Hamiltonian related to TFP?

 

Start from TFP Node Hamiltonian

In TFP (Sections 1–5, 7, 15), each node $i$ carries a bidirectional flow:

$$\Psi_i = F_i^+ + i F_i^-$$

The local Hamiltonian capturing flow energy, coupling, and residual misalignment is:

$$H_i = \frac{1}{2} \left(|F_i^+|^2 + |F_i^-|^2\right) + \frac{\alpha}{2} \sum_{j \in N(i)} | \Psi_j - \Psi_i |^2 + V_\text{misalign}(i)$$

Where:

$$V_\text{misalign}(i) = \frac{1}{2} \beta(l) \, |\Psi_i|^2$$

• First term → kinetic-like term (flow magnitude squared).

• Second term → nearest-neighbor coupling (analogous to chain $k$).

• Third term → residual misalignment / “damping” energy.

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2️⃣ Define Conjugate Variables

TFP has intrinsic flows; define:

$$\Delta F_i = F_i^+ - F_i^-, \quad P_i = i(F_i^+ + F_i^-)$$

• Treat $\Delta F_i$ as “position”

• Treat $P_i$ as “momentum”

Then canonical Hamiltonian mechanics gives:

$$\dot{\Delta F_i} = \frac{\partial H}{\partial P_i}, \quad \dot{P_i} = - \frac{\partial H}{\partial \Delta F_i}$$

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3️⃣ Linearize Around Coherent State

Assume small deviations $\delta \Delta F_i$ around a coherent cluster:

$$\Delta F_i = \langle \Delta F \rangle + \delta \Delta F_i$$

Expand Hamiltonian to quadratic order:

$$H \approx \sum_i \frac{1}{2} (\delta P_i)^2 + \frac{\alpha}{2} \sum_{<i,j>} (\delta \Delta F_i - \delta \Delta F_j)^2 + \frac{1}{2} \beta(l) (\delta \Delta F_i)^2$$

• First term → kinetic

• Second term → harmonic coupling

• Third term → “misalignment potential” = scale-dependent damping

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4️⃣ Derive Equations of Motion

Canonical equations:

$$\dot{\delta \Delta F_i} = \frac{\partial H}{\partial \delta P_i} = \delta P_i$$ $$\dot{\delta P_i} = - \frac{\partial H}{\partial \delta \Delta F_i} = -\alpha \sum_{j \in N(i)} (\delta \Delta F_i - \delta \Delta F_j) - \beta(l) \, \delta \Delta F_i$$

Combine to eliminate momentum:

$$\ddot{\delta \Delta F_i} + \beta(l) \, \delta \Delta F_i = \alpha \sum_{j \in N(i)} (\delta \Delta F_j - \delta \Delta F_i)$$

• Exactly analogous to 1D chain wave equation with damping.

• Nearest-neighbor sum gives discrete Laplacian along chain.

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5️⃣ Specialize to 1D Chain

For 1D chain, $j = i \pm 1$:

$$\ddot{\delta \Delta F_i} + \beta(l) \, \delta \Delta F_i = \alpha (\delta \Delta F_{i+1} - 2 \delta \Delta F_i + \delta \Delta F_{i-1})$$

✅ This is directly the 1D chain equation with:

• Mass = 1 (normalized)

• Coupling = α (from TFP nearest-neighbor flow alignment)

• Damping = β(l) (residual misalignment / decoherence)

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6️⃣ Interpretation

• α → k in chain

• β(l) → η in chain

• ΔF_i → x_i

Residual misalignment β(l) is intrinsic in TFP:

• Emerges from recursive cluster coherence

• Is scale-dependent → longer clusters → smaller β(l)

• Unlike chain, β(l) is not externally imposed, it’s fully dynamical.

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✅ Linearized TFP Chain Summary

$$\boxed{ \ddot{\delta \Delta F_i} + \underbrace{\beta(l)}_{\text{intrinsic damping}} \delta \Delta F_i = \underbrace{\alpha}_{\text{flow coupling}} (\delta \Delta F_{i+1} - 2 \delta \Delta F_i + \delta \Delta F_{i-1}) }$$

• Coherent propagation ↔ α-term

• Decoherence ↔ β(l)-term

• This rigorously shows TFP reduces to a chain-like Hamiltonian system in the linear limit.