Phase space and density function

Sacademy July 12, 2023 2 min read

Phase space and density function

Phase space or 𝚪 space

In classical mechanics the position of a point particles is described in terms of three Cartesian coordinates x, y, z.
And the state of motion of particle is described in terms of velocity component ẋ, ẏ, ż or momentum coordinates p_x, p_y, p_z.
We imagine a 6-d space in which the six coordinates are x, y, z and p_x, p_y, p_z are marked along six mutually perpendicular axes in space.
The combined position and momentum space is known as phase space or Γ space.
A point in the phase space represents the position and momentum of the particle at some particular instant.

Density function

Let a classical system has a large number of molecules (N) occupying a large volume V.
Generally N = 10²³ molecules and V = 10²³ molecular volumes or N → ∞ and V → ∞
N/V = v ; here v = a specific volume, which is a finite number.
The system will be regarded as isolated in the sense that the energy is a constant of the motion.
A state of the system is completely and uniquely defined by 3N canonical coordinates q₁, q₂, …, q_3N and 3N canonical momenta p₁, p₂, …, p_3N.

∴ The dynamics of the system is determined by Hamiltonian H (p, q).

ძH/ძp_i = q̇_i and ძH/ძq_i = – ṗ_i

Since energy E is conserved, therefore the locus of all points in Γ-space satisfying the condition H (p, q) = E defines a surface, which is known as energy surface of E.
The path always stays on the same energy surface.
For a macroscopic system having N particles, V volume and energy lying between E and E + ΔE, a distribution of points in Γ-space is characterized by a density function ρ (p, q, t) defined by

ρ (p, q, t) d^3Npd^3Nq= number of representative points contained in the volume element d^3Npd^3Nq located at (p, q) in Γ-space at any instant t.

According to Liouville’s theorem dρ/dt = 0

$\frac{\partial\rho}{\partial t}+\sum_{i=1}^{3N}\left(\frac{\partial\rho}{\partial q_{i}}\dot{q}_{i}-\frac{\partial\rho}{\partial p_{i}}\dot{p}_{i}\right)=0$

But ძH/ძp_i = q̇_i and ძH/ძq_i = – ṗ_i

$\therefore\;\;\;\;\;\frac{\partial\rho}{\partial t}+\sum_{i=1}^{3N}\left(\frac{\partial\rho}{\partial q_{i}}\frac{\partial H}{\partial p_{i}}-\frac{\partial\rho}{\partial p_{i}}\frac{\partial H}{\partial q_{i}}\right)=0$

Geometrically it states that the distribution of points in Γ-space moves like an incompressible fluid.
In equilibrium state, ρ = ρ (p, q) does not depend on time ⇒ ძH/ძq_i = 0

$\sum_{i=1}^{3N}\left ( \frac{\partial \rho }{\partial q_{i}}\frac{\partial H}{\partial p_{i}}- \frac{\partial \rho }{\partial p_{i}}\frac{\partial H}{\partial q_{i}} \right )=0$

or [ρ, H] = 0

Thus in equilibrium state the Poisson bracket of ρ and H is zero.

To know more about Invariance of Poisson bracket under canonical transformation click here for English and click here for Hindi
To know in detail about the phase space click here for Hindi and click here for English

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