Technique for Deriving New Boltzmann Distribution Functions

 This document outlines a systematic approach for adding new functions to a Boltzmann distribution class by leveraging known data points and scaling techniques.

This class is currently on github at https://github.com/BuckRogers1965/BoltzmannDist

## Step 1: Create a Stub Function

Start by adding a stub function that returns the 50% point scaled to the temperature range:

```python

def new_function_name(self, T):

    return self.point_energy(T)

```

This serves as a starting point, providing a baseline value that's already scaled to the correct order of magnitude for the temperature range.

## Step 2: Compare to Raw Data and Scale

Compare the output of your stub function to the raw data you're trying to match. Determine the scale factor needed to bring your values in line with the target data:

```python

def new_function_name(self, T):

    return self.point_energy(T) * 10e<ScaledToResults>

```

Replace `<ScaledToResults>` with the appropriate exponent to scale your results to the same order of magnitude as your target data.

## Step 3: Adjust for Ratio Differences

If the ratio between your scaled results and the target data is constant across all temperatures, divide by this ratio to achieve a perfect match:

```python

def new_function_name(self, T):

    return self.point_energy(T) * 10e<ScaledToResults> / <ratio>

```

Replace `<ratio>` with the constant ratio you observed. This step often reveals fundamental physical constants, as seen with the derivation of the Stefan-Boltzmann constant.

## Step 4: Compensate for Temperature-Dependent Variations

If the ratio isn't constant but varies with temperature, look for patterns in how it changes. You may need to introduce temperature-dependent terms to compensate:

```python

def new_function_name(self, T):

    return self.point_energy(T) * T**<power> * 10e<ScaledToResults> / <ratio>

```

Experiment with different powers of T or other temperature-dependent functions to match the observed pattern.

## Example: Area Under Curve Function

The area under the curve function provides a good example of this process:

```python

def area_under_curve(self, T):

    return self.point_energy(T) * T**4 * 10e-22 / 50.20444590190353665093425661

```

Here, we see:

- The base `point_energy(T)` scaled to the temperature.

- A T^4 factor, reflecting the temperature dependence of blackbody radiation.

- A scaling factor (10e-22) to match the order of magnitude.

- A precise divisor derived through this process, which turns out to be related to the Stefan-Boltzmann constant.

## Best Practices

1. **Document Your Process**: Keep notes on each step, including the raw data comparisons and the reasoning behind each adjustment.

2. **Verify Across a Wide Range**: Test your function across a wide range of temperatures to ensure it holds up under various conditions.

3. **Physical Interpretation**: Always try to understand the physical meaning behind the factors you introduce. They often relate to fundamental physical principles or constants.

4. **Precision Matters**: Use high-precision numbers in your calculations. Small differences can be significant when dealing with physical constants.

5. **Iterative Refinement**: Be prepared to refine your function multiple times as you discover more subtle relationships in the data.

By following this technique, you can systematically derive new functions that accurately model various aspects of the Boltzmann distribution and related physical phenomena, potentially uncovering fundamental relationships and constants in the process.