Entropy & Thermodynamics

5In 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 1st law, to include the idea of entropy, as defined by the 2nd law of thermodynamics.

[3]      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 deathof 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 ‘useful’ 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

 1

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 characteristics 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 [m1] and [m2] 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]      1

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

[2]      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 [m1] and [m2] 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 of 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 1st law of thermodynamics to an example of the gas in a cylinder, although the applicability of this example might be later questioned.

2

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]      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 1st law of thermodynamics through the concept of heat [Q], which then changes the volume [dV]:

[4]      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 1st 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]      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 consistent 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 1st law of thermodynamics. However, we also need to address another key implication that relates to the definition of entropy, as linked to the 2nd law of thermodynamics, and the description of an adiabatic system. Let us start with a general definition of the 2nd law and 'the idea of entropy':

Embedded in the 2nd 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 steps in the derivation of the Fluid equation starting with [6] below:

[6]      6

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

[7]      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]      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 2nd law of thermodynamics to be reduced to the form show in [9]:

[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]    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 10100 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.

3

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/m3]. 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]    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 some 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]    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 note that 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]    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 [PR] 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]    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]    15

In [15], we see the implication of negative pressure [-PΛ] 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]    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 orientation 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 contributed towards a negative change in volume [-dV], irrespective of the overall expansion of the system.

[17]    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 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 of 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.