Phase space and density function | Statistical mechanics

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

• 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

                         

        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 on the link for English and  click on the link for Hindi

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