On An Approach To A Pre Big Bang Phase Measuring Interparticle Distance For Cosmic Condensate

[Dubbelosix]

 

A simple equation of state is

 

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

 

Where [math]N[/math] is the particle number and [math]R[/math] is the radius of a universe and [math]\Gamma[/math] is the particle production which is also related dynamically to the fluid expansion [math]\Theta = 3\frac{\dot{a}}{a}[/math]. To find the required object to describe a condensate, we divide through by [math]N^2\lambda^3[/math] where [math]\lambda[/math] is the thermal wavelength and [math]\lambda^3[/math] will replace the role of the density term, 

 

[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]

 

In which 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. In this picture, it may make describe the ability to construct a condensate universe from a supercool region that existed before the big bang (the stage in which the universe began to heat up). 

 

The rate of change of the volume of a sphere is, 

 

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

 

Here we have a fluid expansion on the RHS [math]\Theta = 3(\frac{\dot{R}}{R})[/math] and the rate of change of its internal energy is

 

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

 

This is itself a formal analogue to the evolution equation [math]\dot{n} + 3 \frac{\dot{R}}{R}n = \dot{n} + n\Theta = n \Gamma[/math] in which [math]n = \frac{N}{V}[/math] (the particle number density). 

 

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

 

[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]

 

Bringing the density and pressure definitions back, we actually have 

 

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

 

This last equation is not too far from the kind of equation we require for a universe. We make two changes in this equation - the first obvious one is to replace the LHS for the ordinary form encountered for the Friedmann expansion. The second is by noticing the RHS may satisfy the Clausius relationship for entropy, that is, it may possess an irreversible pressure. It has formal similarity to the Clausius entropy:

 

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

 

This means it is possible that not all phase transitions are reversible in a universe. Certainly, not all properties are conserved, such as friction. 

 

In other work relevant to this discussion, it was possible to write the Friedmann equation in the style of a Gibbs equation [math]d(\frac{\rho}{n}) = dq - qd(\frac{1}{n})[/math] to possibly give a mathematical tool in which to describe any phase transitions of a universe. 

 

I arrived at:

 

[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]

 

In which [math]q[/math] measures the heat per unit particle [math]\frac{dQ}{N}[/math].

 

The splitting of the equation into reversible and irreversible form takes the appearance however as

 

[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]

 

From here, it is well noting that the pressure may follow 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]

 

As derived by Motz originally , then later Kraft contributed and updated the paper with Motz. This is a new form of their solution however since it takes into aspects of irreversibility and particle production. 

Edited December 31, 2017 by Dubbelosix