Phase Space, Phase point, Phase trajectory, ensemble
Phase Space (6D space) (r-p space)
The concept of phase space is introduced to establish the theory of statistical mechanics from
the geometrical point of view.
The phase or state of any dynamical particle can be defined by 6 parameters (x, y, z, px, py, pz).
3 (x, y, z) for its position and 3 (px, py, pz) for its momentum (motion)
Let us imagine 6D space having 3D configuration space and 3D momentum space, called phase
space.
Let us consider a system having N particles.
1 particle can be represented by 6 parameters in 6D space called μ-phase space (or, μ-space)
N particles can be represented by 6N parameters in 6ND space called Γ-phase space (or, Γ-space)
So, Γ-space = N × μ-space
Let us divide the phase space into large no of tiny cells, each cell has 6 sides dx, dy, dz, dpx, dpy,
dpz.
The volume of each tiny cell is 𝑑𝜏 = 𝑑𝑥𝑑𝑦𝑑𝑧𝑑𝑝𝑥 𝑑𝑝𝑦 𝑑𝑝𝑧
From Heisenberg uncertainty principle we know that,
𝑑𝑥𝑑𝑝𝑥 ≥ ℎ,
𝑑𝑦𝑑𝑝𝑦 ≥ ℎ,
𝑑𝑧𝑑𝑝𝑧 ≥ ℎ
 𝑑𝜏 ≥ ℎ3  min volume of phase space is ℎ3
When we consider a system of large no of particles, all particles are distributed in phase space
among different cells having minimum volume ℎ3 following Heisenberg uncertainty Principle.
Each particle is located somewhere in the cell having min volume ℎ3 rather than have a discrete
location.
The manner in which the particles are distributed among different cells of the phase space is
called distribution law.
For classical particle, distribution law is called MB distribution law, for Bosons, the distribution
law is called BE distribution law, for Fermions, the distribution law is called FD distribution law.
Each cell of the phase space is associated with a definite value of energy called energy level.
There may be several energy states having same value of energy then that level is called
degenerate level. The multiplicity of the level is called degeneracy.
(r, p)
p
6D space
r
If we gradually reduce the size of the cell in the limit we will approach to a point called phase
point. In rectangular frame of reference the phase point is represented by (r,p).
(r,p) represents the phase of a particle at a given instant of time.
Phase trajectory
At any instant of time (t1) phase or state of any particle can be represented by (r1,p1).
At time t2 phase or state of that particle can be represented by (r2, p2)
At time t3 phase or state of that particle can be represented by (r3, p3) and so on.
The variation of phase points or locus of phase points is called phase trajectory.
(It does not mean the actual path traced by the particle.)
Problem 1:
Consider a 1D simple harmonic oscillator of mass m and spring constant K and total energy E.
Obtain phase trajectory of H.O.
To obtain the phase trajectory we are to start from total energy and we are to obtain an
expression consisting of position and momentum. (Phase trajectory is the locus of phase points
and each phase point is represented by position and momentum).
𝐸=
1 2 𝑝2
𝐾𝑥 +
2
2𝑚
Divide E throughout the equation
𝑥2
2𝐸 ⁄𝐾
+
𝑝2
2𝑚𝐸
= 1  Eq. of ellipse
So phase trajectory of 1D H.O is an ellipse with semi major and minor axis √2𝐸 ⁄𝐾 and √2𝑚𝐸
respectively.
If we consider a damped oscillator, energy is not constant. The energy is decaying exponentially.
What will be the phase trajectory? It will be a spiral.
Problem 2:
Consider a mass m having constant energy E under a vertical motion. At any instant of time,
position and momentum are x and px . Obtain the phase trajectory of the particle.
Total energy = P.E + K.E
𝑚𝑔𝑥 +
𝑝𝑥2
2𝑚
=𝐸

Phase trajectory is parabola.
𝑝𝑥2 + 2𝑚2 𝑔𝑥 = 2𝑚𝐸 (multiplying 2m throughout the eq.)
Problem: A classical particle of mass m is free to move in a cube of side l. If its energy is less than
or equal to E. Find the volume of the phase space available to it.
𝛤 = ∫ 𝑑𝑥𝑑𝑦𝑑𝑧𝑑𝑝𝑥 𝑑𝑝𝑦 𝑑𝑝𝑧
= 𝑉 ∫ 4𝜋𝑝2 𝑑𝑝
𝐸
= 4𝜋𝑙 3 ∫
0
p+dp
p
2𝑚𝐸𝑚𝑑𝐸
√2𝑚𝐸
𝐸
= 4√2𝜋𝑚3⁄2 𝑙 3 ∫0 𝐸 1⁄2 𝑑𝐸
2
= 4√2𝜋𝑚3⁄2 𝑙 3 𝐸
3
=
p
3⁄
2
8√2
3
π𝑚3⁄2 𝑙 3 𝐸 3⁄2
[ For the integration ∫ 𝑑𝑝𝑥 𝑑𝑝𝑦 𝑑𝑝𝑧 let us consider momentum space. Consider two concentric spheres of radii 𝑝 and
𝑝 + 𝑑𝑝 where 𝑝2 = 𝑝𝑥2 + 𝑝𝑦2 + 𝑝𝑧2 . The infinitesimal volume lieing between p and p+dp is 4𝜋𝑝2 𝑑𝑝 which
corresponds to 𝑑𝑝𝑥 𝑑𝑝𝑦 𝑑𝑝𝑧 ]
[
𝑝2
2𝑚
=𝐸
 𝑝2 = 2𝑚𝐸 
2𝑝𝑑𝑝 = 2𝑚𝑑𝐸
 𝑑𝑝 =
𝑚𝑑𝐸
√2𝑚𝐸
]
Ensemble
Ensemble is a French word. The meaning is assembly of systems. Systems of such an ensemble
are macroscopically identical but microscopically different. For instance, if the positions and
motion of two or more molecules are interchanged, a new microscopic state will result; yet from
the macroscopic point of view no change will be detected in the condition of the gas.
Three types of ensemble are important in the application of Statistical Mechanics. The
classification depends on the manner in which the systems interact. The interaction may be in
variety of ways. In thermodynamics they exchange energy or matter or both with each other.
Microcanonical ensemble
The ensemble in which neither energy nor matter is exchanged is called a microcanonical
ensemble.
Canonical ensemble:
The ensemble in which the microsystems exchange energy but not matter is called a canonical
ensemble.
Grand Canonical ensemble
The ensemble in which the microsystems exchange both energy and matter is called grand
canonical ensemble.