The Science of Scale — Greg Egan

2023-03-29 06:19:37

The Science of Scale

The Lightest Mass Units the Scale

What’s it that determines the dimensions of an atom? A hydrogen atom consists of 1 proton and one electron, with equal and reverse electrical cost, however the proton is way heavier than the electron: its mass is about 1836 instances larger. Each
particles obey the identical legal guidelines of quantum mechanics, however the quantum-mechanical wave perform that lets us calculate the chance of
discovering a particle at a given location is rather more unfold out for the electron than it’s for the proton. So whereas the proton accommodates many of the mass of the atom, the electron takes up many of the area that the atom occupies.

There’s a small likelihood of discovering an electron at any distance in any respect from the proton, so we will’t ask for the radius of a sphere by which the electron is certainly contained. Nonetheless, there’s a easy size scale we will use as an alternative,
referred to as the Bohr radius. Within the early historical past of
quantum mechanics, this was conceived of because the radius of a round orbit by which an electron travelled round
the nucleus, however whereas that image of atoms was simplistic, the Bohr radius nonetheless turned out to be a helpful
amount in atomic physics. In trendy phrases, it’s roughly equal to the almost certainly distance between the
electron and the proton.

The picture on the correct plots the chance of discovering the
electron at varied distances from the proton in a hydrogen atom, the place distance is proven as a fraction of the Bohr radius. Be aware that the common distance of the electron is 50% larger than essentially the most
seemingly distance, due to the lengthy tail of chance that stretches out to larger distances.

The Bohr radius is often given the image a₀, and its worth is:

a₀ = ε₀ h² / (π e² m_e)

The fixed ε₀ right here measures a facet of the electrostatic area referred to as the permittivity of the vacuum, whereas h is Planck’s constant, e is the electric charge on a proton or electron, and m_e is the mass of an electron. If we insert the values for these portions into this method, we get:

a₀ ≈ 5.291 × 10^–11 m

If it strikes you as a bit unusual that the mass of the proton doesn’t seem in any respect on this method,
you’re proper, as a result of the primary purpose the Bohr radius is just roughly equal to the almost certainly distance
between the electron and the proton is that the usual definition
of a₀ comes from treating the proton as
being a lot heavier than the electron that it stays fully mounted. However simply because the Solar doesn’t keep completely mounted whereas the Earth and
different planets orbit round it, neither does an atomic nucleus. And in each the celestial mechanics of
two orbiting our bodies, and the quantum mechanics of a hydrogen atom, the best technique to regulate our calculations
to account for that is through the use of what is named the reduced mass. Each time two our bodies are topic to equal and
reverse forces, if we analyse their movement relative to one another — moderately than their particular person trajectories in area — the arithmetic of the issue seems to be equal to that
of a single physique whose mass is the same as the lowered mass, shifting below the affect of the unique drive round an immovable centre of attraction.

For a
hydrogen atom, the lowered mass is:

m_lowered = (m_e m_p) / (m_e + m_p)

the place m_p is the mass of the proton. As a result of m_p/m_e ≈ 1836, we’ve got:

m_lowered = m_e / (m_e/m_p + 1)
≈ (1836/1837) m_e
≈ 0.99946 m_e

So, within the case of a hydrogen atom, the distinction we
get from utilizing m_lowered rather than
m_e is pretty small. This isn’t to say that physicists don’t care about
the distinction; the properties of hydrogen atoms have been calculated and measured with far
larger precision than this. However for the sake of constructing the Bohr radius a sort of normal yardstick for
atomic phenomena, moderately than a selected, exactly measured property of hydrogen atoms themselves,
it’s outlined by the method we’ve given, with no reference to the mass of any particle different
than the electron.

The identical method that provides the Bohr radius additionally yields helpful distances for different atomic-scale techniques if we make some easy adjustments. If, as an alternative of a hydrogen nucleus with a single optimistic cost, we’ve got a single
electron orbiting a bigger nucleus that accommodates Z protons, then making the
substitution e² → Z e², which quantities to dividing a₀ by Z,
offers the almost certainly distance of the only electron from that bigger nucleus. For instance,
if we take a helium atom, with two protons and two neutrons within the nucleus
and two electrons, and strip away one of many electrons,
the remaining electron will probably be almost certainly to be discovered at
a distance of roughly a₀/2
from the nucleus. In different phrases, the optimistic ion He⁺ is about half the dimensions of a hydrogen atom.

One other change we will make is to exchange m_e with the mass of a distinct negatively charged particle. There’s a particle
referred to as a muon, which is analogous to
an electron in lots of respects; it has the identical electrical cost, nevertheless it has a bigger mass, m_μ ≈ 207 m_e. In contrast to the electron, it will possibly decay into different
particles, and it has a imply lifetime of about 2.2 microseconds. Nonetheless, that’s lengthy sufficient for experimenters
to create atoms which comprise muons as an alternative of electrons, and
to measure a few of their properties.

As a result of the muon isn’t as gentle as an electron, the lowered mass:

(m_μ m_p) / (m_μ + m_p)

is considerably
totally different than the mass of the muon itself:

(m_μ m_p) / (m_μ + m_p)
≈ 207 × 1836 m_e / (207 + 1836)
≈ 186 m_e

So an atom of muonic hydrogen is about 186 instances smaller than an atom
of unusual hydrogen, moderately than 207 instances smaller.

Other than the change in measurement, how is muonic hydrogen totally different? Each time optimistic and adverse electrical expenses are introduced collectively, the system has a sure potential energy: the quantity of power it might take to separate the person expenses, pulling them aside towards the enticing drive they expertise. For 2 pointlike particles, this potential
power is proportional to 1/r, the place r
is the gap between them. It additionally is dependent upon the
quantity of electrical cost, however since that’s precisely the identical
in each the unusual hydrogen atom and the muonic
hydrogen atom, the power required to drag a muonic hydrogen atom
aside is about 186 instances larger than for the unusual atom.

This issue of 186 additionally exhibits up within the spectrum of
muonic hydrogen: the set of frequencies of sunshine [and other electromagnetic radiation] which can be absorbed or emitted
when the atom adjustments its quantum power degree. Photons of sunshine absorbed or emitted in these transitions
have frequencies which can be proportional to the distinction in power between the degrees, and since all these
energies are 186 instances larger, all of the frequencies are 186 instances larger too.

Rising the frequency of a wave of sunshine by an element of 186 means making the interval of the wave — the time it
takes to finish one cycle — smaller by an element of 186. Any clock that relied on transitions between electron states in an unusual hydrogen atom would run 186 instances quicker if it was transformed to muonic hydrogen.

This illustrates a really normal precept that comes from the fundamental properties of quantum wave capabilities.
The power of a particle is proportional to the frequency of its quantum wave perform (the variety of cycles the wave matches right into a given time), and the momentum
is proportional to the spatial frequency (the variety of cycles the wave matches right into a given distance). However the power and momentum are each proportional to the mass as nicely, so each the interval of the wave, and the wavelength, should be
inversely proportional to the mass.

Which means if all the plenty in a quantum system have been modified by the identical issue, then all distances and instances associated to the dimensions and behavior of the system can be lowered by precisely the identical proportion.
After all once we swap a muon for an electron in an atom, we’re not fairly doing that: the mass of
the nucleus stays the identical.

Other than the electron and the muon, there’s one other comparable particle referred to as the tau lepton.
That is 3477 instances heavier than an electron, and has a a lot shorter lifetime than a muon, round
3 × 10^–13 seconds. The time period lepton comes from the Greek for “small”, however tau leptons are
extra huge than protons.
In our universe, there are (virtually definitely) simply three generations of elementary particles, together with these three leptons.

Within the novel Scale, nonetheless, the premise is that there are eight leptons, with plenty which can be all powers of two instances the mass of the lightest one: m₀, 2 m₀, 4 m₀, 8 m₀, 16 m₀, 32 m₀, 64 m₀, 128 m₀.
What’s extra, all of them have lifetimes for much longer than the present age of the universe.
We’ll name these particles e₀, e₁, … e₇,
so as of accelerating mass.

What can we are saying about atoms that comprise a number of totally different sorts of leptons, with totally different plenty? In our personal universe, individuals have
shaped atoms that comprise each muons and electrons, so this isn’t only a hypothetical query.
In a helium atom with one muon and one electron, the muon is positioned a lot nearer to the nucleus
than the electron, for 2 causes: first, as a result of it’s extra huge, but additionally as a result of it’s uncovered to the
full cost of the nucleus (+2e), whereas the electron spends most of its time far sufficient away from
each the nucleus and the negatively-charged muon (–e) that the muon successfully “screens” a considerable
a part of the nucleus’s cost. This leaves the electron feeling one thing nearer to the attraction of the
mixed cost of twoe – e = e, so whereas the muon’s location is roughly that
of the muonic Bohr radius with the additional discount that comes from dividing by Z=2, the electron’s location is roughly that of the unusual Bohr radius for the hydrogen atom.

Within the universe of Scale, an electrically impartial atom may comprise any combination of leptons of various plenty, as long as their complete quantity, and therefore the whole adverse cost, balances the optimistic cost of the nucleus. However whereas an atom like this is usually a lot extra sophisticated than the muon/electron helium atom, the identical normal precept applies. The general scale of the atom will probably be decided by the mass of the lightest leptons that the atom accommodates, the variety of such leptons, and the screened cost of the nucleus: its precise optimistic cost, plus the adverse cost of the heavier leptons that sit nearer to the nucleus than the lightest ones.

The picture on the correct exhibits the chance of discovering e₀ and e₁ leptons at varied distances from a shared helium nucleus. As a result of they solely differ
in mass by an element of two, the heavier lepton’s wave perform isn’t as dramatically smaller
as a muon’s, however the screening of the nuclear cost continues to be sufficient that the lighter lepton does
not expertise the complete cost of Z=2, and so its wave perform (blue) is just about 20% smaller than that of hydrogen. Alternatively, the wave perform for the heavier lepton (purple), is smaller
by roughly an element of 4, because of each the attraction of [almost] the complete Z=2 cost of the helium nucleus, and the lepton’s mass being twice as massive.

Because the variety of every sort of lepton within the atom will increase, they are going to refill the
orbitals accessible to them: these are the totally different wave capabilities for particles shifting across the atom, which differ of their
power and angular momentum quantum numbers. Leptons are fermions, which implies no two of them (of a given variety) can occupy the identical quantum state, and every orbital can solely be occupied by a most of two leptons, which distinguish themselves by having totally different instructions of spin. The chemical properties of an unusual atom are largely decided by the variety of electrons, and the ensuing sample of stuffed and partly empty shells. In an atom with a number of totally different sorts of leptons, it is going to be the variety of the
lightest leptons, and the sample of shells they occupy, that’s the dominant affect on its chemical properties. Nonetheless, in contrast to the electron-only case, the place the cost of the nucleus fixes the
variety of electrons, the variety of the lightest leptons will probably be set by each the cost of the nucleus and
the variety of different leptons. So whereas in our universe, every chemical aspect with a set variety of electrons will usually are available a reasonably small variety of isotopes, which differ solely by the variety of neutrons within the nucleus, within the universe of Scale there will probably be a a lot bigger set of “meta-isotopes”, the place solely the screened
cost of the nucleus is similar, however the precise variety of protons, and the actual combination of heavier leptons
current, are free to differ. For instance, a helium nucleus mixed with e₀ and e₁ leptons will probably be a meta-isotope of hydrogen, as a result of though it has
a nuclear cost of +2e and two leptons in complete to steadiness that cost, it solely has
one lightest lepton, making it chemically just like a heavy isotope of hydrogen.

As a result of the lightest leptons set the general scale of an atom, within the novel matter is described as “Scale Zero”, “Scale One”, “Scale Two” … “Scale Seven” if it lacks 0, 1, 2, … 7 of the eight potential leptons. In “Scale S” matter, the lightest leptons within the atoms are those we name e_S.

The lightest leptons current in “Scale S” matter, e_S, have mass 2^S m₀,
the place m₀ is the mass of the lightest of the eight leptons.
The atoms in “Scale Zero” … “Scale Seven” matter are successively smaller by components of two. All else being equal, “Scale S” atoms are roughly ½^S the dimensions of “Scale Zero” atoms.
The portions of power concerned within the chemistry of those atoms
are successively larger by components of two, with the power saved in a molecule, or required to interrupt a chemical bond, being 2^S instances larger in “Scale S” matter than “Scale Zero” matter.
Particular person atoms with the identical sort of nucleus have roughly the identical mass (for the reason that mass of the nucleus continues to be a lot larger than that of the
leptons), so matter turns into denser principally as a result of smaller measurement of the atoms. All else being equal, “Scale S” matter is about (2^S)³ = 8^S instances denser
than “Scale Zero” matter.

Cosmic Abundance of the Leptons

Within the universe of the novel, there are
eight totally different sorts of leptons with totally different plenty, and in contrast to muons and tau leptons in our personal universe,
we don’t have to fret about them decaying away.
However what would we count on the relative
numbers of those totally different leptons to be, after the
universe cooled down within the aftermath of the Massive Bang? Will there be extra of 1 variety than one other?

You may suppose that the heavier leptons can be
scarcer, because it prices extra power to supply them. Nonetheless, it doesn’t prove that manner!

Within the very early universe, the temperature was so nice that particles and antiparticles of all types have been
continuously being produced after which annihilating one another. We don’t but perceive why there was barely
extra matter than antimatter produced in our universe, but when we assume that the identical factor occurs within the
universe of Scale, and that this specific facet of the method doesn’t act in another way for the totally different sorts
of leptons, then what we have to calculate is the variety of leptons and antileptons of every variety that have been
current simply because the universe cooled down sufficient that they have been now not being produced. As long as the
matter-to-antimatter ratio is similar for every sort of lepton, then after all of the antileptons of a given variety have
annihilated an identical variety of leptons, the variety of surviving leptons in every case would be the similar fraction of the unique quantity.

After all, if the universe is infinite it can comprise an infinite variety of leptons, however in that case we
can simply take some finite area and comply with its growth over time, consistent with the general growth of the universe. It might sound difficult to ask how a lot the universe itself has expanded whether it is infinite,
however cosmologists haven’t any bother with this; the best measure is how a lot the radiation from an earlier time
has elevated its wavelength because of cosmic redshift.

When the plasma that fills the early universe is at temperature T, the power that’s accessible
for pair-production (the creation of lepton-antilepton pairs) is E ≈ okay T,
the place okay is Boltzmann’s constant. The temperature itself is inversely proportional to
L, the dimensions of the universe (or the dimensions of the area we’re monitoring).

The
temperature we’re interested by, for leptons of mass 2^S m₀,
is the temperature such that 2^S m₀ c² ≈ okay T, as a result of this
is when the universe cools right down to the purpose that pair-production ceases for these specific leptons.
Right here we’ve got used the well-known mass-energy equivalence method, E = m c².

What this implies is that the measurement of the universe, when this occurs for every specific variety
of lepton, will scale with S as:

L_S = ½^S L₀

However as mentioned within the earlier part, the attribute size scale for leptons of mass 2^S m₀ obeys the identical scaling legislation. So at every transition
temperature, when one sort of lepton stops being created by pair-production, every little thing successfully “appears to be like the identical” to every of the leptons in query. The universe is smaller and warmer when this occurs for the extra huge
leptons, however they’ve proportionately smaller wavelengths, so they are going to be current in the identical quantity, in every case.

We’re making a whole lot of simplifying assumptions right here, however the bulk of the flowery calculations in statistical
mechanics that we might have slogged by, to attempt to quantify the variety of leptons and antileptons in
equilibrium at a given temperature in a universe of a given measurement, find yourself relying solely on the ratios between the lepton mass and the temperature, and between the lepton wavelength and the dimensions of the universe. For the reason that temperature T when pair-production ends is proportional to the lepton mass, and since temperature is inversely proportional to the dimensions of the universe, L, the variations in circumstances for the totally different plenty all simply cancel out.

So, below our assumptions, we count on all eight leptons to be current in equal numbers.

How Are the Leptons Distributed Between Atoms?

We have now seen that the totally different sorts of leptons will all have the identical total cosmic abundance. However does this imply that they can even be evenly distributed? In each atom with Z protons within the nucleus, will
the Z leptons that accompany all of it have a 1-in-8 chance of being every of the eight sorts?

The detailed reply to that will probably be sophisticated, and can depend upon the entire context and historical past
of the actual matter we’re speaking about. In our personal universe, even the refined distinction between isotopes of the identical aspect can result in their native abundance being totally different in numerous astronomical, geological and organic contexts.

One necessary consideration will probably be
the quantity of power it takes to type or disassemble atoms with totally different preparations of leptons.
Let’s contemplate one of many easiest examples. Suppose we’ve got two atoms, shaped from two
unusual helium nuclei, together with a complete of 4 leptons: two e₀ leptons of mass m₀,
and two e₁ leptons of mass 2 m₀. We might prepare issues in both of two methods: we might
have two atoms that every have equivalent leptons, or two atoms that every have dissimilar leptons.
Writing He for the helium nucleus, we’ve got both:

(He e₀ e₀) + (He e₁ e₁)

or:

2 (He e₀ e₁)

Which of those preparations would have the bottom power?

The power ranges accessible to a single lepton certain to a nucleus with atomic number Z are given by:

E_n = –Z² e⁴ m_lowered / (8 h² ε₀² n²)

the place n = 1, 2, 3, …, and the bottom power degree is given by n=1. For instance, for an unusual hydrogen atom, the bottom power, or floor state power, is:

E_H = –2.18 × 10^–18 Joules

What this implies is that it might take 2.18 × 10^–18 Joules to drag the electron away from a hydrogen atom fully, ionising it, so this amount is also called the ionisation power of hydrogen. (There are additional refinements to this method that take account of results past the
non-relativistic Schrödinger equation we’re utilizing, however for our functions this approximation is nice sufficient.)

For atoms with a couple of lepton, there isn’t any easy method like this, however there are strategies we will
use to estimate the power ranges. One technique to estimate the bottom state power of a helium atom is thought
because the variational method. Right here, we take a household of potential wave capabilities,
and differ the parameters that describe their form till we discover the one with the
lowest power inside the household we’re taking a look at. For the case of a helium atom, we regulate the worth of Z, the nuclear cost,
because it seems in two separate variations of the lowest-energy hydrogen wave perform. That is basically the identical as squeezing or increasing every
of the one-particle wave capabilities, till we attain the minimal power that may be achieved this manner.

For an unusual helium atom in our universe, one of the best answer on this household of wave capabilities is discovered by setting:

Z = 27/16 = 1.6875

for each electrons. This worth of Z, lower than the true nuclear cost of two, tells us that every
electron is seeing the nucleus partly screened by the opposite electron.

The related floor state power is:

E_He ≈ –1.24 × 10^–17 Joules

That is inside about 2% of the measured worth.

If we apply the identical methodology to all three sorts of helium atoms that curiosity us, the outcomes are:

Atom	Z₁	Z₂	E_floor
(He e₀ e₀)	1.6875	1.6875	–1.24 × 10^–17 Joules
(He e₁ e₁)	1.6875	1.6875	–2.48 × 10^–17 Joules
(He e₀ e₁)	1.24866	1.95348	–2.00 × 10^–17 Joules

Within the first two circumstances, as a result of the 2 leptons have equal mass, the quantity of screening is equivalent. Within the third case, we’ve got Z for the lighter lepton of 1.24866, indicating that the heavier lepton, being
nearer to the nucleus, is screening much more of the nuclear cost that the lighter lepton sees than when the plenty are equal.
However Z for the heavier lepton may be very almost 2, as a result of the lighter lepton not often comes between
it and the nucleus. We plotted the possibilities of discovering these two leptons at varied
distances from the nucleus earlier.

As anticipated, the helium atom the place we’ve got two e₁ leptons has about twice the power because the one with two e₀ leptons, and the power of (He e₀ e₁) lies in
between. However the essential values are the whole energies for the 2 potential preparations of the elements:

Atoms	Complete E_floor
(He e₀ e₀) + (He e₁ e₁)	–3.72 × 10^–17 Joules
2 (He e₀ e₁)	–4.00 × 10^–17 Joules

So we obtain the bottom power if we share the heavier leptons equally between each atoms.

However let’s have a look at one other instance. Suppose that we’ve got the identical 4 leptons, however as an alternative of two
helium nuclei, we’ve got one helium nucleus and two hydrogen nuclei. For the 2 potential hydrogen atoms, the
floor state energies are:

Atom	E_floor
(H e₀)	–2.18 × 10^–18 Joules
(H e₁)	–4.36 × 10^–18 Joules

The potential preparations of the leptons, and their complete floor state energies, are:

Atoms	Complete E_floor
(He e₀ e₀) + 2 (H e₁)	–2.11 × 10^–17 Joules
(He e₁ e₁) + 2 (H e₀)	–2.92 × 10^–17 Joules
(He e₀ e₁) + (H e₀) + (H e₁)	–2.66 × 10^–17 Joules

This time, the bottom power comes from placing all the heavier leptons into the helium atom, and leaving all of the lighter ones with the hydrogen atoms.

In a state of affairs like this, the heaviest leptons will find yourself certain to the nuclei with the best optimistic cost. Roughly talking,
when a lepton with mass m is certain to a nucleus with cost Z,
there’s a issue of –Z² m within the power. If m₂ > m₁ and Z₂ > Z₁, then:

(m₂ – m₁)(Z₂² – Z₁²) > 0
–Z₂² m₂
–Z₁² m₁
<
–Z₁² m₂
–Z₂² m₁

After all this isn’t the entire story with the power, which can even depend upon what shell every lepton
is in, and the sophisticated results of the presence of all the opposite leptons.
However with these caveats, we will nonetheless make the case for a normal heuristic: all else being equal, the configuration the place the bigger lepton mass is related to the bigger nuclear cost could have the decrease power.

As soon as we get thinking about greater than a handful of particles, there will probably be an enormous quantity
of potential ways in which any assortment of leptons and nuclei may very well be organized, and simply as we usually do
not discover matter in our personal universe having shaped the exact set of chemical compounds that minimises
its chemical power, the distribution of leptons between nuclei in any substantial amount of matter is unlikely to have reached the true minimal power.
Somewhat, as we famous at first of this part, we’ll usually want to think about many extra components than a record of all of the particles which can be current. How lengthy has the actual assortment of atoms and molecules we’re contemplating been collectively? Is it stable, liquid, fuel, or some combination of all three phases? How concentrated is it? What temperature is it now, and what temperature has it been prior to now? Vitality and entropy can provide us
some hints, however the detailed historical past and circumstances will all the time have the potential to make an enormous distinction.

Molecules

Most matter isn’t comprised of single atoms,
however of molecules consisting of two or extra atoms certain collectively
in a roundabout way. When two atoms share a lepton, that is named
a covalent bond. In a covalent bond, the sort of wavefunction that
a lepton can occupy within the neighborhood of a single atom is
prolonged to at least one that binds the lepton to 2 totally different nuclei directly, and though
the nuclei repel one another as a result of they each have a optimistic cost, by the attraction they each have for
the shared lepton (or often, two leptons), they’re certain collectively at a sure distance
the place the attraction and repulsion steadiness.

For the reason that distance between two covalently certain atoms is decided by the lepton wave perform, its measurement will scale
in the identical style as the dimensions of a person atom, with
the identical issue of ½^S. And the power related to the leptons forming this bond will
scale with 2^S, identical to atomic power ranges.

Nonetheless, there are different methods for a molecule to own power. A molecule can vibrate, with the bond
lengths altering periodically, a bit like a toy mannequin of
the molecule might vibrate if the atoms have been stable balls joined along with springs. And a molecule can rotate, turning round its centre of mass, giving it rotational kinetic power.

When a molecule vibrates,
the change within the bond size, say δ, is related to a possible power U:

U = ½ okay δ²

Right here okay is a continuing, and the method is modelled on the potential
power of an elastic materials that obeys Hooke’s law
with a spring fixed of okay. This is similar as
saying that the drive restoring the spring in the direction of its
relaxed size is proportional to the gap δ by which it
has been stretched or compressed. In a
quantum-mechanical harmonic oscillator,
a system the place a mass M is topic to such a drive has power ranges given by:

E_n = (n+½) (h/(2π)) √(okay/M)

for n = 0, 1, 2, … The mass M on this case is said to the mass of the atomic nuclei,
not of the leptons, so it’s unchanged by the scaling of the lepton mass. However how does okay change?

Very roughly, we’d count on U to scale like the standard lepton power when
δ is similar to the dimensions of the molecule. That’s:

okay ~ U / δ²
~ 2^S / (½^S)²
~ (2^S)³

Our method for E_n then offers the outcome:

E_vib ~ (2^S)^3/2

So the vibrational power ranges of a molecule will improve with S,
on the 3/2 energy of the speed at which the lepton power will increase.

What about rotational power? Rotational kinetic power and angular momentum
for classical objects are ruled by the equations:

I = M r²
L = I ω
Ok_rot = ½I ω²

the place I is the momentum of inertia for a mass M at a distance of
r from a centre of rotation, L is the angular momentum, ω is angular
velocity and Ok_rot is rotational kinetic power.

As a result of we’re coping with a quantum system, angular momentum is quantised, and for the
lowest power rotational states it’s of the order of the lowered Planck’s constant:

L ~ h/(2π)

It follows that:

ω ~ h/(2π) / I
Ok_rot ~ I ω²
~ (h/(2π))² / (M r²)
~ (2^S)²

So the rotational power ranges of a molecule will improve with S,
on the sq. of the speed at which the lepton power will increase.

Scales as Ecological Niches

Each time there are mixtures of atoms or molecules that may very well be recombined into
a state with decrease power, however the reactions that may make this occur don’t happen spontaneously
(or solely achieve this very slowly), then
in precept there is a chance for all times to step in and extract some helpful work from
the power distinction. Life on Earth does this on a regular basis — typically in circumstances that
can be hostile to most different organisms, and require particular biochemical variations for a selected area of interest (equivalent to sizzling springs, or hydrothermal vents).

If one of many alternatives within the Scale
universe includes
shifting heavy leptons to higher-charged nuclei — or different energetically beneficial rearrangements — then it’s potential that this type of “lepton shuffling” will probably be exploited in
varied organic pathways.

On the face of it, there isn’t any purpose why this requires any organism to evolve past what is thought within the novel as
“rootlife” (composed primarily of “Scale Zero” matter),
and set about actively eradicating a number of of the lighter leptons from their physique. However there are
definitely some potential benefits in specialising to a single scale:

If there’s a massive provide of power ready to be exploited
by any organism that’s strong sufficient to take care of it, then the primary improvements that faucet into this provide will probably be adopted by choice stress to adapt additional and acquire safer,
extra dependable, extra intensive entry to the identical useful resource.
Coping with the upper energies concerned when manipulating heavier leptons will probably be much less more likely to
harm the organism if all its molecules have chemical bonds at that power scale, moderately than the extra
fragile ones because of lighter leptons.
Predation by rootlife turns into tougher if the organism is “harder”, whether or not
chemically (making it more durable to digest) or bodily (making its pores and skin more durable to rupture).

There are additionally obstacles: scale-specific organisms are going to wish appropriate air and water, and it is going to be simpler if these inputs are already of the required scale.
However on Earth, micro organism and crops have drastically altered the composition of the environment in contrast
to its authentic state, and in addition created numerous niches the place the waste merchandise of 1 set of organisms serve
as meals and power sources for others. The Earth’s environment contained virtually
no free oxygen till the Great Oxidation Event, 2.4 billion years in the past, when cyanobacteria developed photosynthesis.
The 20% of the environment that’s now oxygen is nearly all because of organic exercise.

So the almost certainly state of affairs for all times creating the scales as separate ecological niches would contain single-celled organisms appearing as pioneers, exploring the
biochemical choices and pumping out their very own scale-specific metabolites. As soon as they’d paved the best way, bigger creatures might co-opt among the similar biochemistry, whereas exploiting each the pioneers themselves and their waste merchandise for their very own functions.

Biomechanics and Metabolism

The query of how organisms of various sizes differ of their anatomy, lifespan, metabolism, physique temperature, operating velocity, bodily power, and many others., is already a posh topic when it’s restricted to life on Earth. This space of biology, referred to as allometry, has produced an enormous literature and a large number of “scaling legal guidelines”, some
fairly nicely supported and uncontroversial, some extra contentious. However what the prevailing,
real-world topic definitely doesn’t cowl is the sort of scaling that
lies on the coronary heart of the novel, the place one creature is smaller than one other as a result of
all of their atoms are smaller. So as an alternative of counting on standard
allometry, we might want to return to first rules and assemble our personal scaling legal guidelines.

Mouse vs. Elephant

Right here is one pretty uncontroversial declare, within the context of unusual biology: if a mouse and an elephant each step off a
cliff that’s two metres excessive, the mouse is more likely to be harmed a lot much less by the autumn than the
elephant.

Why? Suppose the linear dimensions of the elephant (its size, breadth and top)
are every about 100 instances larger than these of the mouse, however each animals have comparable
densities. Then the elephant will weigh 100³ = a million instances greater than the mouse, and the kinetic
power it good points by falling the identical distance will probably be a million instances larger.
After all, an elephant additionally has thicker bones, but when their cross-sectional space was solely
100² = 10,000 instances that of the mouse, then as long as the power that wants
to be dissipated is concentrated in a skinny slice of bone,
this nonetheless leaves about 100 instances extra power that every molecule of bone has to deal with.

Actually, the bones of bigger animals are typically proportionately thicker, scaling
up greater than the general measurement of the animal, for exactly this purpose.
This goes a way in the direction of stopping accidents when the animal suffers the sort of falls they’re
more likely to expertise. However even these anatomical adjustments will not be sufficient to compensate fully for the elevated
weight, and the mouse would nonetheless fare higher.

Now, let’s evaluate individuals of various scales (within the sense of the novel), once more each stepping off
a cliff of the identical top. This time, each individuals will weigh roughly the identical, so the
kinetic power they should dissipate once they hit the bottom will probably be comparable.
And although the smaller individual could have smaller bones, there will probably be roughly the identical
variety of molecules in any cross-section.
Nonetheless, all of the chemical bonds within the smaller individual’s physique will probably be stronger
by an element of two^S [where S increases for smaller scales],
so they are going to undergo much less hurt from an equivalent fall.

So we’ve got the identical outcome for mouse vs. elephant and Scale Seven individual vs. Scale One individual, however the causes are fully totally different.

Metabolism and Physique Temperature

How would we count on the metabolic charge of an individual to vary with their scale?
That’s, how briskly would we count on analogous biochemical processes to happen of their our bodies?

This will probably be associated partly to their physique temperature, but when we naively tried to extend the thermal kinetic power of each molecule of their physique to scale alongside
with the chemical power, we must double their physique temperature (on absolutely the temperature scale) with each
halving in measurement! Clearly that may be inconceivable, on condition that these organisms all dwell on
the identical planet with the identical ambient temperature. So, though physique temperature is probably going
to extend with S, the thermal power can’t develop consistent with the chemical power.

At comparable temperatures, molecules of comparable mass could have comparable
velocities, and for the reason that typical distance a molecule must journey to finish
a course of scales with ½^S, as a result of the entire organism is smaller
by that issue, if a molecule merely has to maneuver someplace to do its job, the speed will scale with 2^S.

But when the molecule
has to react chemically, and if it depends on its thermal power to produce the activation energy, the speed of the method will depend upon a amount referred to as the speed fixed, given by the Arrhenius equation:

Charge fixed = A exp(–E_a/(okay T))

Right here E_a is the activation power,
T is the temperature, okay is Boltzmann’s constant, and A is dependent upon the main points of the response. Ordinarily, we’d consider the
charge fixed as being mounted by the temperature and the main points of the response, after which
it might be multiplied by the concentrations of the reactants to search out the speed at which the response occurred.
However we can’t count on A to remain the identical if we shrink the molecules themselves!
As an alternative, we must always think about a film exhibiting the identical complete variety of molecules colliding,
magnified to
look the identical regardless of the precise scale, with the one distinction being that the time between
collisions is shorter by the issue of two^S that comes from the molecules
being nearer collectively, whereas the fraction of collisions that exceed the activation power and permit the response to happen is
managed by the exponential issue within the Arrhenius equation.

If the activation power E_a scaled with 2^S, however the temperature T solely
elevated barely, the speed of the response can be exponentially suppressed.
Nonetheless, in an advanced
biochemical system, the activation energies for the reactions needn’t scale with 2^S simply because the general chemical power behaves that manner. There isn’t any compulsion for various organisms to make use of molecules
which can be all fully equivalent aside from swapping lighter leptons for heavier ones. Somewhat,
evolution might want to determine catalysts that preserve the activation power low sufficient at every scale, and differ the detailed
molecular pathways as wanted to make sure that these reactions can proceed at an inexpensive charge
with out an extreme improve in physique temperature. If this appears like a problem, it
definitely is — however our personal our bodies handle to extract all of the power that comes from
burning carbohydrates, with out requiring the sort of temperatures wanted for inorganic combustion.

In what follows, then, we’ll assume that reactions proceed extra quickly by an element of two^S, and that the energy consumed by an organism
scales by (2^S)², since every response is each quicker and
extra energetic. [There is no contradiction between the energy liberated by a reaction increasing
with 2^S, while the activation energy needed to make the reaction happen grows
much more slowly.]

How can a smaller organism keep away from overheating, whether it is dissipating extra energy? The one
methodology that scales the correct manner is evaporative cooling, the place an appropriate liquid absorbs power because it turns to a vapour on the pores and skin. The full variety of molecules evaporating per unit time will solely have to scale with 2^S, the speed at which different processes
occur, as long as the power absorbed per coolant molecule additionally scales by the standard chemical power issue of two^S. Be aware that the identical complete variety of molecules of sweat
will match on the pores and skin of an analogous organism of any scale; there’s much less floor space, however the
molecules themselves are smaller.

Equally, meals consumption will happen at a charge that scales with 2^S
when expressed as molecules per unit time,
with the power within the meals scaling by 2^S per molecule,
and complete energy consumed per unit time scaling by (2^S)².

Response Occasions

If we undertake the rule that chemical reactions proceed at a charge that scales roughly with 2^S, then it might be constant
to imagine the identical sort of scaling for many physiological and neurological processes. This implies it might be affordable
to count on subjective time to cross extra quickly by an element of two^S, and smaller-scale individuals to give you the chance
to reply to stimuli quicker, each by way of figuring out an occasion and bodily reacting to it.

After all, a smaller-scale setting may give rise to shorter time scales, in a way cancelling out any subjective profit, even when an goal benefit remained. For instance, if individuals of various scales have been all taking part in the identical sort of ball sport (with a separate sport happening for every scale), and the velocity of the ball was impartial of the dimensions, then
the truth that the ball was crossing shorter distances, in shorter instances, would make the sport proceed objectively extra
quickly for the smaller scales, however subjectively the contributors would contemplate every little thing to be taking place at an analogous tempo, relative to their pure response instances, so they might not discover the sport any simpler.

“Water” is Not Water

Life at every scale will want its personal common solvent. Rootlife might nonetheless
depend on H₂O, however for the opposite scales, merely changing all of the lightest leptons with heavier ones can be unlikely to end in a liquid at ambient temperatures. Somewhat,
every scale will want a molecule of its personal that possesses all the correct chemical and bodily properties.

Equally, no matter pigments are utilized by crops and micro-organisms for
photosynthesis,
the molecules that labored at one scale couldn’t merely be
cloned at a smaller scale with any prospect that they might nonetheless perform.
Daylight will solely be accessible throughout a restricted vary of frequencies, so Scale Two crops
couldn’t survive in the event that they wanted radiation at double the frequency that Scale One crops
used.

This will probably be true throughout your entire gamut of biochemical processes: the molecules that carry out comparable roles at totally different scales will not be the results of merely swapping in
heavier leptons.

Energy and Pace

The drive exerted by a spring (or any elastic materials) is the same as the speed of change of its potential
power with respect to its size. If we pack all of the molecules of the spring collectively
in order that its linear dimensions are scaled by ½^S, then even
if the change in power because it contracts by the identical proportion of its size is similar, the drive it produced
would scale with 2^S. This implies that, even conservatively
(permitting for the chance that some organic processes will be unable to
exploit the elevated chemical power that comes from utilizing heavier leptons),
the muscle tissues of smaller scale organisms will be capable to exert forces
that scale by at the very least 2^S.

Beneath the identical conservative assumptions, individuals of all scales must be
capable of soar at the very least equally excessive, which implies smaller-scale individuals leaping to a bigger fraction of their very own physique top.

How would we count on the operating velocity of an organism to vary with scale?
The crudest estimate can be to take a stride that’s shorter by
½^S however extra frequent (extra strides per second) by an element of two^S, leading to a velocity impartial of the dimensions. However what if we enable for the opportunity of utilizing energy that scales as (2^S)^p, the place p ranges from a conservative worth of 1 (no acquire in mechanical power for every muscle contraction, however extra contractions per second) as much as a most of two, consistent with the general metabolic charge?
Drag force because of air resistance takes the shape:

F_D = ½ C_D A ρ v²

the place F_D is the drag drive, C_D is a dimensionless
drag coefficient, A is the cross-sectional space of the physique, ρ is the density of air,
and v is the speed of the physique. The facility expended to beat this drive is given
by:

P = F_D v
= ½ C_D A ρ v³

If P scales as (2^S)^p and cross-sectional
space scales as (½^S)², this could
enable velocity to scale as (2^S)^(p+2)/3.

Nonetheless, that is probably sophisticated by the truth that C_D can’t
all the time be taken as a relentless for objects of a given form; it will also be affected by
the Reynolds number, a parameter related to the airflow. That is given by:

R = ρ v L / μ

the place L is a linear dimension of the item and μ is the dynamic viscosity of air.
For prime sufficient values of R, the coefficient C_D might be taken
as roughly fixed; for low R, C_D turns into proportional
to 1/R. In that case, we’d have:

P = F_D v
~ L v²

If P scales as (2^S)^p and L
scales as ½^S, this could
enable velocity to scale as (2^S)^(p+1)/2.

In both case, the speed will increase for smaller organisms by at the very least an element of two^S.

Gravity

The acceleration because of gravity, g, will in fact be the identical for everybody residing on the floor
of the identical planet. And since individuals of various scales have an analogous complete mass, the whole drive on their physique because of gravity — i.e. their weight — will probably be comparable throughout the scales.

Nonetheless, when this scale-free acceleration and weight work together with different portions that do change with
scale, the outcome can definitely result in totally different experiences for individuals of various scales.

For a begin, an individual’s weight will probably be supported by the realm of the soles of their toes, which is able to scale as
(½^S)², and the stress their weight applies to the bottom will scale
as (2^S)², making some terrain as troublesome to traverse as strolling by sand dunes in stilleto-heeled sneakers. Evolution may give smaller-scale individuals proportionately broader toes to
reduce this impact, however it might be inconceivable to compensate for it fully.

We have now already famous {that a} smaller-scale individual falling from a given top is probably much less more likely to fracture a bone. This benefit is additional consolidated if the sort of heights they
usually fall from scale like their our bodies, as ½^S. A smaller individual
who is just standing on degree floor after which loses their steadiness will probably be falling a smaller distance,
reducing the danger of damage. Evolution
may commerce off this benefit to some extent by making their bones proportionately thinner, as is definitely
the case for a mouse in comparison with an elephant.

The response instances wanted to reply to occasions managed partly by gravity will
be totally different from these whose time scale is about fully by the distances concerned. The sort of constant-velocity ball sport we talked about beforehand, performed over varied distance scales, will probably be subjectively very comparable
for individuals of every scale, as a result of their sense of time will scale in roughly the identical manner because the distances
concerned. But when an individual drops a cup from, say, a top of z that scales as ½^S, then the time it takes for the cup to hit the bottom
is given by:

t = √(2 z / g)
~ √(½^S)

The sq. root right here signifies that though the time to hit the bottom is shorter for smaller scales,
it doesn’t develop shorter as quickly because the neurological and bodily processes for the one that dropped
the cup develop quicker, so a smaller-scale individual will discover it much less demanding to reply to the occasion and seize the
cup earlier than it hits the ground. For any object dropped from a top that scales like ½^S,
the subjective time till it hits the bottom scales like √(2^S), or 8 instances
longer for a Scale Seven individual than a Scale One individual.

If the time scale of an occasion is about wholly by gravity,
there will probably be an excellent larger benefit in being small. For instance, if an object is thrown straight up into
the air at a velocity that’s impartial of anybody’s scale, the time it can spend in movement can even
be fixed, so on this case smaller-scale individuals will get the complete benefit of their quicker response time,
with a subjective time that scales like 2^S.

What about tolerance to elevated gravity (or equivalently, the “g-force” because of acceleration in a spacecraft)? A given drive will compress a smaller-scale materials by a smaller proportion
of its size, with a conservative issue of ½^S simply from packing the
similar change in elastic potential power right into a smaller distance, reducing to
(½^S)² if the potential power additionally will increase. So smaller-scale individuals
might be anticipated to be extra tolerant of upper g-forces.

Suppose two individuals of various scales wished to journey the identical distance by area,
and moderately than being principally in free fall (which is the case for present human area flight),
they have been capable of embark on powered flights with fixed acceleration, restricted solely by their tolerance
of the g-force. How would their subjective journey instances scale?

The time it takes to journey a distance x with an acceleration a is:

t = √(2 x / a)

With the conservative assumption of a ~ 2^S, and subjective time scaling
with 2^S, the sq. root right here means:

t ~ √(½^S)
t_subjective ~ √(2^S)

So, though the journey can be objectively quicker for smaller-scale individuals, it might nonetheless appear longer
for them. At greatest, if we assumed a ~ (2^S)², the journey can be
subjectively the identical for all scales.