Home > The Economics Profession > Ergodicity – the biggest mistake ever made in economics

Ergodicity – the biggest mistake ever made in economics

from Lars Syll


Paul Samuelson claimed that the “ergodic hypothesis” is essential for advancing economics from the realm of history to the realm of science.

But is it really tenable to assume – as Samuelson and most other neoclassical economists – that ergodicity is essential to economics?

The answer can only be – as I have argued here, here, here, here and here – NO WAY!

Samuelson said that we should accept the ergodic hypothesis because if a system is not ergodic you cannot treat it scientifically. First of all, that’s incorrect, although I think I understand how he ended up with this impression: ergodicity means that a system is very insensitive to initial conditions or perturbations and details of the dynamics, and that makes it easy to make universal statements about such systems …

Another problem with Samuelson’s statement is the logic:  we should accept this hypothesis because then we can make universal statements. But before we make any hypothesis—even one that makes our lives easier—we should check whether we know it to be wrong. In this case, there’s nothing to hypothesize. Financial and economic systems are non-ergodic. And if that means we can’t say anything meaningful, then perhaps we shouldn’t try to make meaningful claims. Well, perhaps we can speak for entertainment, but we cannot claim that it’s meaningful.

In what sense would saying something that’s patently false be “meaningful,” or “scientific” rather than “historical”? You can see where I’m going with this. Important models that economists use are not ergodic, so what’s this debate about?…

In finance or economics the situation is different. Take the most basic model of a stock market, Louis Bachelier’s random walk. Is that model ergodic? No. A little later, in the 1950s, maybe starting with M. F. M. Osborne, the popular model in finance became geometric Brownian motion—basically a random walk in log-space …

Since geometric Brownian motion is a mathematical model, you can answer the question of whether that’s ergodic by scribbling a few lines of equations. Of course it is not. It’s a model of growth, after all, so it can’t be ergodic, but you can actually make this completely formal and do the math, and not even the expectation value of the growth rate is equal to the time average of the growth rate. At the end of the day, what’s more important in finance than growth rates?

So Samuelson’s comment makes little sense. A hypothesis is about something we don’t know, but in the case of finance models this is something we do know. There’s no reason to hypothesize—the system is not ergodic. It’s like hypothesizing that 3 times 4 is 0 because it makes the mathematics simpler. But I can calculate that the product is 12. Of course, a formalism that’s based on the 3-times-4 hypothesis will run into trouble sooner or later. In economics, that happens with the ergodic hypothesis when we think about risk, or financial stability. Or inequality, as we’re just working out at the moment.

The reason this is so important is quite simple, and stems from a basic question: what does risk mean if the notion of time is not irreversible? The only reason risk exists is that we cannot go back and make decisions over again. Economics got very confused about the point of dealing with risk, and had to resort to introducing psychology and human behavior and all sorts of things. I don’t mean to say that we don’t need behavioral economics. What I mean is that there are lots of questions in economics that we can only answer behaviorally at the moment, but at the same time we have a perfectly formal natural physical analytic answer that’s very intuitive and sensible and that comes straight out of recognizing the non-ergodicity of the situation.

To be blunter, I’m pointing out that economics is internally inconsistent. I accept all the models that economists have developed. I could critique them, but I’m not worried about that. I didn’t make them up, the economists did. But when the economists treat the models as if they were ergodic, that’s when someone has to say “stop, that’s enough.”

  1. February 12, 2013 at 8:08 pm

    Since I brought the concept of ergoic vs nonergodic systems into economics in 1983-84, I have help developed a Post Keynesian economic model that explains what nonergodicity implies for the need for liquidity, and money contracts to control cash inflows and cash outflows , and even why nations maintain foreign reserves as a security blanket to meet potential future contractual liabilities in a different currency

    All of this is developed at length in my text book POST KEYNESIAN MACROECONOMIC THEORY, secon edition [2011, Edward Elgar, Cheltenham]. The result is an economic model that –, like Keynes’s, — explains why involuntary unemployment can exist even if wages and prices are perfectly flexible.

    I would hope Lars will find this model applicable to the world in which we live. I hope readers of this blog would look at this economic model.


    • greg
      May 26, 2016 at 5:51 pm

      after ingesting your neoclassical verbiage about Keynes and money fed by Phillip Arestis handbook of monetary heterodox economics. The part about the three axioms of mainstream economics neutral money, the ergodic nature of economics and the law of substitution all of which Keynes disagreed with including your self. begs the question: If your contrarian view of the ergodic nature of money is false, then how is it possible for stock market and foreign exchange speculation to be profitable?.

      • Paul Davidson
        May 27, 2016 at 4:04 pm

        how is it possible for people to gamble in a casino wen everyone knows that only the house is a sure winner??? Any “rational” consumer of casino services would now they are going to lose money if they gamble. Yet the casino is full of people gambling and thereby throwing their money away!

        Such action should also explain stock markt speculation!!

  2. February 12, 2013 at 10:11 pm

    In essence, the critique about ergodicity and non-ergodicity is a critique about lack of realism (of course, I agree it has with vast consequences). But in fact even if we accept the primal assumption about ergodicity, things do not go well for mainstream economic theory. Consider a typical Arrow-Debreu model. It assumes ergodicity and yet, it is unable to prove what it sets out to demonstrate, namely that the system possesses global stability! Thus, even after accepting all of the assumptions and restrictive conditions, including ergodicity, the model fails to prove convergence of the price system to an equilibrium vector (except in two absurd cases: gross substitution for all goods or the weak axiom of revealed preferences at the market level). Scarf and the Sonnenschein-Mantel-Debreu theorems are enough to confirm that even with the assumption of ergodicity general equilibrium models are a complete fiasco. In its canonical form, an economic model may or may not need to rely on ergodicity, but it certainly cannot be based on faulty logic.

  3. Geoff Davies
    February 12, 2013 at 10:14 pm

    For a scientist there seems to be a never-ending supply of gob-smacking examples of mainstream economists’ ignorance of any field outside their tiny nineteenth-century playground.

    Vast areas of physics, biology, ecology, systems theory, psychology, anthropology, ethology … we understand so much about ourselves and the world, but this little enclave blithely maintains its monumental ignorance.

  4. February 14, 2013 at 11:22 am

    As another scientist I totally agree, Geoff, but practice often precedes theory. In this case the greatest mistake ever made in the real economy was surely Henry VIII’s re-legalisation of usury (dramatised in Shakespeare’s “The Merchant of Venice”), from whence the euphemising of chrematistics as ‘economics’, fraudulent international banking and the inhuman ignorance of main-stream chrematists followed.

  5. February 16, 2013 at 5:38 pm

    sorry but i dont think you understand ergodicity including the sfi dude you quote—-FINANCE. look it up. random walk aint ergodic? check out 3 toed sloth — shalilizi of sfi/cmu. excercize left to the feederz (BAND FROM CANADA BC– tar sands etc. slave to my my my…).

    • A man
      February 17, 2013 at 9:35 am

      What’s that ishi? Timmy fell down a well?

  6. Bruce E. Woych
    February 17, 2013 at 9:58 pm

    “The term “ergodic” was derived from the Greek words έργον (ergon: “work”) and οδός (odos: “path” or “way”). It was chosen by Boltzmann while he was working on a problem in statistical mechanics.”

    “In statistics, the term describes a random process for which the time average of one sequence of events is the same as the ensemble average, and vice versa.”
    Ergodic theory http://en.wikipedia.org/wiki/Ergodic_theory
    From Wikipedia, the free encyclopedia

    “Ergodic theory is a branch of mathematics that studies dynamical systems with an invariant measure and related problems. Its initial development was motivated by problems of statistical physics.

    A central concern of ergodic theory is the behavior of a dynamical system when it is allowed to run for a long time.”
    It might be applicable to state that “Economists” and Economics” as distinguished from actual “economies” are subject to measures…of ergonomic pathways, random walks, and Ergodic theories… of statics and dynamics. The paradox is that these represent “fixed time sequences” and are subject to complexity.

    But as I mentioned above, the “reality” of expectations and the belief that managerial powers can master decision making over the ocean waves…is professionalized for
    Economists themselves; and the reified expectations of a measured comogenic order that can be controlled: as evidenced in higher institutions of (market) learning:
    “Students in Economic Cybernetics and Informatics are trained as economic analysts to manage and make decisions in the socio-economic life by employing modern information technologies.”
    Read the rest:
    It sounds like high tech marketing to me!

  7. Bruce E. Woych
    February 18, 2013 at 2:29 am

    Paul Davidson: A Response to John Kay
    by Paul Davidson on October 06, 2011
    (excellent discussion)

    • Bruce E. Woych
      February 18, 2013 at 2:32 am

      John Kay: The Map Is Not the Territory: An Essay on the State of Economics
      by John Kay on October 04, 2011
      (the article in question by Paul Davidson)

      • Bruce E. Woych
        February 18, 2013 at 4:28 am

        “This pragmatic thinking, employing many tools, is a better means of understanding economic phenomena than ‘the combined assumptions of maximising behaviour, market equilibrium, and stable preferences, used relentlessly and consistently’ – and to the exclusion of any other ‘ad hoc’ approach. More eclectic analysis would require not just deductive logic but also an understanding of processes of belief formation, anthropology, psychology and organisational behaviour, and meticulous observation of what people, businesses, and governments actually do.”

        From :
        John Kay: The Map Is Not the Territory: An Essay on the State of Economics

  8. Bruce E. Woych
    February 18, 2013 at 6:45 am

    Published on Jan 15, 2013
    One of the fundamental ideas of modern economics – that people have rational expectations, an unbiased, statistically correct view of the future – is, in reality, a simple hypothesis. And despite its prominence in recent economic thought, this hypothesis and the economic models that rely on it have been the subject of serious debate since the advent of the Great Recession.

    Roger Guesnerie leads the charge in this effort to scrutinize the rational expectations hypothesis from diverse angles and propose more realistic alternatives. He is building a network of new economic thinkers and scholars from all over the world to tackle this important economic problem. The idea is to bridge intellectual boundaries and foster communication where communication is absent, namely between highly specialized subfields each singling out one particular aspect of the rational expectations hypothesis and analyzing it from a particular angle. Guesnerie is a bridge builder and pioneer into the economic unknown of non-rational expectations.

  9. paul davidson
    February 18, 2013 at 4:57 pm

    Since 1984,I have provided theoretical arguments of why the efficient market theory is not applicable to the world of experience.in the Journal of Post Keynesian Economics, the Journal of Economic Literature, the Economic Journal, and many books including my book JOHN MAYNARD KEYNES [2007] in the Palgrave/Macmillan Series of “Great Thnkers in Economics” .and THE KEYNES SOLUTION:THE PATH TO GLOBAL ECONOMIC PROSPERITY [Palgrave/Macmillan, 2009]

    INET [George Soros’s Institute for New Economic Thinking” ] posted a feature on their blog regarding my recent talk to the Economics Department at the University of Chicago on why the theory of efficient financial markets can not exist in our world of experience and why the Keynes- Post Keynesian Theory explains the global financial crisis that began in 2007.
    To see the INET posting of my presentation at Chicago click on


  10. February 20, 2013 at 11:05 am

    I’m not sure what Ishi understands or who he is responding to, but thinking that might be me, I looked up C E Shannon’s “The Mathematical Theory of Communication” (source of information theory) and yes, ergodicity is fundamental to it. But that is mathematics; the issue is whether it is being appropriately applied to economics.

    Shannon interprets it significantly differently from physicists and econometrists, whose aim is to establish norms from repeated measurements. In his information theory the norm is given – it is the intended message, the message as transmitted. The effect of “random walks” in the communication channel is “noise”, which distorts the message, and the aim is not to establish but to correct or RE-ESTABLISH the norm: the already known message. Note that the message is not the physical signal but (like the noise) a variation of it, and the signal-to-noise ratio is a key factor in whether the ergodic nature of noise MATTERS: the programme on one’s car radio only becomes distorted as one moves towards the limits of the transmitter’s range. Unfortunately the radio does not distinguish between good programmes and bad, and in cybernetic economies likewise the “invisible hand” determining the aim may be that of God or the devil. (A compass doesn’t point North: it merely aligns itself between North and South).

    Put simply, the trick in being ABLE to correct information is to supply information in two or three ways so if one version goes wrong one can use the others to correct it. The same idea is used in steering, where one has a route and various feedbacks enable one – perhaps via detours – to reach its intended destination). Likewise in reliability engineering, as when three computers are used and if their results differ, likely only one is wrong and the two right.

    Bruce, thank you very much for the link to Paul Davidson’s critique of John Kay, and thence to Kay and other comments thereon. Well worth studying! Paul says “Paul Samuelson [1969] has written that if economists hope to move economics from “the realm of history” into “the realm of science” they must impose the “ergodic hypothesis” on their theory”. So Information Science has done – but at the margins rather than as its basis. Here’s a link to that rather better than most, though (probability creating nothing) not making the necessary Einsteinian advances in physics.


    • Bruce E. Woych
      February 20, 2013 at 6:05 pm

      @davetaylor1: As usual you have expanded the boundaries of my thoughts and potentials.

      I have used the radio tuning analogy (very literal in one sense) to explain signal to noise ratios and I first truly understood it from its physical science origins (see: http://en.wikipedia.org/wiki/Signal-to-noise_ratio). In that regard I have been absent minded in neglecting the fact that “legitimate” noise appears as distortions (unintentional weakening or mutations to the signal itself) and has a place interpreting the translation quality (and quantity) of fact, truth, information input in theoretical frames; and perhaps even random variations that have adaptive function. However, I was literally introduced to the random walks and signal to noise [praxis based] concepts as they appear in social communication; particularly on Wall Street. Random walks imply chance and risk in a context of questioning predictability. Ultimately it is not “noise” per se; as much as direct change that is implied (although I do not mean to dispute the “noise” to signal idea of variable distractions during a walk in the park. (Coming from NYC more or less I am biased to the unpredictable nature of a simple walk in the Park…). I also related more to the direct “intentional” misinformation and manipulation (Hedge Funds & rumor manipulations: Jim Cramer & Co.?)… that prominently depicts signal as facts and noise as intentional market distraction and information as a tool to manipulate capital investments. This is a very practical but real life example of “science” in two realms. Political Noise is about all we really get in media driven information flow, and the true question for “meta-science” is how much “bias” “slant” and “skewing” is actually “intentional noise” in the distribution of power. Especially so for science (Scientia Potentia est) when the “given” is that Scientia est Potentia potentia est Scientia:
      see this discussion: http://www.cryptogon.com/2002_07_14_blogarchive.html

      One more point. In my perspective quantitative science is not “absolutely” exclusive, and while “humanities” can be organic, romantic, emergent, holistic…& so can physics. I also believe that the absolute distinction of a dichotomy between history and science is a false one. History can be science; but it is subject to verification and methodological demands to get rid of the “noise” factor. (Albeit, the fact is that human noise is a factor trait of history).

      Thanks Dave, & regards: Bruce

      • February 20, 2013 at 7:38 pm

        Thanks again, Bruce. Having posted my comment I realised it seemed to undermine history, but in fact I look on that (whether in an anient or a modern idiom) as one of “the other ways of saying things”, and conversely science as another way of telling the human story. The story can be wrong, but so can the scientific method, as witness its application to economics.

      • Bruce E. Woych
        February 21, 2013 at 12:58 pm

        Dave: It occurs to me that economics is not only class-centric and selectively draws upon history to confirm, conform and condition our perspectives on (if not consciousness of…) economic coordinates and organization; but actively operationalizes itself as revisionist historicism.
        In that respect, a return to some empirical scientific methodology would not only be warranted, but would necessitate a critical evaluation of history based emergence as well.
        Samuelson’s “demand” to replace “history” with a pseudo-science is not only wrong, but actually the precise opposite of what is needed. Economics needs to be re-set in a more comprehensive and “objective” history…or it will surely suffer (as you say Dave…) “possibilities of the moon crash-landing or flying off as a result of some as yet unexpected meteoric event.”

      • Bruce E. Woych
        February 21, 2013 at 2:19 am

        Ceteris paribus
        For History & Economics (among other “soft” sciences):
        “A prediction, or a statement about causal or logical connections between two states of affairs, is qualified by ceteris paribus in order to acknowledge, and to rule out, the possibility of other factors that could override the relationship between the antecedent and the consequent.[1]

        A ceteris paribus assumption is often fundamental to the predictive purpose of scientific inquiry. In order to formulate scientific laws, it is usually necessary to rule out factors which interfere with examining a specific causal relationship. Under scientific experiments, the ceteris paribus assumption is realized when a scientist controls for all of the independent variables other than the one under study, so that the effect of a single independent variable on the dependent variable can be isolated. By holding all the other relevant factors constant, a scientist is able to focus on the unique effects of a given factor in a complex causal situation.

        Such assumptions are also relevant to the descriptive purpose of modeling a theory. In such circumstances, analysts such as physicists, economists, and behavioral psychologists apply simplifying assumptions in order to devise or explain an analytical framework that does not necessarily prove cause and effect but is still useful for describing fundamental concepts within a realm of inquiry.”

      • Bruce E. Woych
        February 21, 2013 at 2:38 am

        caeteris paribus is the Latin phrase, literally translated as “with other things the same,” or “all other things being equal or held constant.”

        The critical consideration with fundamental causality (and the virtual reality of a walk in the park along with asymmetrical information…as signal to noise manipulation…is “emergent complexity. As the summary review linked @ :http://en.wikipedia.org/wiki/Ceteris_paribus the concluding discussion adds a caveat in citing the following:

        “W. Ross Ashby wrote in his “Introduction to Cybernetics”
        “Science stands today on something of a divide. For two centuries it has been exploring systems that are either intrinsically simple or capable of being analysed into simple components. The fact that such a dogma as >>vary the factors one at a time<< could be accepted for a century shows that scientists were largely concerned in investigating such systems as allowed this method, for this method is often fundamentally impossible in the complex systems. Not until Sir Ronald Fisher's work in the 1920s, with experiments conducted on agricultural soils, did it become clearly recognised that there are complex systems that just do not allow the varying of only one factor at a time — they are so dynamic and interconnected that the alteration of one factor immediately acts as cause to evoke alterations in others, perhaps in a great many others. Until recently, science tended to evade the study of such systems, focusing its attention on those that were simple and, especially, reducible (S.4/14)."
        Ergo…we might postulate that ergodicity is about as predictive as a dead cat bouncing (another Wall Street gem).

      • February 21, 2013 at 11:12 am

        Bruce, you must remember all this started with Hume misinterpreting Newton’s method in defining the path of the moon’s orbit, which is more or less randomly affected by celestial bodies other than the earth, sun and planets. Hume, following Descartes rather than Locke by denying the possibility of being sure of anything, sought the probability that the moon would follow a particular path, and hence the predictability of the tides. Another way of looking at the problem is to accept that it is in orbit (i.e. recurrently appearing, ceteris paribus, in more or less the same position), tolerancing this by predicting the likely random variation and the residual possibilities of the moon crash-landing or flying off as a result of some as yet unexpected meteoric event.

        Substitute, for this, the money-maker’s view of money orbiting the economy, and one has current economic theory, where the theorists don’t actually look where the money is going and thus haven’t seen it bifurcating between physical and intellectual processes, each involving the real world and its transformation as well as imaginary money.

        Significantly, Adam Smith condemned the circulation of money, in effect suggesting that it should be oscillating between saving and use. Perhaps he had a point. However, since money is now created out of thin air by borrowing, it should be written off after proper use, i.e. facilitating the sharing and grateful regeneration of the wealth of the world: not least the communities, true education and mutual understanding of its peoples.

      • Bruce E. Woych
        February 21, 2013 at 12:47 pm

        Dave: Ancestor worship is not my strong point, but the history of science certainly demonstrates that we walk through the park blindly enough but more importantly, we habituate mistakes and demand conditioned responses that contort both science and its impact on the rest of the intellectual community; hence society in time and space.

        It occurs to me while reading your statement about money circulation (…Substitute, for this, the money-maker’s view of money orbiting the economy, and one has current economic theory,…) that the current theories are not only retroactive and teleological…but that the foundation notions of supply and demand (at the most basic level…) has been distorted.

        The Financial sector (to classify the monetarists) controls supply but what precisely is the demand? The demand is money…profit. Produce nothing,…make money and then scramble to take ownership of some asset base that is real (liquidate the real economy).

        The circularity is that the money makers (supply) also are also the operative “demand” side and price theory is internally flawed from the foundation to its lofty exotic delusions of wealth.

  11. February 22, 2013 at 10:48 am

    Bruce: ‘Worship’ is too strong and one-sided to capture my attitude to our ancestors, but it takes hubris to blind us to the value of little men being willing to stand on great men’s shoulders, and not on the shoulders of those who have dug a hole for themselves and others to fall into.

    On the present economic theories being teleological, there is an aim (self-perpetuation) built into the circulation of money, and another (self-enrichment of the circulators, c.f. burning resources for central heating) unstated or even obscured: Hume’s God-denying philosophy of science positing that real science can’t be teleological.

    Back at #16, I largely agree with you, given the original Baconian form of empiricism (taking things to bits to see how they work) and not Hume’s (repeatedly measuring things to establish what they look like). My take is that radio telescopes have enabled us to take the universe to bits by seeing how it was more or less when it first started. There is obviously some truth in Darwin’s theory of evolution, but there are gaps in the evidence for it and a failure to account for the existence of species in the first place. Standing on great men’s shoulders I’ve chanced on a structure (evident even in electromagnetic waves) which does account for existence and its continuing evolution, and I’ve since been introduced to the work of others with equivalent ideas: notably Arthur M Young’s “The Geometry of Meaning”. Essentially, to the three phases of Hegel’s dialectial logic or Darwin’s evolution one has to add a fourth (representing the context) which can both induce (or fail to) the transitions in the other three, and is itself later changed by the changed activities of objects within it. Hence, anyway, the ambiguity in the direction of economic aims. Is the obective to enrich an object or to change the context (ways of doing things) and thereby enrich all?

  12. May 27, 2016 at 1:54 pm

    The biggest mistake in economics was when John Bates Clark combined the notions of capital with land ownership and thereby gave the green light for land value speculators to borrow from the banks. We are still suffering from this situation today and every 18 years a new business cycle occurs due to cyclic speculation in land values! Look out for 2026 as the next one!

  13. May 28, 2016 at 12:34 pm

    Petitio principii — economists’ biggest methodological mistake
    Comment on Lars Syll on ‘Ergodicity — the biggest mistake ever made in economics’

    You summarize how ergodicity came to economics: “Samuelson said that we should accept the ergodic hypothesis because if a system is not ergodic you cannot treat it scientifically.” (See intro)

    This is pretty much the same way how most other core concepts and foundational propositions came to economics as Mirowski has shown (1995).

    And this eventually resulted in the neo-Walrasian axiom set with these core propositions: “HC1 economic agents have preferences over outcomes; HC2 agents individually optimize subject to constraints; HC3 agent choice is manifest in interrelated markets; HC4 agents have full relevant knowledge; HC5 observable outcomes are coordinated, and must be discussed with reference to equilibrium states. By definition, the hard-core propositions are taken to be true and irrefutable by those who adhere to the program.” (Weintraub, 1985, p. 147)

    Now it is obviously a methodological mistake to take equilibrium into the premises and then to discuss the properties of general equilibrium. The same holds for ergodicity and most other foundational propositions. This mistake — known since antiquity as petitio principii* — is an age old characteristic of incompetent scientists or savants: “These savants, as Galileo put it, first decided how the world should function in accordance with their preconceived principles. … He openly criticized scientists and philosophers who accepted laws which conformed to their preconceived ideas as to how nature must behave. Nature did not first make men’s brains, he said, and then arrange the world so that it would be acceptable to human intellects.” (Kline, 1982, p. 48)

    Now we can argue whether equilibrium, constrained optimization, rational expectations, ergodicity, well-behaved production functions or any other nonentity is the worst petitio principii. But this itself is a methodological mistake: ‘someone has to say “stop, that’s enough”’ (See intro).

    Any discussion about nonentities falls into the how-many-angels-can-dance-on-a-pinpoint category and is merely a pastime of savants and other scientific dilettantes.

    The right thing to do is throw the neo-Walrasian axiom set and all its nonentities and all textbooks from Samuelson to Rodrik without further ado out of the window and to move from microfoundations to macrofoundations. As Joan Robinson put it “Scrap the lot and start again.”

    Egmont Kakarot-Handtke

    Kline, M. (1982). Mathematics. The Loss of Certainty. Oxford, New York, NY: Oxford University Press.
    Mirowski, P. (1995). More Heat than Light. Cambridge: Cambridge University Press.
    Weintraub, E. R. (1985). Joan Robinson’s Critique of Equilibrium: An Appraisal. American Economic Review, Papers and Proceedings, 75(2): 146–149. URL

    * Wikipedia https://en.wikipedia.org/wiki/Begging_the_question

  14. August 31, 2018 at 6:32 pm

    They must have meant advancing economics from questions about people to one about theoretical models of ideal gases.

  1. March 25, 2021 at 9:56 am

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