Diabatic Universe With Non-Conserved Particle Creation

[Dubbelosix]

In a universe, a realistic total entropy should be given as

 

[math]dS = dS_{rev} + dS_{irr}[/math]

 

That is, it consists of two parts, one that is the reversible entropy in a universe and the irreversible entropy. Now, if [math]V[/math] is the volume of a sphere, the rate of change of the volume is

 

[math]\frac{dV}{dt} = V(3 \frac{\dot{R}}{R})[/math]

 

The term in the paranthesis is known as the fluid expansion

 

[math]\Theta = 3(\frac{\dot{R}}{R})[/math]

 

The rate of change of its internal energy would satisfy

 

[math]\frac{d}{dt}(\rho V) = \dot{\rho}V + \rho \dot{V} = (\dot{\rho} + 3 \frac{\dot{R}}{R}\rho)V[/math]

 

If the energy density is replaced with the particle number [math]N[/math], you get back the particle production rate [math]\Gamma[/math]

 

[math]\frac{d}{dt}(N V) = \dot{N}V + N \dot{V} = (\dot{N} + 3 \frac{\dot{R}}{R}N)V = N V \Gamma[/math]

 

which is useful to know, because this quantity is a version of the continuity equation, except in the form when irreversible dynamics are involved, leads to a theory that is diabatic in nature. 

 

The continuity equation is just

 

[math]\dot{\rho} = (\rho + 3P)\frac{\dot{R}}{R}[/math]

 

It looks messy, but that particle production equation can be simplified, making use of the particle number density [math]\frac{N}{V} = n[/math] as well

 

[math]\dot{n} + 3 \frac{\dot{R}}{R}n = \dot{n} + n\Theta = n \Gamma[/math]

 

The reversible and irreversible parts can be written in terms of the first law of thermodynamics

 

[math]\frac{d}{dt}(\rho V) + P \frac{dV}{dt} = (\frac{dQ}{dt})_{rev} + (\frac{\rho + P}{n} \frac{d}{dt}(nV))_{irr}[/math]

 

*Note, also from this last equation, you could rewrite a Friedmann equation involving an extra term describing the reversible dynamics. 

 

Replacing terms we have uncovered for diabatic particle creation we get (and dropping the unecessary reversible and irreversible notation),

 

[math]\frac{d}{dt}(\rho V) + P \frac{dV}{dt} = (\frac{dQ}{dt}) + (\frac{\rho + P}{n})nV \Gamma[/math]

 

Now, let's compare this with earlier work, with a modified Friedmann equation, of various forms. One such for I worked on was

 

[math]\frac{\dot{R}}{R}(\frac{\ddot{R}}{R} + \frac{kc^2}{a}) = \frac{8 \pi G}{3}(\frac{\rho + P}{n})n\frac{\dot{R}}{R}[/math]

 

If we allow the time derivative on the ''almost'' fluid expansion coefficient, it looks identical to the diabatic universe. What you would not have known from the previous equations though, is that this required the third derivative in time, considered as the derivative which leads to non-conservation in the expanding Friedmann universe.

 

The rules have not changed since the original work in my Friedmann model, as it turns out, you can still by definition introduce the heat per unit particle, which would change the thermodynamic law further to suit a Gibbs equation.

 

Heat per unit particle is 

 

[math]dq = \frac{dQ}{dN}[/math]

 

which changes the law into 

 

[math]d(\frac{\rho}{n}) = dq - qd(\frac{1}{n})[/math]

 

This lead to a Friedmann equation of the form

 

[math]\frac{\dot{R}}{R}(\frac{\ddot{R}}{R} + \frac{kc^2}{a}) = \frac{8 \pi G}{3}[(\frac{\rho}{n}) + 3P(\frac{1}{n})]\dot{n}[/math]

 

What we have learned from the following equation:

 

[math]\frac{d}{dt}(\rho V) + P \frac{dV}{dt} = (\frac{dQ}{dt}) + (\frac{\rho + P}{n})nV \Gamma[/math]

 

Is that it fits the general equation of state one would look for to describe the effective density in diabatic universes. 

 

Adiabatic universes, for the particle production equation, would satisfy

 

[math]\dot{n} + \Theta n = 0[/math]

 

Diabatic models satisfy 

 

[math]\dot{n} + \Theta n = n \Gamma[/math]

 

 

 

 

 

 

 

 

 

ref. 

 

 

https://www.researchgate.net/publication/286512912_Gravitationally_induced_adiabatic_particle_production_From_Big_Bang_to_de_Sitter

https://www.researchgate.net/publication/283986394_General_form_of_entropy_on_the_horizon_of_the_universe_in_entropic_cosmology

Edited July 28, 2017 by Dubbelosix

[Super Polymath]

Yeah. If the vacuum medium is photonic matter with a never ending wavelength than those ghost particles will be shortened into matter with normal oscillations given ample heat, id est the background temperatures giving rise to the quark gluon plasma of the CMBR. Its silly to think the universe just emerged from a planck singularity with no cause. There's no evidence of that at all.

 

Evaporating black holes don't give off a lot of heat at once, but given a great enough number of their gamma ray bursts & anti-gravity waves in a small enough space you could get several superluminally orbiting & spinning quark gluon plasma spheres, each equal in surface area to the hubble radius, that fly off in all different directions with enough cyclotron radiation to give rise to stellar expansion.

Edited July 28, 2017 by Super Polymath

[Super Polymath]

Are you planning on presenting these equations the the SC for review?