Temporal Flow Evolution

Temporal Flow Evolution: Formulating the Equation of Change

In this post, I'm considering an equation that models how flows evolve over time. This equation is given by:

$$\frac{\partial f}{\partial T} = L(f, \nabla f, \frac{\partial f}{\partial T})$$

This equation describes how the flow values $f$ change over what we can refer to as temporal progression. It incorporates:

• $f$: the flow values representing the state of the system at any time.
• $\nabla f$: the spatial gradient of flow, which describes how the flow varies across space.
• $\frac{\partial f}{\partial T}$: the rate of change of flow over time.

Now, this equation contains the term $\frac{\partial f}{\partial T}$ on both sides, leading to a potentially recursive structure. To simplify this, we refine it as:

$$\frac{\partial f}{\partial T} = L(f, \nabla f)$$

Here, $L$ is an operator that governs the dynamics of flow evolution, capturing the essential behavior of the system. The next sections will help us define this operator more clearly.

---

The Operator $L$

The operator $L$ is central to understanding how the system evolves. It should account for several key factors:

• Conservation of Total Flow: The flow in the system must remain conserved, implying no creation or destruction of flow.
• Interaction Rules: The operator governs how different flows interact—whether through diffusion, wave propagation, or other types of interaction.
• Threshold Behaviors: There are conditions where flow behaviors change significantly (e.g., near the speed of light).
• Non-local Correlations: Interactions may involve flows at different spatial locations, requiring a non-local term to respect causality.

---

Discrete Evolution in Planck Time

To capture the discrete nature of time in the model, we express the evolution in steps corresponding to Planck time $t_P$. This discretization ensures that changes only occur at sub-Planckian time intervals, as shown in:

$$f(T + t_P) = f(T) + t_P \cdot L(f, \nabla f)$$

---

Enforcing Key Constraints on the Operator $L$

For the system to be physically meaningful, we impose some constraints on the operator $L$. These constraints ensure the conservation of flow, discrete time evolution, and respect for the speed of light:

• Conservation of Flow: We impose that the total flow at any time remains unchanged, leading to the continuity condition:

$$\sum_i f_i(T + t_P) = \sum_i f_i(T)$$

Substituting the evolution equation, we get:

$$\sum_i f_i(T) + t_P \sum_i L(f, \nabla f) = \sum_i f_i(T)$$

This simplifies to:

$$\sum_i L(f, \nabla f) = 0$$

This condition implies that $L$ acts as a divergence operator that redistributes the flow without creating or destroying it. A possible form for $L$ is:

$$L = -\nabla \cdot J(f)$$

where $J(f)$ is the flow current.

• Discrete Evolution in Planck Time: We have already written the equation in terms of discrete time steps, ensuring changes only occur at intervals of Planck time.

• Speed of Light Constraint: We introduce a constraint on the magnitude of the flow to prevent it from exceeding the speed of light $c$:

$$|f| \leq c$$

We modify the operator $L$ to respect this constraint by ensuring the flow magnitude does not violate this limit:

$$|f(T) + t_P \cdot L(f, \nabla f)| \leq c$$

This ensures that as the flow approaches the speed limit, the operator reduces its magnitude.

---

Wave-like Behavior in Certain Regimes

In some regimes, especially for small perturbations, we expect the behavior of the system to resemble wave-like phenomena. In the continuum limit, the equation takes the form of a wave equation:

$$\frac{\partial^2 f}{\partial T^2} = v^2 \nabla^2 f$$

Thus, the operator $L$ must include terms like:

$$L(f, \nabla f) \approx v^2 \nabla^2 f$$

---

Non-local Interactions

In our system, non-local interactions can arise when the flow at a point is influenced by flows at other points. This is incorporated by defining $L$ as an integral over neighboring flows:

$$L(f, \nabla f) = \int K(x - x') f(x', T) \, d^3x'$$

where $K(x - x')$ is a kernel function enforcing non-local interactions. To preserve causality, the kernel satisfies the condition:

$$K(x - x') = 0 \quad \text{for} \quad |x - x'| > cT$$

This ensures that non-local interactions are confined within the light cone, maintaining the principles of causality.

---

Final Refined Equation

Combining all the constraints, the final form of the equation becomes:

$$f(T + t_P) = f(T) + t_P \left( \left( 1 - \frac{|f|}{c} \right) \left[ -\nabla \cdot J(f) + v^2 \nabla^2 f + \int_{|x - x'| \leq cT} K(x - x') f(x', T) \, d^3x' \right] \right)$$

This formulation ensures the system:

• Conserves flow,
• Evolves in discrete time steps,
• Respects the speed of light constraint,
• Exhibits wave-like behavior in certain regimes,
• Incorporates non-local interactions while preserving causality.

---

Expanding the Correction Factor

To ensure the speed of light constraint is respected, we expand the correction factor $\left( 1 - \frac{|f|}{c} \right)$. For small values of $|f|$, this factor approximates:

$$\left( 1 - \frac{|f|}{c} \right) \approx e^{-\frac{|f|}{c}}$$

Alternatively, we can redefine this as a function $\gamma(f)$, where:

$$\gamma(f) = 1 - \frac{|f|}{c}$$

Thus, the equation simplifies to:

$$f(T + t_P) = f(T) + t_P \, \gamma(f) \, L(f)$$

This form is more concise and maintains all necessary constraints, providing a refined model for temporal flow evolution.