Ensemble Theory(classical)-III Micro-canonical Ensemble

P.K. Ahluwalia

1. Learning Outcomes

After studying this module, you shall be able to

• See the importance of Planck’s constant in enumerating the number of allowed microstates in the light of Heisenberg uncertainty principle.
• Recognize the constraints under which micro canonical ensemble is defined and characterized by thermodynamic variables.
• Write the micro-canonical density distribution function.
• Clarify why a small amount of arbitrarily small uncertainty in energy ?? around ? is needed to count total number of allowed microscopic states.
• Calculate allowed number of microstates and hence entropy in an arbitrarily thin shell of a given microscopic ensemble for sample prototype models

o Classical ideal gas.
o A two level system.
o A system of classical harmonic oscillators.
o Magnetization of a system of spin ?/? particles.

2. Introduction

In this module we will study the highly constrained ensemble the so called micro canonical ensemble, representing a system having constant energy ?, constant number of particles ? and constant volume ?. It describes a system which is completely isolated system from its surroundings, a highly idealized situation seldom encountered in real situations. Within these constraints particles of the system can be in any one of the allowed microstates with equal apriori probability. Following the prescription described in module…… an effort is made to count the allowed total number of microstates which helps define the probability of each state followed by calculation of entropy which can then be used to calculate the thermodynamic properties of the system under study. But before embarking on this plan for micro-canonical ensemble we shall underline the importance of Planck’s constant in the light of Heisenberg Uncertainty Principle in counting the microstates in the allowed phase space of the system under study.

3. Phase Space and Number of Microscopic States

As yet, till this point we have described the enumeration of microstates in terms of calculating the volume of the allowed phase space. This enumeration of microstates is possible through Heisenberg’s Uncertainty Principle according to which phase space can be regarded as made up of elementary cells of the minimum volume ?^? where ? is the dimension of the phase space representing single particle quantum state. In the case of an N particle system with phase space having 6N dimensions, ?=??.

4. Micro-canonical Ensemble Distribution Function

5. Applications

5. Summary

In this module we have learnt

• _{That micro canonical ensemble corresponds to a collection of exact replicas of completely isolated system with ?=????????, ?=???????? and ?=????????.}

References:-

1. Pathria R.K. and Beale P. D., Statistical Mechanics, 3rd ed. (Elsevier, 2011).
2. Kubo R., “Statistical Mechanics: An Advanced Course with problems and Solutions,” Amsterdam: North Holland Publishing Company, 1965.
3. Pal P.B., “An Introductory Course of Statistical Mechanics”, New Delhi: Narosa Publishing House Pvt. Ltd., 2008.
4. Panat P.V., “Thermodynamics and Statistical Mechanics,” New Delhi: Narosa Publishing House Pvt. Ltd., 2008
5. Brush S.G., “Statistical Physics and the Atomic Theory of Matter, from Boyle and Newton to Landau and Onsager”, New Jersey: Princeton University Press, 1983.
6. Ma S.K., “ Statistical Mechanics,” Singapore: World Scientific Publishing Co. Pte. Ltd., 1985.