Statistical Ensemble and Basic Postulates.

Statistical Ensemble and Basic Postulates.

Macroscopic State

• Gas left to itself for sufficiently long time, the pressure & temperature of the gas stabilize over the entire volume irrespective of their initial distribution.

• State of the gas is characterized by the parameters P, V & T.

• Macroscopic steady state $\rightarrow$  Equilibrium state.

Microscopic State

• State of a gas is characterized in terms of the states of the constituent particles.

• Microscopic state of a gas at any instant $\rightarrow$  instantaneous position & momenta of the various molecules.

• Each one of the microscopic state pertain to the same macroscopic state.

Ensemble

• It is a very large collection of identical macroscopic systems which are allowed to interact.

• Each system may be multiphase & contain dependent (interacting) particles.

• Stirling's approximation can be applied without sensible error.

Canonical Ensemble

• Each closed system is separated from its neighbors by diathermic walls, so that all systems are in thermal equilibrium.

• System is characterized by constant N,V & T.

Figure

Canonical Ensemble

Grand-Canonical Ensemble

• Each system is separated from its neighbors by diathermic permeable wall $\rightarrow$  both material & energy can be exchanged between neighbours.

• characterized by constant  $\mu$  V & T. 

Figure

Grand-Canonical Ensemble

Micro-canonical Ensemble

• System is separated from its neighbors with a rigid impermeable adiabatic walls $\rightarrow$  neither exchange energy or material with its neighbours.

• It is characterized by constant N,V & E.

Figure

Micro-canonical Ensemble

Basic Postulates-First Postulate

• Each particle possess minute mass & extension $\rightarrow$  point mass.

• At any instant, state of any one particle is not affected by the state of any other prticles.

Basic Postulates-Second Postulate

• Each individual particle is in a definite state Er $\rightarrow$  eigen value of the Schrodinger equation for the single particle.

•  $n_1$  particles have energy  $E_1$ 

•  $n_2$  particles have energy  $E_2$  

• ...........................................................

•  $n_r$  particles have energy  $E_r$  

• These set of numbers determine a microstate of the system $\rightarrow$  Occupation Numbers.

Basic Postulates-Third Postulate

• An energy level is degenerate $\rightarrow$  it my contain more than one quantum state.

• A particle in any of the quantum states  $g_r$   will have the same energy $E_r$   

Basic Postulates-Third Postulate

• For an isolated system in equilibrium, the probability for any one particle to be in a given quantum state is the same.

• Fundamental Postulate $\rightarrow$  'apriori probabilities'