Entropy’s Great... Oversights, PART 1

SECTIONS

Entropy’s Great... Oversights

The Second Law of Thermodynamics

Oh, My Ergodic Hypothesis...

Brownian Motion, “Brownian Entropy” and “Entropo-dynamics”

The “Arrow of Time”

Entropy as “Disorder”

A Gedanken Experiment

Thermodynamic Equilibrium

“Time Reversal”

“Hyper-Dimensional”, “Hyper-Topological” Time

Digression: Maxwell’s Demon

The Second Law of Thermodynamics

The physics of thermodynamics in general, and its 2nd law in particular, are actually an extremely complex study. Almost all of its evolution has occurred in the last 200 years (of the 2nd Millennium). Its names read like a who’s who of physics: Kelvin, Carnot, Clausius, Boltzmann, Maxwell, Gibbs, Planck, Bose, and even Einstein, the Elvis of Science.

One of the principle... oversights in thermodynamics, in fact in entropy, one that remains unacknowledged, is one that most scientists will dispute, at least at first. They will say that it is not an oversight, even of the “ordinary” sort. We will spend a bit of time clarifying this.

One of the simplest of the accepted expressions of the 2nd law of thermodynamics, the law of entropy, is:

•  In an isolated system:  ΔS 4 ΔQ/T > 0
  where S is entropy (a “state function” of the system), Q is heat, and T is the absolute temperature at which the heat is transferred (and the symbol Δ represents change, e.g. in entropy)
  
  A state function does not depend on e.g. any history of system activity, or on outside influences; in thermodynamics, the state of the system is determined by the positions and velocities of all the particles. The qualification of it being an isolated system is needed because increases in the entropy of a system can easily occur by the transfer of heat to-from something outside the system (this is how heat engines work, although it depends on where you draw your system boundaries), unless it is isolated. It is a measure of how much of a systems energy is being converted to heat, which tends to make it less accessible to do work (in the sense of force times distance).
  
  We should also point out that the definitional expression for entropy above depends on its being derived from “equilibrium thermodynamics”, where... this can get confusing... all heat energy transfers take place between two heat reservoirs that are at the “same equilibrium temperature” (which transfers are theoretically impossible), and thus making the single absolute temperature T unambiguous (except e.g. when one takes into account Maxwell’s distribution of molecular kinetic energies; a subtle point, but eventually it will be seen to be essential). One of the reasons this gets sticky, as yet unappreciatedly, is that — classically — heat can’t be transferred when there is no temperature difference; i.e. net heat transfer is classically zero if the temperature difference is zero. There is always some heat energy being transferred each way, even at “equilibrium”, but it “averages out to zero”. (Reminder for emphasis: temperature is a classical thermodynamics concept, i.e. a macroscopic level concept, the average kinetic energy of a sufficiently large ensemble of molecules. Physics has generally ignored the problem of working out how “temperature” needs to be conceived at the microscopic or molecular level, where Maxwell’s distribution won’t give us an average over a sufficiently large macro-ensemble.) So the usual practice has been to say that there is an “infinitesimal” temperature difference, etc. This is on the macro level. BUT... even if we consider only “equilibrium thermodynamics”, on a microscopic level net energy transfer between-among molecules essentially always takes place with a non-equilibrium temperature difference between the molecules (taking the kinetic energy of each molecule as its “average kinetic energy” and thus its absolute temperature), thus making the concept of entropy based on the fundamental equation above locally meaningless and invalid. And, even if we gloss over this fact, this energy transfer is often anti-entropic (since heat is often transferred from a lower temperature molecule to a higher temperature molecule, even though it is more often transferred the other way). This actually means that studies of “non-equilibrium thermodynamics” are essential to studies of  “equilibrium thermodynamics”, which has not yet been noticed, let alone fully appreciated, by physics.

What is perhaps the main... oversight in entropy, the 2nd law of thermodynamics, can be summed up in one word (one likely to provoke controversy in this context) :

•  ergodic

This is where scientists might start to say that “but we knew that already”. The term “ergodic” comes out of statistical mechanics, a branch of thermodynamics where the mathematics of the “disorder” of molecules is related to — bridged to — the classical treatment of thermodynamics (which doesn’t theorize in terms of molecules and atoms). We will attempt to explore this more in a bit, but it is very complex field, even for physicists. So let us first sum up in one word that both scientists and non-scientists are sure to recognize:

•  gambling

Everyone knows what a “house of gambling” is, and everyone is familiar with the fact that you can count on “the house” having an “edge”, an advantage that means that in the long run they will have an “overwhelming propensity” to “win”. But, everyone also knows that there are chances for a “lucky gambler” to “win”, and even to sometimes “break the bank”. This wisdom is not expressed, nor even allowed for, in the standard 2nd law of thermodynamics, the accepted law of entropy.

The standard statement of the 2nd law of thermodynamics is more-or-less equivalent, in gambling terms, to saying that:

•  the gambling house of entropy not only has an edge,
  entropy wins every bet,
  without fail,
  never losing even once.

As much as one may “subjectively” feel this to be true of “one-armed bandits” (an “emotional truth”?!), we generally acknowledge that the ordinary gambler will win at least a few bets before losing... her blouse.

Of course, our usual simple model of gambling has a terminal state for each gambler, who can “go broke”, permanently; the house can also “go broke” permanently, but with greatly lesser probability of doing so than the usual gambler. Unlike in our model of thermodynamics, in our usual simple gambling model these “broke” states do not have the possibility of changing back into “in the game” states. So, in this at least, our gambling simile leaves something to be desired, since in thermodynamics, even if the system achieves any (accessible) highest entropy state, it will still be able to eventually transition back to any (accessible) lowest entropy state, and vice-versa.

This last idea, of being able to transition not only from the lowest entropy states to the highest, but from the highest to the lowest, and what’s more, to do it on a continuing basis, is inherent in the thermodynamic concept of “ergodic”. As we will see below, it is trivially true that:

•  An ergodic process or system is inherently incapable of having a monotonically increasing (or decreasing) state function, such as entropy.

And even though scientists “already know” that this falsifies the 2nd law of thermodynamics, they still hold that the 2nd law is a law of science, and not merely of that pragmatic but ultimately artistic field of study, engineering. Both science and engineering suffer from this attitude, and we all suffer accordingly because every day science is exercising — or justifying the exercise of — ever more total control over “order” and “disorder” in our daily lives. (The Bible has some important things to say about “all rule and all authority and power”. See 1st Corinthians 15:24.)

It sounds satirical because it most naturally is in this situation, but when we try to find what ultimate truth there is in the second law, we will eventually find it to reduce to the statement of the fact that more highly probable events occur with greater probability and lower probability events occur with lesser probability, but they all occur. Let’s make that slightly more formal:

•  Thermodynamic entropy, a state function S of an ergodic process, is actually a (relative?) measure of the relative amount of time that the process will spend in a given state (or perhaps set of states). The reason ΔS seems to increase montonically — or “quasi-monotonically” if we think of the increase in terms of Boltzmann’s  concept that it happens with “overwhelming probability”, not with the strict “inevitability” of the classical second law — is that, since the ergodic process will on the average spend more time in high entropy states, it follows that if it’s in a low entropy state, on the average it must head toward higher entropy states in order to spend more time there. If that sounds a tiny bit circular, it is. This approach to analyzing entropy has yet to be noticed by physicists, and it will take a major but absolutely necessary revision of statistical mechanics to make it satisfyingly formal.

But there is more, much much more. We need to uncover many more... oversights, and start to evaluate their synergistic effect on science.

SECTIONS

Entropy’s Great... Oversights

The Second Law of Thermodynamics

Oh, My Ergodic Hypothesis...

Brownian Motion, “Brownian Entropy” and “Entropo-dynamics”

The “Arrow of Time”

Entropy as “Disorder”

A Gedanken Experiment

Thermodynamic Equilibrium

“Time Reversal”

“Hyper-Dimensional”, “Hyper-Topological” Time

Digression: Maxwell’s Demon

Oh, My Ergodic Hypothesis...

When the term “erogodic” was used above, many people who consider themselves scientists almost certainly said to themselves “we know that already”, and mentally dismissed the whole show, perhaps even failing to read on. This is partly correct, since the “ergodic hypothesis”, which came out of the work of Poincaré and Boltzmann, and is considered to have provided the foundation of the statistical mechanics of Boltzmann and Gibbs, is over a hundred years old.

The Ergodic Hypothesis (usually formally attributed to Boltzmann) has many statements known or thought to be equivalent. One of them is that every thermodynamic system is ergodic, i.e. it either has a large finite number of states or microstates, each of which — if “accessible” — will be revisited infinitely many times as time increases, or is an infinite system in which every microstate will be approached arbitrarily closely any number of times as time increases.

Relatedly, Boltzmann, after some years, changed his earlier stand that his statistical mechanics (based on the work of James Clerk Maxwell; the American Josiah Willard Gibbs Jr. later also developed statistical mechanics independently) showed that entropy would always increase (with the relevant qualifications) to his later stand that there is an “overwhelming propensity” for it to increase. The ergodic hypothesis is considered to be a (an) hypothesis since it is thought that the time needed to effectively observe this phenomenon (as usually conceived) in the laboratory would take more time (perhaps 10³⁰ years, as suggested by Richard Feynman) than the universe is expected to be in existence.

But what is the “ergodic hypothesis”? Let us give a quick series of definitions:

•  a “stochastic process” is a probabilistic process with a parameter that more or less represents time; a stochastic process can be finite, i.e. having only a finite number of states, or continuous, i.e. having effectively an uncountably infinite number of states

•  a “Markov process” is a stochastic process, a combination of a set of states, one of which the process is said to be “in”, with a set of probabilities for transitioning between those states that depends only on the state that the process is said to  be in; (this last condition distinguishes it from the more general concept of stochastic process); any state that has a sequence of state transitions with non-zero probabilities leading to it from the current state is called “accessible”

•  an “ergodic process” is a Markov process in which every state that is accessible (and remains so throughout the “trajectory”, the sequence of states and-or state transitions that the process goes through) will be revisited infinitely many times, with probability 1 (effective certainty, Boltzmann’s overwhelming propensity)

•  the “ergodic hypothesis” in thermodynamics is that thermodynamic systems are ergodic, i.e. a given (accessible) state of the system (given in terms of thermodynamic micro-properties, e.g. particle momenta) will recur any number of times (even if it is a lowest entropy state), even though it may take vast periods of time to do so (Richard Feynman suggested 10³⁰ years for a thermodynamic system such as a room full of air at STP)

This much (just above) is publicly accepted by physicists.

If asked, physicists will say that the ergodic hypothesis does not really falsify the 2nd law of thermodynamics because... well, because of the overwhelming propensity thing, and the 10³⁰ years, and so on. But, in fact:

•  We have a situation that reminds us of that joke:
  “... We’ve just determined that. Now we’re just haggling about the price.”

What has been overlooked? even... oversighted?

•  What has been... oversighted in physics is that, in fact:
  
  Ergodicity really does falsify the 2nd law of thermodynamics,
  
  i.e. of strictly monotonically increasing entropy,
  despite the fact that physicists don’t think of it as doing so.
   

•  It turns out that physicists have... oversighted that:
  
  Ergodicity also implies the overwhelming propensity of change in the opposite direction, that of decreasing entropy;
  
  even if it only seems to be a far distant eventuality, decreasing entropy is an inevitability, ergodically-thermodynamically, and even if the system will spend almost all its time in the high entropy states. A thermodynamic system will ergodicly re-attain even the lowest entropy levels an infinite number of times as the system evolves, and whenever any level is re-attained it means that the sum of the increases and decreases (from any of the previous times it was at that level) is strictly zero.
   

•  Entropy is an
  
  “Inevitability of Spending More Time at ‘High’ Levels Property”,
  
  and not the currently espoused “inevitability of the increase of the entropy state function property” (any more or less than it is an “inevitability of decrease property”).
   

•  The second law of thermodynamics is currently (but only temporarily?!) adequate as an engineering at the classical physics level of existence law,
  but, as a law of science,
  
  The 2nd Law of Thermodynamics is SCIENTIFICALLY FALSE.

We might continue to overlook this fact on the same grounds (of the overwhelming propensity of what we notice, etc.), except that we are dealing here with science, and this overlooking would be — is — a falsification of science. Also, e.g., this... oversight has profound impact on other parts of physics, particularly on the much loved (-not-wisely-but-too-well) concept of “the arrow of time”.

More on this later, after we first go back over some of the basics.

•  An ergodic process is not only a Markov process,
  an ergodic process is a stochastically cyclical process, and

•  a state function of a cyclical process is a cyclic function, and

•  a cyclic function cannot be strictly monotonically increasing;

and it is generally agreed that:

•  a thermodynamic system is an ergodic process,

which means that (since entropy is a state function of such a system)

•  entropy is a stochastically cyclical state function,

which means that

•  entropy cannot be a strictly monotonically increasing (or decreasing) function

If someone tried to make the case that the paradigmatically standard cyclical trigonometric function sin(ax) was a strictly monotonically increasing function, we would question his sanity, or at least the quality of his education. If he then tried to make the case that it would be strictly monotonically increasing if the coefficient a of the variable x were small enough, say a = 10^-30... well, at best we would probably smile at his childlike naiveté (or his childish naïveté, or some such). In any case we would not allow it as either competent mathematics or competent science.

But, if he picked a stochastically cyclical function, with an implicit coefficient a  = 10^-30, and displayed it waving the flag of science, well, then it would be called... the 2nd law of thermodynamics.

Let us note that in general:

•  any ergodic system violates a generalized 2nd Law (as regards a state function such as entropy)
  (It must be generalized to make sense in all cases.)

•  any state function of any ergodic process or system has no long term tendency to increase or decrease;
  pick an ergodic process or system, pick a state function, pick a state, find the value of that state function for that state, and note that since that state is revisited an infinite number of times and the state function takes on that same value an infinite number of times, always decreasing-increasing as much as it increased-decreased to re-attain that value, there can be no long term tendency of that function to increase or decrease

We should also note that the concept of “tendency to increase” in this context is not actually well-defined. No one has carefully delineated what this actually meant. For example, if we put our ergodic process in an arbitrary “lower entropy” state, and do this many times, it might, but not necessarily so, tend to transition to a “higher entropy” state. (How many state transitions from the original state must we observe to calculate this tendency?) We could call this a “tendency to increase”. We could also look at the state function values for all the states of that same lower level entropy, but only one time for each state, and get a different picture of the tendency to increase. And, as we saw just above, we could also put the ergodic process in that first state and watch it for a very long period of time, see it revisit the “lower entropy” state any number of times, and note that this means there is no “long term tendency” for it to increase or decrease.

•  The “tendency to increase” of a state function of an ergodic process is NOT WELL-DEFINED.

•  We have also overlooked that we have not (well-) defined how we assign a given entropy state function value to a given state. We may think we have, but does it accurately reflect either the “tendency to increase”, which is obviously not well-defined, or the “tendency to spend time in that state”, which is “more well-defined”.

It can also be noted in passing that technically:

•  An ergodic process is a theoretical variant of a classical perpetual motion machine.
  (Although thermodynamic systems model molecules, or at least atoms of an idealized noble gas, that theoretically go on bouncing around forever, this is not considered useful work, required at least implicitly in most statements of perpetual motion. But if a thermodynamic system — isolated, for obvious reasons — re-attains any state, then it re-attains infinitely many times 1 or more states associated with useful work, or its possibility, even if the whole kit-n-kaboodle also eventually goes down a black hole or into a “big crunch” infinitely many times.)

There is another essential factor in this picture. Above we have looked at 2nd law violating entropy decreases over the long run, but there is a short run, as well. This short run can be summed up in one word:

•  Brownian

SECTIONS

Entropy’s Great... Oversights

The Second Law of Thermodynamics

Oh, My Ergodic Hypothesis...

Brownian Motion, “Brownian Entropy” and “Entropo-dynamics”

The “Arrow of Time”

Entropy as “Disorder”

A Gedanken Experiment

Thermodynamic Equilibrium

“Time Reversal”

“Hyper-Dimensional”, “Hyper-Topological” Time

Digression: Maxwell’s Demon

Brownian Motion, “Brownian Entropy” and
       “Entropodynamics”

Another... oversight in the standard 20th (and now early 21st) Century concept of entropy is Brownian motion. When we have small visible particles in a clear liquid, we can see them move in some “orderly” fashion with the flow of the liquid (usually no flow, of course, when we look for Brownian motion), but also erratically, randomly in “small” movements. By serendipity, luck or God’s design, e.g. the sizes of atoms and particles, the distances moved, and the sensitivity of our eyes to detect these movements, we can even see these movements unaided by microscopy, although we could have considered ourselves very lucky even if they could only have been detected at all.

Rarely, but inevitably, these random motions can take a given particle a great distance in a short time. By the same token, large macroscopic particles (think of lots of very small ones, however distributed), although they usually experience only small changes in momentum-energy, can experience very large changes very quickly. These changes are almost always, but not always, damped by e.g. viscosity-like substances too quickly to be observed. And, to top it off, any such motion that was also observed would most likely be attributed to something else (e.g. to hydrodynamics rather than “entropodynamics”), which is probably all too common in science. Chaos theory in part deals with some of the wilder extrapolations of these types of combinations of lots of such seemingly small events, e.g. from the motions of a butterfly’s wing to the “precipitation” (more humorous, but also probably much more accurate than “causation”) of a hurricane (uhh, tornado, actually), i.e. from, or at least through, Brownian thermodynamics to classical hydrodynamics.

•  To anticipate, somewhat: there is an essential set of relationships among 1) 2) thermodynamics, 3) Brownian motion, 4) entropy with its overlooked “Brownian Entropy”, and 5) chaos theory, synergistic relationships that have been stoutly overlooked by the scientific community.

Brownian motion is considered to be fairly well understood by science, and is held to verify the atomic and thermodynamic nature of matter. We have ignored some quite obvious implications of Brownian motion for thermodynamics, but at least it should be easy to begin applying that theory to e.g. any probabilistic kind of entropy.

Just as the position of a small particle in a liquid is made to jiggle so erratically that some scientists model the trajectory (here of the particle) as having its first derivative discontinuous almost everywhere, so entropy (even in an isolated system) rapidly and erratically alternates direction (here in 1 dimension), decreasing for an instant, increasing for an instant, etc., even — or especially — if the system is at “thermal equilibrium”.

It is somewhat as if we placed a particle experiencing Brownian motion in a river flowing to the sea: the particle often slows its downriver journey, and even goes upriver briefly but often, but on the whole we will usually see that its course is down to the sea. Or will we?! It depends on how we look. Sometimes the coin that is weighted to show heads 99.99% of the time will come up tails tens of thousands of times in a row, millions, billions... and just so, our particle will — on rare occasions to be sure, but with statistical certainty — make it back upriver all the way to the source of the Nile.

Except for the damping, these “quantinuous” Brownian changes (in both directions) in entropy are a lot like cumulative noise on top of the signal of the main flow of the increasing entropy of the 2nd law. The “cumulative” is important because, even if the Brownian changes are great, although the system may-will try to damp them micro-viscously as they are happening, just as with standard viscosity, after they have occurred the system will not try to “put ’em back the way they was.” (Actually, this last will need to be questioned carefully when “Brownian entropy” is eventually studied carefully. There might very well be quantum mechanical effects that are... paradoxical.) But, too, we should not fail to think of chaos theory and its extravagant exceptions to those viscosity-like damping effects.

•  Repeated for emphasis: there is an essential set of relationships among thermodynamics, Brownian motion, entropy with its overlooked “Brownian Entropy”, and chaos theory that has been seriously overlooked by the scientific community.

Gedanken Experiment: Let us look at a common type of collision of idealized particles with the same mass, but different velocities, momenta and energies. This kind of collision is studied in physics classes, but usually in the context of a class on mechanics. If we study it only from the standpoint of mechanics, we do not find any problem. But if we study it from the standpoint of thermodynamics, i.e. thinking of the 2 spheres as atoms (e.g. of helium), it is clear that it has never been fully appreciated publicly for what it is, a counterexample to the second law of thermodynamics and a clear statement from Mother Nature that we need to start a new field of study: “entropodynamics”.

Figure 1a.
In Figure 1a, we see a higher kinetic energy particle-body (body 2, with velocity 2v) approaching a lower energy body (body 1, with velocity v).

Figure 1b.
In Figure 1b, we see them collide like billiard balls in such a way that the center of mass of body 2, the body with the greater kinetic energy, is in the direct line of motion of the center of mass of body 1, the less energetic body. If body 2 had no velocity and therefore no kinetic energy at this point, we would expect a standard billiard ball collision that would transfer all the momentum and kinetic energy from body 1 to body 2, leaving body 1 with a kinetic energy corresponding roughly to “absolute zero” (if we didn’t take into account anything else).

Figure 1c.
In Figure 1c, we see that particle-body 1 is (almost) motionless, with its “average kinetic energy” placing it at a temperature near absolute zero (if it were e.g. an atom of helium).

It is clear that there exist ranges of collision angles (over 3 dimensions) that will yield anti-entropic mechanical transfer of kinetic energy from a lower energy particle to a higher energy particle (not to mention e.g. quantum mechanical possibilities). These will have a lower “cross-section” than the ranges of angles that yield entropic exchanges, thus favoring the increase of entropy (up to the system’s “maximum accessible entropy”), but this still allows for the system wide sum in a given time interval to itself be anti-entropic (especially for smaller time intervals, of course).

•  Not even the simplest model — spherical-atomic, ideal gas laws, etc. — of the micro-dynamics of “Brownian entropy” has ever been explicitly studied by scientists, e.g. to the extent that kinetic energy distributions are modeled in Maxwell’s distribution equation. How did Maxwell, Boltzmann, Gibbs, et al, overlook the particle dynamics shown in Figures 1a-c? How did they overlook the immediate implication that (at least something like) Brownian entropy, with its entropy change direction reversing “almost everywhere”, must exist? How did they overlook that entropy does NOT truly have a tendency to increase, but instead to spend more time in some sets of states than in others? The answer may be that they couldn’t bring themselves to take the decidedly Politically Incorrect stand that the 2nd Law of Thermodynamics, which already had a sacred cow status, was... false. Newton may have had a similar problem with the fact that his theory predicts that lighter and heavier bodies fall at different rates.

One of the accepted statements of the 2nd law of thermodynamics is that “there exists no device that can transfer energy from a cooler body to a hotter one with no other result”, but the above gedanken experiment disproves this:

•  There does exist a “device that can transfer energy from a cooler body to a hotter one with no other result”,
  in violation of the 2nd law of thermodynamics:
  
  LUCK...
  
  (and another concept disfavored by science: “vitalism”. Mechanism may give us propensity, but “vitalism” — not well-defined, to be sure — gives us essence, and I for one feel sure that we will soon find that “vitalism” can change those entropically preponderant probabilities and their emergent behaviors. Although we will need to revamp our concept of Life being a local reversal of increasing disorder-entropy, the question of the possibility of “vitalism” re-deeming and re-directing the emergent behaviors of mechanism is still valid. In other words, the behaviors that emerge from mechanical systems that adapt and become ever more complex themselves become the vehicle (in a sense reminiscent of the “vehicles” we find in Buddhism: Hinayana, Mahayana, Vajrayana, etc.) Who knows? Perhaps we will soon find that Life can make Luck as well as Love...)

•  This study of the micro-dynamics of “Brownian entropy” should have been, and today should be, a top priority in physics. It is prerequisite e.g. to modeling the quantum mechanical aspects of molecular-atomic interactions.

•  It should also be obvious that this (Figures 1abc) is a non-equilibrium microscopic sub-situation, variants of which can and must often take place in macroscopic equilibrium situations. (At classical equilibrium Maxwell’s distribution guarantees an arbitrarily wide range of molecular kinetic energies.) What is the “temperature” at which the “heat” energy transfer takes place?! At the average kinetic energy of the 2 (in our case) atoms is a simple answer, but non-equilibrium thermodynamics is not always so simple. This is a perfect point of study for basic non-equilibrium thermodynamics that has application to equilibrium thermodynamics, and-or vice-versa.

•  This brings up another... oversight relating to entropy: the usual definition of entropy has a non-zero net flow of heat at the equilibrium temperature of 2 bodies, and, classically, heat cannot have a non-zero net flow of heat if there is no temperature difference. We can talk all we want to about “infinitesimal temperature differences” and “limiting cases”, but we are here dealing with a Gedanken Convenience Concept that doubles as a theoretical concept. When it does so, we are in grave danger of reasoning from false premises within our theory and therefore as part of our theory. “That’s a bad way to fly.”

It should be obvious from this simple analysis that this kind of reverse entropy collision is very common. In fact, without these law-of-entropy-violating kinetic energy transfers, Maxwell’s distribution of kinetic energies could not be maintained since the kinetic energy of each particle would move (erratically) toward the average and (eventually) stay there. I.e. in each 2 particle collision, the kinetic energy of each particle would have to monotonically (but not strictly) move toward the average of the kinetic energy of the 2 particles, inevitably reducing the highest energies toward the average, and likewise with the lowest energies.

•  Brownian Entropy is a fundamental underlying mechanism for the maintenance of Maxwell’s distribution of kinetic energies, which otherwise would collapse toward the average.

•  A generalization of Brownian Entropy will eventually be accepted as an underlying mechanism in the various “power law” distributions, (power as in exponential, not as in work per unit time; the value of 3/2 seems to always magically appear as an exponent, as it does in Maxwell’s distribution equation). E.g. lower energy earthquakes are far more frequent than high energy earthquakes, and the distribution stochastically follows a decreasing exponential. When the micro-dynamics of earthquakes are studied more carefully, the release and storage of the energy will be seen to undergo Brownian Entropy style “quantinuous” reversals throughout the fault system spatially, and throughout time as well, instead of the current concept of “continuous” buildup to stochastically determined “continuous” release with a power law distribution of the size of buildup before release. The usual model ignores that the buildups and releases are stochastically overlapping in both space and time.

Digression: single collisions (as opposed to averaged larger numbers of collisions) of atoms and molecules (as opposed to much larger particles) are a good place to look for quantum effects taking precedence over our simple Newtonian mechanical picture.

We can conceive of the Brownian motion particle as being distributed through the whole volume, e.g. of an ideal gas. (The same arguments can be extended to liquids and even solids.) When entropy-violating collisions occur, their contributions to system entropy are negative. If we get enough such collisions — which, remember, instead of striking a single larger particle as in standard Brownian motion, are distributed throughout the volume of e.g. gas — at the “same” time, we get an entropy-violating reversal of the entropy in the system as a whole.

Just as a standard particle is kicked around in 3 dimensions by standard “Brownian motion”, so our distributed particle is kicked around by our distributed “Brownian motion”, although not always in such a way as to reverse entropy. But the “Brownian motion” of the distributed particle corresponds to changes in the value of the state of the system, and these correspond to changes in its entropy, a state function of the system, and just as with standard Brownian motion, the entropy will be kicked into at least small excursions in the reverse direction quite frequently (which might show up mostly as an erratic slowing of the increase of entropy if entropy is increasing rapidly enough on the average).

So, it is highly likely that standard analysis methods for Brownian motion can be adapted, perhaps readily, to the study of:

•  “Brownian entropy”

the study of which, along with ergodic entropy, we can term:

•  “Entropodynamics”

as it is a thermodynamic variant of hydrodynamics (one of many possible).

This new concept of “Brownian entropy” suggests strongly that, just as we have Maxwell’s equation for the distribution of energies in an ideal gas at equilibrium, we need to develop:

•  “entropodynamics” — a new subfield of thermodynamics —
  to study the equilibrium and non-equilibrium distributions of changes in thermodynamic entropy (or perhaps a revised concept of it), which will in general depend greatly on systemic peculiarities (in much the same way that hydrodynamics does).

•  For example, just as the Brownian motion of a particle occurs in 3 dimensions, we should look for the spatially distributed Brownian motion of entropy to have higher dimensionality, i.e. for entropy to be a multi-dimensional as opposed to a scalar state function.

The probability of a large random reversal of entropy in an isolated system may be small in a small interval of time, but the difference between Einsteinian and Newtonian gravity is “small”, too. In both cases that “small” is (usually) negligible to the engineer, but essential to the scientist. Today (at the beginning of the 21st Century), engineering — but not science — can maintain the 2nd law of thermodynamics as an approximation with suitable strictures concerning the validity of the scope of its application, and can do so without shame. E.g., an engineer today need not have a care to the cosmic time scale needed to produce a reasonable probability of a cosmic scale entropic reversal. Likewise, a suitably small thermodynamic system can cycle entropically many times a second, but today this is still of little practical engineering significance.

So, where is this result of “Brownian entropy” important? It is important in any case theoretically-scientifically, and for the engineer it may be soon that “Brownian entropy” becomes a pragmatic consideration in the nanoengineering that is all the rage these days. “Brownian entropy” is most likely to be observed in systems with small numbers of particles, and with values averaged over very short periods. Physicists will eventually, and hopefully soon, try to observe these “Brownian” entropy fluctuations experimentally and model them in detail.

Although engineers may not be there today in terms of needing to engineer very small or anti-2nd law systems entropically, they will be tomorrow. Eventually, one practical field of application for “entropodynamics” will be the rapidly evolving science and engineering of nanotechnology, where this non-quantum mechanical “Brownian entropy” will likely join synergistically with quantum mechanical effects to make for “living in interesting times”.

It should now be obvious that, both theoretically and actually, “Brownian” fluctuations in entropy must occur. Brownian motion is considered to be fairly well understood by science, and is held to verify the atomic and thermodynamic nature of matter. We have ignored the obvious implication of Brownian motion, that it is almost a description of the microcosmic level of the macrocosmic ergodic actuality of thermodynamic entropy. It should be easy to begin applying the theory of Brownian motion to any flavor of entropy (even non-physical flavors).

“Brownian entropy” is very important scientifically yesterday, and it is sure to be important for engineering in the future, but for now we can note its inevitability and the necessary violation of the 2nd law of thermodynamics that it entails.

Boltzmann et al to the contrary notwithstanding, the concept of entropy has not made a successful migration from the classical-macroscopic level of thermodynamics to the quantum atomistic-molecular (not to mention quantum mechanical) level of our modern understanding of matter and heat. Not at all. It was first conceived of in terms of classical (non-molecular) “equilibrium” thermodynamics, which meant that there was in affect only an “infinitesimal” temperature gradient (or whatever corresponds to that at the molecular level), and only an “infinitesimal” ambiguity or vagueness in the absolute temperature T in the expression:

•  ΔS 4 ΔQ/T

Not Boltzmann nor Maxwell nor any physicist since has even begun to sort out what should happen to our concept of entropy when we approach the fundamental unit of entropy change, the “elastic” (quantum theory might eventually turn that term on its head) collision of 2 molecules (perfectly spherical, of course, so we have a chance to make it seem like equations will do the job). No one has even noticed that it was essential that we re-conceive entropy in these terms. (Statistical mechanics still totally ignores the existence of Brownian Entropy.) And this is just classical elastic collisions. Statistical mechanics will have to be extended to quantum theory in the wave-particle sense, not just the atomic-molecular sense, when it is extended to “Entropodynamics”.

We can notice that, as is invisibly common in science, the classical concept of entropy (ΔS 4 ΔQ/T) lacks “scalability”, so crucially important in the computer world. There it usually means being able to take a system and get it to work without problems on a much larger scale, but the concept works in both-all directions. (Some computer concepts work well for large systems, but become unwieldy when trying to use them for small systems.) The classical concept of entropy cannot be scaled down to atomic-molecular size levels (see Figures 1a-c), down to very small (non-averaged) time increments and their small (non-averaged) changes in entropy, or up to very large (non-averaged) time increments and their (again, non-averaged) changes in entropy. We may also eventually find that it cannot be scaled up to cosmological size levels.