1 Introduction
Ideal gas equations can be applied to non-interacting dilute gases. Molecules and atoms in gas phase are so far away from each other, it makes a little difference if we ignore the molecular interaction. These molecules and atoms are further subject to Heisenberg uncertainty principle [
1]. When dealing with the universe as a whole, we see that the molecules, nucleons and other subatomic particles are not uniformly distributed over the space. There exists various scales in the universe. The nucleons are particles in a star and at a larger scale the stars can be treated as particles of a galaxy and so on. In order to incorporate all scales one is required to generalize the ideal gas equation. In this paper, we have derived generalized gas equations for such a universe. In an earlier paper, based on configuration space, we addressed future orientation of time with the growth of entropy [
2]. In the present paper we have extended the same model to phase space. In this model we assume that the universe is structureful which consists of clusters. Each cluster contains subclusters. These subclusters can further be divided into even smaller ones and so on. Such a description is called the universal self-similarity [
2]. It should be noted that, for instance, cluster of galaxies are bound, virialized, high over density system, held together by the cluster self gravity [
3]. In this perspective we can say that our model intrinsically incorporates gravity.
2 Model
We can assume that on large scale the universe is homogeneous, isotropic [
4] and unique [
2,
5]. We can further assume that all forms of the matter in the universe are contained in clusters. Each cluster contains sub-clusters and so on. We treat these clusters, at their corresponding scales, as particles of ideal gas. We confine ourself to nonrelativistic regime.
We can represent the universe by a set
Gn [
2]
where
represents the
ith cluster and
Nn is the number of clusters in the
Gn. We can regard that
Nn is the cardinal number of set
Gn. Similarly we can write
as
We can continue this to sub-atomic level and finally we get
where
’s correspond to constituent particles which we assume, have no further substructure.
The spatial distribution of accessible states for this system can be written as [
1,
2,
6]
where
α stands for configurations space.
is the spatial volume of
Gn and
is the volume occupied by
. We assume that
is not arbitrarily small. It is much smaller as compared to
at the scale of
Gn whereas at the scale of
, it is much larger than
. Therefore the corresponding accessible states at the scale of
where
is the volume occupied by
. Putting
from eq.(2) into eq.(1) and on re-arranging we get
We can generalize it as follow
As we know that
where
s is the entropy and
kB is the Boltzmann’s constant. Using eq. (4) and eq. (5), we get
where
sα is the spatial entropy and
is the spread or uncertainty in volume of the constituent particles.
It is important to note that a relation describing average value of physical quantities such as entropy, mass or thermal energy of an element of
Gi with an element of
Gi−1 can be written as
where
x can be the average value of entropy, mass or thermal energy etc. Using eq.(7) for entropy into eq.(6) and after iteration we get
where
is the average of spatial entropy of the constituent particle and
It is worth mentioning the last term in the series of
is much large as compared to other terms in the series.
It follows from eq. (7)
where
X corresponds to the average value of entropy, mass, thermal energy or any other additive physical quantity of the universe
Gn.
Nt stands for total number of massive particles in the universe. In the above equation, we have further assumed that each
Gi have equal number of
Gi−1 elements.
If we treat
X as entropy in eq. (10) and using this equation and eq.(8), we can obtain the spatial entropy of the universe which can be written as,
Next we consider momentum space. The momentum distribution of accessible states for
Gn can be written as [
1]
Here [
1,
7]
where superscript
β stands for momentum space.
mn−1 is the average mass of
and
un is the average kinetic energy of
Gn.
Similarly we can write
Following the same procedure as for eq. (4), we can write
where
and ∆
px is the uncertainty in momentum of the constituent particle in one dimension. As we reach smaller and smaller
G’s, then we are ultimately in quantum regime (i.e. nucleons in a star). A factor of
N1! appears in the denominator of r.h.s of eq. (15) in order to avoid over counting of
N1 momenta of identical nucleons in stars [
1].
After doing some straight forward calculation we can write an expression for entropy due to momentum distribution as
For large
Nn [
7]
where
e = 2.718... i.e. base of natural logarithm. Using above approximation we can rewrite eq. (16) as
Now using eqs. (11) and (18), we can write the combined entropy of our universe in phase space as
As from eqs. (7) and (10), we can write
mn−1Nn =
mn ≡
M and
un ≡
U, where
M is the mass of the universe,
U stands for the total thermal energy of the universe. We can finally write entropy of the universe as
Here
Nn is the number of top most clusters (say the number of superclusters in the universe) which makes the universe and
N1 is the average number of particles in the bottom most cluster (say the number of nucleons in a typical star). In the above equation we have also dropped the indices
α and
n over the spatial volume of the universe for brevity.
3 Implication of the Model
We can now find the thermal energy and the equation of state by using the following thermody-namics relations.
and
where
P is pressure and
T is temperature. From eqs. (20) and (21), we get
This equation gives us the thermal energy of our structureful universe. From eqs. (20) and (22)
Eqs. (23) and (24) can be treated as generalized equations for an ideal gas for the structureful universe.
To verify validity of our system of equations, the ideal gas equations for molecules/particles must be deduced from eqs. (23) and (24). These equations can be obtained, if we take
n = 1 as a special case. In this case
Nn =
Nt, where
Nt is the number of particles in a gas. Further we find that for such a system
= 1, which can be neglected as compared to
Nt in the denominator of r.h.s. of eq. (23). We finally get
and
The last two equations give us the thermal energy and the equation of state for an ideal gas respectively.
We can get interesting results from these equations. Let us denote the thermal energy (given in eq. (23) of the structureful universe by
UA and the pressure by
PA (given in eq. (24), and denoting the thermal energy and pressure for structureless universe (given in eqs. (25) and (26) by
UB and
PB respectively. For same temperature and volume
As we can see that
UA «
UB, and
PA «
PB. It suggests that expansion of structureful universe can be much slower as compared to a structureless universe.