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National Bird Day: Why Birds Matter to Us

National Bird Day is celebrated every year on 5 January . This day is observed to spread awareness about birds, their importance in nature, and the need to protect them. Birds are beautiful living beings and play a very important role in keeping our environment healthy. Origin of National Bird Day National Bird Day was first celebrated in the year 2002 . It was started by bird lovers and environmental groups to protect birds from dangers like deforestation, pollution, and illegal hunting. The main aim of this day is to teach people, especially students, why birds are important and how we can help save them. Why Is National Bird Day Celebrated Every Year? National Bird Day is celebrated every year because many bird species are disappearing due to human activities. Cutting trees, using plastic, pollution, and climate change are harming birds and their homes. This day reminds us that: Birds need protection Nature should be respected Everyone has a responsibil...
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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 px, py, pz.
  • We imagine a 6-d space in which the six coordinates are x, y, z and px, py, pz 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, …, q3N and 3N canonical momenta p1, p2, …, p3N.
  • ∴ The dynamics of the system is determined by Hamiltonian H (p, q).
      ძH/ძpi = q̇i and ძH/ძqi = - ṗi
  • 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) d3Npd3Nq= number of representative points contained in the volume element d3Npd3Nq 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/ძqi = 0
                         

        or         [ρ, H] = 0
    • Thus in equilibrium state the Poisson bracket of ρ and H is zero.

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