Mar 29, 2018 - questions, to be discussed elsewhere. I. REALITY ... is strongly objective, quantum mechanical statements can be seen as ... ments are predictive of observations, even when they go ... Recognizing bias, and ensuring no conflicts of in- terest ... we con- sider limiting cases, Blue for external appearance, and.
On Thermodynamics Misconceptions, Part I Edgardo V. Gerck∗ Pasadena City College (Dated: 29 March, 2018. This version supersedes the previous version, dated 12/8/2018.) To study reality modes in physics, we set the stage in two limiting cases, called Blue for external appearance, and Red for underlying reality, critically studying a popular interpretation of thermodynamics. We use the lenses of classical thermodynamics, statistical thermodynamics, information theory, game theory, mathematical finance, biology, psychology, and literature, trying to obtain a more diverse view. We report a set of macro-world processes (not quantum mechanical) that “always win” in the limit, which is, mutatis mutandis, a result in quantum mechanics. The behavior of their dynamical laws, governing micro and leading to their macro, links Shannon’s entropy with thermodynamical entropy. This study may reflect on “Arrow of Time” and thermodynamic reversibility questions, to be discussed elsewhere.
I.
REALITY MODES IN PHYSICS
Almost 85 years ago, Breit and Wheeler [1] were the first to suggest that it should be possible to turn light into matter by smashing together only two photons, to create an electron and a positron, the simplest method of turning light into matter ever predicted. To an observer before that time, it would seem as if matter is coming out of nothing. However, quantum mechanics states that energy is conserved, but mass is not. Mass comes out of energy, giving a new meaning to the notion of objectivity, of reality in natural sciences. Is there a limit to objectivity in natural sciences? Recognizing the importance of reality modes in physics, Bernard d’Espagnat [2] asks, “Is reality something meaningful and is science steadily coming closer to a true depiction of it?” Contrary to classical physics, which is strongly objective, quantum mechanical statements can be seen as only weakly objective [1, 2], i.e. which could be interpreted as intersubjectively valid [2]. However, one sees that quantum mechanical statements are predictive of observations, even when they go against conservation of mass [1], restoring the notion of observer-independent reality. This consideration should be seen in addition to the principle of equivalence in general relativity, explaining how, out of light with no visible mass, gravitational effects (associated with the presence of mass) may appear [1]. In interpretation terms (e.g., subjective, intersubjective, objective, and abstract) of such reality [2], bias often appears as a distortion of reality, and maybe the ultimate challenge facing science as a description of facts.
∗
https://www.researchgate.net/profile/Edgardo Gerck
Recognizing bias, and ensuring no conflicts of interest, are important in science, in order to double-blind or just minimize the influence of bias; the lesson is that we cannot avoid bias, the multifarious manifestations of bias can be pervasive and often undetectable, so one must always be on watch. These two considerations add importance to this study also in terms of reality modes in physics: (1) if science is to steadily come closer to a true depiction of reality; and (2) if bias is to be reduced as close as possible to zero. To better understand this two-prong test, we consider limiting cases, Blue for external appearance, and Red for underlying reality. With (Blue), one takes the Blue Pill and stays in the known but superficial reality; or with (Red), one takes the Red Pill and tumbles down the “rabbit hole,” going deeper into the real world. The names for the two cases come from a scene in the 1999 film “The Matrix.” In the “safe” but fake world (Blue), as popularly written, life seems to follow a collective conception of thermodynamics. There, it seems that one is already victimized by the laws of that nature, one can never win, the most one could do is break even, but even that is forbidden by that nature. The film’s viewpoint (Blue) seems to be that with the Blue Pill, the world — qua that nature, slightly tinted blue as well — is “unwinnable,” because its thermodynamics says there, efficiency cannot be 100% and that a gain must be less than the loss. Do nothing, one loses less. But, let us here take the Red Pill, instead, and go deeper into reality. This work is linked to a current and old discussion in physics, why does one observe an “Arrow of Time”? All physical processes are reversible in the micro-world, in quantum mechanics, but not for systems in the macro-world above, which are not reversible. One cannot undo diffusion, omelets, or any process
2 governed by entropy. As Sean Carrol at Caltech says, “Irreversible processes are at the heart of the arrow of time.” [3] As Carrol explains, the principle underlying irreversible processes is summed up in the second law of thermodynamics: the entropy of an isolated system either remains constant or increases with time. We will continue this topic elsewhere. This work is divided in two parts, with further material to be considered in Part II. This Part will motivate the answer in different areas, not just physics, using a diversity of approaches in STEM, biology, and humanities. In the following, we will examine reality using the lenses of classical thermodynamics, statistical thermodynamics, information theory, game theory, mathematical finance, biology, psychology, and literature, trying to obtain a more diverse view. We report a set of macro-world processes (not quantum mechanical) that “always win” in the limit (affording 100% efficiency in winning, as closely as desired), which is, mutatis mutandis, a result in quantum mechanics, where efficiency is 100% and there are no irreversible processes [3]. Cautionary Note: It is not our purpose to advocate or to criticize the popular interpretation or other approaches: we adopt a philosophically neutral stance. Rather, our aim is conceptual unification in terms of physics, thermodynamical entropy, game theory, and Shannon’s information theory, by generalizing the concept of entropy beyond the thermodynamic framework, and potential applications to understand the “Arrow of Time”, and other questions. II.
CLASSICAL THERMODYNAMICS
Classical thermodynamics [4] is an area in physics, involving the transformation of energy from one type to another. It is out of scope in physics to talk about, e.g., “The perversity of the universe tends to a maximum” [5] — there can be no such law in classical thermodynamics. Because entropy is an extensive property [6], it is not dependent on the amount of material in a system; it is an additive, scalar quantity, where it follows that the principle of maximum entropy must hold for each individual, however infinitesimal, segment of the world-line [6]; I ∂Q ∆S = (1) T where S is entropy, Q is heat, and T is temperature, in the Clausius integral above. This section suffers from the limitation of materials already discovered, so one cannot approach, e.g., zero
loss “as closely as desired” in a limiting process. One does not know if it is because of a temporary absence of current materials, or due to a physical principle limiting their existence permanently. This will be resolved in the next sections, where material limitations are removed. However, classical thermodynamics (macroscopicallybased) does say that efficiency can approach 100%, as closely as materials permit. With materials existing today, efficiency can already be close to 99.99% [4]). The macroscopic thermodynamic loss can, thus, approach zero, theoretically as close to zsero as current materials permit. Therefore, the popular hypothesis is not supported, that, e.g., “Nature is unwinnable.” Research on materials and processes may increase macroscopic thermodynamic efficiency, already close to 100%.
III.
STATISTICAL THERMODYNAMICS
In statistical thermodynamics, entropy is usually defined as [4]: S = kB · ln(P ) + cte.
(2)
where S is entropy, kB is the Boltzmann constant, P is the probability of the microstate, and cte. is a non-negative constant. The conjecture is that (since the macroscopicallybased and the microscopically-based branches of classical thermodynamics must lead to the same results [6]), one will likewise see that the popular hypothesis is not supported also by the statistical approach. We will be able to stop here and prove, in the next sections, a stronger and broader result, approaching in the macro-world a result that has been only reported in a quantum process, achieving zero loss as closely as desired. This could apply to the current discussions on the “Arrow of Time” [3], and reversible thermodynamic processes.
IV.
INFORMATION THEORY
Claude Shannon [7], introduced the mathematical concept of the “entropy” of a random variable or sequence. He defined information theory in a narrow way [7], “In Information Theory, information has nothing to do with knowledge or meaning. In the context of Information Theory, information is simply that which is transferred from a source to a destination, using a communication channel. If, before transmission, the
3 information is available at the destination then the transfer is zero. Information received by a party is that what the party does not expect – as measured by the uncertainty of the party as to what the message will be.” Preceded by the efforts of Szillard [8], who in 1929 identified the unit or ”bit” of information when dealing with entropy and the Maxwell’s Demon problem in physics, by Hartley in 1928 [9] and by Nyquist [10] in 1924, Shannon took a different approach than just positing a behavior for information, as commented in [11]. Shannon’s zeroth-contribution [11], was to recognize that unless he would arrive at a simplified but real-word (valid in nature) model of information to be used in the electronic world, no logically useful information model could be set forth. History of science does not justify science. This defined a way to correlate information theory (i.e, electric signal statistical distribution) with the statistical entropy from thermodynamics, with the intuition that it measures how well behavior can be predicted. This intuition by Shannon, together with the provable existence of universal encoders, was proved in his Tenth Theorem, that there should be an algorithm for predicting the next symbol in a sequence, which is assured to come as accurately as desired: close to 100%, on an open interval. Thus, the results using Shannon’s Tenth Theorem in information theory [7], afford error-free transmission of signals, where signal loss can approach zero, as low as desired; further denying the popular hypothesis.
V.
CHANCE AND GAME THEORY
The information theory definition of entropy has the further advantage of being used in chance considerations, on maximum Shannon entropy versus minimum risk and applications to some classical discrete distributions [12], in game theory, games where one can win or lose. Mass or energy are not relevant here, only the chance. This eliminates the concern expressed in a previous section, from a possible, yet theoretically unknown, limitation of materials. In game theory, one can win any betting game — through the martingale strategy [13]. “This betting strategy has you doubling your bet every time you lose a hand. When you finally win a hand, your bet returns to your minimum and you recover your losses with one win.” [13]
The martingale strategy has been applied to card games, and roulette as well, where the probability of hitting either red or black is close to 50%. It works on limited runs, but seems to fail if the gambler does not have some “infinite” time and “infinite wealth” — which is a mathematically valid argument, but not in physics, in the “Red Pill” real-world, where money itself must be finite, as any resource. Rather, a gambler should be spending money securely (without care, as he is sure to win when he leaves the game after just a single win), not boisterously, playing yet another hand, saying, “I’m lucky!” To make matters more precise, let us consider a bet of $1 on one drawing of event A, with 1:1 payoff, and P(A) = 40%, where P(A) is the probability of winning on event A. It would seem that one most likely loses. However, consider two events in a row, where the probability of an event is independent of the draw order (no history assumption: ergodic, the same chance in every event), where player Z follows a martingale strategy and ends the game unilaterally (e.g., walks away) after a single win. The probability of player Z winning the first time is 40%, ending the game and leading to $1 profit; losing first and winning second has a chance of 24%, where player Z ends the game and leads to $1 profit; losing both has a probability of 36%, and is deferred to a next round, as many rounds as needed until a win occurs, when winning once recoups all the intermediary losses, with player Z ending the game with $1 profit. Using the martingale strategy, player Z enjoys a probability in the open neighborhood of zero, meaning as close to zero as desired, for losing on the total bet, ending with $1 profit in all cases. The game can be repeated in parallel and in series, and $1 could be $1M, or any amount desired. With sufficient capital (always possible; infinity is not an occurrence in reality, although considered in mathematics), usually OPM (other people’s money), player Z can defer a loss until its probability is small (being just compounded of the previous result), allowing enough time for player Z to win, with increasing probability, as close to 100% as needed. It is not that each successive failure has to be reduced in probability by some “magic,” relaxing to a result; there is a “no history” assumption, and one can have a string of failures, at any time. This is often an argument against using a martingale strategy [13]. But, as one proceeds into the strategy, in the conditions noted above, parallel success channels start to open up, at the rate of 1, 3, 7, 15, ... while there is no
4 competing growth on the number of failure channels, which remain at a single value. This ‘path engineering,” chosen by player Z, adds up to the success probability tending to 1, while the total probability (success and failure) must remain at 1, squeezing out the only failure probability channel to zero, as closely as desired.
and that the log prices follow a Brownian walk with normally distributed innovations. The formula is not affected by any linear drift in the random walk. Usually, the formula [16] is expressed as, C = N (d1 )S − N (d2 )Ke−rt , 2
For example, one already reduces the probability of losing from 60% to 36% in two steps, to 13% in four steps, and to 1.7% in eight steps, at the simple investment of more capital, while the capital is secured by playing again if needed, all but assuring a return of 100%, with a single win. This strategy is not openly allowed in casinos [13]. But, this strategy can be used covertly, and even used openly in capital markets with instruments such as financial derivatives, with the additional tax and “merit” advantages of betting being legally disguisable as hedging [14], attracting investors. Thus, the popular hypothesis is further denied. In thermodynamic terms, the microscopic uncertainty, qua entropy, stays the same – but the path changes, to a desired outcome. In physics, one could use the following conjecture (e.g., by Arieh Ben-Naim [15]): ∆S = entropy := uncertainty
(3)
to link the behavior of macroscopic systems in terms of the dynamical laws governing their microscopic parts, linking thermodynamical entropy with Shannon’s entropy. This could apply to the current discussions on the “Arrow of Time” [3], and reversible thermodynamic processes, to be pursued elsewhere. There is a further strategy, leading to even higher probable gains in shorter time, automatically, to be discussed next, which can also be of interest in physics. VI.
BLACK-SCHOLES FORMULA
It is possible to reduce risk to near zero by dynamic hedging (e.g. the Nobel Prize Black-Scholes formula [16, 17]), where one bets on both sides, pro and con, and one may break-even if all fails. Presumably, one can collect a fee that offsets expenses, so one has a profit even in the case where all bets are losses. As J¨ urgen Franke, et. al. [18], explain, “Due to a dynamic hedge-strategy the portfolio bears no risk at any time.” This means that losses due to stocks do occur, but are neutralized by profits due to the calls. The formula is derived under the assumption that the time interval between observations is very small,
S ln( K ) + (r + s2 )t √ d1 = s· t √ d2 = d1 − s · t
(4)
where C is the call premium, and others as defined in [16]. Therefore, one can always win when dynamically hedging properly, which can be seen as a “disguised betting.” That disguise, which would be factually illegal in the US, is where a covert deceit can be added, leading to further monetary wins — the US tax code could see it as hedging, but the investors see it as “legalized” betting. The intent to deceive can be hidden not only in betting versus hedging, but also in passing along the financial derivative as something credible, which the buyer has no way to verify — although the buyer would be the legal relying party. [17] One could ask, “Doesn’t this require one has different odds depending which outcome one is betting on? If the odds are the same both ways, one seems to break even, except for the overhead.” [5] The overhead is payable as expenses, and it is seldom that it is exactly balanced. In general, if there is no conflict there is no interest, as one can not hedge when there is no conflict. So one should pick cases where the odds are somewhat unbalanced, even against, to maximize winning. This is a financial oversimplification, as one also has to take into account leveraging and other factors. If one is using OPM (other people’s money), if one loses, one is not exactly losing, and one can write it off, and get a tax break over years, although many people may become losers of large sums. “However, with finite runs the house wins on the average.” [5] That is a popular view, but with dynamic hedging, one (the house, or anyone with enough capital) can win every time against other players, and never actually lose. One can walk out, with a large profit, even after a single win or, if one loses, try another place, and do so recursively. The “second” place (and/or stock), does not know where the person came from, and the strategy can continue without detection, assuring fault-free operation and deniability. Thus, the strategy can remain covert, at will. This technique is used everyday in the stock market because it gives results, notwithstanding the loss it causes to others.
5 This can be seen as a savage idea, even perverse in human terms, it seems to fit within the Blue pill reality; it needs clueless payers, fools who mostly lose. And those, who mostly lose, will tend to confirm qualitatively the popular hypothesis, their very presence and anecdotes confirming the real existence of those others, who mostly win.
Literature may be viewed as a recursive, multivalued, group computation of a final state, given an initial state and sufficient time. Speakers of the same natural language communicate with one another, in recursive group expressions. They trade contents, not uninterpreted strings of symbols. In other words, one expects that there must be communicable content which is conveyed in discourse, especially of literary nature.
This is the reality, however. Is it fair? Yes, it is faced by all. Even the savvy players can be “gamed”, equalizing the playing field. There is no final watcher, in “who watches the watcher?” — everyone may be up-played, outfoxed, won by someone else. And the world, as a whole, is not in one game, nor in a particular zero-sum game. New resources are found all the time, new technologies developed, and so on; this expansion creates new markets, new consumers, and new opportunities. Thus, the popular hypothesis is denied, in this case as well. The reality provides for fair outcomes, if not today, tomorrow.
VII.
BIOLOGY, PSYCHOLOGY, AND LITERATURE
Qualitative reasoning, although not part of physics explicitly, may be useful in adding uncertainty of two quite different, albeit unknown, components: what we know we ignore, and what we ignore we ignore [19]. This will also help reduce bias. In this vision, science is not a collection of immutable facts, but evolving sets of NOT YET FALSE (“true”), MAYBE FALSE (“false”), and WHO KNOWS? [19] Regarding biology, something (more) comes out of a lesser set, as a family of ten living farmers come out of their two parents — more life seems to come out of less, to multiply. There are more chickens, dogs and cats too. We can even go against the “Arrow of Time”, by remembering the future, as in d´ej` a vu, if we accept the evidence. In terms of psychology, thinking that “life’s choices are always against you” or perverse, can be a psychosis, a mental illness characterized by loss of contact with reality. One could become psychotic, to believe that life is hopeless, perverse, that there is some sort of worldwide conspiracy against oneself. No one is that important! Thus, the negation of the antecedent can be inferred. Also, observing Nature, the Sun does not hide itself in the sunset in order to allow for crimes, nor does it rise to reveal crimes! Aristotle remarked the same in relation to rain and seeds — rain does not fall in order to make seeds grow, or to spoil seeds when laid outdoors to dry. In the balance, one has a fair shot — if not now, later, and mutatis mutandis, everyone else.
Here, we find in positive for the popular hypothesis, irrespective of its truth value. As a literary expression, it could be a valid vehicle for expressing angst, of some action-reaction to what seems unavoidable, of consequents that must remain a permanent disaster, of unsolvable situations. However, notwithstanding the value of its literary exploration, the world does not end in an abyss, as feared by early sea explorers, and reported in novels. Further, as Shakespeare wrote, in Julius Caesar, “Men can control their destinies. The reason that we are oppressed, dear Brutus, is not a matter of fate, but because we don’t do anything about it.”
VIII.
CONCLUSIONS
Physics does not consider Nature to be intersubjective [2], willy-nilly, without direction or planning. In this work we attempt to further clarify the role of physics in understanding Nature, in two limiting cases, called Blue for external appearance, and Red for underlying reality, critically studying a popular interpretation of thermodynamics. Contrary to the popular hypothesis, Blue option, this work shows, using a diversity of macro and micro methods, that the more likely result becomes that one can win with proper encoding (e.g., martingale strategy, proper dynamic hedging, or a correction channel with enough capacity), and win as closely to 100% as desired. This is the Red case, the observer-independent reality. Current material limitations notwithstanding, this work finds no basic thermodynamic limitation in opposition to objectivity in science: there is an absolute rule of right versus wrong, and of mathematical ordering in-between, in physics [19, 20] — right is what mostly works, and wrong is what mostly does not. Both, as shown here, can be used, objectively, to “always win,” as closely as desired, even taking into account local rules, such as tax code differences on hedging vs. betting. Thus, we find in the positive, under the Red option, regarding Bernard d’Espagnat’s [2] question, “Is reality something meaningful and is science steadily coming closer to a true depiction of it?”
6
[1] G. Breit and J. A. Wheeler. Collision of two light quanta. Phys. Rev, 46:1087–1091, 1934. [2] B. d’Espagnat. Quantum Physics and Reality. Foundations of Physics, 41:1703–1716, November 2011. [3] Sean Carrol. The Arrow of Time. Engineering and Science, 73 (1):20–25, 2010. [4] J.W. Jewett and R.A. Serway. Physics for Scientists and Engineers with Modern Physics. Thomson Brooks/Cole, 2008. [5] Prof. Ken Cheney at PCC. Private Communications, 2017. [6] J. Kestin. A Course In Statistical Thermodynamics. Elsevier Science, 2012. [7] C. E. Shannon. A mathematical theory of communication. Bell Systems Technical Journal, 27:623–656, 1948. [8] L. Szilard. On the Decrease of Entropy in a Thermodynamic System by the Intervention of Intelligent Beings. Zeitschrift f¨ ur Physik, 53:840–856, 1924. [9] R. V. L. Hartley. Transmission of Information. Bell Systems Technical Journal, page 535, July 1928. [10] H. Nyquist. A mathematical theory of communicationCertain Factors Affecting Telegraph Speed. Bell Systems Technical Journal, page 324, April 1924. [11] E. Gerck. Certification: Intrinsic, Extrinsic and Combined. ResearchGate, https://www.researchgate.net/
[12]
[13]
[14] [15] [16] [17] [18]
[19]
[20]
publication/286459966_Certification_Extrinsic_ Intrinsic_and_Combined, 1997. F. Topsøe. Maximum entropy versus minimum risk and applications to some classical discrete distributions. IEEE Trans. Inform. Theory, 48(8):2368–2376, Aug. 2002. Nikki Katz. The Book of Card Games: The Complete Rules to the Classics, Family Favorites, and Forgotten Games. Adams Media, 2013. Stefan Andreev. Quanto Credit Hedging, Lecture 23, MIT OpenCourseware, 2013. Arieh Ben-Nalm. A Farewell To Entropy. World Scientific Publishing Co. Pte. Ltd., 2008. Black and Scholes. The Pricing of Options and Corporate Liabilities. J. of Political Economy, 81:637–654, 1973. Merton. Theory of Rational Option Pricing. Bell J. of Econ. and Mgt. Sci., 4:141–183, 1973. Hafner C.M. Franke J., H¨ ardle W. Black-Scholes Option Pricing Model. Statistics of Financial Markets. Universitext. Springer, Berlin, Heidelberg, 2004. E. Gerck. Science and the Search for Truth: The Scientific Method. http://ollidev.ucsd.edu/videos/ videoPlayer.cfm?vid=200267324, 2017. Not to be confused with local rules of right vs. wrong, e.g., in politics, societal rules, language, ethics, law, or tax codes.
