Emergence of Fluctuation
NKBorsuk
N. K. Borsuk1
February 2017 (draft manuscript 1.b)
The Exotic Nature of Thermodynamic Equilibrium
The Emergence of Microscale Fluctuation from Insignificance
to Thermodynamic Dominance
[draft 1.b]
ABSTRACT
We know that the state of thermodynamic equilibrium is an exotic and extreme physical
and thermodynamic condition: It is where the fundamentally-understood and universally
powerful forces and fluxes of entropy production vanish. But it is not an inactive state.
Within innumerable material systems, the equilibrium state involves
incessant microscale fluctuations in the motions and distributions of the atoms
within; these physical fluctuations are concurrently microscale thermodynamic
fluctuations involving changes in entropy and Gibbs free energy which are
contrary to macro thermodynamics: ΔS < 0 and ΔG > 0.
These exotic thermodynamic reactions are spontaneous, random, extremely small in
magnitude and duration, and are normally insignificant in equilibrium thermodynamics. All
this has long been acknowledged, beginning with the foundational work of Gibbs in
statistical mechanics.
I find that, within simple aqueous salt solutions kept very near the state of perfect
equilibrium and at rest within a gravitational field, solute-cluster fluctuations are ephemeral
physical objects that develop gravitational potential energy — and, although putatively
inconsequential, they emerge to powerfully dominate the thermodynamics of the equilibrated
macro system which will, consequently, exhibit unprecedented and invaluable properties.
Electrochemical 'gravity-cells', which were tested within environs highly conducive to the
devices attaining thermodynamic equilibrium, were precluded from equilibrium and instead
exhibited steady-state voltages in strong correspondence to quantitative modeling
incorporating the gravitational potentials of equilibrium solute-concentration fluctuations.
These empirical results indicate a valuable technological application with globally important
environmental impact: Electrical power generation from novel and benign thermodynamic
processes.
The novel and key insight here is this: The gravitational potential of solute-concentration
fluctuation is a new and consequential addition to our scientific roster and accounting of
energies in the universe.
1
[email protected]
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www.nkborsuk.net
Emergence of Fluctuation
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PREFACE
we know the rules...
We know energy inexorably disperses within our universe:
chemical potential spontaneously decreases; entropy always increases.
the only exceptions...
Insofar as we know and have inscribed in our literature,
equilibrium particle fluctuations are the only exceptions to the thermodynamic rules.
Fluctuations in particle concentration—what I call 'particle-cluster fluctuations'—are the particular
variety of equilibrium particle fluctuation at the forefront of my studies. For example, within a glass of
saltwater, the incessant random motions of the atoms and molecules create fluctuations in salt
concentration — chance come-and-go clusterings of solute particles within microscale subvolumes
within the bulk volume of saltwater. In instances of these naturally occurring particle-cluster
fluctuations within fluid solutions that are at or very near equilibrium, the universal rules are upended:
chemical potential spontaneously increases; entropy always decreases.
the history...
These thermodynamic exceptions have been an understanding-in-the-making for over 100 years;
the insight started with Gibbs, one of the master originators of statistical mechanics. Over the decades,
the nature of equilibrium fluctuations has become well understood. We've come to recognize that
microscale particle-cluster fluctuations are actually ubiquitous, they are omnipresent within
innumerable material systems at equilibrium and far-from-equilibrium, they are extremely small in
magnitude, and they occur randomly and are short-lived. Although they nibble at the rule of universal
entropy increase (the most well-known rule), it has been reasonably presumed that equilibrium particlecluster fluctuations are inconsequential. In truth, although theoretically interesting, they have never
been significant in thermodynamics until now...
my research...
My research finds that microscale equilibrium particle-cluster fluctuations can have astoundingly powerful
impact on the tangible thermodynamic nature of things... Gravity makes the difference. The "entropy always
increases" rule is certainly valid within its domain. But, in view of my studies, there is a Second Domain of
thermodynamics in which entropy spontaneously decreases. The two domains are coterminous around the
equilibrium state. They influence one another. They connect.
the key notion...
The key notion in my studies is that
equilibrium solute-concentration fluctuations have gravitational potential energy
within a column of saltwater that is at rest within earth's gravity.
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("solute-concentration" fluctuations are the variety of particle-cluster fluctuation important within the
electrolytes in my studies.) The 'key' notion, it seems to me, is at most a modest insight. It seems a
reasonable step forward upon our existing knowledge of physics and equilibrium fluctuation. All
things made of atoms have gravitational potential energy in a gravitational field, solute-concentration
fluctuations are no exception in this regards.
Two corollary thoughts develop: 1. Solute-concentration fluctuations are physical objects.
2. There are now more energies in a column of saltwater than previously presumed. The
gravitational potential of solute-concentration fluctuation is a new addition to our scientific roster and
accounting of energies in the universe.
the gist of the study...
Equilibrium particle-cluster fluctuations become powerful when the column of saltwater is kept
very close to perfect thermodynamic equilibrium, and kept at rest within a gravitational field.
Powerful? Why? —The state of thermodynamic equilibrium is an exotic condition (in a similar
sense as extremely high or low temperatures, or exceedingly small size tend to be exotic physical
extremes). It is exotic because the normally powerful thermodynamic forces and fluxes that drive our
known reality of 'increasing entropy', vanish at thermodynamic equilibrium. In that equilibrium state, particle-cluster fluctuations and gravity persist. It is there and then, within
that column of saltwater, that the weak gravitational and exceptional thermodynamic potentials of
solute fluctuations become ascendant.
The normally inconsequential thermodynamics of equilibrium solute-cluster fluctuation will
dominate the thermodynamics of the macro system that is at equilibrium and at rest within
earth's gravity. The effect of the normally insignificant but now dominant microscale
fluctuation energies is an equilibrium macro distribution of chemical potential that
contradicts—that far exceeds—the classical expectation. This is a powerful result.
I refer to this thermodynamic domination of a macro system—from the dominance of spontaneous
microscale entropy decreases and chemical potential increases—as 'Second Domain' thermodynamics.
Our familiar entropy-increasing reactions are "First Domain".
In that 'Second Domain' state, the equilibrium distribution of chemical potential within the macro
system will compel a steady-state voltage from an electrochemical gravity-cell built upon the
equilibrated saltwater column. The gravity-cell's steady-state voltage is readily anticipated from
application of a nascent Equilibrium Fluctuation model (or EF Hypothesis) to basic equilibrium
electrochemistry. The steady-state voltage alludes to an astounding possibility of directly converting
second domain thermodynamic reactions to electrical power generation.
Decisively invaluable in my studies have been the experiments. Results from two sets of
electrochemical experimentations were compared against quantitative expectations from the EF
Hypothesis. The test results are signature of the model — this outcome is, to me, the most powerful
and satisfying of the entire research. It is still fascinating to recall the voltages and contemplate their
origination from microscale equilibrium fluctuations.
There is a bit more... The gravitational and chemical potentials — that come and go with the
microscale cluster fluctuations — are coupled energies. That coupling is the dynamo of the steadystate electrochemical potential. The First and Second Domains of the 2nd Law, mentioned a couple of
paragraphs above, are coterminous around the state of thermodynamic equilibrium. To my thinking, it
all connects. No doubt there is much much more to discover.
Norm Borsuk
February 2017
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by any other name...
equilibrium particle-cluster fluctuation
subvolume
Vf
column of saltwater
a test-tube column of saltwater... with representation of equilibrium soluteconcentration fluctuation within a micro
subvolume Vf of the fluid column.
(1887) — Planck2 apparently thought they were "unnatural" ...they never happen.
(1902) — Gibbs3 called them "anomalies" and he calculated their magnitude.
(1933) — Guggenheim 4 took-up Planck's position on them.
(1961) — Prigogine5 discovered the fundamental play of microscale fluctuation in far-from-equilibrium thermodynamics. Yet, for equilibrium
thermodynamics... he called them "inconsequential".
(1967) — Guggenheim 6 still stayed with Planck, but included calling them "negligible".
2
Planck, Ann. Phys. Lpz. 1887, 30, 563 (ref from Guggenheim; 1967, p12)
3 J. Willard Gibbs; Elementary Principles In Statistical Mechanics Developed With Especial Reference To The Rational Foundation Of
Thermodynamics; 1902; throughout.
4
Guggenheim; Modern Thermodynamics by the Methods of Willard Gibbs; 1933; page 4.
5
Prigogine and Kondepudi; Modern Thermodynamics; 1998; Preface.
6
Guggenheim; Thermodynamics; 1967; pp. 12 and 65.
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historical notion of...
voices past...
The blue color of the sky is due in part to
fluctuations of air particle number-density within small
volumes in the atmosphere.
Such 'particle-cluster
fluctuations' are real and influence our world. Richard Tolman7 and many others since 8 have
written of it.
continuity...
Solute-concentration fluctuations are also real...
ubiquitous transients of mass density, entropy,
and chemical potential in fluid solutions.
My work makes use of them.
7
Tolman; The Principles of Statistical Mechanics; 1938; p.647
8
for example: Van P. Carey; Statistical Thermodynamics and Microscale Thermophysics; 1998; pp. 133-134; see "Example 4.4".
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a first notion of...
flickering clusters of atoms → physical, ephemeral, cloud-like things
equilibrium particle-cluster fluctuation
subvolume
Vf
column of saltwater
a notion of the physical...
Figure A. Saltwater driblets gently dribbling from tip of eyedropper
immersed in a glass of saltwater.
The salt concentration in the driblets is only slightly greater than the bulk
solution; and so they have a greater mass-density than the surrounding fluid;
and so they have a gravitational potential that compels them to drift towards
the bottom of the vessel. They eventually dissolve and disappear and that's
that.
Solute-concentration fluctuations are also ephemeral physical objects, but
of microscale size — much smaller than the macro driblets.
Importantly, particle-cluster fluctuations have gravitational potential in
earth's gravity and are omnipresent in their incessantly reoccurring way.
Their thermodynamic properties are entirely different from the driblets.
my favorite lines...
Equilibrium particle-cluster fluctuations are chance clusterings... ephemeral, come-and-go cloud-like gatherings of atoms and molecules.
They are transient physical objects and exotic thermodynamic reactions. Anomalous? No.
They are omnipresent.
They are ubiquitous.
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the entropic nature of...
ΔS = δ2S/2 < 0
fluctuated
subvolume Vf
subvolume Vf
⇒
column of saltwater
− earth −
− earth −
subvolume Vf
⇒
− earth −
Prigogine9...
..."The random motion of molecules causes all thermodynamic quantities such as
temperature, concentration, and partial molar volume to fluctuate. ... In general, the
entropy can be expanded as a power series in these parameters, so we have
S = Seq + δS + δ2S/2 + ...
... At equilibrium, since S is a maximum, the first order term δS vanishes.
Fluctuations can only decrease S, that is to say δ2S/2 < 0, and spontaneous, entropyincreasing irreversible processes drive the system back to the state of equilibrium."
"...10 But at equilibrium or near equilibrium, these fluctuations are inconsequential."
9
Prigogine and Kondepudi; Modern Thermodynamics; 1998; chapter 12.
10
ibid. Preface.
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solute-concentration fluctuation and gravity...
fluctuated
subvolume Vf
subvolume Vf
⇒A
− earth −
subvolume Vf
⇒B
− earth −
− earth −
ni + Δni
Vf
ni
+ Δni
Vf
ni
+ Δni
Vf
Δµi, f = δ 2 µi, f > 0
⌣
⎛ miθ α ⎞
⎝⎜ n1 2 ⎠⎟
i
f
the basics...
Fluctuated subvolume Vf represents a solute-concentration fluctuation.
It embodies...
n + Δni
• a spontaneous increase in solute i particle-concentration beyond the equilibrium mean ⇒ i
Vf
• a spontaneous increase in chemical potential of solute i within Vf ⇒ Δµi, f > 0
✴ a spontaneous gravitational potential of subvolume Vf as a unit object 11 ⇒ EV f
α
β
=
⌣
−Mi θα − θ β
(
12
i
n
✴ the radical element in my studies is this...
within simple saltwater solutions at equilibrium and at rest in earth's gravity,
solute-concentration fluctuations Vf have gravitational potential energy:
α
EV
f β
=
⌣
−Mi θα − θ β
(
)
12
i
n
• ni is the number of solute particles of species i.
β
α
• θ and θ are gravitational potentials in phases 𝛼 and 𝛽 of the system (i.e. different vertical positions in the column).
•
⌣
M i = M i − (Vi Vo ) M o It is from Guggenheim 12. I call it the 'effective' molar mass of the solute in solution. More about it here on page 15 in Table 1.
11
From 2012 manuscript, section 2.3.3 "Gravitational Potential Energy of a Fluctuated Subvolume", line 5.
12
Guggenheim; Thermodynamics; 1967; line 9.09.2.
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)
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the radical element...
—worth repeating—
𝛽
(V ,θ β )
𝛼
(V ,θ α )
f
⇒
− earth −
− earth −
f
the radical element in my studies...
within simple saltwater solutions at equilibrium and at rest in earth's gravity,
solute-concentration fluctuations Vf have gravitational potential energy:
α
EV
Page 9 of 34
f β
=
⌣
−Mi θα − θ β
(
12
i
n
)
Emergence of Fluctuation
NKBorsuk
So what ?
fluctuated
subvolume Vf
subvolume Vf
⇒
⇒
− earth −
ni
+ Δni
Vf
subvolume Vf
− earth −
− earth −
ni + Δni
Vf
ni
+ Δni
Vf
Δµi, f = δ 2 µi, f > 0
⌣
⎛ miθ α ⎞
⎜⎝ n1 2 ⎟⎠
i
f
so what ?
Gibbs 13 said the end result of equilibrium fluctuation is zero.
"Of especial importance are the anomalies of the energies, or their
deviations from their average values. The average value of these
anomalies is of course zero. The natural measure of such anomalies is the
square root of their average. Now..."
gravity makes a difference...
Mostly, I agree with Gibbs. But I believe his assessment is tremendously misleading...
Fluctuation reactions A and B are sequential in time. I find that, in gravity, the two
reactions can become temporally discontinuous and spatially distant. True, the average
will ultimately tend to zero, but the 'equilibrium-restorative' reaction B can become
indefinitely delayed and far remote from reaction A for systems at or near
thermodynamic equilibrium and at rest in gravity.
These ideas are at the heart of my studies and this paper.
13
Gibbs; Elementary Principles In Statistical Mechanics; 1902; pp. 71-72
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microscale fluctuation → macro distribution
gravity makes a difference
the metric14...
Fluctuated subvolume Vf embodies an ephemeral object with two energies:
(1) the subvolume's gravitational potential, and (2) the chemical potential of the solute within it.
The gravitational and chemical potentials are coupled; Kf is a metric of correspondence:
distribution in gravity
𝛽
(V ,θ β ) (δ µ ) β
𝛼
(V ,θ α ) (δ µ ) α
f
spontaneous Fluctuation Reaction A
2
i, 1
⇒
− earth −
− earth −
Δµi, f = δ 2 µi, f > 0
Kf =
2
f
i, 1
⌣
− M i (θ α − θ β )
α
Δµi, 1 β
The solute chemical potential within a microscale fluctuated subvolume is metered to the
macro system by the fluctuated subvolume's gravitational potential.
•
Δµi, f
α
β
is the distribution of solute chemical potential, attributable to a standard fluctuated subvolume Vf,1 which
contains a single particle, within the column within the gravitational field.
14
This metric is an idea prompted by an idea Gibbs had mentioned. Page 27 has a bit more detail.
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Equilibrium Fluctuation Hypothesis (EF Hypothesis)
the equilibrium distribution of solute chemical potential...
(
⌣
Δµi classical = − M i θ α −θ β
)
⌣
⎛ 1
1
1
1 ⎞
Δµi, EFH = − M i θ α − θ β ⎜ 3/2 + 3/2 + 3/2 ...+ 3 2 ⎟
2
3
ni ⎠
⎝1
(
)
⌣
Δµi, EFH = − M i θ α − θ β
(
n
Δµi, EFH = K f ∑
n=1
) ∑ n1
n
n=1
3/2
i
⌣
= −2.6 M i θ α − θ β
(
)
α
Δµi, 1 β
ni3 2
• All EF Hypothesis statements are equivalent to one another.
• They are premised on the summation of energies of fluctuated subvolumes containing increasing
numbers of particles within. (detailed derivation begins at page 13).
• Each fluctuated subvolume Vf is a distinct physical, chemical, and thermodynamic component in the
fluid solution. Their are innumerable species of Vf , each is speciated by the number of solute
particles ni within.
• The summation is a hyperharmonic series and converges towards a value of about 2.6 assuming at
least 108 particles in the system under study.
• The Equilibrium Fluctuation Hypothesis subsumes the classical model. In the EF Hypothesis, the classical value for the solute chemical potential is but the first energy entry
in the summation; it is of a single species of fluctuated subvolume—the standard fluctuated
subvolume containing but a single solute particle.
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Equilibrium Fluctuation Hypothesis — into the details...
physical reactions and thermodynamic reactions
— At Thermodynamic Equilibrium —
• all thermodynamic forces and fluxes vanish.
• solute-concentration fluctuations persist.
• Gravity persists.
Premises:
The First Law of Thermodynamics is valid in both domains.
I use the Gibbs free energy equation to organize things: ΔH − TΔS = Δμi
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Equilibrium Fluctuation Hypothesis — into the details...
physical reactions and thermodynamic reactions
subvolume
Vf
fluctuated
subvolume Vf
⇒
solute-concentration fluctuation ⇒
ni
+ Δni
Vf
A
subvolume
⇒
ni + Δni
Vf
B
ni
+ Δni
Vf
(1)
Δµi = δ 2 µi > 0
Δµi = δµi < 0
ΔS = δ 2 S < 0
ΔS = δ S > 0
ΔH = δ 2 H = ?
ΔH = δ H = ?
ΔH − T ΔS = Δµ i, f > 0
Second Domain
Reaction A: (1) Subvolume Vf gains Δni 'extra' particles. (2) entropy decreases. (3) solute chemical potential increases.
Reaction B: (1) Subvolume Vf restores towards equilibrium. (2) entropy increases. (3) solute chemical potential decreases.
Page 14 of 34
Vf
ΔH − T ΔS = Δµ i, f < 0
First Domain
Emergence of Fluctuation
NKBorsuk
Equilibrium Fluctuation Hypothesis — into the details...
one measure of Gibbs' equilibrium 'anomalies'
fluctuated
subvolume
Vf
⇒
ni
+ Δni
Vf
A
ni + Δni
Vf
A measure of the Gibbs "deviation" in particle-fluctuated subvolume Vf :
fractional deviation of particle number-density
(ν
from Gibbs15:
1
ν1
2
>
2
(n − n )
(n )
2
.
12
ν1
=
i
from Carey17:
1
2
i
i
from Tolman16:
)
− ν1
σN
1
= 12
N N
1
ni
.
.
A valuable quantity for working with
particle-concentration fluctuation.
It will set the metric for the mass density and
gravitational potential of the subvolume.
I use Carey's formula for the fractional standard deviation of particle number. 𝜎N is the standard deviation of N, where N is the typical or mean value.
( let N = ni and 𝜎N = Δni where ni is the number of solute particles of species i.).
15
Gibbs; Elementary Principles In Statistical Mechanics; 1902; line 542
16
Richard C.Tolman; Principles of Statistical Mechanics; 1938; line 141.45
17
Van P. Carey; Statistical Thermodynamics and Microscale Thermophysics; 1999; line 4.147
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Equilibrium Fluctuation Hypothesis — into the details...
physical things
Evaluating the mass-density deviation of fluctuated subvolume Vf
subvolume
Vf
Fluctuated subvolume Vf is different than its local fluid environment: the number of solute particles
per unit volume is increased which gives Vf a greater mass density.
Calculating its mass-density
deviation will lead to an evaluation of the subvolume's gravitational potential in gravity, and then to
evaluating the macro impact of the chemical potential deviation of the solute within it.
Molar mass-density deviation18, 19 of subvolume Vf :
⌣
M
Δρ f = 1 2 i
ni V f
(2)
⌣
M i = M i − (Vi Vo ) M o is the 'effective' molar mass of the solute in solution20 .
Mi is the molar mass of the solute species i.
Mo is the molar mass of the pure solvent,
Vi is the partial molar volume of the solute,
Vo is the partial molar volume of the pure solvent,
⌣
M i is presumed constant throughout fluctuation reactions A & B.
⌣
M i = 0.22 kg/mol using values in Table 1 for AuCl3 in aqueous solution.
Table 1: Parameters for the 'effective' molar mass of AuCl3 in solution.
note: Line 2 here, is from equation 4 in my 2012 manuscript. The derivation there, I believe, is too detailed and distracting for this paper.
Definitely it is consistent with Hill's observation that the 1/n1/2 order of magnitude for the fluctuation "... is the standard result in statisticalmechanical formulas." If you are interested in the details of the equation's derivation, I can forward the material.
18
19
Tolman; 1938; line 141.60 for the density fluctuation in a perfect gas is essentially identical to line 2 here.
20
Guggenheim; Thermodynamics; 1967; line 9.09.2. " 'effective' molar mass of the solute in solution" is my wording.
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Equilibrium Fluctuation Hypothesis — into the details...
equilibrium particle fluctuation.... the system in gravity
At thermodynamic equilibrium...
• all thermodynamic forces and fluxes vanish.
• solute-concentration fluctuations persist.
• Gravity persists.
Let E g be the i
⌣
Eg = − M i (θ α − θ β )
molar gravitational potential of the solute21
in solution between phases 𝛼 and 𝛽 in which
(3)
i
θ α and θ β are the gravitational potentials.
⌣
For the AuCl3 cells using Table 1: − M i θ α − θ β = −0.0218 J/mol ⋅ cm.
(
)
Equilibrium Fluctuation Hypothesis (EFH) 22
Molar gravitational potential of any fluctuated subvolume Vf containing ni number of solute particles between phases 𝛼 and 𝛽...
α
EV
The
denominator is consistent with our understandings of equilibrium fluctuation. The gravitational potential of Vf becomes insignificantly
small when the fluctuations involve larger and larger numbers of particles. From line 1.5b in 2012 manuscript.
n1/2
Molar gravitational potential of any fluctuated Vf per typical number of particles within Vf ...
f β
EV f
ni
I call line 5... the vertical distribution of solute chemical potential
innate to any fluctuated subvolume Vf within the gravitation field...
α
Δ µi
Summing all contributions to the chemical potential
from all species of solute-concentration fluctuations from line 6, using ni as the unit of speciation, gives the total gradient of the solute chemical potential...
within the macro system.23
f
=
ni
β
α
n
Δµi, EFH β = ∑ Δ µi
α
EV f
n=1
f
β
=
α
=
β
α
=
β
⌣
−Mi θα − θ β
(
)
ni1 2
(4)
⌣
− M i (θ α − θ β )
(5)
32
i
n
⌣
− M i (θ α − θ β )
n
⌣
1
= − M i (θ α − θ β ) ∑ 3 2
n=1 ni
Line 7 is one statement (the first) of the Equilibrium Fluctuation Hypothesis.
21
From Guggenheim; 1967; line 9.09.3
22
Equations here are selected from 2012 manuscript, section 2 (for the most part).
I've never found comfortable wording for equation (7). It is a simple thing. It fits the experimental results. So I use it as is. A different
approach to it, taken from an idea Gibbs mentioned, was introduced on page 10. Page 26 has a bit more to say about it.
23
Page 17 of 34
(6)
ni3 2
(7)
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Equilibrium Fluctuation Hypothesis (EFH)
Repeating line 7...
α
α
n
Δµi, EFH β = ∑ Δ µi
n=1
f
β
n
⌣
1
= − M i (θ α − θ β ) ∑ 3 2
n=1 ni
(7)
The focus for now...
⌣
Δµi, EFH = − M i θ α − θ β
(
n
) ∑ n1
n=1
(8)
32
i
Expanding the series gives
⌣
⎛ 1
1
1
1 ⎞
Δµi, EFH = − M i (θ α − θ β ) ⎜ 3/2 + 3/2 + 3/2 ...+ 3 2 ⎟ .
2
3
ni ⎠
⎝1
(9)
The hyperharmonic series is convergent.
For systems containing at least 108 solute particles the numerical value of the series converges to approximately 2.6 so that
⌣
Δµi, EFH = −2.6 M i θ α − θ β .
(
)
Lines 7 - 10 are equivalent expressions for the distribution of total solute chemical potential in the
system at rest in a gravitational field according to the Equilibrium Fluctuation Hypothesis.
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(10)
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— conflict —
Classical vs 'Equilibrium Fluctuation Hypothesis'
conflict in a chemical potential nutshell...
From Guggenheim24, the classical expectation is
⌣
Δµi classical = − M i (θ α − θ β )
while the EFH expectation is
⌣
⎛ 1
1
1
1 ⎞
Δµi, EFH = − M i θ α − θ β ⎜ 3/2 + 3/2 + 3/2 ...+ 3 2 ⎟
2
3
ni ⎠
⎝1
(
)
a couple of insights...
In the EFH model in lines 7 thru 10:
• The classical expectation is but the first of many energy entries.
• The Equilibrium Fluctuation Hypothesis subsumes the classical model.
• The total solute chemical potential comprises innumerable 'solute species'
• Those 'innumerable solute species' are solute-concentration fluctuations.
24
E.A. Guggenheim; Thermodynamics; 1967, lines 9.04.1 and 9.09.3
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(11)
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— the decisive criterion —
Thermodynamic Equilibrium
equilibrium without gravity...
Staying with Guggenheim, the equilibrium criterion for a chemical species i in a system at
uniform temperature and without influence from an external field is such that the component's
chemical potentials have the same value in any two phases 𝛼 and 𝛽 so that
µiα = µiβ
(12)
in gravity...
Where
⌣
Mi
is the effective molar mass of species i in solution (line 2) and
θα
and
θβ
are the
gravitational potentials in different phases (vertical positions) 𝛼 and 𝛽 of the system, then
thermodynamic equilibrium for the system within a gravitational field is
⌣
µiα − µiβ = − M i (θ α − θ β )
(13)
The Equilibrium Criterion...
I trust that line 13 is a valid principle
— a useful and valid criterion of thermodynamic equilibrium —
for both the classical model (e.g., Guggenheim) and for the Equilibrium Fluctuation Hypothesis.
The term on the far right in line 13 is constant and immediate in the saltwater systems in
gravity that are under study here; there is no conflict over it between the classical model and my
Equilibrium Fluctuation Hypothesis.
a good question...
The gradient of chemical potential in the field,
µiα − µiβ , is contentious.
I frame the matter
thus: What is the total magnitude of the solute's chemical potential gradient:
µiα − µiβ = Δµi total = ?
Which is the correct value of
Page 20 of 34
Δµi total ?
EFH (lines 7-10) or classical (line 11) ?
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Emergence of Fluctuation
NKBorsuk
comparing equilibrium solute gradients...
Classical vs EFH
a first step towards resolving the conflict...
The relationship between the solute's total molar chemical potential and molar concentrations
α
i
c
and
ciβ
in our system of interest is
Δµi total =
µiα
− µiβ
= RT
α
i
ln β
c
i
c
(15)
a measurable criterion...
So, for our column of saltwater at rest in earth's gravity, a measurable criterion of
thermodynamic equilibrium is
Δµi total = RT ln
ciα .
ciβ
(16)
The steepness of the equilibrium gradient depends on the magnitude of
Which is the correct value of
classical model
( from line 11 )
α
i
β
i
c
=e
c
(
⌣
− M i θ α −θ β
Δµi total ?
EFH (lines 7-10) or classical (line 11) ?
EFH
( from line 10 )
𝛽
)
α
i
β
i
c
=e
c
RT
(
⌣
− M i θ α −θ β
𝛼
—earth—
The Equilibrium Fluctuation Hypothesis anticipates a steeper equilibrium salt gradient than the classical model.
ciα and ciβ — molar concentration of solute i in gravitational phases 𝛼 and 𝛽
R —gas constant
T —Temperature
Page 21 of 34
Δµi total .
RT
)⋅
n
∑ ni3/2
n=1
1
(17)
Emergence of Fluctuation
NKBorsuk
measuring the equilibrium gradient...
an electrochemical gravity cell
𝛽
Vcell, equilibrium = −
⌣
Δµi total + M i (θ α − θ β )
neF
(18)
𝛼
—earth—
a valuable meter ...
At equilibrium, the voltage of an electrochemical gravity cell is ideally a measure of two
energies existent in the vertical electrolyte column between the two gravitational potentials:
Vcell, equilibrium = −(total solute chemical potential + solute gravitational potential) / neF .
That is...
Vcell, equilibrium = −
⌣
Δµi total + M i (θ α − θ β )
neF
Since the solute gravitational potential is constant, the gravity cell is then a valuable meter of Δμi, total .
an important experimental consideration...
The most important (and demanding) experimental criterion in testing the equilibrium state of the
electrochemical cells was in maintaining the cells within a well insulated, energy quiescent environment;
i.e. an environment strongly conducive to a cell-in-test closely approaching perfect thermodynamic
equilibrium (I set the cells into a water-tight canister and immersed the canister into a water bath; stirred
and temperature controlled ± 0.002 °C for weeks at a time).
Vcell
— cell Voltage
ne
—ionic charge of the solute cation
F or F —Faraday constant
Page 22 of 34
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Emergence of Fluctuation
NKBorsuk
what to expect, electrochemically...
the equilibrium voltage
Repeating line 19...
Vcell, equilibrium = (total solute chemical potential + solute gravitational potential) / neF :
Vcell, equilibrium = −
classical expectation...
The classical total chemical potential is
⌣
Δµi total + M i (θ α − θ β )
(19)
neF
⌣
Δµi total = Δµi classical = − M i (θ α − θ β ) .
Inserting it into line 19 gives the classical expectation...
Vcell, classical equilibrium = −
⌣
⌣
− M i (θ α − θ β ) + M i (θ α − θ β )
neF
= 0 volts
(20)
EFH expectation...
The EF Hypothesis, from line 9, anticipates the total chemical potential to be
⌣
⎛ 1
1
1
1 ⎞
Δµi total = Δµi EFH = − M i θ α − θ β ⎜ 3/2 + 3/2 + 3/2 ...+ 3 2 ⎟
1
2
3
n
⎝
⎠
i
(
or, equivalently, line 10
)
⌣
Δµi total = Δµi EFH = −2.6 M i (θ α − θ β ) .
Inserting it into line 19 gives the EFH expectation...
Vcell, EFH equilibrium = −
Vcell, EFH equilibrium =
⌣
⌣
−2.6 M i (θ α − θ β ) + M i (θ α − θ β )
neF
⌣
1.6 M i (θ α − θ β )
neF
≠ 0volts .
So, when the cell's electrolyte column is at thermodynamic equilibrium, the electrochemical cell
would exhibit a steady-state nonzero voltage according to EFH.
Next, line 22 is used to give a range of values expected in actual test of the Equilibrium Fluctuation
Hypothesis for cells using aqueous AuCl3 electrolytes.
Page 23 of 34
(21)
(22)
Emergence of Fluctuation
NKBorsuk
quantitative expectations...
EFH projections for the AuCl3 electrochemical cells...
When the electrolyte column is equilibrated, the EF hypothesis anticipates a steady-state cell
voltage. Test results for the AuCl3 cells were compared against calculated expectations using line 22
based on two sources:
1. Inserting tabulated parameters from Table 1 into line 22.
This calculation projects a steady-state voltage/cm of 0.12μV/cm.
⌣
It is derived by using M i = 0.22 kg/mol, 9.8 m2/s2 for the gravitational potential, and ne =3 for the Au+3 cation (all from Table 1, page 7) in line 22.
Line 22 then develops:
Vcell, EFH equilibrium =
⌣
1.6 M i (θ α − θ β )
neF
V ⋅ cm −1cell, EFH equilibrium =
1.6 × 0.22 kg/mol × 9.8 m 2s 2
1
⋅
3 × 96, 487 C/mol
100cm
V ⋅ cm −1cell, EFH equilibrium = 0.12 µV ⋅ cm −1 ⌣
2. Using experimental measurements of the 'solute gravitational potential' Eg = M i θ α − θ β of the
i
(
AuCl3 salt in the cells' 0.5 molar aqueous
)
electrolyte25.
The average value of the voltage spikes produced upon inverting a 7.5 cm tall cell implied a value of about -.011 μV/cm with the bottom
electrode having an immediate negative-going polarity with each inversion.
Again using line 22, inserting the experimentally derived value of .011 μV/cm , and multiplying by the 1.6 factor gives...
Vcell, EFH equilibrium =
⌣
1.6 M i (θ α − θ β )
neF
V ⋅ cm −1cell, EFH equilibrium = 1.6 × 0.11µV ⋅ cm −1 = 0.18 µV ⋅ cm −1
Rapid inversions of cell to measure the voltage 'spikes' driven by Eg.
range of 'equilibrium' voltage anticipated from EF hypothesis...
Together, 1 and 2 above give a range of the expected voltage-per-centimeter as well as polarity of
the cell (the bottom electrode will always become the cathode or, the more positive of the two
electrodes). Here then is the range of calculated expectations from my Equilibrium Fluctuation Hypothesis for
the AuCl3 gravity cells:
+0.12 μV/cm to +0.18 μV/cm
25
from section 3.2 and Figure 6 in the 2012 manuscript. Sections 3 and 4 in that manuscript have all the calculations here in greater detail.
Page 24 of 34
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Emergence of Fluctuation
NKBorsuk
— the experiments 26 —
Testing a cell involved setting it upright and keeping it undisturbed for prolonged periods, often for days or weeks at a time,
in an environment as free as possible of extraneous influences, and constantly monitoring its voltage with a nanovoltmeter.
Eventually a cell-in-test was inverted (180 degree rotation). Each inversion took about 5-10 seconds to complete, and afterwards
the cell would again rest undisturbed until the next inversion.
Figure 3. 14½ week test of a 2.5 cm cell. Top: Record of
test during days 23–103. Bottom: Overlay of voltage
traces following inversions on day 63 and day 83. Trace
(a) indicates the cell’s voltage with the working electrode
at the bottom position (starting on day 63). Trace (b)
indicates cell voltage with the working electrode at the
top (starting on day 83). The 0.8 µV separation between
the inverted positions implies +0.16 µV/cm from a
common midpoint or offset.
26
Figure 4. 5½ week test of a 7.5 cm cell. Top: Record of
test during days 21–40. Bottom: Overlay of voltage traces
from inversions on day 26 and day 30. Trace (a): the cell’s
working electrode at the bottom position (starting on day
26). Trace (b): the cell is inverted so that the working
electrode is at the top (starting on day 30). The 2.8 µV
separation between the inverted positions implies +0.19
µV/cm from a common midpoint or offset.
much additional write-up of the original Au-AuCl3-Au 'gold-cell' tests is readily available.
Page 25 of 34
Emergence of Fluctuation
NKBorsuk
comparing calculations to test results...
The range of calculated expectations from EF hypothesis (line 23) is
+0.12 μV/cm to +0.18 μV/cm.
The actual experimental findings ranged from
+0.16 μV/cm to +0.19 μV/cm.
The classical expectation, from line 20, is zero volts.
the final experimental word...
• Results of equilibrium experimentations (line 24 and Figures 3 and 4) were entirely
inconsistent with expectations from the classical model.
• Yet, the results were signature of the Equilibrium Fluctuation Hypothesis:
When the electrolyte column attains thermodynamic equilibrium, then the
electrochemical cell will exhibit a steady-state voltage commensurate to the 'steep'
equilibrium solute-concentration gradient extending throughout the height of the
column.
an important outcome...
The correspondence of the carefully measured test results to the quantitative EFH
model is, in my opinion, a singular and compelling outcome.
Page 26 of 34
(24)
Emergence of Fluctuation
NKBorsuk
metric Kf
In his considerations of the magnitudes of the statistical deviations from the mean (fluctuations), Gibbs wrote27
"To get an idea of the order of magnitude of these quantities, we may
use the average kinetic energy as a term of comparison..."
Instead of kinetic energy, I use the gravitational potential energy of the fluctuated subvolume. The objective is to find a
correlation of the two principle energies... and thus, perhaps, a deeper insight into the distribution of micro chemical
potential fluctuation within the macro system.
relating potentials...
The saltwater column in gravity...
distribution in gravity
β
(V ,θ ) (δ µ ) β
𝛽
2
f
spontaneous Fluctuation Reaction A
i, 1
⇒
− earth −
Figure 5
− earth −
Δµi, f = δ 2 µi, f > 0
(V ,θ α ) (δ µ ) α
𝛼
2
f
i, 1
Fluctuated subvolume Vf embodies an ephemeral object with coupled energies: its gravitational potential and the chemical
potential of the solute within it. I propose that the ratio kf is a valuable correspondence of the two energies:
(
⌣
− mi θ α − θ β
kf =
α
)
(25)
δ µi, 1 β
2
where δ 2 µ α = ⎡δ 2 µ ⎤α − ⎡δ 2 µ ⎤ β . The subscript '1' refers to a standard Vf involving only a single particle.
i, 1 β
i, 1 ⎦
i, 1 ⎦
⎣
⎣
micro to macro...
This still unfinished approach leads to a molar metric Kf that seems generally valuable towards quantifying all
thermodynamic fluctuations in reaction A, not just the solute chemical potential fluctuation. Without elaboration here, the
coupling metric appears akin to...
Kf =
⌣
− M i (θ α − θ β )
α
Δµi, 1 β
It distributes the microscale solute-concentration fluctuations within the macro system as identified in line 8:
n
Δµi, total = Δµi, EHF = K f ∑
n=1
27
Gibbs; 1902; page 203
Page 27 of 34
α
Δµi, 1 β
32
i
n
⌣
= −Mi θα − θ β
(
n
) ∑ n1
n=1
32
i
(26)
Emergence of Fluctuation
NKBorsuk
— Synthesis —
solute-concentration fluctuation...
Fluctuated subvolume Vf represents a solute-concentration fluctuation. It embodies...
• a transient increase in solute i particle-concentration beyond the equilibrium mean.
• a transient increase in chemical potential of solute i within Vf .
• a transient gravitational potential of subvolume Vf as a unit object.
subvolume
fluctuated
subvolume Vf
Vf
⇒
solute-concentration fluctuation ⇒
ni
+ Δni
Vf
subvolume
Vf
⇒
ni + Δni
Vf
A
B
ni
+ Δni
Vf
Δµi, f = δ 2 µi, f > 0
α
EV
f β
=
⌣
−Mi θ α − θ β
(
)
ni1 2
correlating the two potentials...
The chemical and gravitational potentials are coupled; Kf is a metric of correspondence:
distribution in gravity
𝛽
spontaneous Fluctuation Reaction A
(V ,θ β ) (δ µ ) β
f
2
i, 1
⇒
− earth −
Kf =
− earth −
Δµi, f = δ 2 µi, f > 0
𝛼
(V ,θ α ) (δ
f
2
µi, 1 )
⌣
− M i (θ α − θ β )
α
Any microscale fluctuation of solute chemical potential — occurring within any
fluctuated subvolume Vf — will distribute within the macro system proportionally
to the gravitational potential of Vf .
Page 28 of 34
α
Δµi, 1 β
Emergence of Fluctuation
NKBorsuk
— Synthesis —
the equilibrium distribution of solute chemical potential...
(
⌣
Δµi classical = − M i θ α −θ β
)
⌣
⎛ 1
1
1
1 ⎞
Δµi, EFH = − M i θ α − θ β ⎜ 3/2 + 3/2 + 3/2 ...+ 3 2 ⎟
2
3
ni ⎠
⎝1
(
)
⌣
Δµi, EFH = − M i θ α − θ β
(
n
Δµi, EFH = K f ∑
n=1
) ∑ n1
n
n=1
3/2
i
⌣
= −2.6 M i θ α − θ β
(
)
α
Δµi, 1 β
ni3 2
The Equilibrium Fluctuation Hypothesis subsumes the classical model.
Fluctuated subvolume Vf is a distinct physical, chemical, and thermodynamic
component in the fluid solution. Their are innumerable species of Vf , each is
speciated by the number of solute particles ni within.
equilibrium electrochemical experimentations...
Their outcomes were signature of the Equilibrium Fluctuation Hypothesis.
Range of calculated expectations from EF hypothesis:
+0.12 μV/cm to +0.18 μV/cm
from (23)
𝛽
Vcell, equilibrium = −
⌣
Δµi total + M i θ α − θ β
(
e
nF
⌣
) = 1.6 M (θ
n
α
i
e
nF
−θβ
) = −K ∑
f
n=2
e
α
Δµi, 1 β
ni3 2
nF
𝛼
—earth—
Range of actual AuCl3 gravity-cell experimental findings:
+0.16 μV/cm to +0.19 μV/cm Page 29 of 34
from (24)
Emergence of Fluctuation
NKBorsuk
— Synthesis —
... at thermodynamic equilibrium, First Domain thermodynamic Forces and Fluxes vanish.
Second Domain Thermodynamics...
• Energy is conserved in Second Domain thermodynamics.
• The Gibbs Free Energy Function is valid in Second Domain thermodynamics.
(albeit with a twist: second domain reactions are spontaneous when Δμi > 0 )
δ 2 H − T δ 2 S = δ 2 µi
wherein Tδ2S < 0 and δ2μi > 0 are spontaneous in Fluctuation Reaction A.
⌣
⎛ 1
1 ⎞
⌣
n
⌣
α
β
α
β
= −2.6 M i (θ α − θ β )
3/2
• Δµi, EFH = − M i (θ − θ ) ⎜⎝ 13/2 + 2 3/2 + 33/2 ...+ ni3 2 ⎟⎠ = − M i (θ − θ ) ∑
n
n=1 i
1
1
1
is the energy supporting...
(1) the 'steeper-than-classical-expectation' equilibrium concentration
gradient of the solute in the electrolyte column in gravity and
(2) the steady-state electric potential of the electrochemical gravity cell: Vcell, equilibrium =
⌣
1.6 M i θ α − θ β
(
nie F
)
.
Δμi, EFH originates spontaneously in Fluctuation Reaction A.
It is distributed throughout the column in the presence of a gravitational field.
• δ2H − T δ2S > 0 is the primal energy source. It produces δ2μi in Fluctuation Reaction A. It originates spontaneously in Fluctuation Reaction A.
• T δ2S < 0 always obtains... ...since δ2S < 0 and T > 0 always prevail in Fluctuation Reaction A.
• Entropy is reduced in Second Domain thermodynamics in electrochemical gravity cells in
proportion to the metric Kf. δ 2S
<0
32
n=2 ni
n
ΔStotal = −K f ∑
It originates spontaneously in Fluctuation Reaction A.
• Is δ2H in Fluctuation Reaction A less than, more than, or equal to zero?
I do not know the magnitude of thermal energy change 'q' in Reaction A. My hunch: δ2H = q such that q > 0; the fluctuation is endothermic and the
electrolyte will decrease in temperature in proportion to the work output of the cell. If Reaction A is such that q ≤ 0, then the work output of the electrochemical cell is
supported solely by TΔS < 0 since ΔS < 0 is innate and T > 0 prevails:
− TΔS = Δμi, f
which is consistent with Δμi, f > 0.
Page 30 of 34
Emergence of Fluctuation
NKBorsuk
— Synthesis —
whatever became of Fluctuation Reaction B ...
Fluctuation Reactions B become the output of the electrochemical cell. They become the
gravity-organized outcome of the primal fluctuation 'Reactions A' relaxing back towards
equilibrium, but delayed in time and remote in space from when and where they originally
occurred. Reactions B are finally realized in the First Domain output of the electrochemical
cell; they are indefinitely delayed and made remote from their originations with Reaction A.
Second Domain
𝛽
Reaction A
energy output = First Domain
⇒
Δµi, f = δ 2 µi, f > 0
𝛼
Page 31 of 34
Emergence of Fluctuation
NKBorsuk
— Salient Matters —
Vf
column of saltwater
⇒
⇒
equilibrium solute-cluster fluctuation
existing knowledge...
a.
equilibrium is exotic: thermodynamic forces and fluxes vanish; cluster-fluctuation and gravity persist.
b.
solute-cluster fluctuations Vf are discrete, speciated, physical objects:
Vf : {ni , Δ𝝆f }
( ni is the number of solute particles of species i )
( Δ𝝆f = 𝝆f ⋅ ni −1/2 is the standard density deviation of subvolume Vf )
c.
entropy spontaneously decreases in solute-cluster fluctuations: δ 2S < 0
d.
chemical potential spontaneously increases in solute-cluster fluctuations: δ 2μ i > 0
e.
relative gravitational potential of solute i in solution:
Ei = Mi ⋅g⋅h
( Mi is derived from Guggenheim; Thermodynamics; 1967; line 9.09.2).
findings from study...
A. fluctuations Vf have gravitational potential in gravity: Ef = Ei ⋅ ni −1/2
B. solute chemical potential is coupled to fluctuation Vf gravitational potential: δ2μi = −Ef /ni = −Ei ⋅ ni −3/2
C. the sum of coupled potentials dominate the macrosystem at equilibrium: −3/2
Δμi = −Ei ⋅ (1
−3/2
+2
−3/2
+3
+ ... ni −3/2)
conflict — the equilibrated saltwater column in gravity...
classical expectation: Δμi = −Ei
my finding: Δμi = −Ei ⋅ (1
−3/2
−3/2
+2
−3/2
+3
+ ... ni −3/2 )
𝛽
gravity-cell Voltage — with equilibrated saltwater column...
classical expectation: V = 0
my finding: V = Ei ⋅ ( 2
−3/2
−3/2
+3
+ ... ni −3/2 )/neF
V
− earth −
𝛼
Page 32 of 34
Emergence of Fluctuation
NKBorsuk
Final Thoughts
everybody is right...
Gibbs is right; equilibrium fluctuations are insignificant.
Prigogine is right; equilibrium fluctuations are inconsequential.
Guggenheim is right: equilibrium fluctuations are negligible.
ΔS always increases.
But only in First Domain Thermodynamics.
there is more...
When the electrochemical gravity cell is at or very near the difficult-to-attain and exotic extreme of perfect thermodynamic equilibrium where First Domain thermodynamic Forces and Fluxes vanish
and
the system is at rest in a gravitational field,
then Second Domain Thermodynamics prevails:
from micro...
Equilibrium solute-concentration fluctuations are fluctuations in mass density; this is consequential.
Equilibrium solute-concentration fluctuations couple to gravity; this is consequential.
Equilibrium solute-concentration fluctuations involve a spontaneous decrease in entropy; this is significant.
Equilibrium solute-concentration fluctuations involve a spontaneous increase in solute chemical potential; this is significant.
to macro...
Two 'fluctuation energies' are coupled: 1) the gravitational potential of the fluctuated subvolume and 2) the chemical potential of the solute within.
The coupling of the two energies distributes the microscale chemical potential fluctuations
throughout the macro length of the column in gravity.
The resultant equilibrium solute-concentration gradient is steeper than classically anticipated.
Electrochemical gravity cells will exhibit a steady-state voltage due to it.
Second Domain Thermodynamics can impact our First Domain electrochemical reality; this is a powerful outcome.
Page 33 of 34
Emergence of Fluctuation
NKBorsuk
a final note...
In my opinion, the most radical element in my work is the proposition of the gravitational
potential of solute-cluster fluctuations. But the proposition isn't really all that radical; at most, it is a
modest idea. It should rankle no one.
I feel no distress concerning potential losses of knowledge or of fact as a result of my studies
since there are no losses of the sort happening, insofar as I know. Actually, it seems to me that there
is only gain. Solute-cluster fluctuations—long known to theory—are now physical objects, and
their gravitational potentials are new energies in the scientific accountings of such things. The
Second Law of Thermodynamics is bicameral: The first domain, the familiar one, encompasses
reactions in which entropy spontaneously increases; the second domain is new, wherein entropy
spontaneously decreases. The two domains are coterminous at the equilibrium state, and they
connect, of course. Gravity makes the difference as to which thermodynamic domain dominates the
macro system under study.
NKBorsuk 28
February 2017 28
www.nkborsuk.net
Page 34 of 34
[email protected]