Statistical Mechanics

Microcanonical ensemble

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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, …

Remarks
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

A simulation

Phase space, ensemble and Liouville’s theorem

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Phase space, Ensemble and Liouville’s theorem.

Topics covered in this lecture

a. Ensemble and average — thermodynamic systems

b. Phase space — a classical system

c. Liouville’s theorem

Ensemble and average in thermodynamic systems

For a given “macrostate” (N, V, E) a statistical system, at any instant of time, t, is likely to be found in any one of an extremely large number of distinct “microstates”.

When time passes, the system evolves into different microstates. In due course of time the system exhibits an average behavior of all microstates it passes through. 

We can equivalently depict this behavior by envisaging a large number of mental copies of the system, with the same macrostate as the original system, but all the possible microstates, in which the system can exist, all at once. Such a collection of hypothetical or mental copies of the given system is known as an ensemble. 

Thus the average behavior of the ensemble is expected to be identical with the time-averaged behavior of the actual physical system. In fact this is one of the fundamental requirements for statistical mechanics to be valid. No matter which mathematical avenue we prefer to meander through we must in the end reach our unique destination of physical validity. 

To understand the deeper aspects of this ensemble theory we need to define what is known as “phase space” of a statistical system.

Entropy, probability and equilibrium in thermodynamic systems.

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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.