# STATISTICAL MECHANICS | Ensembles | Equilibrium | Third law of thermodynamics

## ENSEMBLE

### Ensemble

An ensemble is a collection of large number of  macroscopically identical but essentially independent system.

*System means – collection of large number of particles

Macroscopically identical means – satisfy the same

macroscopic conditions such as – i) energy (E)

ii) no. of particles (N)

iii) Volume

iv) temperature (T)

There are three types of ensembles —

i) Micro canonical ensemble

ii) Canonical ensemble

iii) Grand canonical ensemble

### Micro canonical ensemble

The collection of large number of independent systems those have- same energy (E), same volume (V) and same number of particles (N) but individual systems which are separated by insulated, impermeable and rigid walls such that E, V, N for a particular system, are not affected by the presence of other, called micro canonical ensemble.

### Canonical ensemble

The collection of large number of independent systems having – same number of particles (N), same volume (V) and same temperature (T) but individual systems are separated by impermeable, rigid and conducting walls such that V, T, μ for a particular system are not affected by the presence of others, called canonical ensemble.

### Grand canonical ensemble

The collection of large number of independent systems having same volume (V), same temperature (T), and same chemical potential (μ) but individual systems are separated by rigid, conducting and permeable walls such that , V, T, μ  for a particular system are not affected by the presence of others, called grand canonical ensemble.

***** E constant – insulated wall

N constant – impermeable

V constant – rigid

T constant – conducting

μ constant –  permeable

## Statistical equilibrium

An ensemble is said to be in statistical equilibrium if the probabilities of finding the phase points in the various regions of the phase space and the average values of the properties of its systems are independent of time.

Mathematically,

for all values of q and p

Hence an ensemble of systems will be in statistical equilibrium if

i)  The density (ρ) is independent of time at all points in phase space.

ii) The density (ρ) is a function of some property of the ensemble which depends only on p’s and q’s value.

Properties of a system in equilibrium stage :

To find the conditions for equilibrium in a system of two inter – connected sub-systems, we consider that initially the sub-systems are separated from each other by a rigid, insulating and non-permeable barrier.

There are three types of equilibrium

1) Thermal equilibrium

2) Mechanical equilibrium

3) Particle equilibrium

### Thermal Equilibrium

The barrier allows the transition of energy but the condition of two sub-systems 1 and 2 does not change, then They are said to be in thermal equilibrium ·

If  σ1 and  σ2 are the entropies of the two sub-systems 1 and 2, then

Also, we have

and

For transfer of energy from one system to another one

For thermal equilibrium,

as

Condition of thermal equilibrium.

The quantity

Where τ is called statistical temperature

From equation 6, we may write

Hence the condition of thermal equilibrium  is that The statistical temperature of two sub-systems should be equal.

### Mechanical equilibrium

The barrier allows the transmission of energy the barrier can move itself. Then

Also,

For thermal equilibrium,

So, in mechanical equilibrium , equation 2 yields ,

So, total volume is constant,

Hence equations 3 , becomes ,

Now, we define a quantity π by

Where π is called the statistical pressure

Hence for a system in thermal equilibrium, the condition of mechanical equilibrium is

### Particle equilibrium

Let  us suppose the barrier allows through it the diffusion of the molecules, called particle equilibrium,

Again

In this case,

As 6= constant in equilibrium

In thermal equilibrium,

In mechanical equilibrium,

And in thermal and mechanical equilibrium, equation 2 reduces to

Since total number of particles are constant,

Hence equation 3 becomes,

Let us now define a quantity,

Where μ is called chemical potential. Hence for a system in thermal and mechanical equilibrium the condition of particle equilibrium is

#### Statistical definition of entropy

In equilibrium state both the entropy (s) and thermodynamical probability (w) have their maximum values. their 3

According to thermodynamics, entropy (s) of a system is related wilt temperature by the relation,

According to statistical mechanics

From equation 1 and 2 , we may write

which is the statistical definition of entropy.

Statistical definition of pressure:

Elementary work done

where z is the partition function.

#### Statistical definition of chemical potential

where

is called generalised force.

called  statistical chemical potential.

### Third law of thermodynamics

“Every system has a finite positive entropy, but at absolute zero temperature the entropy may become zero”.

The entropy of a system tends to a constant value when its temperature is approaching to absolute zero.

This constant value cannot depend on any other parameters characterising the closed system, such as pressure or applied magnetic field, etc. At absolute zero temperature the system must be in a state with the minimum possible energy. Entropy is related to the number of accessible micro-states, and there is the ground state with minimum energy. Then, the entropy at absolute zero will be exactly zero. If the system does not have a proper order ( glassy, for example), then there may remain some finite entropy as the system is brought to very low temperatures. The constant value is called the residual entropy of the system.