# Biocosmology: Biology from a cosmological perspective

Marina Cortês,1,2 Stuart A. Kauffman,3 Andrew R. Liddle,1,2 and Lee Smolin1

1Perimeter Institute for Theoretical Physics, 31 Caroline Street North,
Waterloo, Ontario N2L 2Y5, Canada
2Instituto de Astrofísica e Ciências do Espaço, Universidade de Lisboa,
Faculdade de Ciências, Campo Grande, PT1749-016 Lisboa, Portugal
3Institute for Systems Biology, Seattle, WA 98109, USA

###### Abstract

The Universe contains everything that exists, including life. And all that
exists, including life, obeys universal physical laws. Do those laws then give
adequate foundations for a complete explanation of biological phenomena? We
discuss whether and how cosmology and physics must be modified to be able to
address certain questions which arise at their intersection with biology. We
show that a universe that contains life, in the form it has on Earth, is in a
certain sense radically non-ergodic, in that the vast majority of possible
organisms will never be realized. We argue from this that complete
explanations in cosmology require a mixture of reductionist and functional
explanations.

###### Contents

  1. 1 Introduction
  2. 2 Three kinds of cosmological statistical systems
    1. 2.1 Type I systems
    2. 2.2 Type II systems
    3. 2.3 Resolving the early Universe thermodynamic paradox
    4. 2.4 Type III systems
  3. 3 How do we explain the phenomena of Type III systems?
    1. 3.1 The failure of determinism and reductionism in Type III systems.
    2. 3.2 The need for functional explanations in Type III systems
  4. 4 What is different about Type III systems?
    1. 4.1 Extreme non-ergodicity and its consequences
    2. 4.2 Type III systems can be recursive or excursive and also universal
    3. 4.3 The three arrows of dynamics
  5. 5 Proposal for a fourth law of thermodynamics
  6. 6 A definition of life
  7. 7 Summary

##  1 Introduction

This is the second in a series of papers, starting with Ref. [1], that asks
whether and how the kinds of explanations we employ in cosmology must be
modified if we are to understand biological organisms as having a natural
place in the Universe. How must the role of physics and physical explanations
change if we live in a biological universe — a biocosmos?

The purpose of this article is to present a detailed argument which leads us
to a definite answer to those questions. The main argument of this paper is
methodological, and it rests on the answers to two questions:

  1. 1.

What kind of explanations are necessary for elucidating biology? and,

  2. 2.

How can we revise the kinds of explanations we use in cosmology so as to make
possible biology as a science of parts of the Universe?

The starting point of our investigation is the affirmation that statistical
thermodynamics can be extended to provide the physical basis for explaining
how biological systems organize themselves. However, we point out that the
existence of life is due to the existence of thermodynamic subsystems whose
thermalization times are much greater than the Hubble time, or rather no
thermodynamical equilibrium is ever reached while the system is alive and
thriving.

These are systems whose descriptions cannot ignore the fact that they are
embedded in a dynamical, expanding Universe. An even larger departure from our
expectations of how statistical thermodynamics governs physics is the
discovery that all living organisms are radically anti-ergodic, in the sense
that only a tiny fraction of their state spaces will ever be realized, even if
all the baryons in the Universe were to be available for many orders of
magnitude of Hubble times. Because of the vast capacity of phase space, these
systems will never be in equipartition and will never approach an equality of
time and ensemble averages (so long their circumstances remain the same),
which is the ergodic hypothesis to be satisfied by standard thermodynamical
systems in equilibrium. This arises because the number of types of living
organism that could exist, even if the Universe were crammed full of life, is
vastly smaller than the number of organisms that could be coded in DNA or RNA
or catalyzed and built with proteins. The state space is then extraordinary
sparsely occupied. The main question that must be asked concerning any
organism or biological molecule is why does it exist, while vastly more
alternatives do not.

Such questions have answers, but bottom-up, purely-reductionist, explanations
are not enough — we will never understand life by evolving it from initial
conditions, no matter how well we understand the fundamental laws. What is
needed to incorporate biology into physics and cosmology is a combination of
reductionist and functionalist explanations. Neither alone suffices.

The methodological arguments get their relevance from the conspicuous and
suggestive fact that the life emerges and evolves on cosmological timescales.
To put this simply, during the roughly 3.8 billion years that life on Earth
has taken to evolve to its present state, the Universe has grown and altered
significantly.

##  2 Three kinds of cosmological statistical systems

We start by affirming that the right framework to situate the study of large
macroscopic subsystems of the Universe is statistical thermodynamics [2].
Whether we are interested in systems in equilibrium or far from it, in gases
or the most complex of systems, we want to formulate descriptions in terms of
probability distribution functions on phase space or state space. These
provide coarse-grained descriptions of how macroscopic systems evolve under
the fundamental laws. Note that when we name the state of equilibrium of a
system we need to specify the interactions with regard to which the system is
in equilibrium. That is because the equilibrium state of a hot gas looks very
different from the equilibrium state of a living system.

This leads to our first and fundamental point, which is that if we wish to
apply notions from statistical thermodynamics to subsystems of the Universe,
we must distinguish three different kinds of systems, which we will call Type
I, Type II, and Type III systems. These are defined in terms of the
thermalization time $t_{\rm thermal}$, which is the time it takes for the
system in question to return to a stable or metastable state of thermal
equilibrium after an order unity perturbation.

Additionally, ordinary statistical systems in equilibrium are typically
ergodic, i.e. they satisfy Boltzmann’s ergodic hypothesis, which says that
over long periods of time, the portion of time spent in any region of the
accessible phase space is proportional to the volume of that region. Closely
related to this is equipartition: the postulate that in thermodynamic
equilibrium all accessible microstates are equally probable.

Metastable states, in which the system can be trapped in a false ground state
with a long decay time $t_{\rm constraints}$, are common in astronomy; typical
examples are red or brown dwarfs, which are the lowest mass stars that undergo
nuclear fusion. Red dwarf masses extend down to about 0.08 Solar masses, so
their lifetimes extend up to $10^{13}$ years (about a thousand Hubble times).
Because they may burn meta-stably for Hubble times or longer, these metastable
states play an important role in cosmology. Conversely, cosmology, i.e. the
fact that they exist in an expanding Universe, may be important for gaining a
complete description of stars which live many Hubble times.

Metastable states may occur due to the presence of constraints or high
potential energy barriers that either drastically slow down, or prevent
altogether, the approach of the system to its true equilibrium state. An
example of the latter situation is a white dwarf star, whose final reserve of
stored energy can only be released via accretion or collision with external
objects taking it over the Chandrasekhar limit. We will refer to this as a
system in (metastable) equilibrium, subject to constraints.

Systems like these are very common. There are two relevant timescales. First
$t_{\rm constraints}$ is the timescale over which the constraints hold; on
longer timescales they may fail or leak. For these systems we will use $t_{\rm
eq-con}$ to refer to the time it takes to evolve to equilibrium on the
constrained subset of states, i.e. the states allowed by the constraints.
Provided

$t_{\rm eq-con}\ll t_{\rm constraints}$ (1)

a system will, over intermediate time scales $t$ defined by $t_{\rm eq-
conl}\ll t\ll t_{\rm constraints}$, behave in some ways like an ordinary
equilibrium system. In particular it may, given a sufficiently-long time, be
ergodic on the restricted state space created by the constraints. With the
same restrictions, it may achieve equipartition.

An important example of such a system is a white dwarf star, which we will
discuss below.

Cosmological evolution is characterized by the Hubble timescale

$t_{\rm H}\equiv H^{-1}\quad\mbox{where}\quad H\equiv\frac{\dot{a}}{a}\,,$ (2)

whose present value $t_{\rm H}=14.5$ billion years is also an excellent proxy
for the current age of the Universe, 13.8 billion years. We will also use the
timescale of the stellar era, over which galaxies form stars which then burn
through nuclear fusion, which at about $10^{13}$ years is three orders of
magnitude larger than the present Hubble scale [3].

We can now define the three system types.

###  2.1 Type I systems

We start with ordinary systems in statistical-mechanical equilibrium. We will
categorize Type I systems as systems that satisfy Boltzmann’s ergodic
hypothesis [4] and for which the thermalization time $t_{\rm equil}\ll t_{\rm
H}$.

Following Ref. [4], an isolated system satisfies the ergodic hypothesis if,
for a sufficiently-long timescale $t_{\rm ergodic}$, averages over times
longer that $t_{\rm ergodic}$ are equivalent to averages over an ensemble.
These ensemble averages are defined by integration of a probability measure
over phase space. It is also the case that these averages do not depend on
what starting time is taken.

A system can fail the ergodic hypothesis for several reasons [5]. It can fail
to be isolated, or there may be a spontaneous breaking of a symmetry. There
may also be constraints or barriers, which either lengthen the thermalization
time or define metastable states.

The Universe was a Type I system early in its evolution, when the temperature
of the radiation component was large compared to nuclear binding energies
(together with all other interactions in the Universe, gravity, vacuum energy,
etc., which were still switched off at those temperatures) and material was
distributed approximately homogeneously. Whilst this was true it existed as a
gas or plasma of elementary particles, first quarks and gluons and, after a
phase transition, protons and electrons. For the most part equilibrium
thermodynamics suffices to describe the statistical physics of the Universe in
that era.

###  2.2 Type II systems

The second kind of system is that where the timescale to come to equilibrium,
subject to constraints, is significantly larger than the Hubble time,

$t_{\rm eq-con}\gg t_{\rm H}\,.$ (3)

That is, Type II systems will eventually come to (possibly metastable)
equilibrium, subject to constraints, although they will take longer than the
Hubble time to do so.

One might have thought this is an empty category, but we already pointed out
that such systems are common. The category includes many stars and galaxies
[6]. Very remarkably, the time for a star to come to equilibrium is vastly
larger than one would naively expect from nuclear timescales. This is due to a
combination of two circumstances. First, there is a landscape of several
hundred stable nuclear bound states, with binding energies in the MeVs. This
generates a complicated construction of constraints and barriers to evolution
to equilibrium, which results in a large number of metastable states that trap
and sequester nuclear potential energy. The second is that gravitationally-
bound systems, including stars and galaxies, have negative specific heat. Such
systems heat up when energy is removed, so that they do not come to unique or
homogeneous equilibrium states.

The reason this is significant is that, after an initial period of cosmic
nucleosynthesis when the temperature first falls below that of the binding
energy of helium and the existing thermal population of neutrons is
incorporated into helium nuclei (plus trace amounts of a few other light
elements), the subsequent burning of the remaining hydrogen to helium, and
then up through the heavier nuclei, takes place in stars [7]. For most of
their lifetimes, stars establish a dynamical equilibrium between the
gravitational binding and the nuclear burning; this is sustained by a feedback
loop which drastically slows the timescale for the nuclear matter in stars to
burn [6].

As we described above, most stars, and most of the baryonic mass, never reach
the equilibrium specified by their nuclear binding energies. They do reach
metastable ground states of the nuclear interaction, which have the properties
of equilibrium states when restricted to the states left accessible to them by
constraints. For most stars, roughly those lighter than a Solar mass, this
implies a time-to-equilibrium several or many times greater than the Hubble
time. Indeed a white dwarf left in isolation would be supported by electron
degeneracy pressure even after cooling and persevere essentially indefinitely,
disrupted ultimately only via quantum tunnelling through nuclear reaction
barriers.

Such systems are but one example of those satisfying Eq. (3).

Finally we note that large classes of non-equilibrium systems, such as the
dissipative systems studied by Prigogine and Nicolis [8], those governed by
Morowitz’s cycle theorem [9],111This states that “In steady-state systems, the
flow of energy through the system from a source to a sink will lead to at
least one cycle in the system.” or more generally that “The flow of energy
through a system acts to organize that system.” Kauffman’s autocatalytic
networks [10], and Turing’s model of pattern formation [11], are all examples
of Type II systems. Because the flow of negative entropy222Throughout, this
refers to a decrease of entropy of a subsystem via an external flow of energy
with very low entropy, e.g. the Earth receiving low-entropy photons from the
Sun. through the system is slow, there is time to establish steady states in
non-equilibrium systems, and this by itself drives the formation of structures
and patterns. These are Type II because, while they are not in equilibrium,
they flow to systems which are in equilibrium subject to constraints [6].

###  2.3 Resolving the early Universe thermodynamic paradox

This subsection bridges the gap between Type II and Type III systems, which we
will define in the next section. In particular we will address a point of
confusion in the literature regarding states of equilibrium in the early
Universe. During the early Universe its energy budget is strongly radiation-
dominated, with a distribution very close to homogeneous, and in thermal
equilibrium. At that time the Universe can be well described as a Type I
system, and as such, as the Universe continues to expand and cool down one
would expect that it stays in thermodynamical equilibrium. And yet, after
roughly three minutes, the Universe will spontaneously move away from this
equilibrium, and ends up finding itself in a region of phase space which is
far from it. Under thermodynamical laws this behaviour is impossible if no
external agents are at play. As such this spontaneous move away from
equilibrium appears to form a paradox that violates the laws of
thermodynamics. Let us analyse it in detail.

Equilibrium was established at very high temperatures, and is maintained as
the Universe expands and the radiation cools. The Universe then undergoes a
phase transition, big bang nucleosynthesis (BBN), when the temperature falls
below $T_{\rm nucleo}$, and is cool enough for the strong nuclear force to
switch on. This means that, now, stable nuclear fusion of the first nuclei
will survive disruption by high-energy photons, and the formation of the first
(stable) light elements can take place. A new phase of the Universe’s
evolution begins here: a very slow progression towards the newly-unveiled
state of equilibrium. Whereas before equilibrium meant equipartition of the
available energy amongst individual particle species, it now means minimising
the nuclear binding energy, which is the interaction that was switched on
during BBN. The state of maximum entropy will be (if we ignore gravity for a
moment) a universe of stable/long-lived iron nuclei (the most bound atomic
nucleus, per nucleon). This will be achieved by the formation of heavier and
heavier elements of the periodic table; the lightest elements formed around 1
to 200 seconds after big bang and the heavier ones formed much later in
stellar nucleosynthesis.333Ref. [12] studies how different the laws of physics
would have to be for the Universe to emerge from nucleosynthesis directly into
the iron-dominated state.

So, during BBN the Universe is suddenly disrupted from its previous
equilibrium to a state which is very far from equilibrium. This doesn’t mean
that the entropy goes down during BBN, but rather that a new ceiling of
maximum entropy, much higher than the previous, has opened up. Now the
Universe will begin its evolution towards this new ceiling of entropy, which
it will achieve when the newly-formed elements have been burned in the cores
of stars and their end stages to the constrained equilibrium distribution.

It would appear that there is a contradiction here: the Universe was already
in thermal equilibrium and had maximal entropy before cosmic nucleosynthesis.
How did it then come to find itself, spontaneously and without external
influence, so far from equilibrium? How can there, after BBN, occur a boost in
the (newly) maximum attainable entropy that will be responsible for the
creation of stars, the burning of their fuel in nuclear reactions, and all of
the Universe’s subsequent evolution until today? In other terms, given that
total entropy was already extremized, how could the Universe spontaneously
move away from equilibrium? Nucleosynthesis appears to violate the second law,
with its prompt, unaided, displacement from equilibrium and the corresponding
boost in the maximum available entropy, providing the Universe, which was on
its way to a fate of heat death, with a new breath of life. We shall see that
no such violation is required for understanding the phenomenon at hand.

There is a simple answer to this conundrum: the computation that attributes
equilibrium and maximal entropy to the pre-BBN universe is incomplete. The
pre-BBN universe was indeed at equilibrium, subject to the constraint $T_{\rm
nucleo}<T$, but it was an equilibrium state of the high-energy regime of the
full theory, the quantum chromodynamics Lagrangian, ${\cal L}_{\rm QCD}$. At
those high energies bound states of hadrons, like neutrons and protons, are
not possible. The paradox exists only if we don’t include all known
interactions in our estimation of the entropy.

At low energies these bound states become stable, and the nuclear interaction
becomes possible. With these new bound states included in the accounting of
the entropy budget of the Universe, the apparently-spontaneous move away from
equilibrium ceases to be surprising or paradoxical, and it not only can be
explained but is also expected and predicted.444Plus notice that we are
leaving out gravity out of the full Lagrangian for now, since it is still
switched off at high temperatures. However, as we will see, it will switch on
soon after and will rule the subsequent evolution of the entire Universe
during most of its history up to the present time. After all, had the
distribution of matter and radiation been already in the equilibrium of the
full theory, it would have remained in that state till today, with the cooling
of the Universe leading to its heat death.

We call the pre-BBN universe a ‘local’ equilibrium, in the sense of locality
in energy and temperature: once the Universe moves away from this particular
temperature, and falls below $T_{\rm nucleo}$ new equilibrium states will be
unveiled that are different from the equilibrium state prior to cosmic
nucleosynthesis. The probability density of the Universe, which was near
equilibrium at high temperatures before nucleosynthesis, has evolved into a
region of the state space that was inaccessible beforehand, and entered a
small corner, of states which are far from equilibrium and will henceforth
evolve towards it.

However note that, unlike as argued in Refs. [13, 14], the states in this
region of configuration space are not novel additions to Hilbert space, nor do
they constitute an expansion of the volume of configuration space. Rather,
they are states that were always present in the statistical ensemble of the
full theory, but were unoccupied, and unavailable to the Universe at high
energy, in regimes in which the temperature was too high for their bound
states to survive.

This means that when the temperature is above $T_{\rm nucleo}$ the
completeness relation of the Hilbert space sums over all states, both bound
nuclei and free. The bound states are part of the completeness relation but
unoccupied at high temperature.

One can describe this by saying that the nuclear bound states consistent with
the constraints of the fundamental theory were ‘unveiled’ because the system
at low temperatures is now in a different effective regime, but they don’t
expand the Hilbert space of the fundamental interactions (nor that of the
Standard Model).

We call this distinction between equilibrium states in different effective
field theories true equilibrium and false equilibrium in an analogy to the
true and false vacua of field potentials in cosmology and particle physics.
Just like the false vacuum of potential energy, the state of false equilibrium
at a given temperature is unstable, or metastable. It will continue to evolve
towards the true equilibrium, once constraints allow, which in the BBN example
happens when $T<T_{\rm nucleo}$. The state of true equilibrium is the state
which maximizes entropy in the thermal ensemble in a calculation that includes
all known interactions at play in the system.

Most importantly here we want to emphasise that equilibrium and entropy are
theoretical concepts and not observable, measurable quantities of Nature.
While observables describe the state of the system, entropy describes the
observer’s knowledge of the system.555This is with the exception of well-
controlled purely-thermodynamical closed systems where the sole contribution
to entropy comes from thermal interactions. Under such particular
circumstances entropy can be accurately estimated through measurement via the
first law of thermodynamics. Both the state of equilibrium and the value of
entropy of a system result from theoretical deduction, and therefore are
dependent on the assumptions going into the derivation. In particular if we
leave interactions that are present in the system out of the calculation, we
will no longer be able to explain the system’s long-term behaviour, as we have
seen in the example of the BBN phase transition. This has created the
misunderstanding of a paradoxical viewpoint, due to the fact that major
interactions at play in the Universe were neglected in the estimation.

We can summarise this as,

The value of entropy calculated for a system is not an observable, and depends
on the number and nature of the interactions included in the calculation. An
observable is a function of the state of the system, but entropy is a function
of the observer’s knowledge of the system.

A good example of how entropy is not a measurable quantity can be found in the
cosmologists’ evolving understanding of the Universe’s entropy, in recent
decades. In Ref. [1] we present in detail an account of this evolution,
following Egan and Lineweaver [15]. The first estimates of the (observable)
Universe’s entropy date from around 1980 and stemmed from thermodynamics,
attributing the bulk of the Universe’s entropy to the blackbody radiation of
the cosmic microwave background, $S_{\rm relativistic}\sim 10^{90}$ [16],
expressed in natural units, $G=c=\hbar=k_{B}=1$. Some years later the
realisation that the entropy of gravity could not be overlooked in the overall
budget led to this value being revised by Frautschi [17], and later firmed up
by Penrose [18]. The conclusion is that the formation of supermassive black
holes has led the gravitational contribution to today dominate over that from
the microwave background by around 15 orders of magnitude, $S_{\rm BH}\sim
10^{104}$.

More recently this estimate has had to be revised yet again to account for the
fact that the vacuum energy $\Lambda$ dominates the Universe’s energy density
today, so that the corresponding entropy that $\Lambda$ generates at the
cosmic horizon has to be considered. It turns out our previous estimate of the
Universe’s entropy due to gravity is wrong yet again, and altogether
negligible when compared to the entropy of the vacuum which dominates over the
gravity value by a further twenty or so orders of magnitude, $S_{\Lambda}\sim
10^{124}$ [15].

The point we are making here, with the example of the evolving understanding
of the Universe’s entropy, illustrates our earlier premise that the estimated
entropy for a system is a function of the observer’s knowledge of the system.
Had the entropy of the Universe been an actual observable, these erroneous
estimates for total entropy would have been ruled out by observation alone.
Instead, each of the values $S_{\rm relativistic}$ and $S_{\rm BH}$ were
revised when they were discovered to be subdominant in light of new
theoretical knowledge on the composition and the nature of interactions
present in the Universe. With regard to today’s entropy of the Universe,
$S_{\Lambda}$ is our most reliable estimate. However nothing forbids that,
should our understanding of the composition of the Universe continue to
evolve, further additions can potentially have to be considered, and the
entropy value revised once again.

This subsection conveys the message that, in order to avoid apparent
paradoxes, when we estimate the equilibrium state of a system, we need
consider, to the best of our knowledge, the single microscopically-complete
Hilbert space. It then follows that entropy is maximized or not at a given
time depending on constraints intrinsic to the system.

###  2.4 Type III systems

Both Type I and Type II systems arrive eventually at a stable or metastable
equilibrium. Type I are those that take substantially less than the Hubble
time, while Type II take at least the Hubble time to come to equilibrium, so
that the expansion of the Universe may influence their evolution.

But we are not yet done. For there are interacting thermodynamic systems that
seem to never come to equilibrium, stable or metastable. That is, they do not
reach an equilibrium state so long there is a slow steady flow of energy,
which, as we shall see, is a requirement for this sort of systems to
evolve.666Our definition includes the caveat ‘interacting’ in order to exclude
systems that do not evolve into complex bound states, but fail to ever reach
equilibrium simply because their interactions are too weak for the particles
to interact any time in the future of an expanding Universe [19, 20]. Without
this, systems such as gravitational waves would fall under Type III. These are
the Type III systems. How are such systems possible?

As the Universe expands it cools below the threshold, $T_{\rm mol}$, where
bound states of atoms become stable. This opens a landscape of possible
molecular states so large that the number of actual states that could be
realized concretely, making use of the materials on the surface of a planet,
or for that matter all the material in the approximately $10^{22}$ solar
systems in our observable Universe, is vastly fewer than the number of
possible states.

At this point ergodicity and equipartition are simply not possible and
equilibrium will never be reached, subject to the presence of a steady flow
from the energy reservoir. These systems are then radically non-ergodic, in
that the vast bulk of their state spaces will never be explored or realized,
even were all the baryons in the Universe available for many orders of
magnitude of Hubble times. These systems will never be in equipartition and
they will never approach an equality of time and ensemble averages.

Such a system, which is vastly non-ergodic because the Universe cannot in its
whole expanse of space and time populate any more than a tiny fraction of the
space of states, is what we call a Type III system. Their state or
configuration spaces are so vast that in any fixed region, such as the
biosphere of a planet, the number of particles is utterly insufficient to fill
out all but a tiny fraction of the possible bound states.

This implies that, subject to the condition that the system remains in the
situation where the number of possible states vastly exceeds the number of
realizations, there is no reasonable timescale over which it might be
thermalized, provided the slow flow of energy persists. Thus, we can say that
there is no thermalization time.

Biological systems, including those of Earth’s biosphere, are prime examples
of Type III systems, with the Sun providing a slow and steady flow of energy.
Even more, biological systems are to date the only example of Type III we know
of. Alternatively, and perhaps more appropriately, one could consider the
entire Earth–Sun system to be the Type III system, as that combination is then
closer to an isolated system for the purposes of satisfying the second law.
Even then this combination cannot be considered a fully-isolated system as
both Sun and Earth radiate photons, and hence both energy and entropy.
Particularly in the case of Type III dynamics we need to take into account
that the Earth receives very low entropy radiation from the Sun during the day
and radiates higher-entropy photons at night into outer-space. So the
inclusion of the Sun makes clear that there is no paradox of perpetual low-
entropy cycles in Type III systems on Earth. After all our biosphere would be
very short-lived without the Sun.

This does not mean that Type III systems live forever. In the case of the
living organisms, the biochemical systems that sustain them, such as the
ribosomes and other machinery of the cell, are always subject to thermal
fluctuations, as are the delicate chemical bonds themselves. There are many
ways that a Type III system can cease to be Type III — we would say die or
that the whole system becomes extinct. If a Type III system dies it later
reaches the state of equilibrium of a Type I or Type II system.

In the case of the biosphere, the existence of Type III systems is due to the
vast number of possible stable biological molecules. These are template
synthesized according to the genetic code, which can represent many more
possible molecules than can actually exist. Each is a set of bound states, and
hence we have to consider them to add to the dimensionality of Hilbert space.
If we arrange the molecules which exist in Nature by atomic weight, we find
that there are a large number of small- and medium-sized molecules, made up of
the full variety of atoms, according to the laws of chemical bonding. The
Universe is thought to be ergodic up to 500 AMU [21].

But there is a huge landscape or configuration space of biological molecules,
with vastly more states, which are the polypeptides and the nucleic acids.
These only exist because they are made in biological cells, through
specialized and exceedingly-complicated processes. But since life exists we
have to count them as part of Nature.

Essentially, all molecules on Earth larger than a certain size are
biological.777There are a few exceptions, which are however molecular products
of human inventions that would not occur otherwise such as synthetic polymers;
these may also, for present purposes, be classified as biological. There are a
vast number of these due to the fact that they are built as chains of a small
number of interchangeable modules. These are the 4 base pairs which are
components of RNA and DNA and the 20 standard amino acids, chains of which
comprise proteins.

A typical protein has a few hundred to a few thousand amino acids; this gives
a factor of, say, $20^{1000}$ different protein types. This is vastly more
than the entire number of different proteins in the biosphere [22]. It is also
vastly more than the number of baryons in the observable Universe, and indeed
greater than the spacetime volume within our causal horizon in Planck units.
This latter is the maximum number of bit-operations that could be realized in
the parts of the Universe that could causally affect us.

Each of these proteins can be coded for by a segment of RNA or DNA. Moreover
there is no limit to the length of a protein, so there is no upper bound on
the dimension of the Hilbert space of this Type III system. This is the case
both for the Hilbert space of potential bound states and if we limit the
description to the Hilbert spaces of actual bound states. The former increases
dramatically whenever a falling temperature goes over a threshold; the latter
follows, and may increase the dimension of its state space indefinitely,
subject only to the availability of resources.

Of course, the second law has not been suspended when a system is in the non-
ergodic Type III regime, but only temporarily postponed, until the low-entropy
energy reservoir is exhausted. The second law continues to be perfectly valid
without exception when the entire system is considered. For the case of life
on the planet, this means including the Sun in the system (which of course
cannot be excluded when studying the dynamics of living systems). The random
fluctuations of heat and the accumulation of errors mean that a continual
application of work is required to make and repair the long chains of nucleic
and amino acids that keep the system far from ergodic, and hence Type III.
This requires being situated in a non-equilibrium steady-state environment,
where continual flows of energy and cycles of materials are available. When
they no longer are, the system degenerates, and returns to Type I or II, i.e.
dies.

That is, it seems that Type III systems require a source of negative entropy
near or within them, and these sources may be Type II systems, as is the case
of the Sun.

Living organisms are all Type III systems. In addition, the biosphere as a
whole is a vastly non-ergodic Type III system.

As we discussed, the era of fusion-burning stars is expected to last roughly
$10^{13}$ years, after which the stars become cold burnt-out remnants [3]. A
Type III system could thus survive at least this long. But, even after all the
stars have died there remain stores of nuclear potential energy, for example
locked into the white dwarfs. This energy may be released in collisions or
mergers of white dwarf remnants; or as a result of Type I supernovae caused by
the inflow of materials onto a white dwarf. Thus, as Dyson [20] and others [3]
have found, a sufficiently-intelligent and resourceful Type III system might
survive in the Universe indefinitely, by making use of the energy and negative
entropy radiated by these sources.

##  3 How do we explain the phenomena of Type III systems?

We now discuss the kind of methodologies required to understand Type III
systems. Here we will refer mainly to our own planet and its biology, since
the biosphere and subsystems within are the only known examples of Type III,
though our discussion is intended to be valid within a broader context of Type
III phenomenology.

We first affirm that the aim of science is to discover the best explanation
for natural phenomena [23, 24]. By the best we mean the closest to sufficient
reason in Leibniz’s sense, i.e. a good explanation gives us a convincing
account as to why the Universe has feature or property $X$, rather than some
alternative, $X^{\prime}$. The better or best explanation does that more or
most convincingly.

###  3.1 The failure of determinism and reductionism in Type III systems.

Let us start by supposing that the standard physicalist methodologies based on
determinism and reductionism can be applied to the biosphere. Then given some
fine-grained initial conditions we should be able to compute, at least in
principle, the value of any well-defined physical observable. Let us consider,
for every protein $C$, defined as a chain of up to around 1000 amino acids,
the number of molecules $N[C]$ present in the biosphere at time $t$. Indeed,
as this is a macroscopic quantity we expect that it can be considered a
classical observable that has a definite value. So it must be computable from
applying the laws of physics to the initial condition. But $N[C]$ is not
computable, for several closely-related reasons we now discuss.

Let us take an even simpler question: for which amino acid sequences $C$ is
the number, $N[C]$, larger than, say, a million? We can’t predict this either,
for the same reasons.

There are $N_{\rm P}=20^{1000}$ possible proteins, and of these only around
$10^{9}$ are realized at any one time. This is a tiny fraction of those that
could exist. An explanation of the occupation numbers of proteins requires us
to understand what is special about the tiny fraction of possible proteins
that do get to exist.

It is easy to characterize the ones that do exist; these are the proteins that
are coded by the DNA of all the living species at the time $t$. That is, a
protein exists if it is coded in the DNA of a species that has been successful
at finding a niche in which it prospers. These are very, very few compared to
the species that might have existed.

How do we determine which organisms exist? There is an extraordinarily
sensitive dependence on initial conditions. As has often been said, if we ran
evolution again, after the same time we would have a completely different
collection of creatures. How do we know this? There have been five mass
extinctions in the past 550 million years. Each is followed by the emergence
of novel species, genera, and even families. This is accompanied by a rapid
divergence in the proteins that come to be made after each mass extinction.
And each time a different mixture of species proliferates.

Let $\\{P_{\rm poss}\\}$ be the set of possible sets of proteins. For every
time, $t$, let $P_{\rm now}(t)$ be the member of $\\{P_{\rm poss}\\}$
consisting of all protein types represented in the biosphere time $t$. For
each of these we have a number of each kind.

Given $P_{\rm now}(t)$, a complete knowledge of the microscopic laws of
physics, and the locations and velocities of all the molecules, could we
predict the future value of $P_{\rm now}$ for $\Delta t=1,100,10^{6},10^{8}$
etc. years in the future? We don’t think this is plausible, for several
reasons.

First, we should subject our notions of computability to a reality test. It is
known that digital computers do not keep a uniform accuracy; even special-
purpose integrators show large uncontrollable errors after some time [25, 26].
Moreover there are limits on the size and speed of computers that would fit on
Earth or even in the Universe [27].

Additionally, one of the drivers of mutation is radioactive and cosmic-ray
disruption of DNA which happens by quantum chance, so even if we could
integrate from initial conditions specified to arbitrary accuracy, we still
wouldn’t be able to generate a unique prediction of outcome. So there is no
pragmatic — in other words actual physically-embodied — computation that is
accurate enough to realize the fantasy of integrating the initial conditions
of the biosphere forward from initial conditions.

###  3.2 The need for functional explanations in Type III systems

Unlike Type I systems, a complete explanation of a Type III system is not
given when we describe the motions of the atoms in the physical phase space.
The reason is we have to explain why a tiny fraction of the possible proteins
exist in Nature, while most do not. The same goes for nucleic acid sequences
and so on. The small fractions that exist have almost identical physical
properties as the ones that do not. How are we to explain the differences?

It is good to take a minute here to emphasize that as scientists, we have
stronger commitments than we do to the principles of reductionism and
determinism. These include the more basic commitment to finding the truth.
Each of these might fail and science will go on, so long as we continue to
fulfil our quest as scientists which is, as Deutsch reminds us [24], to always
seek the best explanation for every phenomenon in Nature.

Following Deutsch (and Leibniz [23]), the best explanations are both
sufficient and complete. They suffice to explain why $P$ rather than NOT$-P$,
and nothing more is needed to explain either why $P$ or NOT$-P$.

We have no reason to challenge the assumption that the basic laws of physics
govern how Type III systems evolve in time. But it is also the case that they
do not give an explanation which is sufficient to genuinely explain the fact
that certain proteins exist in the biosphere, while most do not.

The problem is that if the laws plus certain initial conditions explain why
$P$ exists, but $Q$ does not, there are many very small changes in the initial
conditions that, when evolved by the same laws, will lead to the opposite
conclusions. This makes it clear that even if we had infinite precision,
evolution of the initial conditions by the laws cannot give by themselves
either a complete nor a sufficient explanation for the outcomes [25, 26, 28].

What kinds of explanations suffice? There is only one kind of explanation that
can fill the gap: functional explanations [30, 31, 32, 33, 22].

What is a functional explanation? Let us consider an example and work towards
a general definition. We want to explain why the molecule $P$ exists in the
biosphere, while almost no others do. We propose that a sufficient reason has
two parts: 1) $P$ is in the class of molecules that can, according to the laws
of physics, be made by a cell, 2) $P$ codes for an enzyme that catalyzes or
regulates a chemical reaction that contributes to the survival and fitness of
a larger organism it is a part of.

Both assertions are necessary. The first is reductionist and ”bottom up”, the
second is ”top down” and refers to a function.

A functional explanation for why some process or component $X$ exists, is one
where the reason $X$ exists is because $X$ contributes to the survival or
well-being of a larger system. What $X$ does to ensure the well-being of $S$
is its function (in the context of $S$). The point is that explanations in
terms of the function of a part or a subsystem play a necessary role in
explanations in Type III systems, of which biological systems are the most
important example [22].

Several different ideas are wrapped up in this definition of a function. A
function is a process, carried out in time. We can talk of an organism, a cell
or an animal, as a bundle of processes — many subprocesses coordinated and
tied together. The life of a protein, from formation in the ribosome to its
disassembly, is a subprocess within an organism that carries out a function
which contributes to the success of the whole.

This is a functional explanation. Note that functional explanations are needed
for giving sufficient explanations for many features of Type III systems. This
is because only in Type III systems is there such a radical non-ergodicity
that much of what has to be explained is why a structure like an enzyme
exists, out of vast numbers that don’t.

In each case the explanation is given in terms of the function of the part in
a whole system. Consider this question: why are there hearts in the biosphere?
Or why is there a particular kind of enzymes? Reductionism plays a role, since
some of the explanation for how the properties of the part — the heart or the
enzyme — contribute to the well-being or survival of the larger organism or
system, is because the component is made of atoms which follow the basic laws
of physics.

But the opposite is also part of the explanation; the heart exists because the
cat exists. This is what has been called a Kantian Whole [29, 22, 34]. The
point is that explanations in terms of the function of a part or a subsystem
play a necessary role in explanations in biological systems. Functional
explanations rely on what are called downward causations.

Instances of downwards causation, which are all around us, appear to some
reductionists to be illicit. But they misunderstand reductionism to mean that
the explanatory arrows must always point one way; from the microscopic laws
upward to properties of macroscopic systems [35]. However, in an evolutionary
context which gives precise, historical explanations for how functions and the
subprocesses that carry them out arise, the common-sense idea of reductionism
is perfectly compatible with both functional explanations and downward
causation. This is because a necessary aspect of functional explanations is
that the larger organism thrives because the function in question contributes
to the organism’s survival and fitness [10]. We can’t emphasize this basic
point enough: in biology there is both downwards and upwards causation.
Reductionist explanations work hand in hand with functional explanations: both
are present, and necessarily so.

Indeed, stable and metastable far-from-equilibrium structures are stabilized,
as shown by Morowitz’s theorem [9], because the flows of material and energy
develop closed loops, around which they cycle, controlled by feedback
processes. These cycles, as well as the feedback loops that stabilize them,
combine processes which are the result of downwards causation with processes
which are the result of upwards causation.

Now for each such loop, there is a rate of flow around it of energy and the
various materials involved. These are concrete examples of new degrees of
freedom which emerge in the special circumstances where one has subsystems
such as hearts and enzymes that exist because they contribute to the thriving
of a larger Kantian whole.

##  4 What is different about Type III systems?

###  4.1 Extreme non-ergodicity and its consequences

How do we define statistical physics for Type III? We easily see that the
usual methods of equilibrium statistical mechanics don’t apply. There is no
ensemble average because anytime you start the system off in even an
infinitesimally-different initial state, the evolution through the space of
states or configuration space will be radically different. There is no time
average, for the same reason. There is no timescale over which a time average
will yield repeatable results. The statistical description of a classical
system $S$ must be given, as in Types I and II, in a space of possible
microscopic configurations ${\cal P}$ in which the configurations ${\cal C}$
live. In Type I there is a static and uniform probability distribution
$\rho[{\cal C}]$. In Type II the system evolves towards such an equipartition
over a timescale longer than the Hubble time. In Type III the set of possible
configurations is so vast, it dwarves the number of actual realizations of
these possible configurations. The dynamics is chaotic so that nearby initial
conditions diverge. It is not possible to define or predict a single
probability distribution function.

The basic issue is that existence is a very rare property of any particular
possible configuration or subsystem. As a consequence a new methodology is
needed which is centered on a new kind of explanation, which we argued in
Subsection 3.2 is functional explanation. This is required to explain why
those very few that exist, do, while many others do not.

The formal definition of a function goes like this. We work in a Darwinian
framework in which a species has a fitness, which is proportional to the
chance that it is able to reproduce itself. We define function in terms of
Kantian Wholes [22, 34].

In a Kantian Whole, the Parts exist in the Universe for and by means of the
Whole. All living organisms are Kantian Wholes. You exist in the Universe for
and by means of your heart, liver, kidney. These parts exist in the Universe
for and by means of you, the Whole. The function of a Part is that subset of
its causal properties that sustain the Whole. The function of the heart is to
pump blood, not to make heart sounds or jiggle fluid in the pericardial sac.
The Parts of Kantian Wholes evolve to create ever-new functions that better
sustain the Whole, and they are passed along in evolution by propagation of
the Whole.

More precisely, in a Kantian Whole:

  1. 1.

The occupied states or configurations of a Type III system may be
characterized as follows: they contain inter-connected bundles of
subprocesses, ${\cal K}$, which are connected in the sense that many of the
outputs of the subprocesses go into inputs of other subprocesses.

  2. 2.

These subprocesses can perform tasks and functions.

  3. 3.

A task is a change in the states of subprocesses which requires work, chemical
and/or physical.

  4. 4.

A function is a more general alteration of ${\cal K}$, its subprocesses or
environment, that contributes to the fitness of the Whole, ${\cal K}$.

  5. 5.

A bundle of subprocesses ${\cal K}$ is then called a Kantian Whole, in the
case that many of its sub-processes, ${\cal P}$, perform functions or tasks
that increase or contribute to the fitness of the Whole ${\cal K}$. Each of
these ${\cal P}$ exists because it is part of, and contributes to, the fitness
of ${\cal K}$, which, due to the contributions of all its subprocesses,
thrives. Thus there is a symbiosis among the many subprocesses, $P_{I}$ (with
$I=1,2,\ldots$) that make up ${\cal K}$, which exist because by their combined
contributions they make ${\cal K}$ viable.

  6. 6.

The subprocesses exist in the Universe because they make the Kantian Whole
viable in its world. The Parts exist in the Universe for and by means of the
Whole.

Kantian Wholes can be arranged hierarchically. A given subprocess may be part
of more than one Kantian Whole at the same time, and the Kantian Whole it is
part of may be different at different times.

The existence of Kantian Wholes implies a radical change in how we understand
and explain Nature. Once we have a Kantian Whole it makes sense to say that a
process, event or alteration of its environment, is “good for $\cal K$”, which
means “increases ${\cal K}$’s fitness or survival.”

###  4.2 Type III systems can be recursive or excursive and also universal

Note that a Type III system can realize a function which is recursive, or
self-referential. That means it refers to the larger Kantian Whole, $\cal K$,
to perform a task that increases the fitness of the Whole, $\cal K$.

It does this because it has a code that provides a complete mapping from all
possible chains of nucleic acids to all possible catalytic functionalities. To
see why, note that there has been estimated to be roughly $10^{8}$ different
shapes or catalytic tasks which a set of enzymes would need to cover to be
universal [10].

A coding is a process that allows $\cal K$ to store information about its own
structure. This involves an abstraction, in the sense that a sequence of
letters made of nucleic acids on a strand of DNA can represent a sequence of
amino acids. A code is an abstraction if the function that reads and writes
information to a subcomponent involves an arbitrary map, which gives a
representation of, but is not an instantiation of, that structure.

At this point something very interesting happens. The same coding that gives
the information needed to construct itself can be altered to code any of a
vast space of alternatives. That is, the organism in its construction
explicitly refers to itself (through the coding) and implicitly it refers to a
vast set of adjacent possibilities. The first is recursive, we call the second
excursive.

###  4.3 The three arrows of dynamics

If one takes a Type I system a bit out of equilibrium and lets it evolve from
there, an arrow of dynamics emerges which reflects that it is more likely that
entropy will increase until the system returns to equilibrium, after which the
system fluctuates around equilibrium and there is no longer an arrow of
dynamics. Nor was the arrow of dynamics, while it existed, coupled to
cosmological degrees of freedom.

We will call this a Type I arrow of dynamics. It is simply the manifestation
of the tautology that the most likely outcome of evolution of a closed system
is to one of the most probable macro-state. As Paul and Tatiana Ehrenfest
showed in 1909 [36] this statement is time-reversal symmetric; it is equally
likely for the entropy to have been higher in the past as it will be to be
higher in the future.

In Type II systems, the return to equilibrium can take cosmological
timescales, for the reasons discussed above. The result is an arrow of
dynamics that is present from the Big Bang to at least three orders of
magnitude past the present Hubble time. This second arrow of dynamics might be
defined in terms of the lifetime of the Sun or a red or brown dwarf.

But the Type III systems in the Universe have the property that they don’t
reach equilibrium, subject to the existence of a source of negative entropy
such as the Sun. Hence their search for the most probable state defines a
third kind of arrow dynamics that can continue indefinitely, or at least for
as long as they can find resources.

The existence of an arrow of dynamics that is not reversible by a finite
process raises some interesting questions. Is this compatible with fundamental
laws that are reversible, or does this require that the fundamental laws are
time-irreversible?

Finally, we briefly note there is in Nature another system which would have
fallen under Type III, in the sense that it never approaches equilibrium or
equipartition and hence is highly non-ergodic, were it not for our caveat
restricting Type III to interacting systems. This is gravitational radiation
[19], both classical and quantum. However, a defining feature of Type III
systems, and their main reason of interest, is their thriving in dynamical
cycles within pockets of phase space with very low entropy, the goal being the
study of the complexity that results from that. Gravitational waves do not
satisfy these criteria since they do not form complex bound states as living
systems do, hence are less attractive from the point of view of this
study.888Cosmic photons and neutrinos also behave as non-interacting systems
after their respective decouplings, in that any subsequent energy injection
never re-thermalises, though they differ from gravitational waves in having
previously been in thermal equilibrium with other particles.

##  5 Proposal for a fourth law of thermodynamics

There have been several attempts to formulate a fourth law of thermodynamics,
which would characterize how and why steady-state far-from-equilibrium states
are able to self organize, in a way that at first look suggests that the
second law is being transcended. These attempts were formulated mainly for
what we call here Type II systems. From the perspective we present here, the
irreversibility in these systems is not due to a transcendence of the second
law, but rather to a channelling of negative-entropy flows to delay, sometimes
by very long times (as long the slow steady source of energy is active), the
approach to equilibrium.

We would like, however, to propose that Type III systems offer a new
perspective on irreversibility, which opens up the possibility for a fourth
law which is not a re-channeling of the second.

Let us consider a Type III system, $\cal T$, perhaps a biological organism or
the whole biosphere. As discussed above, a Type III system has subsystems
which carry out a number of functions for $\cal T$. Let $F_{\rm A}$ stand for
the number of distinct, actual functions being used by $\cal T$ at a given
time. We want to distinguish the actual functions from the set of possible
functions, which includes those functions that $\cal T$ already has the
possibility of expressing by presently-unexpressed codons, or which could be
created in a few steps of mutations or transmutations of the DNA. $F_{\rm P}$
is the number of possible functions which the system might make use of.

Define

$R=\frac{F_{\rm P}}{F_{\rm A}}$ (4)

The fourth law we conjecture states that for any Type III system, as long as
the non-equilibrium conditions persist which allows the Type III system to
maintain its main functions (see below), then

$F_{\rm P},F_{\rm A}\mbox{ and}\;R\mbox{ all have a tendency to increase.}$
(5)

In fact, in the one case of Type III systems we have so far treated
numerically [1], we have found that the ratio $R$ does not merely increase. It
does so at such an unprecedented rate of growth that we needed to resort to
the mathematical operation of tetration (involving, at each step, an
exponentiation of the argument of the function) to be able to describe the
numbers encountered for the increase of phase space, or in this case $F_{\rm
P}$. While both increase in Type III systems, the difference between $F_{\rm
P}$ and $F_{\rm A}$ is staggering for the example we studied.

We want to emphasise here that the fourth law is by no means a violation of
the second law. If we include the source and outflow of energy as part of the
system, as needs be for a consistent treatment, it becomes clear that the
total entropy of the system is always increasing. What Type III systems are
doing is rather exploring pockets of phase space of exceptionally-low entropy,
which they can reach via the exchange of energy with exceptionally-low entropy
from the energy source. By exploiting this almost unlimited source of low
entropy — e.g. the Sun — Type III systems can persist almost indefinitely in a
suspension of a low-entropy cycle, subject to the availability of that source.

##  6 A definition of life

We consider how a definition of Type III systems may augment our familiar
definitions of life. These familiar definitions include “an open non-
equilibrium thermodynamic molecular system capable of self-reproduction and
evolution”. This minimal definition is often augmented by the concepts of: i.
Replicating genetic material, ii. A metabolism, iii. A bounding membrane.
Further augmentations include that of thermodynamic work and agency [37], and
constraint closure [38] by which cells do thermodynamic work to construct
themselves.

These features ensure the existence of Kantian Wholes that can evolve and
construct ever-new functions in the Universe. The construction of such novelty
requires the non-ergodicity extant in Type III systems.

Note next that the maintenance of a Kantian Whole requires functions which
support identity, such as controlling the exchanges of materials as well as
the ion concentrations on each side of its membrane. Since these involve
moving charges or other molecules against gradients, each of these functions
requires work. Hence there must be a flow of energy through the system.

Work is the constrained release of energy into a few degrees of freedom [39].
A cannon, cannon ball, and exploding powder are examples. The cannon is the
constraint, the boundary condition. No constraint, no work. But work can
construct constraints [37]. In a constraint-closed system [38], the set of
constraints constrain the release of energy in a set of non-equilibrium
processes that construct the very same set of constraints. We construct our
artifacts, while cells construct themselves. By heritable variation and
natural selection, cells continually evolve to construct boundary conditions
that are new to the Universe [22, 34]. The new boundary conditions create new
phase spaces, and hence new possibilities in the evolution of the Universe
[22, 34]. This is the behavior of evolving life in Type III systems.

Then our definition of life is:

> A living organism is a Kantian Whole within a Type III system that is a non-
> equilbrium self-reproducing system with a metabolism, an identity and a
> boundary, and which is capable of open-ended evolution by heritable
> variation and selection or drift.

##  7 Summary

Ever since Schrödinger raised the question of What is life? [40] it has been
evident that statistical thermodynamics is the right setting from which to
approach an understanding of life in the context of fundamental laws and
principles. It has long been suspected that non-equilibrium statistical
mechanics should provide a general understanding of pattern formation and
self-organized systems, which would underlie biology [8, 9, 10, 11].

Thermodynamics also plays a central role in cosmology. It is then natural to
approach the question of why the Universe is hospitable to life using the
tools and concepts of thermodynamics. Here we uncover some conceptual and
physical distinctions that turn out to be crucial if we want to discuss how
biology fits into cosmology from a physics perspective.

The first is that we have to distinguish three kinds of statistical systems.
The first two come eventually to thermal equilibrium, but we distinguish them
depending on the ratio of the thermalization time to the Hubble timescale:

$r=t_{\rm thermal}H$ (6)

We know how this plays a role in nucleosynthesis, but it turns out to have a
more general significance.

At one end of the scale, for small $r$, are the ordinary equilibrium
thermodynamic systems, which satisfy the familiar principles of equipartition
and the equality of time and ensemble averages. These apply universally on
small scales as well as in the early Universe, when the temperature is so high
that it dominates over the other mass and energy scales. We call these Type I
systems. They have no coupling to cosmology because the time required by them
to thermalize is much less than the Hubble scale.

Type II systems are those whose thermalization times are on the order of or
larger than the Hubble time. This is the intermediate case for which

$r\gtrsim 1\,.$ (7)

These are surprisingly common and include many of the main-sequence stars and
disk galaxies. These have been somewhat well studied and their role in setting
up steady-state non-equilibrium sources of negative entropy that make life
possible have been much discussed, as summarised in Ref. [6].

Lastly, here we introduced a new third class of statistical-mechanical system,
which we call Type III systems. These are cases for which the rate of
expansion and addition of novel states to Hilbert space is so vast and
explosive that we cannot foresee the expression of all the states in that
Hilbert space within a time of order of the finite lifetime of the Universe.
An alternate characterization of a Type III system is that it has at least one
subsystem or component, $Q$, that comes in a vast number of alternative
versions, $N_{Q}$, which are continually added onto, and are roughly
equivalent energetically, and where

$N_{Q}\gg N_{\rm p}.$ (8)

where $N_{\rm p}\sim 10^{80}$ is the number of protons within the Hubble
distance. This implies a startling difference from equilibrium systems, of
Type I, for which the equality of time and ensemble averages quickly obtains.

As a consequence of Eq. (8), in Type III systems there is not enough time,
space, nor material in the Universe for more than a tiny fraction of the
allowed possible states of those subsystems ever to be realized. There are
such a vast number of states within narrow energy ranges that equipartition or
equality of time and ensemble averages will never be achieved or relevant,
while the system is still of Type III, that is, while it still ‘alive’. If the
system dies it will cease to be of Type III and becomes Type I or II and will
also most likely ‘die’ to a state of equilibrium. The recognition that Type
III systems are very different than others thermodynamically is essential for
understanding why there is life in the Universe.

An important example is the possible proteins, which number something like

$N_{\rm protein}\sim 20^{1000}\gg N_{\rm p}\sim 10^{80}\,.$ (9)

Only a small fraction, perhaps $10^{9}$ of these, are realized in our
biosphere

It is even the case that the number of possible proteins is much greater than
$10^{120}$ so even at a rate of expression of one protein per Planck volume
per Planck time, the vast majority of proteins would never be expressed.

Thus, simply existing is a very rare property of the states of these
subsystems. Consequently, as we discussed in detail above, we require an
adequate explanation of why some states do exist, while almost all the others
do not. As we explained, the narrow pure-reductionist style of explanation
does not suffice to give a complete explanation to that type of question. One
problem is that the effects of noise, thermal and otherwise, would overwhelm
any attempt to answer this kind of question by integrating initial data, which
is itself going to be noisy. If we continue to search, nonetheless, for a
better account of such a why me and not them? kind of question, it must
involve a new kind of explanation. We proposed above that this must be a mix
of reductionist and functional explanations.

Very briefly, in the context of Darwinian natural selection, a functional
explanation as to why some particular molecule or protein, $\alpha$-bazoodle,
exists in the biosphere, is that it exists partly because it contributes to
the viability or fitness of a larger system $\cal K$, of which it is a part.
In Ref. [1] we characterised this condition of viability for existence with
the expression “The name of the game is getting to exist”. Our
$\alpha$-bazoodle does this by performing a function, which is a sequence of
operations during which some chemical or mechanical work is done on $\cal K$
or its environment, the result of which is to increase the viability or
fitness of $\cal K$. Of course, the explanation for the existence of
$\alpha$-bazoodle, must also include an argument that the laws of physics are
compatible with its both existing and performing the function it does for
$\cal K$. Consequently, any adequate explanation for the existence of a sub-
system or sub-process in a Type III system must involve a combination of
bottom-up reductionist explanations and top-down functional explanations.

To spell this out:

  * •

Functional explanations are necessary to give a satisfactory explanation for
many questions in Type III systems including our biosphere. This is especially
true when we wish to understand living systems from a physics point of view,
which we began to do in the research programme we launched in Ref. [1] and
continue here.

  * •

Functional explanations rely on knowledge of the basic physical laws that
govern the components and processes of an organism. In this sense, functional
explanations are compatible with physicalism.

  * •

But functional explanations answer questions that purely reductionist
explanations cannot, because they rely on downward causation, e.g. “hearts
exist because they perform a function of pumping blood, which is necessary for
animal life.”

  * •

Thus, functional explanations, which rely on specific instances of downward
causation, complete explanations that naïve reductionism leaves incomplete.

  * •

Living things, and the biosphere as a whole, are sustained by stable cycles of
energy and materials, controlled by feedback. These cycles combine functions
which express downwards causation with ordinary upwards causation.

Thus, we can say that the reductionist account is valid, but does not give a
complete explanation of how and why these systems come to exist. We may say
the same for the purely-functional explanation. Both are needed and indeed,
both are true. Nor is this a case of complementarity, for they are perfectly
consistent with each other. We provide a complete summary of the arguments in
this paper in Table 1.

|  Type I |  Type II |  Type III
---|---|---|---
Ergodicity: |  Ergodic. |  Ergodic and well mixed on times of order the Hubble time and only when taking into account constraints. |  Highly non-ergodic, never returns, to exist is rare, over arbitrarily long times only a very small portion of the state space is sampled.
Methodology: |  Reductionism and upward causation are sufficient. |  Reductionism and upwards causation have to be supplemented by downwards causation because constraints and channels lead to feedback and cycles. |  We must admit functional explanation, which is a form of downward causation. This goes hand in hand with a weak form of reductionism.
Ensemble type: |  All of state or Hilbert space. |  Dominated by capture by limit cycles. |  Dominated by capture by Kantian Wholes.
Sensitivity to small changes in initial conditions: |  None, quickly averages away. |  Moderate. |  Extreme, can result in a very different biosphere or none at all.
Equality of time and ensemble averages respecting constraints: |  Yes. |  Only on much greater than Hubble times. |  Never.
Correlations of fluctuations: |  Gaussian. |  Gaussian over very long timescales, and subject to constraints. |  Small fluctuations can produce arbitrarily large deviations.
Emergent phenomena: |  None. |  Steady-state non-equilibrium systems, dissipative structures, cycles (Morowitz cycle theorem), limit cycles. |  Kantian Wholes, biological cells, the biosphere.
Characterized by: |  Boltzmann, Maxwell, Gibbs. |  Prigogine, Morowitz, etc. |  Darwin.
What we count: |  All possible microstates, consistent with macrostate. |  All possible microstates, consistent with macrostate and constraints. |  Possible novel functions of possible Kantian Wholes.
How to count states: |  Equilibrium thermodynamics. |  Non-equilibrium thermodynamics, subject to constraints. |  Use TAP equation to count functions.
Table 1:  The complete argument. The TAP (‘Theory of the Adjacent Possible’)
equation [41, 42] is a model for the growth of systems that are defined by
functions, either intrinsic or extrinsic, and can grow by combinations, such
as inventions or patents. It is described in detail in Ref. [1].

We turn, briefly, to the payoffs for science of adopting this kind of
methodology, which gives functional explanations a necessary role alongside
physicalist explanations.

The first payoff is a new kind of definition of life. We introduced it briefly
here and will return to it in future work. We believe that no purely
reductionist characterization of life will succeed and that every useful
characterization of life will involve a mixture of physicalist and functional
criteria.

The second payoff is a proposal for a fourth law of thermodynamics, which we
presented briefly in Section 5.

If biology requires this hybrid of reductionist and functional explanations to
provide adequate explanations from a physics perspective, and if the science
of cosmology is going to fully explain all that goes on in the Universe thus
including biology — an assertion we made in Ref. [1] — then this kind of mixed
functionalist explanation must be part of the methodology of cosmology.
Reductionism is required to understand the microscopic degrees of freedom of
living Type III systems and functional explanation are required to explain the
complex, structured degrees of freedom.

Some readers may reply that the argument we are tracing here can be disrupted
by assuming that the multiverse exists, and that within it, there are enough
copies of a Type III system that we can apply to it a usual microcanonical
argument. We could dispute this, but we don’t need to, because this argument
just proves our point that the kind of methodology and explanation we can
apply to such systems depends on assumptions we make about the Universe at
large. In other words, there is a coupling between cosmology and biology.

The existence in the Universe of systems whose thermalization times, even when
subject to constraints, are much larger than the Hubble time, is going to have
to be significant for the thermodynamics and evolution of the Universe. And
the full description of such systems must take into account cosmology, because
they exist for timescales on which the Universe evolves significantly. Thus we
see there is a need for a hybrid of biology and cosmology, which we call
biocosmology.

## Acknowledgements

We are grateful to Stephon Alexander, Barbara Drossel, George Ellis, Carlo
Rovelli, and Simon Portegies Zwart for helpful discussions. This research was
supported in part by Perimeter Institute for Theoretical Physics. Research at
Perimeter Institute is supported by the Government of Canada through Industry
Canada and by the Province of Ontario through the Ministry of Research and
Innovation. This research was also partly supported by grants from NSERC and
FQXi. This work was supported by the Fundação para a Ciência e a Tecnologia
(FCT) through the research grants UIDB/04434/2020 and UIDP/04434/2020. M.C.
acknowledges support from the FCT through the grant
SFRH/BPD/111010/2015 and the Investigador FCT Contract No. CEECIND/02581/2018
and POPH/FSE (EC). A.R.L. acknowledges support from the FCT through the
Investigador FCT Contract No. CEECIND/02854/2017 and POPH/FSE (EC). M.C. and
A.R.L. are supported by the FCT through the research project EXPL/FIS-
AST/1418/2021. We are especially thankful to the John Templeton Foundation for
their generous support of this project.

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