Field Equations
Draft Formulation for the SET Resonance Substrate#
This document provides a working draft of the evolution equations for the Spin (S), Charge (C), and Temperature (T) fields in the resonance‑substrate model. The goal is to define a minimal, testable system that can be implemented in numerical solvers and compared against experimental data.
Quicklinks#
- docs README
- docs api integration examples
- docs api README
- docs api schema overview
- docs api using the schemas
- docs experiments faraday paradox experiment
- docs experiments README
- docs experiments replication checklist
- docs experiments resonance alignment tests
- docs experiments rotating conductor tests
- docs methods dimensional layers
- docs methods operator definitions
- docs methods README
- docs methods substrate dynamics
- docs methods triadic fields
- docs onboarding model map
- docs onboarding reading guide
- docs onboarding triadic quickstart
- docs onboarding verification tests
- docs overview comparison to gr models
- docs overview glossary
- docs overview introduction
- docs overview README
- docs overview resonance primitives
- docs overview theoretical background
- docs simulations boundary conditions
- docs simulations numerical methods
- docs simulations README
- docs simulations solver_architecture
- docs simulations validation metrics
- docs simulations core README
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1. Notation and Conventions#
- Spatial coordinate: (\mathbf{x} \in \mathbb{R}^3)
- Time: (t \in \mathbb{R})
- Spin field: (S(\mathbf{x}, t) \in \mathbb{R}^3)
- Charge field: (C(\mathbf{x}, t) \in \mathbb{R}) (scalar form for baseline model)
- Temperature field: (T(\mathbf{x}, t) \in \mathbb{R})
- (\nabla): spatial gradient
- (\nabla^2): Laplacian
- (\partial_t): time derivative
2. Spin Field Evolution#
The Spin field encodes local rotational alignment and coherence.
2.1 Baseline Equation#
[ \partial_t S = D_S \nabla^2 S - \gamma_S S + \lambda_{SC} \nabla C - \lambda_{ST} \nabla T ]
Where:
- (D_S): spin diffusion coefficient
- (\gamma_S): spin damping coefficient
- (\lambda_{SC}): coupling of Spin to Charge gradients
- (\lambda_{ST}): coupling of Spin to Temperature gradients
Interpretation:
- (D_S \nabla^2 S): smooths spin inhomogeneities
- (-\gamma_S S): relaxes spin toward zero alignment
- (\lambda_{SC} \nabla C): aligns spin with charge gradients
- (-\lambda_{ST} \nabla T): destabilizes spin in high‑T gradients
3. Charge Field Evolution#
The Charge field encodes interaction bias and potential gradients.
3.1 Baseline Equation#
[ \partial_t C = D_C \nabla^2 C - \gamma_C C + \beta_{CS} \nabla \cdot S ]
Where:
- (D_C): charge diffusion coefficient
- (\gamma_C): charge relaxation coefficient
- (\beta_{CS}): coupling of Charge to Spin divergence
Interpretation:
- (D_C \nabla^2 C): smooths charge gradients
- (-\gamma_C C): relaxes charge bias
- (\beta_{CS} \nabla \cdot S): generates or modulates charge bias from spin structure
4. Temperature Field Evolution#
The Temperature field encodes stochastic and dissipative contributions.
4.1 Baseline Equation#
[ \partial_t T = D_T \nabla^2 T - \gamma_T (T - T_0) + \eta_S |S|^2 + \eta_C |\nabla C|^2 ]
Where:
- (D_T): thermal diffusion coefficient
- (\gamma_T): relaxation toward background temperature (T_0)
- (\eta_S): heating from spin activity
- (\eta_C): heating from charge gradients
Interpretation:
- (D_T \nabla^2 T): spreads thermal energy
- (-\gamma_T (T - T_0)): relaxes toward ambient
- (\eta_S |S|^2): spin‑induced heating
- (\eta_C |\nabla C|^2): gradient‑induced heating
5. Resonance Envelope Condition#
A resonance envelope is defined as a region where SET gradients exceed a threshold:
[ \mathcal{R}(\mathbf{x}, t) = \left( |\nabla S| + |\nabla C| - \alpha |\nabla T| \right) - \Theta ]
Resonant region:
[ \mathcal{R}(\mathbf{x}, t) > 0 ]
Where:
- (\alpha): dissipation weighting factor
- (\Theta): resonance threshold
6. Dimensionless Form (Optional)#
For numerical work, the equations can be non‑dimensionalized by introducing characteristic scales:
- Length scale (L)
- Time scale (\tau)
- Field scales (S_0, C_0, T_0)
Resulting in dimensionless parameters:
- (\tilde{D}_S, \tilde{D}_C, \tilde{D}_T)
- (\tilde{\gamma}_S, \tilde{\gamma}_C, \tilde{\gamma}_T)
- (\tilde{\lambda}{SC}, \tilde{\lambda}{ST}, \tilde{\beta}_{CS}, \tilde{\eta}_S, \tilde{\eta}_C)
A separate note can specify the chosen scaling for a given simulation.
7. Implementation Notes#
- Discretization: finite difference, finite volume, or spectral methods.
- Time integration: explicit or implicit schemes (e.g., Runge–Kutta, Crank–Nicolson).
- Boundary conditions: periodic, Dirichlet, or Neumann, depending on experiment.
These equations are intended as a minimal, testable starting point. Coefficients and coupling terms can be refined based on experimental calibration and further theoretical development.