# Entropy & Thermodynamics

In
order to get a better perspective of some of the issues being
discussed, it might be useful
to expand the definition of both the work done [W] and the input energy
[Q], as defined by the 1^{st} law, to include the idea of entropy,
as defined by the 2^{nd} law of thermodynamics.

[3]

In the last equation in [3], we now see the potential relationship
between the change in entropy [S] against the change in the ratio of
the net energy [E] and temperature [T] of a given system, e.g. the universe.
If we are to pursue the assumption of the Big Bang model that leads
to the *‘heat death’ *of the universe
due to increasing entropy, albeit over trillions of years, there would
have to be a non-linear increase in energy [E] with respect to temperature
[T]. However, as an initial assumption, we might consider the suggestion
that energy associated with high temperatures has lower entropy simply
because it tends to be more *‘useful’* than the same amount of
energy at lower temperatures.

In practise, the relative changes to the parameters in [3] are
too speculative to pursue in terms of the actual universe, therefore
the initial discussion of entropy will continue more by way of an analogy
to a much simpler example. As such, we will try to substantiate some
of the ideas by considering an example in which entropy increases in
a ‘*system*’ defined by a glass of ice cold water (0°C) and its
‘*surroundings*’ defined by a large room (25°C). Due to the implied
temperature gradient, some thermal energy [dQ] will flow from the surrounding
room into the glass of ice-water. As a result, the entropy of the system,
i.e. the ice-water, will change by the amount [dS=dQ/T=dQ/273K] and,
in so doing, the water in the glass will change from a solid to a liquid
state. By the same token, the entropy of the surroundings, i.e. the
room, will change by an amount [dS=-dQ/298K], such that the entropy
of the *‘system’* increases, whereas the entropy of the ‘*surroundings*’
decreases. However, it might be realised from the previous ratios that
the decrease in the entropy of the ‘*surroundings*’ is less than
the increase in the entropy of the ‘*system*’. To find the total
entropy change of this '*micro-universe*’ comprising of the ‘glass’
and ‘*room*’ we have to add the two entropy changes, i.e. system
+ surroundings, which then appears to suggest that there would be a
net increase in entropy.

If we pursue this example to its logical conclusion, the surroundings
in the form of the room and the system, consisting of the glass and
its contents, will reach a point of thermal equilibrium, i.e. equal
temperature. At this point, nothing else can happen, even though thermal
energy would still exist in the room and, in fact, the amount of thermal
energy is actually unchanged in this example, because we are conceptually
describing a closed system. As such, we might describe this system as
unable to do ‘*usefu*l’ work and it would therefore remain in
this state for all eternity or until somebody opens a door to a
larger universe! So while we might want to initially model the real
universe as a closed system, without any external surroundings with
which to transfer energy, it is an assumption that may require
further scrutiny as the underlying energy processes of the universe,
as a whole, are only poorly understood at this time

## The Implications of Thermodynamics

In many ways, the study of thermodynamics is the application of statistics
within classical physics. In classical physics, we might start by attempting
to describe the motion of individual objects in terms of kinetic and
potential energy, putting aside rest mass energy for the moment. However,
applying this approach to a volume of gas containing billions of ‘*particles*’
is not practical and, to some extent, not the object of the exercise,
when considering the overall characteristic of the gas as a system.
In this respect we are considering a transition of scale in which the
terminology of kinetic and potential energy normally gets replaced by
the ideas of internal energy and work-done supplemented by the concepts
of density, pressure and temperature. However, this appears to be a
practical limitation of the mathematical modelling and not necessarily
a description of the fundamental physics. Therefore, the purpose of
this discussion is to try to anchor some of the concepts of thermodynamics,
which underpin the Friedmann equations, to the conservation of energy
as defined by classical physics. We might start by defining the simplest
of systems comprising of just 2 masses [m_{1}] and [m_{2}]
and describing the status of this system in the following terms:

The change in the kinetic energy of this system is equal to the work done on the system by the external and internal forces. |

Of course, even in isolation, these 2 masses would have some dynamic centre of mass against which the potential energy was changing, although this energy might implicitly be considered as internal to the system. However, it would appear that the statement above is confined to just kinetic energy and would not therefore be a full description, if potential energy has to be taken into account. At this point, we might try to consolidate both these positions as follows:

[1]

As such, we might formulate an overall expression of the internal
energy [U] of this system in terms of the kinetic [E_{K}] and
potential [E_{P}] energy, but noting that the latter is quantified
in terms of a negative energy:

[2]

It is also worth noting that while individual particles can have kinetic energy, the definition of potential energy always requires a minimum of two particles. However, at this point, we might want to introduce, and consider, the implications of the conservation of energy in terms of an isolated system:

The internal energy [U] of an isolated system remains constant. |

As such, there can be no external work done on this system and any
change in the kinetic energy of [m_{1}] and [m_{2}]
must be balanced by a change in the potential energy, internal to the
system. Of course, should the internal energy change, the implication
that follows from this fact can be summed up in the next statement:

Any change to the internal energy [U] of a system is equal to the work done [W] on the system by external ‘forces’. |

While the above definition of [1] and [2] was far from rigorous,
the main purpose of this introduction was simply to highlight that the
description of a ‘*system*’ in terms its internal energy and work
done can still anchored in kinetic and potential energy. However, the
behaviour of the ‘*system*’ also depends on the scope of its definition
as either an open or closed system. For example, should an isolated
system actually turn out to exist within some larger system, the internal
gravitational potential energy implied by [1] and [2] would change and
be defined by the ‘*centre of mass’* of the extended system.

*So how might this description change when applied
to a larger system? *

As indicated, the mathematical technique for dealing with larger
systems is described in terms of ‘*statistical mechanics’*. Within
this process, the behaviour of the system is defined by its internal
energy [U] and work done [W] on, or by, the system. In this context,
the kinetic energy is now represented by the temperature [T] of the
system and the ‘*flow*’ of energy between the internal and external
systems now described in terms of heat energy [Q] and work done [W].
At this point, we might scale the description of the 1^{st}
law of thermodynamics to an example of the gas in a cylinder, although
the applicability of this example might be later questioned.

In the diagram above, we might describe the ‘*internal system’*
as the gas trapped in the cylinder by the piston and initially assume
that this system is in equilibrium with its environment, i.e. the ‘*external
system’*. Of course, another way of defining the equilibrium of this
collective system is to say that the pressure in the cylinder, which
creates the force on the piston, is equal to some pressure-force on
the other side of the piston.

*How might we quantify the macroscopic nature
of a gas? *

The ideal gas law is the equation of state of a hypothetical ideal gas and although it is a good approximation of the behaviour of many gases, it may have a number of limitations when applied to a cosmological model.

[3]

In the equation above, [P] is the pressure of the gas, [V] is the
volume, [N] is the number of particles in the gas, and [k] is Boltzmann’s
constant that relates temperature [T] to the kinetic energy of the particles
moving in the gas. However, at this point, we are primarily interested
in the general relationship between the product of pressure [P] and
volume [V] with respect to temperature [T], as it affects the model
in the diagram above. In this context, the injection of heat energy
[Q] into the cylinder causes the temperature [T] of the gas to rise,
which expands the volume [dV] of the gas, while maintaining constant
pressure [P]. This behaviour is often integrated into the 1^{st}
law of thermodynamics through the concept of heat [Q], which then changes
the volume [dV]:

[4]

In [4], we see the relationship between the change in internal energy [dU] due to the sum of the change in the heat energy [dQ] and the work done [dW], but there is clearly some potential for confusion concerning the polarity of these quantities.

**Heat Energy [Q]**- Heat is input into the system: +Q
- Heat is output from the system: ─Q

**Work Done [W]**- Work done on the internal system: +W
- Work done on the external system: ─W

Again, with reference to the diagram above, the implication of this model is that heat [+Q] is input into the internal system, which then increases the temperature [+T] of the gas, and based on the assumption of the ideal gas equation in [3], allows the gas to increase in volume [+dV], while maintaining a constant pressure [+P]. The result of this expansion defines the work done [W] on the external system, which appears to be inferred from the overall sign of [dW] in [4], not directly from the product of [P.dV].

As a slightly tangential line of thought, consider a 'system' of matter particles floating as dust with no discernible velocity with respect to the CMB radiation, but which are still drifting apart due to expansion. In terms of particle physics, the work done on these particle equals the distance moved times the force on the particles. So what kinetic force [F=ma] is acting on these particles? What is the full scope of the gravitational force [mgh] in this system of particles? Is the work done [W] energy 'lost' to expansion, as implied in [4], simply the energy gain in potential energy? |

Anyway, returning to the main thread, it would appear that we have
a description and some justification of the 1^{st} law of thermodynamics,
which we might ultimately trace back to the conservation of energy in
terms of the kinetic energy of all the individual particles and potential
energy between them. However, we might like to see whether our definition
of the polarity of the various quantities remains consistent, if we
reverse the heat flow in the example above. In this case, the heat flows
out of the internal system, i.e. [-dQ], which causes the temperature
[T] to fall. Again, based on the assumption in [3], we shall assume
the pressure [+P] remains constant, while the change in volume [-dV]
is falling. So interpreting [4] for this case:

[5]

So, at face value, the interpretation appears to remain consistent with the negative value of [-dQ] signifying the flow of heat out of the internal system, while the negative rate of change of volume [-dV] would appear to lead to positive work done [+dW] on the internal system.

However, it is not clear whether the direction of the work done [W], as described above, will remain consistency when the equations of state for radiation, i.e. [ω=+1/3], and dark energy, i.e. [ω=-1], are used to define the pressure [P] derived from the energy-density [ρ]. However, we shall defer this issue until a later point in this discussion, as for now, we are primarily trying to understand some of the basic principles assumed to underpin the Fluid equation. |

So, at this point, we have attempted to define some of the basic
concepts linked to the ideal gas equation and the 1^{st} law
of thermodynamics. However, we also need to address another key implication
that relates to the definition of entropy, as linked to the 2^{nd}
law of thermodynamics, and the description of an adiabatic system. Let
us start with a general definition of the 2^{nd} law and
'*the
idea of entropy*':

Embedded in the 2^{nd} law
of thermodynamics is the suggestion, that over time, differences
in temperature and pressure, in a system, as a whole, must move
towards thermodynamic equilibrium. As a consequence of this
tendency towards equilibrium, it follows that the entropy of
the system is always increasing and defines the irreversibility
of the ‘arrow of time’. |

In this context, the description of an adiabatic system would appear
to be restricted to a sub-system, which can reverse entropy, but only
at the expense of increasing entropy in the system as a whole. Therefore,
we may need to revisit some of the key step in the derivation of the
*Fluid equation* starting with [6]
below:

[6]

Based on [6], we can substitute for the values of [dU] and [dV] previously
obtained in the derivation of the ‘*Fluid equation’ *:

[7]

Again, we can rationalise [7] by adopting the dot notation for the differential terms [dρ/dt] and [da/dt], while recognising that the latter is actually an expression of the expansion velocity [v]:

[8]

As previously indicated in the original derivation, the form of [8]
can be described as a general solution of the Fluid equation, which
is then further rationalised by the assumption that the system is to
be described as adiabatic, i.e. reversible, which allows the 2^{nd}
law of thermodynamics to be reduced to the form show in [9]:

[9]

Of course, what [9] also underlines is that the universe, as described, is a closed system in which no energy [Q] flows in or out. Therefore, based on these assumptions, the Fluid equation in [8] can be reduced to its more common form shown in [10]:

[10]

So, in summary, the original derivation of the fluid equation
proceeded on the basis of the 1st law of thermodynamics, which in-turn
was rooted in the principles of energy conservation. However, its final
form appears to make the additional assumption that because the universe
is a closed system, there can be no net flow [Q] of heat into
or out of the universe. So, if we were to compare this model with the
initial analogy, we would presumably equate the closed universe
to *'the room*' and all the stars to the *'glass of water*',
which will ultimately obtain thermal equilibrium some 10^{100}
years into *the future.*
However, if we cannot model the universe as a closed, adiabatic system,
the simplification of [8] via [9] to [10] cannot be justified and we
are left with the problem of trying to quantify the input energy [dQ].
Of course, at this point, the most salient question would appear to
be:

*
Do the Friedmann equations really describe the totality of the universe?
*

Clearly, any answer to this question will depend on the level of
speculation you are prepared to accept regarding the *‘totality’*
of the universe within your cosmological model. However, at face value,
there would appear to be a number of question marks being raised against
the universe, when described in terms of a closed, adiabatic system.
Therefore, it is possibly not that unreasonable to indulge in a little
speculation in the form of the following diagram.

The speculative model above was originally introduced in the discussion
'*The
Cause of Expansion*' that was also considering '*The
Need for Inflation*'. Within this model, our observable universe
is only a small part of a larger '*bubble universe*', as represented
by the yellow circle in the diagram above, which in-turn might only
be part of a potentially infinite '*quantum universe*'. In this
context, the first assumption being made with respect to both models,
as shown above, is that the observed universe is not a closed
system, but rather part of the internal system defined by the '*bubble
universe*'. In (a), work is done on the internal system such that
the net internal energy [U] increases due to both the input energy [Q]
and the work done [W] on the system. In (b), work is done by the internal
system such that the net internal energy [U] is the sum of the input
energy [Q] input and the work done [W] by the system.

The assumption that energy [Q] is always being input into the system is made based on the fact that the total energy [ρV] defined in terms of the comoving volume of the universe appears to be increasing due to the introduction of dark energy – see Energy Graph. |

However, in order to make some cross-reference to potential energy,
it is also being assumed that the internal system, as defined above,
does have a centre of gravity, although this may be far from obvious
to any observer, whose *cosmological
horizon* is defined by the small yellow circle shown. It might
also be worth highlighting that our observer has no real idea as to
the actual size of the internal system, or the nature of the external
system, but assumes the internal system to have homogeneous energy-density
[ρ=8.52*10^{-10} joules/m^{3}]. Equally, based
on measurements, our observer believes that the internal system is expanding
at a rate defined by [H], which is a function of time. So, our observer
having assumed that he is located within an internal system, which is
not closed, starts with 2 thermodynamic possibilities shown in the diagram:

[11]

Ignoring the amount of energy [Q] flowing into the system, if work
is done on the system, its internal energy would tend to increase and
by the same token, if work is done by the system, its internal energy
would tend to decrease. Again, we might like to simply cross reference
this requirement of thermodynamics with the concept of potential energy.
In expansion, the work done [W=F.dx] might be described as the energy
*'lost'* to potential gravitational energy, while in contraction,
energy would be '*gained*' from gravitational potential. So,
on the basis of the observed expansion, we might assume that [3b] is
more applicable to the cosmological model under discussion:

*What implications might follow from this assumption?
*

If we use [3b], then the work done [dW] by the internal system will
cause a reduction in the internal energy of the system, although there
might be spme speculative implication that the the work done energy
is simply converted to potential energy with respect to some centre
of gravity. However, if we still require a net increase in the internal
energy [dU], the energy '*lost*' to work done [dW] would have
to be offset by the energy [dQ] input into the system. We might also
have to consider the applicability of any assumptions linked to the
ideal gas equation in [3], e.g. constant pressure [P] under expansion,
as within the cosmological model pressure would appear to be linked
to energy-density [ρ] via the equations of state [ω].

[12]

In [12], we start with the basic relationship of mass [M] being defined in terms of the density [ρ] times the volume [V] that is then converted into some form of energy equivalent, which corresponds to the internal energy [U] of the system as defined by some comoving volume [V]. As such, the implication from [12] is that the energy-density [ρ] is inversely proportional to volume [V] and, ignoring the implications of dark energy for the moment, the pressure [P] must also share the same relationship to volume [V], if the equations of state [ω] that binds pressure [P] to the energy-density [ρ] is a constant.

At this point, we might just noted the original suggestion that the temperature [T] of a system also reflects the kinetic energy within the system. |

The final equation in [12], based on the ideal gas law in [3], might suggest that it is temperature [T] that remains constant, if [P] and [V] are inversely proportional. Of course, this suggestion would not align to the most basic assumption of the ΛCDM model, i.e. temperature is falling with expansion. However, if pressure [P] does not remain constant under expansion, we might have to consider the implications on work done [W]:

[13]

Based on [13] and the original description of the gas-cylinder model
above, it would appear that the direction of the work done [dW] will
depend on the nature of the pressure [P] and the direction of the change
in volume [dV]. Now, in the context of an expanding universe, we might
initially assume that the change in volume [dV] has always been positive
at all points in time. However, the implication of the various equations
of states associated with each energy-density implies that the net pressure
[P=ωρ] within the universe will change as a function of time
- see *previous graph*.
In the radiation dominated early universe, it would seem that the nature
of the pressure [P_{R}] would have been determined by the equation
of state [ω=+1/3], such that we might infer the 1st law of thermodynamics,
as defined in [4], in the following fashion:

[14]

As such, it might appear that the net result of [14] aligns with
model (3b) above, i.e. work is done on the external system, which might
also be interpreted in terms of the expansion converting work done energy
into potential energy of gravitation. However, it is unclear whether
this is a true reflection of cause and effect, because [14] is implicitly
making the assumption that it was the radiation pressure, in isolation,
which '*caused*' the change in volume [dV]. In contrast, the
ΛCDM model assumes that the net effect of the radiation density and
its associated pressure will only contribute to the slow-down of expansion.

*What about the inclusion of dark energy?*

With the doubt about the cause and effect of radiation pressure still
in mind, let us now consider the implication of the
*previous graph*,
when the pressure [P] becomes dominated by dark energy. The equation
of state [ω=-1] associated with dark energy suggests that the pressure
[P] eventually becomes negative and, in this context, the ΛCDM
model implies that it is this pressure that drives the acceleration
of expansion, i.e. [+dV].

[15]

In [15], we see the implication of negative presure [-P_{Λ}]
and a positive increase in volume [+dV] aligning with model (3a) above,
i.e. positive work is done on the internal system. However, this interpretation
of the work done by dark energy would then appear to contradict the
idea that dark energy is a cause of expansion, i.e. model (3a) seems
to suggest contraction.

Might we question the description of dark energy having negative pressure?

In the orientation of the 2 models above, it is difficult to reconcile the description of dark energy having negative pressure, when the expected result is expansion. So, given that the description of the pressure may depend on the orientation of the model, let us reverse [15] so that it gives a more meaningful answer, i.e. we will describe dark energy as having a positive pressure [+P] leading to positive expansion [+dV]:

[16]

So, based on [16], we now have a description of the dark energy era that would reflect the work done by the internal system on an external system in the form of expansion. Of course, if we make the change in orientiation for the dark energy pressure, we would also have to apply the same logic to radiation in [14]. However, if we now describe radiation in terms of a negative pressure, we would also be led to the conclusion that it contribute towards a negative change in volume [-dV], irrespective of the overall expansion of the system.

[17]

How experts in the field of thermodynamics and cosmology actually resolve such issues will have to be tabled as an open issue, but for the moment, we might simply consider [16] and [17] to be a more logical description of the thermodynamic processes associated with the radiation and dark energy pressures.

But are we losing sight of the physical processes at work in the expansion of the universe?

At a basic level, there seems to only be 2 possible descriptions of expansion; either objects are moving away from each other by physically moving through through pre-existing space or the space between objects is expanding. In this context, the former description seems to run into problems regarding the kinetic energy of objects moving with some velocity [v], while remaining stationary with respect the CMB radiation, plus its raises the issue of superluminal velocities. As a result, we might again return to the question:

So how does the 'fabric' of space expand?

Well, based on the inference of dark energy being a cause of expansion, we might simply speculate that the expansion od space is driven by some form of quantum energy process, at the Planck scale, which in terms of any large-scale thermodynamic model is simply generalised in terms of the input energy [Q]. Based on the thermodynamic model in (3b), we might also speculate as to whether the work done [W=F.dx] could be equated to the change in the potential energy of gravitation within this system, as a whole. However, in many ways, we have simply returned to another previous question:

*Do the Friedmann equations, which appear to underpin
the ΛCDM model, really describe the totality of the universe?
*

However, having now raised some basic issues of concern regarding
the assumptions and implications of the ΛCDM model, we will now
turn our attention on how this model is thought to have evolved as a
*function of time*. Of course, without
a resolution of the issues discussed, the overall validity of this model
may still have to be questioned.