Authors : Muhammad Umar Jabbar
DOI : 10.2139/ssrn.6746438
Volume : 1
Issue : 1
Year : 2026
Page No : 11
The Virial Theorem asserts kineticpotential equipartition for conservative harmonic systems in the innite-time limit. For an N-body system of coupled viscously damped harmonic oscillators observed over a nite window [t0, t0 + τ ], the normalised energy departure Φ(t0, τ) = |⟨T ⟩-⟨V ⟩|/(⟨T ⟩ + ⟨V ⟩) is generically non-zero and has not previously been expressed analytically for the deterministic N-body case. We derive a closed-form expression for Φ(t0, τ) valid for arbitrary N , arbitrary initial conditions, and arbitrary observation window in the weak-damping regime ε = γ/(2ωmin) ≪ 1. The result (Theorem 1) expresses Φ as the absolute ratio of two exactly evaluated window integrals over the normal-mode solution; no numerical quadrature is required. A rigorous O(ε 2) absolute error bound (Proposition 1) with constant Cq expressed through the spectral invariants of M-1/2 KM-1/2 is proved via an operator-norm argument. A time-rescaling symmetry identies γτ as the fundamental dimensionless parameter. The geometric content of Φas a normalised projection onto the energy-imbalance subspace of the 2N-dimensional phase spaceis established, and its exponential convergence to zero is derived as a consequence of Liouville volume contraction at rate N γ via an explicit Lyapunov function. A biorthogonal complex-mode framework extends the result to non-proportionally damped systems. Numerical verication for N = 4 conrms the formula to within 3.44 × 10-5 across eight test cases, consistent with the proved bound (Fig. 2). Three applications are identied: an exact ergodic relaxation-time criterion τ ≳ γ-1 ln(1/δ) for Φ < δ; a harmonic baseline for molecular-dynamics virial-pressure validation; and a transient diagnostic for pump-probe spectroscopy. The spectral-measure representation of Φ connects its convergence to the RiemannLebesgue lemma in the thermodynamic limit N → ∞, unifying deterministic phase-space contraction with ergodic-theory arguments.