Ensemble Theory (classical)-I (Concept of Phase Space and its Properties)

P.K. Ahluwalia

1. Learning Outcomes

After studying this module, you shall be able to

• Examine the relationship between an n particle system and its time evaluation in phase space
• Construct phase space trajectory and phase space volume of some interesting physical systems
• To recognize phase space as a hyperdimensional space with interesting geometrical properties.
• Appreciate the significance of phase space as a discrete space of consisting of cells each of minimum volume h^3N as per the Heisenberg’s uncertainty principle, where N is the number of particles in the system

2. Introduction

In the microscopic description of a macroscopic system, made up of a large number of particles (~ 1023), state of the system at any instant t requires specification of instantaneous position and momentum of each of the constituting particles making the system. This leads us to so called work bench of statistical mechanics, a kind of a conceptual space, where all the action in statistical mechanics happens for its statistical study. Phase space provides us the methodology for counting the number of possible states (microstates) for a given system of interest. In this module we study this idea leading to a hyper-dimensional space of momentum and position co-ordinate space called phase space. In this module we look at properties both classically and quantum mechanically of this hyperspace, which shall be useful later in statistical description of the system. The key notions are phase point, phase space trajectory and dimensionality.

3. Degrees of Freedom

In classical mechanics degrees of freedom, f, corresponds to a number of independent parameters needed to completely define position and motion of a mechanical system. Consider a system made up of a single particle allowed to move in all three directions. The state of such a system at any instant can be described by three position co-ordinates or three momentum co-ordinates, amounting to 3 degrees of freedom. If we constrain this particle to move in a plane, only two position co-ordinates or two momentum co-ordinates are needed to describe its state, reducing the number of degrees of freedom to 2. If we further constrain to move the particle along a straight line, one position or one momentum co-ordinate are needed to describe its state, thereby system having 1 degrees of freedom.
If a system has N particles, for its complete description we require 3N position co-ordinates or 3N momentum co-ordinates a total of 6N co-ordinates, i.e. f = 3N.

4. Phase Space

5. Visualisation of Phase Space

6. Properties of Hyperspace

7. Phase Space and Quantum Mechanics

Summary

In this module we have learnt

• About the important concept of degrees of freedom for a N particle system
• About the concept of phase space, the work bench of statistical mechanics, and the meaning of dimensionality of phase space, phase point, phase trajectory and constant energy surface.
• About how phase trajectory for a system evolves and determined by Hamilton’s equations of motion.
• How phase trajectory can be visualized in the case of a particle confined to move along a straight line of length L with constant energy E ?
• How phase space accessible to a particle executing simple harmonic oscillator with energy lying in the interval E and E + 8E ?
• How to calculate volume and surface area of a hypersphere in an n dimensional space? A useful result in counting the number of probable states available for a given state of a macroscopic system
• How in going from classical mechanics to quantum mechanics we discretize the phase space into cells with each cell with a minimum volume of h^3N guided by Heisenberg’s uncertainty principle which has an important implication of counting the number of probable states in a system?

References:-

1. Pathria R.K. and Beale P. D., Statistical Mechanics, 3rd ed. (Elsevier, 2011).

2. Landau L.D., Lifshitz E.M., “ Statistical Physics Part 1,” 3rd Edition, Oxford: Pergamon Press., 1982

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. Ma S.K., “ Statistical Mechanics,” Singapore: World Scientific Publishing Co. Pte. Ltd., 1985.