B. Lukács
President of the Matter
Evolution Subcommittee of the Geonomy Scientific
Committee of HAS
CRIP RMKI, H1525 Bp. 114,
Pf. 49,
lukacs@rmki.kfki.hu
ABSTRACT
S. M. Stirling is
writing books in Alternate History. Besides the obvious evolutionary
connections, these books suggest lots of interesting scientific or scholarly
problems, some discussed in the books, some not. In the “Emberverse”
stories mighty aliens perturb terrestrial physics stopping detonations and flow
of electricity since
0. PROLOGUE
S. M. Stirling is
producing a big variety of Alternative History scifi’s. Alternative History is a subclass of scifi’s, where the fantasy does not enjoy its unlimited freedom. In an
ideal AH story only one event has other outcome than on our timeline (that is
the PoD, Point of Divergence), especially an event
which "might have happened as well otherwise", by other words, where
even now we do not know if the particular outcome was necessary or not. (Here I
omit philosophical discussions of Free Will, determinism/indeterminism, the
Everett Theory of Quantum Mechanical Measurement and such, for simplicity.) So
at PoD one event has another outcome, and then the
author continues the story, with as
strict "causality" as possible. Of course the PoD
acts then as an initial condition different than in Our TimeLine (OTL). Then
the author elaborates this Alternative History.
Now, we do not yet
know if such concurrent timelines can be realised "synchronously", or
not. (Against the a priori
impossibility of the alternative see
The machinery can be
demonstrated by means of, e.g., Poul Anderson's
works. In Eutopia [2] a modern (c. 1970 AD) Greek
social researcher from a timeline where Alexander III of Macedon did not drink himself exactly into death but
after some time recovered, so Aristotle
did not have to leave the Lyceum in 322 AD, can switch timelines,
and explores one where Alexander did die prematurely, but the ToursPoitiers battle in 732 AD
had an outcome opposite than in OTL. So the Frankish Kingdom became weaker, the PippinCharlemagne line could not
substitute the Merovings, so in the next 200
years Dane Vikings & Magyars destroyed the Western Christian civilisation.
In 1970 AD Danes can be found in Canada & Minnesota (
S. M. Stirling works
in the AH scheme. In one sequence (Lords of Creation [4], [5]) the PoD was 200 My in the past; a
superhuman civilisation terraformed Venus & Mars,
and from time to time they imported new animals & plants to the other two
planets. Terrestrial history was practically not influenced until 1947 AD, when
astronomers detected the habitable planetary surfaces. Therefore Cold War went
to space. The Draka sequence of Stirling puts the PoD to somewhere at the second half of the XVIII^{th}^{ }century, Northern American
loyalists go to South Africa, so the XXth century
will be very, very different. (But this sequence is par excellence Military SF,
so I ignore it completely.)
Now, Stirling &
Flint invented a somewhat different new AH scenario. (I think
The idea is
"simple". A tremendously superhuman ET civilisation sends back a
negligible part of our contemporary Earth into the past, and replaces the
surface with the past one. In
While this opus is not
yet comparable in size and details with that of Honoré
de Balzac, it may be later; and while Balzac had to know only his present society and its immediate past, Stirling
is confronted with scientific problems, and with a number of different
societies. True, the scientific problems are not direct. Earthpeople
do not understand the Lords of Creation; not even the bioscience of Martian hominids.
But
So far he generally
fulfils the expectations. But the opus is a challenge for somebody (I mean
myself) who is the President of a Matter Economy Subcommittee. Evolution of
sciences, evolution of languages, evolution of hominids...
I am going to make a
series of comments of that opus. That
will not be criticism. Nobody is
interested if I like or dislike a book. Rather I am going to discuss different
items in the books from the viewpoint of the actual science (or, sometimes,
scholarship).
Sometimes this is
really a challenge. E.g. Lords of Creation definitely use a
physics unknown for us. And our present physics is not the Physics. For 2000 years it was not only elementary
experience but also Fundamental Theory that there is no Motion (at least,
except for transients,) without Force. People generally believe that Aristotle
was stupid and the later generations did not dare innovate; but Aristotle did
know that his theory was not yet complete. See his own words about the problem
of ballistics (spears and arrows) [16]. In Chap. 32 of Mechanics he writes:
"Why is it that an object which is thrown eventually comes to a
standstill? Does it stop when the force which started it fails, or because the
object is drawn in a contrary direction, or is it due to its downward tendency,
which is stronger than the force which threw it? Or is it absurd to discuss such questions, while the principle escapes
us?" (Italics are mine.) The paragraph is clear enough even it
contains only questions. The Stagirite did know that
ballistics was at best very difficult to describe even in his theory; and he
was not able to do. Still, missiles flew; and later somebody would improve even
his description.
Before Einstein's
success not only everybody in physics was convinced that Flow of Time was
independent of spatial motions; everybody believed that even a doubt would be
meaningless. Special Relativity appeared a mere 103 years. Similarly, until the
formulation of Quantum Mechanics (more or less 1926) physicists were convinced
that any object has its sharp momentum and
position, although maybe we do not know them; and similarly that one object
cannot be simultaneously here and there. It seemed defetism
to question the first statement and unscientific magic the second. That was 82
years ago.
Our present science
will be so primitive and untrue for the XXV^{th}
century as the Aristotelian physics for us. Still, Aristotle was a true
scientist (while Platon, Marcion,
Plotinos and Melanchton
were not), as a true physicist as e.g. Galileo,
Here we are going to
discuss the thermodynamic problems arising in
1. WHAT IS ETL, THE EMBERVERSE SCENARIO?
Emberverse
Scenario is one half of a timeline; but for all practical purposes we can call
a timeline the Ember TimeLine or ETL, because on one of who knows how many
alternative TL's "the Fire Dies" on Earth in March 1998 and "only
Embers remain".
Well, this is
metaphoric speech and exaggeration. Fires continue to burn quite well after the
Change, but intense fires are slightly less intense. But this means that
gunpowder only burns, does not detonate. And internal combustion engines do not
work either. This two facts are observed seconds after
the Change. Later it turns out that the efficiency of steam engines decline
too.
Electricity stops at
the Change as well. Not only the electric networks stop; that might be a
consequence of the stopping of power stations. Instead, electric wristwatches
stop immediately too.
Something drastic
happened; about physical laws. The question is not if that is possible; perhaps
the whole thing was made by Alien Space Bats, superhuman mighty aliens. Also,
the important question is not: how? The technology of the Alien Space Bats is
so above ours that we cannot answer the how until a new civilisation is not
built up and in it lots of research is not made. The reasonable question is: which laws have changed. If one can
answer this, one can start to build up optimally the new civilisation.
In the Embervere sequel people sometimes guess that the laws of Thermodynamics have changed.
I do not believe this. An Earth with a new Thermodynamics would be much more
exotic than the Emberverse sequel reports it. But
this needs some exposition and that will be done here.
The Emberverse sequel (so far Refs. [11][15]) is one half of the story of one Alternate TL. The
Change, made by the unknown aliens, was centered on
the
2. ON THE CHANGE AND THE RESULTS
The Change, which was
global, is described in details in [7] (
It seems that the
Change was not instantaneous. In
Just then no research
happened; there was simply a mass panic
The Change arrived at
1998 AD civilisation
has been impossible to continue (no electricity, no engines), but lots of
information remained, mainly in books. Some top administration and its
structure survived in the
Experiments of Larsson
and the events mentioned in the Emberverse sequel
have told us so far the following stationary changes:
Effect 
Mention 
Gunpowder does not detonate anymore, only burns fast 
[11] 
Electricity does not work, down to transistor radios; however neural
systems are uneffected 
[11] 
Internal combustion engines do not work 
[11] 
Steam engines in principle work, but the efficiency is negligible 
[11] 

[12] 
Combustion has become slower 
[12] 
Napalm &c. still burn fast enough 
[11] Chs. 28 & 31 
Nuclear chain reactions are somehow inhibited, but radiation still
kills 
[12] 
We can follow a
substantial discussion of Ken Larsson, Scientific Advisor to Lord Bear and Lord
Bear himself in [12], Ch. 2 where they concluded from experiments that on
postChange Earth
The Gas Law has
changed.
There is some "glueing together" in gases.
In a cylinder of a
steam engine pressure p cannot go above a moderate limit, either T is
increasing or V is decreasing.
Still
"concentrated heat" is possible, so foundries &c. work.
The energy
conservation is still valid, but the extra work goes into heat.
Ken Larsson concludes
that somehow Thermodynamics has changed ("It simply fucks parts of the
laws of thermodynamics..."); in the next Chapter we discuss this
possibility. But I tell the result already here: this solution is extremely
improbable and surely reflects the engineers' picture about Thermodynamics.
Namely, it is much more probable that "only" the equations of state
of gases have changed.
3. ON THE STRUCTURE OF THERMODYNAMICS
OTL Thermodynamics is
mainly a theory/paradigm speaking about energy balances and probabilities. We
have been collecting many empirical facts about that area in the last 2500
years (yes; our Thermodynamics continues to be Aristotelian for structure
[17]), and now it is roughly as follows [18][21]:
There are extensive
parameters, whose minimal but sufficient set (finite)
fully determine the equilibrium state of a thermodynamic system (apart from
phase transitions when two or three different states may exist with the same extensives, but one is the stablest).
You may think e.g. of a simple system of a single particle component without
magnetism &c.; then the independent extensives
are maybe {V,E,N}. For moderate gradients in the system you may switch to densities E/V
and N/V and hope that the local state
is still determined by the local
densities (the principle of local equilibrium); in many cases this is true.
Explanation:
there is an energy distribution of maximal probability in the system. In or
near to equilibrium it is probable that the internal state of the system is in
the neighbourhood of the state of maximal probability. This state is determined
by a finite set of macroscopic
variables, as the total energy, the volume in which it is distributed, the
number of particles on which it can be distributed &c.
For large enough
systems internal forces of short range ("surface forces) are negligible (a
large container has "much more volume than surface"), so let us
neglect them. Longrange interactions are mainly electromagnetism &
gravitation. Electromagnetism can be roughly classified into electric force and
magnetic force. Electric force is indeed of infinite range (or 1/r^{2}
law), but the problem has negative feedback (in a neutral system fluctuations
can produce positive and negative clusters, but the opposites attract each
other, so this separation is transient). Not being magnetic monopoles (or,
anyway, not being found), magnetism is formally not of infinite range. As for
gravity, it is strong for really large systems as stars & galaxies, but the
selfgravity of a steam engine is negligible. So we ignore volume forces.
Comment:
Thermodynamics of plasmas (with charged clusters) is possible but quite
difficult. The behaviour of systems with internal magnetism was manageable up
to now but surprises are not ruled out. Newtonian gravity is really a possibility
of positive feedback, and this is the reason for e.g. star formation when the
final configuration is more inhomogeneous than the initial one was (so no equilibration). In General Relativity
Gravity is not an interaction at all but Geometry, and
it is not yet completely settled how to handle it together with Thermodynamics.
There is a thermodynamic potential, meaning that
its extremum acts as a variational principle. So local functional dependences (as between pressure
& energy & such) are determined from the functional form of the
potential, for spontaneous processes the potential changes into a definite
direction &c. For a simple system of independent extensives
{V,E,N} this potential is maybe the entropy S, and
then
S = S(V,E,N) (3.1)
This S takes its maximum for equilibria in a
closed system.
Comment:
this S must be in some connection with the distribution of the energy in the
system, otherwise its functional form could not determine the particular
functional forms amongst the quantities in the system. Indeed, Planck showed
that for ideal gases
lnS
~ kW (3.2)
k being simply the Boltzmann
constant and W the number of microstates belonging to the same macrostate; there are good arguments that the connection
remains true for almostideal gases and it is used also for strongly nonideal
ones as well, for which our description is not yet two reliable, anyway. And
then look: the spontaneous increase of S can be verbalised as: it is probable
that a state is followed by another state even more probable.
As told then earlier,
the volume interactions are absent and the surface ones can be ignored for a
large enough system. Now, consider a doubly large one in equilibrium. None will
change if you separate it into two halves by a wall, either virtual or real. So
the thermodynamic potential must be halved too, otherwise it would give changed
forms of connections between halfenergy, halfvolume, halfnumber &c. By
other and more unequivocal words, the potential is a homogeneous linear
function of its extensive variables
S = ∑S,_{R}X^{R} (3.3)
where X^{I}'s are the independent extensives,
the comma means partial derivative, and the sum sign is left out henceforth
according to the Einstein convention (i.e. that the same index in strict pairs,
above and below, indicates summation). So entropy is an extensive too. The
partial derivatives of the potential are called the canonically conjugate
intensives, and are denoted by Y's, so
Y_{I} = S,_{I} (3.4)
Comment:
being Y's the derivatives of the homogeneous linear potential with respect to extensives, the values of intensives are insensitive of the
size of an equilibrium system.
The change of the
potential can of course be expressed via the changes of its all independent
variables. So
dS
= S,_{R}dX^{R} = Y_{R}dX^{R} (3.5)
This is the differential form of the First Law.
Explanation:
For anybody but people really in Thermodynamics it is more natural to formulate
the First Law for energy not for entropy. However when there is a single
internal energy in the system then the two conventions are equivalent, and when
ther are more than one internal energies
in the system then only (3.5) is correct.
But now from (3.35) it follows that
X^{R}dY_{R}
= 0 (3.6)
This is called historically the GibbsDuhem
relation, but it is a consequence of the above formulation.
Comment:
so there is no thermodynamics without (3.6). You might, on the other hand
modify the First Law (3.5); but that may simply mean that you had forgot about
some variables and now you are going to repair this. I am not discussing holy
“conservation”; eq, (3.5) simply means that S can
change only through its variables.
Now, we are almost
ready. In the usual axiomatic construction the 0th Law is the existence of the
finite set of extensives and intensives, the First
Law is (3.5), the Second Law is a consequence of the fact that S is a
thermodynamic potential, so it fulfils some extremum
principle, and the further Laws fix zeros or infinities of intensives.
Generally only the Third Law, or Nernst's Law is
explicitly formulated, establishing something about the absolute zero of
temperature, or the thermodynamic behaviour of the system when approaching
ground state, however for the fixing of convention the asymptotic behaviours in
the infinities also need further Laws or Axioms. See e.g. [18], [22], [23].
Explanation:
if you do not understand a word of the last few sentences, you can remain
happy. These sentences were a shortcut of several pages, which would not be
interesting at all, except for cca. 1 of a million (estimation on Hungarian data).
And now comes the last, hardly obvious, but very
important point. The rank of the Pfaffian form of our Thermodynamics is 2.
The exact, punctual
and absolutely correct formulation would belong to Differential Geometry. I
could reproduce it (after a halfday reading), but maybe it would be boring for
you. However leaving off some details it goes as follows. Assume that you have
already defined the reversible and irreversible parts of energy exchange, or
defined somehow the natural or spontaneous or adiabatic processes or such. Then
dE
= dW + dQ (3.7)
where the strikethrough in the
last term will get its explanation soon. Here dW is the reversible work (say, work of pressure) and
dQ is the contribution of irreversible processes,
generally called "heat transfer", or "noncompensated
heat", although nobody ever has proven that only heat transfer can be
irreversible.
Now, the Pfaffian form is connected with dQ.
But first observe that dQ is not unequivocally
defined [21]. Namely, you may add some part of dW to dQ and still
the new d'W and d'Q
will remain reversible and irreversible, respectively. People generally do not
discuss this problem and so let us believe now that this freedom is irrelevant
for the rank of the Pfaffian.
Obviously
dQ
= G_{R}(X^{I})dX^{R} (3.8)
being the set of the independent
extensive variables complete, and also all the coefficients G_{I} are
functions of homogeneous zeroth order. Otherwise they
depend on the details of, e.g., the intermolecular interactions. But if we know
the function S(X^{I}) and we know which change
is reversible and which is not, then we know the G's.
Now, let us introduce new variables Q^{i}
Q^{i} = Q^{i}(X^{K}),
1≤i≤n≤N, 1≤K≤N (3.9)
Then eq.
(3.8) takes the new form
dQ
= g_{r}(X^{k})dQ^{r} (3.10)
Different transformations result in different values of n (the length
of summation). Let us take the one with minimal
n. (If there are two transformations with the same n, we may take any of them.)
And now dQ can be written as one of the possibilities


K=1 


K=2 


K=3 
and so on. And K is the rank of the Pfaffian.
Now, in the case of
K=1 there is an extensive variable Q, the
heat, so then dQ = dQ,
the noncompensated heat is just the change of an
extensive variable. In this case there would be irreversibilities
in thermodynamics, but there would not be a temperature. This is not the OTL
world, but not the ETL one either.
For K=2 one gets dQ=TdS, and this is OTL
physics.
For K=3 Thermodynamics
would be "more complicated" than ours. But before anybody would be
happy, note that if K>2, perpetua mobilia of second type are possible [19]. One such perpetuum mobile works as follows. You have lots of water
of the Pacific of, say, 70 F°. Now, you separate half and half of the volume,
cool the first to 50 F° and heat the other to 90 F°. No change in total energy.
But then you can drive a heat engine with the temperature difference! After
some time you are back at the initial situation, but the heat engine performed
some work!
In OTL this does not
work; it takes more energy to cool and heat than the work you get back
afterwards, and the difference is just the "noncompensated
heat". You may generalise the long experience for the impossibility into
an Axiom that K≤2. In contrast, in
a K=3 world some really ingenious engineers could build perpetua
mobilia even with energy conservation. (For the
possible physics in the remote past of the Universe see [24]; but I do not
argue for it, only state the possibility.) Now, the reports [11][15] do not
suggest such a situation after Change, ergo K is still 2 in ETL. It seems that
nothing mentioned in this Chapter changed in ETL on
4. IF NOT THERMODYNAMICS
CHANGED THEN WHAT?
The discussion so far
was to demonstrate that, contrary to Ken Larsson's opinion [12] very probably
the Change has not changed the Laws of Thermodynamics. OK, Ken Larsson is an
engineer, not a physicist, so he did not formulate the statement as Callen, Landsberg or Martinás would have made it. However he did get some
intuition, see again Chap. 2 of [12].
At a point he tells:
"...instead of producing work, the heat energy or the work put into
mechanical compression gets locked into some weird form of potential energy". Even if this statement is somewhat
indefinite (it is far not the same case if heat or mechanical compression goes into weird potential energy), we can
formulate the nebulous hypothesis behind.
Assume that you have
two disjoint energies in thermodynamic sense, E^{1} and E^{2},
in the system. Then
S = S(V,E^{1},E^{2},N) (4.1)
For physical examples see e.g. [25] & [26]. This means two energy reservoires. Of course, only the sum has conservation or
any other balance law.
To explain this, let
us take the example discussed in [25]: a LiF crystal
in external magnetic field. The ions are near to the points of a cubic lattice,
and oscillate around. This is the familiar thermal motion. However the spins of
the nuclei of the ions can take up/down, so +/ positions relative to the
external magnetic field.
Now, the two ways of
excitation are only weakly coupled. The spatial motion around the ideal lattice
point would not influence too much the up/down position of the magnetism of the
nuclear spin and vice versa. The spatial motion has a squared average of
velocity <v^{2}> = 3MT_{1}, and the spins will be
populated as N_{+}/N_{} = exp{2Hμ/T_{2}}.
All experiments show that after some time T_{1}=T_{2}; but it
is very easy to disturb the equipartition. The
simplest way is to reverse the external magnetic field H. Then the spin distribution becomes of inversely populated, or,
by other words,
T_{1} = T_{2} = T_{lab}
= 293 K (4.2)
And indeed the negative spin temperature, followed by a gradual
equilibrium, first with T_{2} going to ∞ then from +∞
towards +293 K, can be clearly observed. The process takes minutes [25].
So the two reservoirs
may have two different temperatures. However, this is a consequence of the weak
coupling between nuclear magnetic momenta and the
spatial motion of the ions, and not of a declaration
of two independent reservoirs, or saying "Abrakadabra".
If you push down the piston, the gas in the cylinder will become denser. A gas
is a bunch of molecules, with or without forces between. If we knew the
interactions, we could calculate the reaction of the gas to compression. If we
do not yet know them, at least some characteristic data of the extra reservoir
can be determined.
Perform a slow
compression. We are told that the temperature is higher than before Change,
even if not much higher. So it is possible we see Equipartition
in Action. The energy of the compression work goes first into the "weird
reservoir" (E^{2}) and then seeps
into E^{1} as well. I would
be very, very surprised if the evolution equation of the process were not
approximately
dE^{1}/dt = (other trivial
thermo terms) (E^{1}E^{1}_{,eq})/τ (4.3)
where E^{1},eq is the value belonging
to
T_{1}(E^{1},E^{2},V,N) = T_{2}(E^{1},E^{2},V,N) (4.4)
In simple cases the equilibrium state is somewhere at E^{1} ~ E^{2}. Here τ is the
relaxation time, and that can be observed even with unaided eyes. For a similar
problem but in cosmology see [27], for a better elaborated
analogy see [28].
If the Alien Space
Bats established (or much enhanced) an intermolecular interaction damping the
thermal motion of gas molecules or sticking them transiently together and
releasing them with the relaxation time τ, that τ must be at least in
the seconds range, otherwise internal combustion engines would work. If it is
really in the seconds range then wind guns still work. If it is in minutes
range then even they do not work. However, it is rather a very serious assumption
that mere intermolecular forces could keep lots of energies for minutes. In our
OTL physics these forces are most often van der Waals forces from the tails of the electromagnetism in
molecules. One way to feel them is the polarisation behind solvation
processes (which could not seriously change otherwise preChange Biology would
have stopped at change). Another example is the Hbridges, e.g. in peptide
chains establishing the tertiary structures of enzymes, or between the two
chains of a DNA "ladder". In these cases the OTL intermoleculary
forces are not of very long range and the characteristic energy of the interaction
is a mere ~0.05 eV, which is ~500 K in temperature.
Such forces are behind
the familiar vapourliquid phase transition as well. And we can produce
nonequilibrium phase transitions without any hightech, simply by heating,
cooling, compressing or decompressing the matter really fast.
So maybe the Alien
Space Bats can be circumvented by means of tricky
heat engines. Or, maybe they have not created a "second reservoir" at
all. There is an even more traditional way; of course I, here in OTL, cannot
perform the decisive experiments instead of Ken Larsson or the Physics
Department of OSU at
5. ON THE CHANGED EQUATIONS
OF STATE
Assume now that some
interactions have indeed changed (otherwise the Fire would not have died on
s = s(e,n) (5.1)
Thermodynamics does
have a canonic structure, so we can substitute one or
more intensive variable instead of the canonical conjugate extensive, if we introduce an appropriate new
potential as well. Also, if there is a
single energy variable, we can switch from entropy as potential and energy
as variable to energy as potential and entropy as variable. So, let us make
this, second, transformation first, and then let us substitute entropy with the
conjugate temperature; and finally let us go to densities [19]. Then the
potential density is the density of
the free energy f:
f = f(T,n) (5.2)
s = f,_{T} (5.3)
p = nf,_{n}
 f (5.4)
e = f  Tf,_{T} (5.5)
Obviously f(T,n)
carries all the quasistatic information about the
thermodynamic behaviour of the system.
Before the Change at 1
atm pressure and 293 K temperature all simple real
gases (as air, water vapour, carbon dioxide and such were quite near to the
ideal gas limit, i.e.
f ≈ nTln(n)  (3/2)nTln(T) + CnT (5.6)
where C is some combination of
molecular weights, statistical factors & such and is not important for
mechanical behaviour. From (5.6) you get two
equations of state:
p = nT (5.7)
e = (3/2)nT (5.8)
(5.7) would not be true if the molecules occupied a substantial part of
the whole volume (which is definitely not true: at usual circumstances air is
almost 3 orders of magnitude less dense than water), or if the intermolecular
interactions were quite strong (which they were not). I do not believe that the
Aliens changed the volumes of the molecules (then, e.g., the lungs would
operate strangely); but they might change the intermolecular interactions (I do
not yet know, how; but that knowledge is not needed for discussing the
thermodynamic behaviour).
Now, Ken Larsson's
observations resulted in:
lim
_{n→0} p(n,T) ~ n
lim
_{n→n}_{o} p(n,T) = ∞
lim
_{T→0} p(n,T) ~ T
lim
_{T→∞_} = nT_{o} (5.9)
where n_{o} is sime characteristic
density well above normal air density but well below the characteristic
densities of liquids; and T_{o} is a temperature probably below the
preChange critical temperature of water (640 K).
One could try with Ansätze for approximate free energy functions without any
guess for the intermoleculary forces behind, even
with only paper and pen. To demonstrate this, here I start with the simplest
form
p(n,T) = a(n)b(T) (5.10)
Maybe it will not be enough (alas, the information is on an other TimeLine) but then it is rather straightforward for
everybody to generalise the Ansatz.
With (5.10) eq. (5.4) can be integrated. Let us call A(n)
as follows:
dA/dn
≡ a(n)/n^{2} (5.11)
Then
f(n,T) = n[A(n)b(T) + c(T)] (5.12)
Now, for the internal energy
density eq. (5.5) yields
e(n,T) = nA(n)(b(T)Tdb/dT) + n(c(T)Tdc/dT) (5.13)
where the constraints (5.9) give
similar ones for a(n) and b(T).
Obviously A(n) diverges approaching n=n_{o}, but this is not a
surprise at all, and simple van der Waals systems do it (although they do it after phase transition). It is also
obvious that the system must have a divergent thermal capacity at high
temperatures otherwise the compression would increase the temperature; however
now eq. (5.13) shows that the single energy reservoir
may take extra energies if c(T) is
tricky enough.
Maybe the simple Ansatz (5.10) will be enough, maybe not. Even, you may
choose different f(n,T)'s
leading to different fates of the Emberverse TL after
CY 23, which may be fun.
Thermodynamics is
almost solely phenomenologic, but in its own
territory it is very strong.
ACKNOWLEDGEMENT
Earlier illuminating
discussions with K. Martinás are acknowledged.
REFERENCES
[1] H. Everett:
[2] P. Anderson: Eutopia. In: H.
Turtledove & M. H. Greenberg (eds.): The Best Alternate History Stories in
the 20^{th} Century.
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