LEDO 5 Force - Itterative Resonance Modelling

Absolutely—let’s make this concise, structured, and simulation-ready.

Names & Abbreviations:

• $F_AR$ — Axial Resonance Force “Coaxial resonance” between nearly collinear Rotons with similar spectral paramameters. Axial Distance-independent axial magnitude (in axial 1D direction).
• $F_CR$ — Centric Resonance Force “Cocentric inward lockin force between nearly co-centric and co-axial Rotons with resonant frequency params. Centric force decays 2D linearly proportional to distance $~ 1/d$.
• $F_PR$ — Planar Resonance Force “Coplanar/parallel resonance” when both axes are parallel and the separation lies in their common rotation plane; decays in 2D linearly proportional to distance $~ 1/d$.
• $F_GR$ — General Resonant Attraction (“fallback” attractive term) Weak, omnidirectional resonance pull based on overall similarity; decays into 3D space proportional to the distances power of two $~ 1/d^2$.
• $F_EDP$ — Energy-Density Pressure Repulsive pressure from the gradient of total field energy density (sum of all waves); independent of detailed similarity, scaled by the roton’s own energy. Individual pressure typically decays into 3D space with $~ 1/d^2$.

Considerations:

The $F_CR$ force is separated from the $F_AR$ force, because it is a lock-in force (inwards and outwards) to the magnitude/radius of the rotation (or to their subparticles that is). It specifically applies, when the angular velocities are coherent (multiples of each other) and therefore have a common resonance frequency.

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Clean definitions (with compact formulas)

Let $i \neq j$. Notation: • Positions $\mathbf{x}i, separation \mathbf{r}{ij}=\mathbf{x}i-\mathbf{x}j, r{ij}=|\mathbf{r}{ij}|, \hat{\mathbf{r}}_{ji}=(\mathbf{x}_j-\mathbf{x}i)/r{ij}$. • Axes $\mathbf{t}_i,\mathbf{t}j (unit)$, axis parallelism $c{ij}=\mathbf{t}i!\cdot!\mathbf{t}j$. • Planarity factor (separation lies in the plane ⟂ to the axes): $\Pi{ij}=\big(1-(\mathbf{t}i!\cdot!\hat{\mathbf{r}}{ij})^2\big)\big(1-(\mathbf{t}j!\cdot!\hat{\mathbf{r}}{ij})^2\big)$. • Chirality match (spin orientation, clockwise vs counterclockwise): $\chi{ij}=\operatorname{sign}(\boldsymbol{\Omega}i!\cdot!\mathbf{t}i);\operatorname{sign}(\boldsymbol{\Omega}j!\cdot!\mathbf{t}j)\in{-1,+1}$. • Spectral similarity (radius/frequency match; tune widths $\sigma_a$,$\sigma\omega$): $Q{ij}=\exp!\Big(-\frac{(a_i-a_j)^2}{2\sigma_a^2}-\frac{(\omega_i-\omega_j)^2}{2\sigma\omega^2}\Big)$. • Soft core $r_0>0$ to avoid singularities; $\bar a{ij}=\tfrac12(a_i+a_j)$.

1. F_AL — Axial Locking (distance-independent magnitude)

Active when axes are almost collinear and spectra match. \boxed{ \mathbf{F}^{\mathrm{AL}}{j\leftarrow i} = \kappa{!AL},Q_{ij},c_{ij}^{,p},\chi_{ij};\hat{\mathbf{r}}{ji} \quad \text{for } r{ij}\le R_{AL}\ \text{(or always, if you want truly distance-independent)} } • p\ge 1 sharpens preference for parallel axes (c_{ij}\to 1). • \chi_{ij} lets you flip sign for opposite spin sense (same chirality attracts; opposite repels), per your note.

2. F_PR — Planar Resonance (~1/d)

Requires axes parallel and separation lying in their plane. \boxed{ \mathbf{F}^{\mathrm{PR}}{j\leftarrow i} =\kappa{!PR},Q_{ij},c_{ij}^{,p},\Pi_{ij}; \frac{1}{\max(r_{ij},,\bar a_{ij})};\hat{\mathbf{r}}{ji}\times \chi{ij}^{(\text{optional})} } • \Pi_{ij} suppresses pull if the line of centers points along the axis (not coplanar). • Optional \chi_{ij} factor if you want PR to also depend on spin sense.

3. F_GR — General Resonant Attraction (~1/d^2)

A weak, isotropic “baseline” attraction that only cares about being somewhat similar: \boxed{ \mathbf{F}^{\mathrm{GR}}{j\leftarrow i} = \kappa{!GR},\tilde Q_{ij},\frac{1}{(,r_{ij}+r_0,)^2};\hat{\mathbf{r}}{ji}, } • \tilde Q{ij} can be a looser similarity (e.g., only frequency), or even set to 1 for a pure background term.

4. F_EDP — Energy-Density Pressure (repulsive, gradient of total energy density)

Define a local energy density field (no phases needed; use envelopes only): \epsilon(\mathbf{x})=\sum_{k} \underbrace{\alpha_k,A_k^2}_{\text{source energy}};G!\big(|\mathbf{x}-\mathbf{x}_k|;,a_k\big), with a smooth, radial kernel (e.g., Gaussian·soft-1/r^2): G(r;a)=\frac{e^{-r^2/(2a^2)}}{r^2+r_0^2}. The pressure P=\gamma,\epsilon. The roton at \mathbf{x}_j feels: \boxed{ \mathbf{F}^{\mathrm{EDP}}j = -,\nabla P\big|{\mathbf{x}j} = -,\gamma,\nabla \epsilon(\mathbf{x}j) = -,\gamma \sum{i\ne j} \alpha_i A_i^2,\nabla G(r{ij};a_i). } • This is repulsive from high-energy regions (pressure pushes down the gradient). • Scale it relative to the roton’s own energy if desired: \gamma\to \gamma,f(A_j) (e.g., f(A_j)=1+\beta A_j).

Gradient (closed form): \nabla G(r;a)=\Bigg(\frac{-r}{a^2(r^2+r_0^2)}-\frac{2r}{(r^2+r_0^2)^2}\Bigg) e^{-r^2/(2a^2)},\hat{\mathbf{r}}.

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One unified force vs. three separate forces?

Mathematically, yes, you can embed F_AL, F_PR, and F_GR in a single anisotropic potential with angular kernels and different radial laws—think of a multipole-like expansion: \mathbf{F}{j\leftarrow i} = \sum{n} \kappa_n,\mathcal{A}n(\mathbf{t}i,\mathbf{t}j,\hat{\mathbf{r}}{ij},\chi{ij});\frac{1}{r{ij}^{,\nu_n}};\hat{\mathbf{r}}_{ji}, choosing \nu_n\in{0,1,2} to emulate AL (constant), PR (1/r), GR (1/r^2). But: • It’s harder to tune/interpret (angular kernels couple nonlinearly). • You’ll likely want different cutoffs and similarity widths for AL/PR/GR anyway. • For exploration and clarity, the three-term model is better. You can always collapse them later.

Recommendation: keep F_AL, F_PR, F_GR separate; leave F_EDP separate by design (it’s a field gradient, not pairwise).

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Final total force & torques (what you’d sum in code)

\boxed{ \mathbf{F}j =\sum{i\ne j}\Big(\mathbf{F}^{\mathrm{AL}}{j\leftarrow i} +\mathbf{F}^{\mathrm{PR}}{j\leftarrow i} +\mathbf{F}^{\mathrm{GR}}_{j\leftarrow i}\Big) +\mathbf{F}^{\mathrm{EDP}}_j. }

Alignment torque (optional, same idea as before): \boldsymbol{\tau}j = \lambda{\tau}\sum_{i\ne j}\Big[c_{ij}^{,p} ,(\mathbf{t}j\times \mathbf{t}i),\Theta{\text{AL/PR}}\Big] -\gamma\Omega,\boldsymbol{\Omega}j, with \Theta{\text{AL/PR}} picking which resonance channels (AL, PR) produce alignment.

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Notes on parameter choices (good defaults) • Exponents: p=2 (stronger bias for parallel axes). • Radii/frequency widths: \sigma_a \sim 0.3,\mathrm{median}(a), \sigma_\omega \sim 0.2,\mathrm{median}(\omega). • Soft core: r_0 \sim 0.05,\min a. • Magnitudes: choose \kappa_{!AL} \gtrsim \kappa_{!PR} \gtrsim \kappa_{!GR} (AL strongest, GR weakest). • Pressure scale \gamma: dial until clusters don’t over-collapse.