So to fulfill what I said in the last post, here is the first part that will lead to the description of my model, coming together in some... pretty novel ways. This first part is largely instructional, part 2 will come tomorrow.

If [math]L[/math] and [math]\Delta V[/math] do not change very much over a certain temperature rage than integration yields:

[math]P_2 - P_1 = \frac{\mathbf{U}}{\Delta V}\ \ln \frac{T_2}{T_1}[/math]

Now I go back to a blog investigation into similar equations which use this temperature gradient to measure anistropies, except it uses a time derivative in a conventional application which removes the need of subscripts (ie: [math]\frac{\dot{T}}{T}[/math]) - so lets take a look at those entropy equations.

https://physics.stac...peyron-equation

Now, let's take into consideration an excerpt from my blog issue on entropy:

The relativistic heat energy equation was from Fourier’s law of heat induction was:

[math]Q = -k\Box T = - k \nabla T + \frac{ik}{c} \frac{\partial T}{\partial t}[/math]

(where [math]k[/math] is conductivity - while this has been for years a conventional notation, when we introduce the Boltzmann constant we will change it to [math]\mathbf{k}[/math] with the Boltzmann constant in usual notation [math]K_B[/math]).

In which the heat energy density is defined in the following way first in simple Cartesian coordinates,

[math]\mathbf{Q} = - k \nabla^2 T= -k (\frac{\partial^2 T}{\partial x^2} + \frac{\partial^2 T}{\partial y^2} + \frac{\partial^2 T}{\partial z^2})[/math]

However, the d'Alembert operator just involves an extra term and is nothing too complicated,

[math] \mathbf{Q} = - k \Box^2 T= - k (\frac{\partial^2 T}{\partial \tau^2} - \frac{\partial^2 T}{\partial x^2} + \frac{\partial^2 T}{\partial y^2} + \frac{\partial^2 T}{\partial z^2}) = k( \frac{\partial^2 T}{\partial \tau^2} + \nabla^2T)[/math]

I noticed that an entropy can be formed in the following way: The heat energy from Fourier's law is just

[math]\delta Q = -\int k\Box^2 T\ dV[/math]

As we already established, but we have changed it slightly for the squared d’Alembertian - this allowed us to have the definition of the element volume. And it was noticed there may be a definition of the entropy from this

[math] \Delta S = \frac{\delta Q}{T} = -\int \frac{k}{T}\Box^2\ T dV[/math]

and an irreversible entropy production as

[math]\dot{S} = -\int k\Box^2 \ln \frac{\dot{T}}{T}\ dV[/math]

and production density

[math]\dot{\mathbf{S}} = -\int k\Box^2 \ln \frac{\dot{T}}{T}[/math]

Without the time derivative and in the simple Cartesian coordinate system the construction simply looks like

[math]dS = - \frac{k}{T} \int \nabla^2 T\ dV= -\frac{k}{T} \int (\frac{\partial^2 T}{\partial x^2} + \frac{\partial^2 T}{\partial y^2} + \frac{\partial^2 T}{\partial z^2})dV[/math]

Now we have to move into a new subject, so that we can piece this altogether in a coherent way, which is certainly avoiding any ad hoc assumptions.

A pressure term, like the first features in the OP, would always be a component of the density parameters located in the squared brackets as demonstrated

[math]\frac{8 \pi G}{3}[\rho + 3P ... including\ other\ terms][/math]

Since change of pressure could be a very important dynamic feature of a universe, we should implement it, but with an additional physics taken from the blog conversations:

[math]\dot{P}_2 - \dot{P}_1 = \frac{\mathbf{U}}{\Delta V}\ \ln \frac{\dot{T}}{T}[/math]

What happens if pressure becomes dependent on the density? (ie. [math]P(\rho)[/math])

Again, third derivatives in a Freidmann equation will yield a power equation, and third derivatives to have an interesting applicatio within this context specifically since it breaks energy conservation - as Motz and Kraft elegantly put, the Friedmann equation having a constant energy was an ''unfounded assumption,'' but I investigated a bit deeper, it may not have been unfounded in Friedmann's eyes since he may have been aware of the Noether energy conservation theorem. Despite all that, Motz is still right, a Friedmann equation can be diabatic and adiabatic, depending on the physics happening.

**The Key Equations**

As I said before on another post, I like to find similarities in physics and see what they may say about each other. The three equations we will focus on is:

The diabatic form of the first equation was obtained with the addition of a time derivative, this will become clear soon.

[math]\dot{P}_2 - \dot{P}_1 = \frac{\mathbf{U}}{\Delta V}\ \ln \frac{\dot{T}}{T}[/math]

(maybe others will interpret this different, but this is telling me that the change in pressure depends on the temperture anistropy gradient)

From Fourier's law of heat conduction, I was able to in a very simple way, obtain the entropy production density

[math]\dot{\mathbf{S}} = -\int \mathbf{k}\Box^2 \ln \frac{\dot{T}}{T}[/math]

And of course, Fourier's law of the form

[math]\frac{\delta Q}{T} = -\int \mathbf{k}\Box^2 \frac{T}{T}\ dV[/math]

In the last two equations here, we have changed convention as promised, because we need to cover a small few details about entropy.

**Into Friedmann Territory In A Pseudo-de-Sitter Space**

To be continued.....

**Edited by Dubbelosix, 14 April 2019 - 11:35 AM.**