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Keyboard Shortcuts

Φ(t₀, τ) = |Σ_α w_α I_C(α, t₀, τ)| / [I_E(t₀, τ) · Σ_α w_α] I_E(t₀,τ) = [e^{−γt₀} − e^{−γ(t₀+τ)}] / (γτ) H = Σ p²/2m + ½Σ k_ij(q_i−q_j)² ⟨T⟩ = −½⟨Σ r_i · F_i⟩ Ω = Σ_α ω_α · w_α / Σ w_α

Key Equations

TRVIT Theorem 3.1:

$$\Phi(t_0,\tau) = \frac{|\sum_\alpha w_\alpha I_C(\alpha, t_0, \tau)|}{I_E(t_0, \tau) \cdot \sum_\alpha w_\alpha}$$

Energy Decay:

$$I_E(t_0,\tau) = \frac{e^{-\gamma t_0} - e^{-\gamma(t_0+\tau)}}{\gamma\tau}$$

Hamiltonian:

$$H = \sum \frac{p^2}{2m} + \frac{1}{2}\sum k_{ij}(q_i-q_j)^2$$

Virial (Clausius 1870):

$$\langle T \rangle = -\frac{1}{2}\langle \sum r_i \cdot F_i \rangle$$

GOMT Scalar:

$$\Omega = \frac{\sum_\alpha \omega_\alpha \cdot w_\alpha}{\sum w_\alpha}$$

Publications

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Exact Modified Virial Identity for Driven-Damped Non-Harmonic Oscillators in NESS

Author list removed for double-blind review

Submitted to Physica Scripta, 2026

Exact identity for the time-averaged energy partition of a periodically driven, damped, non-harmonic oscillator in its non-equilibrium steady state. Introduces the virial defect Δ, verified to within 0.13% across 30 parameter combinations.

Read Summary Full PDF Under Double-Blind Review
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TRVIT v2.2: Time-Resolved Virial Imbalance Theorem

Muhammad Umar Jabbar

Zenodo, April 2026

Exact closed-form expression for normalised kinetic-potential imbalance in N-body coupled damped harmonic systems. Error bound ≤ 3.44 × 10⁻⁵, O(ε²) proved.

DOI: 10.5281/zenodo.19994556 Preprint — Zenodo Open Access
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JAP: The Jabbar Adal Principle

Muhammad Umar Jabbar

Zenodo, February 2026

Two formally proved theorems in non-relativistic Hamiltonian mechanics establishing structural symmetry constraints on phase-space trajectories.

DOI: 10.5281/zenodo.19815385 CC BY 4.0
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GOMT: Global Oscillator Mode Trace Framework

Muhammad Umar Jabbar

Zenodo, January 2026

Single-scalar spectral fingerprint for N coupled harmonic oscillators. Enables mode identification and structural comparison with O(1) storage.

DOI: 10.5281/zenodo.19798586 CC BY 4.0