy range E and E+dE,

————-5

The function f(E) is called density of quantum states .

where f(E)=g(E)/V

For electron.

Since an electron has two independent directions of spin orientation, hence for electron above equation can be formed as

#### problem- A particle is moving in XY plane . What is the dimension corresponding phase space?

Answer:

The particle may have position coordinates

and momentum coordinates

Dimension of corresponding phase space = 4

### Phase trajectory

The locus of the phase points of a particle on the phase plane is called phase trajectory.

Actually, the curve Showing the variation of px with x represents the phase trajectory.

Determine the phase trajectory of a one dimensional linear harmonic oscillator of constant energy E moving along x axis.

Also find the region of states accessible to like oscillator.

Sol :

The energy of one dimensional linear harmonic oscillator moving along x axis

where, m is mass of the particle

Above equation represents en equation of ellipse, whose semi major length a = √2E/k and semi minor length b=√2mE. Thus the phase trajectory of the oscillator (i.e. x vs Px) in phase space is an ellipse.

Region of the states accessible to the oscillator with energy E is equal to the area of the ellipse

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