Introduction to Statistical Mechanics

Topics Covered : Introduction to Micro-sates, Macro-states of a system. To know about Distinguishable and indistinguishable particles. The Boltzmann relation.  

    

Micro-states & Macro-states | Boltzmann Relation    

Micro-states:  

    The microscopic state or micro-state of a system is defined by specifying the microscopic parameters of the individual particles of the system.

    [ Position, Velocity, Momentum are the microscopic parameters of the system. ]

Macro-states:

    The macroscopic state or macro-state of a system is defined by specifying the macroscopic parameters of the system. 

    [ The parameters which characterize the system as a whole and not the individual particles of the system are called macroscopic parameters. Such as - Presser, volume, temperature. ]

    A large number of microstates can produce a single macro-state.

    Now, we have talked about some system of particles of particles in the above section. Those particles of the systems may be of two types.  

 

Distinguishable Particles:

If the molecules of the system of particles can be distinguished i.e. can be labeled in any way by which we can separate them from others in the assembly of molecules then they are called distinguishable particles.

Indistinguishable Particles:

    If the molecules of the system of particles can not be labeled in any way then they are called  indistinguishable particles.

  Example :   Let us take an example of three spin-half particles each having magnetic moment μ which can be oriented up or down. Now, how their micro-states and macro-states are arranged ?

Here, we can label the particles only on basis of their spin, thus they can make the following macro-states and micro-states.

Possible Macro-states	Possible Micro-states	Total Magnetic Moment
Three Up-Spin	↑↑↑	μ+μ+μ= 3μ
Two Up and one down spin	↑↑↓ ↑↓↑ ↓↑↑	μ+μ-μ= μ μ-μ+μ= μ -μ+μ+μ= μ
Two down and  one up spin	↓↓↑ ↓↑↓ ↑↓↓	-μ-μ+μ= -μ -μ+μ-μ= -μ μ-μ-μ= -μ
Three down spin	↓↓↓	-μ-μ-μ= -3μ

Relation between Statistical-mechanics and Thermodynamics (Boltzmann Relation) :

    Let us consider two physical systems A₁ and A₂ , each in equilibrium condition separately, have microstates Ω₁(N₁,V₁,E₁) and Ω₂(N₂,V₂,E₂) respectively. They are separated by rigid walls so that the volume(V) and the number of particles in the system (N) is constant. Now, we bring them to thermal contact with each other, thus the energy(E) can be exchanged between the two systems.  

A1 (N1,V1,E1)

A2 (N2,V2,E2)

(The two systems A₁ and A₂ are in Thermal contact with each other only.)

When the exchange of energy is stopped i.e. reached at equilibrium condition then the total energy $E^0$ of the composite system is given by- 

------------------------(1)  

    Now, the possible number of microstates of the composite system is given by,

   

------------------------------(2)  

                                                     

 is constant. Thus, Ω₀(E₁,E₂) can be expressed as a function of E₁ only _.

    _{Now, at equilibrium condition the number of microstates is maximum. Let, at equilibrium condition E1=E'1 and}  . At equilibrium condition the number of microstates is maximum.   

Then, maximizing Ω₀ = Ω₀(,E₁)     we get, 

----------------------(3)  

From thermodynamics we know ,

        

 [ Here, S= Entropy, T= temperature in S.I units]

Now, putting this in equation (3) we get,

 or,              

 [ where , $ k_\beta $ is Boltzmann Constant ]

       or,     

This equation relates statistical mechanics and thermodynamics. Again, it gives the statistical interpretation of Entropy.

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