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Bhupal Nobles' University, Udaipur Convocation | भूपाल नोबल्स विश्वविद्यालय, उदयपुर दीक्षांत समारोह

भूपाल नोबल्स विश्वविद्यालय दीक्षांत समारोह महाराणा प्रताप स्टेशन रोड, सेवाश्रम सर्कल, उदयपुर। भूपाल नोबल्स विश्वविद्यालय उदयपुर द्वारा वर्ष 2018 से 2024 तक की स्नातक एवं स्नातकोत्तर परीक्षा में उत्तीर्ण एवं विद्यावाचस्पति (Ph.D.) उपाधिधारियों के लिए दीक्षान्त समारोह 27 मार्च 2025 गुरूवार को प्रातः 10:30 बजे आयोजित करने का निश्चित हुआ है। दीक्षान्त समारोह में 2020 से 2025 तक की विद्यावाचस्पति की उपाधियों तथा स्नातक एवं स्नातकोत्तर परीक्षाओं में वर्ष 2024 तक प्रथम स्थान प्राप्त करने वाले छात्रों को उपाधि एवं स्वर्ण पदक प्रदान किए जायेंगे। अतः जो उपाधिधारी उक्त समारोह में उपाधि प्राप्त करने के इच्छुक हों, वे समारोह में उपस्थित होने की लिखित सूचना के साथ स्नातक एवं स्नातकोत्तर प्रथम वरीयता प्राप्त छात्रों हेतु, पंजीकरण शुल्क ₹500 व उपाधि शुल्क ₹5000 (कुल ₹5500) एवं विद्यावाचस्पति (Ph.D.), शोधार्थी पंजीकरण शुल्क ₹500 व उपाधि शुल्क ₹5000 (कुल ₹5500) नकद अथवा डिमाण्ड ड्राफ्ट भूपाल नोबल्स विश्वविद्यालय, उदयपुर के नाम बनाकर कुलसचिव, भूपाल नोबल्स विश्वविद्यालय, उदयपुर को दिनांक 17.03.2025 तक ...
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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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