On An Irreversible Pre-Big Bang Phase giving Rise to Our Universe: As a Radiation Vapor

[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]. Based on the new perspective of reversible and irreversible dynamics, the Friedmann equation is now

 [math]\frac{\dot{R}}{R}(\frac{\ddot{R}}{R} + \frac{kc^2}{a}) = \frac{8 \pi G}{3}(\dot{\mathbf{q}}_{rev} + [(\frac{\rho}{n}) + 3P_{irr}(\frac{1}{n})]n\Gamma)[/math] 

I argue that irreversible dynamics have gone on in the universe - it is possible to view the universe in a pre-big bang model. So the reversible and irreversible investigation is actually part of a much larger investigation into my cosmology studies, which was related to a richer plethora of ideas that circled around a universal primordial rotation term attached to a Friedmann equation. Those early equations are still the basis of my arguments to explain dark energy and even dark flow, including several other topics of interest, such as an interesting relationship to the antimatter problem. In a later post, I will show that work that attempts to view the universe with the all-important Poincare symmetry: This work is still related to those previous investigations, because we deal with a different kind of de Sitter spacetime, a type only I can classify as a pseudo de Sitter spacetime, but what makes it different? Well, a third in time derivative Friedmann equation does not conserve the energy of an expanding universe. 

In this work, we adopt a solution from Motz and Kraft for a reversible isothermal Gibbs-Helmholtz phase change of an all matter liquid degenerate gas into a radiation vapor. It is assumed in our model, that a realistic phase change involves neither reversible or isothermal phase changes. 

To help explain a diabatic anisothermal phase change from some super cool pre-big bang phase, we introduce a Friedmann equation which has been rewritten in the style of a Gibbs equation - this specific equation can be argued in a number of different ways: The basic way to view it is that the Friedmann equation is related to the entropy of a universe insomuch that it consists of two parts, a reversible and irreversible particle creation dynamics.

[math]\frac{\dot{R}}{R}(\frac{\ddot{R}}{R} + \frac{kc^2}{a}) = \frac{8 \pi G}{3}(\dot{\mathbf{q}}_{rev} + [(\frac{\rho}{n}) + 3P_{irr}(\frac{1}{n})]n\Gamma) = \mathbf{k}nT \dot{S}[/math]

where [math]S[/math] has dimensions of [math]k_B[/math] and [math]\mathbf{k}[/math] is the Einstein factor.

in which

[math]nk_B T \dot{S}  = \dot{\rho} + (\frac{\rho + P}{n})n \frac{\dot{T}}{T}[/math]

which also justifies the following form as a fully thermodynamic interpretation:

[math]\frac{\dot{R}}{R}(\frac{\ddot{R}}{R} + \frac{kc^2}{a}) = \frac{8 \pi G}{3}(\dot{\mathbf{q}}_{rev} + [(\frac{\rho}{n}) + 3P_{irr}(\frac{1}{n})]n\frac{\dot{T}}{T}) = nk_B T \dot{S}[/math]

[math]P_{irr}[/math] is known as the irreversible pressure, and inside of it, we can talk about the Gibbs-Helmholtz free energy equation for an irreversible phase change from a liquid particle creation phase to vapor for some infinitesimal change in volume,

[math](P_{irr}(\frac{1}{n}))\dot{n} = -(\frac{1}{4 \pi R^2}\frac{dU_2}{dR})\frac{\dot{n}}{n} = -(\frac{dU_2}{dV}(\frac{1}{n}))n\Gamma[/math]

Where, [math]n[/math] is the particle number density. The pressure is irreversible because particle production through the phases has happened in this case, in an irreversible way in which we take [math]n\Gamma \ne 0[/math]. The irreversibility of particle creation is expected to happen when the term [math]\frac{kc^2}{a}[/math] is large, implying a large cosmological curvature which would have been present during these phase changes; it is a result of particle creation in curved spacetimes that yields possibilities for non-conservation in the universe. Any reversible dynamics [math]\dot{\mathbf{q}}_{rev}[/math] in the universe I speculate as probably a post big bang phenomenon, whereas I expect this conversion from the two phase states as related to chaotic and irreversible dynamics. 

In this model, the universe is expected to have a cold dominated era (the pre big bang phase) as a degenerate gas of particles in a very high condensed gravity-dominated region with little thermodynamic freedom. Some collapse underwent in the cold dominated region leading to the radiation vapor phase - which can be thought of as the heating of the universe leading to a big bang scenario. We live inexorably, in this vapor phase of the universe. 
 

 

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

[notes]

Now that we have a definition of the entropy, we can argue the reversible and irreversibility in another type of form, the Clausius entropy production equation will provide such a possible form:

[math]N = S - S_0 - \int \frac{dQ}{k_BT}[/math]

When [math]N=0[/math], it means a thermodynamic process was reversible and [math]n > 0[/math] for some irreversible process where [math]S[/math] is the final state and [math]S_0[/math] is the initial state of entropy. So yes, it would be possible to write a Freidmann equation to satisfy this equation as well. In fact, this is doing the same thing as our initial approach in a way as it is measuring the irreversible dynamics, if there is any. In the previous form, we looked at the entropy per unit particle and the heat per unit particle.

It can be written as

[math]\dot{S} = \sum_k \frac{\dot{Q}_k}{T_k} + \sum_k \dot{S}_k + \sum \dot{S}_{ik}[/math]

with [math]\dot{S}_{ik} \leq 0[/math]

we can see how this is formally similar to how we treat the reversible and irreversible dynamics separately within the same equation. If any terms in the last equation have a subscript of ''i'' indicates it is an irreversible process, so it can be written in the following way to express which parts are reversible and which are not

[math]\dot{S} = (\sum_k \frac{\dot{Q}_k}{T_k} + \sum_k \dot{S}_k)_{rev} + (\sum \dot{S}_{ik})_{irr}[/math]

Again, the structure of the equation reveals a global entropy must consist of reversible and irreversible dynamics. Irreversible dynamics occur in strongly curved spacetimes. 

https://en.wikipedia.org/wiki/Entropy_production

https://www.researchgate.net/publication/1922810_Thermodynamics_of_Friedmann_Equation_and_Masslike_Function_in_General_Braneworld

Edited September 22, 2017 by Dubbelosix

[Dubbelosix]

I noticed later, that the theory did have formal similarities to the Ekpyrotic theory - except for two crucial points

 

1) There is no bounce in this theory, because the phase change from the super cool state into the warm radiation vapor stage is an irreversible process in the universe. Extended, this has serious implications, because it would mean in the far-reaching sense, information isn't conserved in the early universe when curvature was dominant. 

 

2) There is no need for other universes, something the Ekpyrotic theory relies on as a mechanism... it is also something inflation theories lead to, which is why Steinhardt, one of the original creators of the theory, is now a vocal opponent. 

 

Also we avoid singularities, because of the negative pressure. This was Motz' original motivation for looking at a modified Friedmann equation, which we have only partially taken. We have formulated an entirely modern view of his theory which has been adapted in some crucial ways.

Edited September 22, 2017 by Dubbelosix

[Mordred]

Wouldn't the above be better described using Maxwell Boltzmann for multiparticle mixed system, Einsten and Fermi statistics for the individual fields? Mainly you are adding effective degrees of freedom. What is the effect upon the above equations.

May I also suggest looking into the Hamiltons on your three threads, it would fill in details, that may aid other readers.

see equation 37 of this link for the temperature particles start to dissapear to ground state

http://hep.ph.liv.ac.uk/~hock/Teaching/StatisticalPhysics-Part5-Handout.pdf

though there is readily better articles, this was a quick grab article. (though it has the pertinant equations and proof of those equations.)

You will find this one of much greater interest. (you did mention you understand group hehe)

Second Law-Like Inequalities with Quantum Relative Entropy: An Introduction

https://arxiv.org/pdf/1202.0983.pdf

note 62 pages long lol....

the first 15 pages  intro stuff essentially (intro to involved theories) the meat starts to hit at page 15 3.2 directly to quantum relative entropy

Edited September 23, 2017 by Mordred

[Dubbelosix]

Yes I have looked into thermal statistics for the model as well. As well as creation and annihilation operators as well. I will take a read of the links before saying anything else. 

Those papers are interesting, and the last one is helpful for a better investigation into those statistics, which I have looked at in some ... relatively basic forms. 

[Mordred]

No problem I recognize the amount of time needed to address the points I made in your three threads and that all three are under development.

Glad to see your finding the articles useful to your goals.

(you have no idea how glad I am seeing someone properly modelling a speculative model) I am more than willing to assist in a proper model development regardless of personal feelings opinion on feasibility.

Edited September 23, 2017 by Mordred

[Dubbelosix]

Thank you for the compliment. And I will consider help if I need it. Of course, other people can fiddle with my equations and start their own investigations.

[Dubbelosix]

The paper on condensates is proving very interesting. It seems related to the thermodynamic interoretation of a wavelength (thermal wavelength). Those wavelengths measure whether a system follows Fermion or Bose statistics. 

 

The section on thermodynamic properties vanishing at the ground state, is also very interesting - it would tie in with the idea that effects of fluctuations are negligable on the cosmological scale. May answer for a cold, radiationless pre-state of a universe as well. Finding an exact model has been difficult, but it seems like you have offered one.

Edited September 24, 2017 by Dubbelosix

[Dubbelosix]

I'd be careful though to assume the ground state implies zero energy as the article suggested - I strongly disagree with this interpretation. The ground state of a field is not zero, it is given by [math]\frac{1}{2}\hbar \omega[/math] - people perhaps, naively assumed that you can count this up in space as an amount of something. This isn't the case, the fluctuations are on a scale that is not observable and the effects of virtual particles seem to be negligible on the macroscopic scale - even photons do not seem to couple to the vacuum ground state of fluctuations. It's not that they do not contain energy, but rather they contain a lot of it when these fluctuations exist for very short time periods. Equally, they have not really travelled much space either to respond to its environment or the medium it is in. 

 

Whatever ''ground state'' really means, it certainly does not mean zero energy in physics.

Edited September 24, 2017 by Dubbelosix

[Mordred]

Ground state is best treated as whatever the global metric field value is.

You set that non zero value as zero to maintain the symmetry relations of your group dynamics.  Also for the potential energy calculations in relation to the field. ( potential energy arising due to position relative to the field.)

Another field with non zero ground state for an example is the Higgs field. 

(always look for the compared to) However you are right in that zero does not always mean zero, it is the set value so one can compare one object to the other.

You already noted that once you account for field fluctuations due to HUP no field is truly zero, Zero point energy is the effective minima, it is simply set as zero.

 

Edited September 24, 2017 by Mordred

[Mordred]

19 hours ago, Dubbelosix said:

The paper on condensates is proving very interesting. It seems related to the thermodynamic interoretation of a wavelength (thermal wavelength). Those wavelengths measure whether a system follows Fermion or Bose statistics. 

 

The section on thermodynamic properties vanishing at the ground state, is also very interesting - it would tie in with the idea that effects of fluctuations are negligable on the cosmological scale. May answer for a cold, radiationless pre-state of a universe as well. Finding an exact model has been difficult, but it seems like you have offered one.

I thoroughly enjoyed that paper myself, 

A large part of it ties into your numerous threads, figured you would find it as highly informative as I did

[Dubbelosix]

Equation 3. appears to have a typo, an extra Gamma term on the LHS, no worry, it is really the following:

 

 

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

 

I want to use this object to create another object which can describe the physics of the condensate in a more appropriate way. To start off, we simply divide by [math] N^2[/math]

[math]\frac{d}{dt}(\frac{NV}{N^2 \lambda^3}) = \frac{\dot{N}V}{N^2\lambda^3} + \frac{N \dot{V}}{N^2\lambda^3} = (\frac{V}{N\lambda^3})\Gamma[/math]

In which we can measure the statistics from the interparticle distance where

 

[math]\frac{V}{N \lambda^3} \leq 1[/math]

 

Then the interparticle distance is smaller than its thermal wavelength, in which case, the system is then said to follow Bose statistics or Fermi statistics. On the other hand, when it is much larger ie.

 

[math]\frac{V}{N \lambda^3} >> 1[/math]

 

Then it will obey the Maxwell Boltzmann statistics. The latter here is classical but the former, the Bose and Fermi statistics describes a situation where classical physics are smeared out by the quantum. This is one such approach to describing a condensate from the model. Remember, we want a pre big bang universe which was very cold, - Bose condensates are a million times colder than open space. The pre big bang model, may be just as cold or colder.

 

Edited September 26, 2017 by Dubbelosix

[Mordred]

Sounds to me like the Gross-Pitaevskii equation is applicable to the above.

https://www.google.ca/url?sa=t&source=web&rct=j&url=https://cmt-qo.phys.ethz.ch/wp-content/uploads/teaching/qg/Chapter_03.pdf&ved=0ahUKEwiC6N-03MHWAhVV7WMKHZMkDzAQFggiMAA&usg=AFQjCNFbn6Xd3XZfuxxzadu8nqLrXZAjCA

 

The application is the condensate wave function mean field, which will tie into the elastic cross scatterings and the above statistics. note already applies QM and HUP

 

Edited September 26, 2017 by Mordred

[Dubbelosix]

I will take a look. Thanks. 

[Mordred]

I've run into it in a few of my studies, not too terrifically familiar with its proof. However usually see it applied to numerous condensate applications.

[Dubbelosix]

Yes, there seems to be correlation.

The total wave function minimizes the expectation

[math]\int dV |\psi|^2 = N[/math]

It does link the physics. Extract from wiki

'' If the average spacing between the particles in a gas is greater than the scattering length (that is, in the so-called dilute limit), then one can approximate the true interaction potential that features in this equation by a pseudopotential. The non-linearity of the Gross–Pitaevskii equation has its origin in the interaction between the particles. This is made evident by setting the coupling constant of interaction in the Gross–Pitaevskii equation to zero (see the following section): thereby, the single-particle Schrödinger equation describing a particle inside a trapping potential is recovered.''

 

Will require more reading. 

 

This looks like a nice paper as well 

 

https://arxiv.org/pdf/1301.2073.pdf

[Mordred]

Might I also suggest you tie in Pati-Salam for its methodologies regarding charge, parity etc and its applications under guage groups for helicity etc.

en.m.wikipedia.org/wiki/Pati%E2%80%93Salam_mode

Here is Pati-Salam a la SO(10) though Salam also applies to the SM groups via [latex]\mathbb{Z}^2[/latex] as applied to the SU(2) groups left and right hand states.

(arxiv)

https://www.google.ca/url?sa=t&source=web&rct=j&url=https://arxiv.org/abs/hep-ph/0204097&ved=0ahUKEwi8_M235cHWAhUUTmMKHY73CAwQFggdMAA&usg=AFQjCNGACL5BvGOGBQARSFfQxUM-Rd3npQ

 

[Dubbelosix]

I will certainly take a look.

[Mordred]

Either way simply understanding Pati-Salam is incredibly useful in cosmology and particle physics related studies. In particular the symmetry groups.

Hehe 

[Dubbelosix]

I feel you are a bit a head of me. You should know, I don't actually consider myself a mathematician. Being a drop out, I have been mostly self-taught over the years.

But I do feel you are a bit ahead of me at this time and you'll need to give me time to read up on these two things.I do appreciate the help though.

[Mordred]

Glad to be of help, no worries I'm often ahead. Simply due to years of active study. I fully understand how long it truly takes to fully understand the materials I provide.

 I've read tons of various papers, textbooks etc. So much of it is simply being aware of different applicable treatments.

Though much of it also applies to my studies.

Edited September 26, 2017 by Mordred

[Dubbelosix]

15 hours ago, Dubbelosix said:

Equation 3. appears to have a typo, an extra Gamma term on the LHS, no worry, it is really the following:

 

 

ddt(NV)=N˙V+NV˙=(N˙+3R˙RN)V=NVΓ

 

I want to use this object to create another object which can describe the physics of the condensate in a more appropriate way. To start off, we simply divide by N2

ddt(NVN2λ3)=N˙VN2λ3+NV˙N2λ3=(VNλ3)Γ

In which we can measure the statistics from the interparticle distance where

 

VNλ3≤1

 

Then the interparticle distance is smaller than its thermal wavelength, in which case, the system is then said to follow Bose statistics or Fermi statistics. On the other hand, when it is much larger ie.

 

VNλ3>>1

 

Then it will obey the Maxwell Boltzmann statistics. The latter here is classical but the former, the Bose and Fermi statistics describes a situation where classical physics are smeared out by the quantum. This is one such approach to describing a condensate from the model. Remember, we want a pre big bang universe which was very cold, - Bose condensates are a million times colder than open space. The pre big bang model, may be just as cold or colder.

 

 

 

In the same way, the interparticle distance is found from an effective density through some very simple and short manipulations. Without reversible and irreversible notation, the effective density is

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

Divide off [math]N\lambda^3[/math]

[math]= \frac{d\mathbf{q}}{dt}(\frac{1}{N}) + (\rho + P)\frac{V}{N \lambda^3} \Gamma[/math]

This is a formulation that can be easily understood as having meaning and context within a Freidmann equation, in which the interparticle distance is (still) satisfied by the physics of Bose or Fermi condensates contained in the term [math]\frac{V}{N \lambda^3}[/math]. The [math]\lambda^3[/math] has also been absorbed by [math]Q[/math] making [math]\mathbf{q}[/math]. At least, this is how I would have imagined it would enter the theory.

I've found a way to describe the spin within the theory like you wanted me to look into as well, so will write that up later.

Edited September 26, 2017 by Dubbelosix

[Mordred]

no prob looks good, just renemember a scalar field follows spin zero, while gravity follows spin 2 statistics that has been largely confirmed via G-wave detection. 

Won't interfere with your methodology as any orthogonal group is reducible to the unitary Hilbert spaces.

Edited September 26, 2017 by Mordred

[Dubbelosix]

You do know that spin 2 gauge theory came from the quantization of the graviton? We haven't actually measured any spin, or as far as I am aware. 

I am not happy with gauge theory, especially within the context of gravity. I have some serious problems accepting it - like for instance, from first principles, gravity isn't even a force. I have been a vocal opponent against quantum gravity in the context of some quantization of a field - I see no evidence and have no reason to think gravitons actually have to exist. 

As for the spin thing, what was found was the following:

[math]\frac{N}{V} = \int^{\infty}_{0} n_{BE}(E) g(E)\ dE[/math]

[math]g(E)[/math] is the density of states per unit volume. For non-interacting particles, which have a kinetic energy you can have

[math]g(E)\ dE = (2S + 1) (\frac{2m}{\hbar^2})^{\frac{3}{2}} \sqrt{E}\ dE[/math]

And of course, this relates to the investigation form the term [math]\frac{V}{N \lambda^3}[/math]. This is the same. So if for instance, 

[math]\frac{V}{N \lambda^3} \leq 1[/math]

then

[math]\frac{V}{N} \leq \lambda^3[/math]

 

 

http://web.pa.msu.edu/people/duxbury/courses/supercon/Kishore-Kapale-BEC-talk.pdf

Edited September 27, 2017 by Dubbelosix

[Mordred]

You don't require the gravition to give rise to the quadrupole nature of a GW wave which arises from the EFE treatments. Which will also correspond to spin 2 statistics. Just because a dynamic follows a certain symmetry relation does not necessarily mean a particle is involved.

Edited September 27, 2017 by Mordred

[Dubbelosix]

Ok, was just checking.