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Microcanonical ensemble

Microcanonical ensemble
Lecture IV; This lecture, the 4th in the series of statistical mechanics lectures, a paper for the physics honors degree class, was delivered on the 10th of January this year (2018).

You can find the previous lectures here ( Lecture — I, II ) and here ( Lecture — III ).

Topics covered in this lecture

a. Recapitulation of some previous ideas and  — important remarks

b. Microcanonical ensemble — definition and properties 

c. Some basic parameters and formalism 

Recapitulation and remarks
In our previous lecture we defined the phase space density or distribution function rho (q, p; t) for a classical statistical system with an aim to connect it to a thermodynamic system.

We saw that an ensemble system would be stationary if rho does not have any explicit time dependence, …

The type of general ensemble we defined as mental copies of actual system occupying each possible microstate can be called a Gibbsian Ensemble.
The above condition of statistical density as a stationary or time-independent variable would represent conditions of equilibrium.
We defined ensemble average of a physically measurable quantity

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Entropy, probability and equilibrium in thermodynamic systems.

The current lecture numbered lecture – I and II, is intended to be an introduction to the statistical mechanics paper of a Physics honors degree. It was delivered to the same class, on 22 November 2017.

Topics covered: 
i. Micro and macro state. 

ii. Entropy and thermodynamic probability and thermal equilibrium.  

Thermodynamic limit. 

Lets consider a physical system which is composed of N identical particles, in a volume of V. N is an extremely large number, typically in the order of 10^{23}.

Lets confine ourselves to the “thermodynamic limit”. i.e. N goes to infinity, V goes to infinity so that; n = N/V is fixed at a value chosen.
Important note: The ratio n is known as number density or particle number density — also concentration is sometimes used instead of density. One can distinguish them by referring to mass concentration vs number concentration. In a similar way one must distinguish number density from the not so unrelated parameter by the name mass-density. 

Extensive properties. 

In the thermodynamic limit, the “extensive properties” of the system such as energy E and entropy S are directly proportional to the size of the system, viz. N or V.

Intensive properties. 
Similarly the “intensive properties” such as temperature T, pressure P and chemical potential (mu) are independent of the size.  

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