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Phase space, ensemble and Liouville’s theorem
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 timeaveraged 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.

Aerial view of Bhabanipatana
The video shows a beautiful video of the city of Bhabanipatana — alternated to Bhawanipatna due to influence of Hindi, in the district of Kalahandi of the state of Odisha. The name Kalahandi might invoke a stark sense of depravity as it brings to memory the notoriety of poverty associated with the KBK districts and the perpetual image of the state itself as a residence of the poorest of poor. But reality is as startlingly contrasting as this video shows.

The plight of Paralakhemundi
Once the citadel of the mighty Gajapati, who ruled from the Ganga to the Godavari: Paralakhemundi of the Gajapati district, languishes in the peril of apathy from the system, read mostly, the Indian Railways here.
26 years is a long time in democratic memory and forcing our own citizen to live in the darkness like paupers is nothing less than sort of a genocidal crime.

Crystal structure: introduction to lattice properties.
Lattice and crystals.
A lattice is a threedimensional periodic array of identical building blocks. The building blocks are atoms or groups of atoms. The crystals usually come with imperfection of structure and impurities.The periodicity of crystals is well established by the experimental studies of Xray, neutron and electron diffraction patterns.
A solid is a crystal if the positions of the atoms in it are exactly periodic. Here is a diagram that represents this property ideally.
i. Distance between two nearest neighbors is ‘a’ along xaxis and ‘b’ along yaxis, where x–, and y– axes are not necessarily orthogonal.
ii. A perfect crystal maintains the periodicity for infinity < x < infinity and infinity < y < infinity . The points A, B, C are equivalents. That means for an observer at A, the environment at A is exactly the same as it is for an observer at B or C.
This is expressed by saying crystals have translational symmetry. e.g. if the crystal is translated by a vector R — joining two atoms, the appearance of the crystal remains unchanged.The atoms have no restrictions as to which location they preside over, as long as that position can be occupied by any atom, it can be taken over by any other given atom, and all others would relent.
Imperfection in crystals.
There are no perfect crystals though, defined the above way. All crystals have some degree of imperfections. There are 3 basic examples of imperfections.i. Atoms near the surface have a different environment than atoms deep inside the crystal.
ii. Due to thermal vibrations, equilibrium position of atoms are distorted, which depends on temperature T.
iii. Atoms always contain foreign elements known as impurities.
The effect of imperfections can be neglected in very ideal crystals. Imperfections lead to interesting physical properties of crystals. E.g. Resistivity of metals is a result of thermal vibrations of atoms. — We will discuss this at a later time, in this course.
When atoms are replaced by geometrical points, geometrical patterns depicting the periodicity of the crystals are obtained. They do not have any physical contents. Such geometrical patterns are known as "Lattice" or "crystal Lattice".
Bravais and nonBravais lattices.
There are two classes of lattices, Bravais and nonBravais lattices.Bravais Lattice: In a Bravais lattice all lattice points are equivalent, hence all atoms of the crystal are of the same kind.
NonBravais Lattice: In a nonBravais lattice some of the lattice points are not equivalent.
This is easily understood by the following diagram.

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 massdensity.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.