Maxwell's demon is famous for threatening the second law of thermodynamics. But does it? Here we will look at Maxwell original writing, and how we can interpret Maxwell's thought experiment.

Thermodynamics is the branch of physics that deals with energy management. Thermodynamics, like all other branches of science, identifies laws that we believe nature must abide by. Without a doubt, the most important law of thermodynamics, is the second.

There are various statements that derive from the second law. The simplest and most intuitive of these states that it is impossible for a heat engine to produce energy if it exchanges heat only with bodies at a single fixed temperature.

To understand the second law, let us consider a simple engine, consisting of two heat reservoirs and a piston in a cylinder full of air. Imagine one moves the piston from one reservoir to the other. If the reservoirs are at different temperatures, the air in the cylinder expands and contracts. This drives the piston up and down, just like a piston in a steam engine or an internal combustion engine.

But if the two reservoirs are at a single fixed temperature, the volume of the gas is the same. This means the piston does not move up and down when we move the cylinder. If someone could invent a way to produce energy using such a system, they’d have solved humanity’s energy problems. We could in fact take two points in the sea and use them as "reservoirs". In this way, we could transform part of the thermal energy stored in the ocean into mechanical energy.

The second law of thermodynamics states that this is impossible.

But is it?

The little demon

In the late 1800s, Clerk Maxwell was laying the cornerstones of physics. But he also appeared to challenge the second law in a "thought” experiment. Maxwell does not speak about a demon in reality, and winds up his “Theory of heat” as follows:

“… the second law of thermodynamics … is undoubtedly true as long as we can deal with bodies only in mass, and have no power of perceiving or handling the separate molecules of which they are made up. But if we conceive a being whose faculties are so sharpened that he can follow every molecule in its course, such a being, whose attributes are still as essentially finite as our own, would be able to do what is at present impossible to us.

For we have seen that the molecules in a vessel full of air at uniform temperature are moving with velocities by no means uniform, though the mean velocity of any great number of them, arbitrarily selected, is almost exactly uniform. Now let us suppose that such a vessel is divided into two portions, A and B, by a division in which there is a small hole, and that a being, who can see the individual molecules, opens and closes this hole, so as to allow only the swifter molecules to pass from A to B, and only the slower ones to pass from B to A. He will thus, without expenditure of work, raise the temperature of B and lower that of A, in contradiction to the second law of thermodynamics.”

Maxwell concludes by explaining that his theory does not necessarily clash with the laws of thermodynamics: “This is only one of the instances in which conclusions which we have drawn from our experience of bodies consisting of an immense number of molecules may be found not to be applicable to the more delicate observations and experiments which we may suppose are made by one who can perceive and handle the individual molecules”

Maxwell and Socrates' demons

In practice, Maxwell says: "We cannot know the velocity of each molecule. Therefore the second law is valid for the macroscopic world". The second law is a “statistical” law. It is valid if we consider many molecules as if they were one single body. We use statistical instruments because we do not have sufficient information on the position of each single molecule. We are ignorant.

But you won’t find many who agree with this. Maxwell’s door-opening being was called a “demon”, a very appropriate name. The city accused Socrates of believing more in his own personal demons than in the Gods. Maxwell was attacked merely for having dared to imagine his demon challenging the "divine laws" of thermodynamics.

In reality, Maxwell was only recognising his (and our) ignorance. And like Socrates was attacked for that: the city indeed accused Socrates of believing what the oracle of Delphi said, that he was the wisest of all men. And he explained he actually believed he was the wisest because he was the only one recognising and accepting his own ignorance: "I am better off than he [a conceited politician] is - for he knows nothing, and thinks that he knows. I neither know nor think that I know. In this latter particular, then, I seem to have slightly the advantage of him".

Attacking the demon

On 28 February 1879, Lord Kelvin gave a lesson at the Royal Institution. Soon later, the American Popular Science Monthly published this editorial: “There is a certain class of minds whose efforts to explain things generally leave them more obscure than they were before. … A marked illustration of this is afforded by a lecture delivered … by the eminent physicist and mathematician, Sir William Thomson [Lord of Kelvin], who announced as his topic of discourse the curious subject, ‘Maxwell's Sorting Demons.’”

The editor of the American monthly, today published as Popular Science, didn’t really understand Kelvin’s lesson. He thought Maxwell used the demon to explain diffusion processes. ??.

In his confused interpretation, the editor stumbled on something that scientists later connected to the demon: the demon had a hand in “the origin of terrestrial life”.

The concept of the demon, in fact, seems to suggest that the origin of life derived from the capacity to take another approach to nature and therefore extract energy in ways in which other, less capable beings, had failed.

But how can this have been possible? Does the second law only apply to us, but not to microscopic life forms?

It was actually the concept of entropy that required more in-depth study, and Maxwell’s note on the differences between macroscopic and microscopic is crucial.

Entropy and probability: what is the connection? (future post)

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Non ci vuole matematica avanzata: così com'è, l'essere umano è lontano dalla sostenibilità. La quantità di energia disponibile non è sufficiente a sostenere la società tecnologica.

La potenza (energia per unità di tempo) media usata dagli esseri umani, $1.8\cdot 10^{13}$ watt, è un decimillesimo di quanto arriva sulla terra, $1.6\cdot 10^{17}$ watt. Può sembrare poco, ma non lo è. Gli organismi che trasformano l'energia solare in energia "da bruciare" (petrolio, legno, carbone, gas etc) assorbono, sia pure molto approssimativamente, il 5% dell'energia solare (Hall, 1999), quindi noi assorbiamo il 2 per mille di quello che viene effettivamente assorbito dalle piante.

Sembrerebbe sostenibile. Ma. Considerando che buona parte di questa energia da sintesi clorofilliana non viene usata nella sintesi di combustibili (carboidrati), si stima che consumiamo dal 30 al 40% dell'energia che, attraverso le piante, entra a far parte della catena alimentare del pianeta (Christian, 2011).

Altro che sostenibilità: siamo di gran lunga oltre il limite della sostenibilità. La nostra sopravvivenza è possibile nel breve periodo –qualche secolo– solo perché stiamo consumando gli idrocarburi fossili, le fonti appunto non rinnovabili.

Il sapiens e la formica

La quantità di energia che l'Homo sapiens estrae dall'ambiente è enorme, soprattutto se comparata ad altre forme di vita.

Un formicaio di 60 kg, il peso medio di una persona, consumerebbe quasi 10.000 chilocalorie al giorno (Macom, 1995) –più delle 1.500 che consumiamo in cibo.

Detto con altre unità di misura, noi consumiamo come una lampadina di 100 watt scarsi, un formicaio del nostro peso come una da 500 watt. Il singolo umano consuma meno del superorganismo composto da formiche.

Ma se andiamo a guardare l'organizzazione umana, i numeri cambiano radicalmente. Le società umane consumano –in energia elettrica, combustibili etc– $1.8\cdot 10^{13}$ watt.

Questo vuol dire, considerando una popolazione al 2012 di 7 miliardi di persone, quasi 2.500 watt a testa. Cinque volte le formiche e ben 25 volte una singola persona.

Perché pesiamo come 1000 miliardi

Consideriamo solo l'Unione Europea a 28. Nel 2016 abbiamo consumato una potenza media di 4.300 watt a testa. In pratica, un sapiens europeo oggi richiede all'ambiente quasi quanto 50 sapiens arcaici.

Detto in altri termini, dal punto di vista dello sfruttamento dell'ambiente è come se i primi colonizzatori del continente, i cosiddetti "European early modern humans", identificati normalmente con i Cro-Magnon, si fossero moltiplicati fino a raggiungere una popolazione di 25 miliardi in Europa, e non il mezzo miliardo effettivo.

Su scala mondiale, considerando che si stima che arriveremo ad plateau demografico di 20 miliardi di individui, se il consumo di energia pro capite in tutto il mondo sarà vicino a quello europeo, peseremo sul pianeta come mille miliardi di individui.

Homo sapiens: abbiamo davvero avuto successo?

Negli ultimi centomila anni, l'Homo sapiens ha modificato il pianeta tanto da far proporre alla presente era geologica il nome di antropocene. In pratica, l'uomo dice che è tanto importante da poter intitolare una sua creazione a se stesso... 

Forse abbiamo ragione.  Ma dal punto di vista della quantità di biomassa presente sulla terra l'Homo sapiens non è poi così speciale: esseri umani e formiche nella loro totalità pesano più lo stesso.

Anche il punto di vista del "gene egoista" lascia un po' a desiderare. In termini numerici, il nostro DNA rappresenta un infinitesimo di tutte le catene genetiche presenti, e come se non bastasse siamo forse noi più dipendenti dai batteri che ospitiamo (dieci per ogni cellula con DNA "puro") che non viceversa.

Quantità di biomassa per diverse forme di vita. (Bar-On, 2018).

Quello che al momento ci rende unici è, appunto, l'impatto che abbiamo sul pianeta grazie alla quantità di energia che manipoliamo.

(Bar-On, 2018) Bar-On, Y. M., Phillips, R., & Milo, R. (2018). The biomass distribution on Earth. Proceedings of the National Academy of Sciences, 115(25), 6506–6511.

(Christian, 2011) Christian, D. (2011). Maps of time: An introduction to big history. University of California Press.

(Macom, 1995) Macom, T. E., & Porter, S. D. (1995). Food and energy requirements of laboratory fire ant colonies (Hymenoptera: Formicidae). Environmental Entomology, 24(2), 387-391.

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Although the entropy (and therefore the amount of information stored) of a network can be computed mathematically (see my attempt here: Graph entropy: a definition and its relation to information), it's much more sensible, in my opinion, to derive the amount of information from the amount of energy the brain is able to extract from the environment.

Information measures, in the end, how much you decrease uncertainty. And that is not easy to define. How much uncertainty our brain decreases? Let's say enough to exploit systems in an unstable state, where a small quantity of energy releases a big amount of energy, so-called metastates (e.g. a match, which with just a scratch gives you fire).

Deduction through Physical principle

In physics, information and use of energy are related to each other, and if we know the latter, we know the former. Therefore, if we know how much energy we use during our whole life we can now how much information we store.

This principle is used by Leó Szilárd in his "On the decrease of entropy in a thermodynamic system by the intervention of intelligent beings", where he quantifies the equivalence between information and the "ability to extract energy": he concluded that one needs about 40 exabytes to extract one calorie from a system at room temperature.  As we burn about 2,000 calories a day, and live for about 80 years, the amount of calories burnt in a lifetime is:

a life's energy = 2,000*365*80 kilocalories ~ 60M kilocalories

That's the minimum our body –from proteins, to cells and organs and up– must be able to get from the environment. That means that it must store at least  $40 exabytes \cdot 60M = 2.4 \cdot 10^{30} bytes$, or:

capacity of brain (physics) = 2,5 million of yottabytes

As said, this is the information stored in the whole body, not just the brain. Each of the $10^12$ cells of our body actually store about 1Gb of information, from the same principle above.

Books

Here book you can read if you are interested in the subject. Minds behind the brain is the history of neuroscience through all the people, from ancient Greek to today, who studied the brain. Enjoyable and rich of information.

Information Theory, Evolution, and the Origin of Life, by Hubert P. Yockey is a must for anyone wanting to apply Information Theory to biology –I just quote a paragraph from the epilogue:

Galileo believed that the language of Nature is inherently mathematical and is essential to describing natural phenomena. Although there are many fields of biology that are essentially descriptive, with the application of information theory, theoretical biology can now take its place with theoretical physics without apology.

At the end a Biography of Claude Shannon which (shame on me!) I haven't read. Yet. If you do, please comment a review:)

1. Neuronal Connections and the Mind, The Connectome
2. erg/bit : http://link.springer.com/article/10.1007%2FBF02477767#page-1
3. consumption in c. elegans: http://www.ncbi.nlm.nih.gov/pubmed/15809072
4. weight and lifespan in c. elegans: http://bmcecol.biomedcentral.com/articles/10.1186/1472-6785-9-14
5. energy consumption in chimps: http://www.ncbi.nlm.nih.gov/books/NBK53561/

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In Bayesian statistics, probability has a nice meaning –it means how strongly you believe a certain outcome will happen. If this probability is normalized ($p_i > 0$ and $\sum_i p_i = 1$) we can also define a function which measures our surprise to when a measurement is "$i$": $Surprise(i) = -log(p_i)$. The more we thought the event was rare, the more we are surprised by the outcome.

The entropy can be defined as the average surprise:

$<surprise> = \sum_i q_i \cdot (-log(p_i))$

where $q_i$ is how often the measurements actually gave $i$, and $p_i$ how strongly we believed (prior to running the experiment) that the measurement could be $i$. It's easy to prove the Gibb's inequality:

$\sum_i q_i \cdot (-log(p_i)) >\sum_i p_i \cdot (-log(p_i))$ if there is at least one $p_i \neq q_i$

which says: your average surprise is smaller if your model –your prior distribution– is closer to the "reality". If you are using a biased coin, your average surprise to the results is bigger than the surprise from someone who knew the coin was biased and used the appropriate distribution. The opposite is true of course –someone thought the coin was bias but it was not.

The Kullback-Leibler Divergence measures exactly that difference:

$D_{KL} =\sum_i q_i \cdot (-log(p_i)) - p_i \cdot (-log(p_i)) =$

$D_{KL} =\sum_i q_i \cdot log(q_i/p_i)$

The $D_{KL}$ can be big for two reasons: we are using the wrong distribution (e.g. Poisson instead of binomial) or the "right" distribution but with wrong parameters (e.g. the binomial but with the wrong probability of success).

Given that $D_{KL}$  measures how "more" our measurement surprise us, it's a good way of measuring how much we are wrong with our model (usually we think we have the right distribution and want to find the best values of the parameters).

Usually, we simply take the cross-entropy  $\sum_i q_i \cdot (-log(p_i))$, i.e. our actual average surprise given the frequency $p_i$ of our data and the expected frequency $q_i$ given our model:

$H(q, p) =\sum_i q_i \cdot (-log(p_i))$

Below some books on the subject, plus the autobiography of Edward Thorp –a must if you are interested in entropy and information!

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Microsoft scientists have demonstrated that by analyzing large samples of search engine queries they may in some cases be able to identify internet users who are suffering from pancreatic cancer, even before they have received a diagnosis of the disease.

The scientists said they hoped their work could lead to early detection of cancer.

Little below you read:

The researchers reported that they could identify from 5 to 15 percent of pancreatic cases with false positive rates of as low as one in 100,000.

Before you run to Bing with the hope of having some miraculous diagnosis, ask yourself: what's the probability that, being "positive" at the bing test, you actually have pancreatic cancer?

(Spoiler: it's 50%, and in the indented paragraphs below there is some math you can skip.)

There is a simple formula for that, called the Bayes formula (or theorem):

$P(Cancer given Positive) = \frac{P(Positive given Cancer)\cdot P(Cancer)}{P(Positive)}$

$P(Positive)$ is the probability that you score positive at the bing test, regardless if you have or not the cancer. This the probability that you score positive and have the cancer ("true positive" probability times probability of having the cancer) plus the probability that you score positive without the cancer ("false positive" probability times 1 - probability of having the cancer).

The article states that the "false positive rates [are] of as low as one in 100,000". The "true positive" is reported as 5-15%, which I will approximate to 10%.

The probability that a random person will be diagnosed with pancreatic cancer is about 10 out of 100,000 (see cancer.org).

We now have all the numbers to compute the probability that, having microsoft diagnosing the cancer, a user actually has it:

$P(Cancer given Positive) = \frac{0.10 \cdot 0.0001 }{0.0001*0.10+0.00001} = 50\%$

This is not difficult to understand. It's common sense. The probability of being rightfully diagnosed by bing is 1 out of 100,000 (10 out of 100,000 people have the cancer, and of these 10 only one receives the diagnosis). Of the remaining healthy ones (practically 100,000), 1 person is diagnosed by mistake. Therefore, bing, out of 100,000 people, will diagnosed 2 people: one is the "true positive", the other the "false positive". The probability of being a true positive is 50% then.

Now, there is some difference between the original article by the two Microsoft researchers (Ryen White and Eric Horvitz) and the NY Times' one. White and Horvitz are not sensationalist in their publication. For a reason –in 2008, they wrote an article on "cyberchondria", or "unfounded escalation of concerns about common symptomatology, based on the review of search results and literature on the Web". Their belief, I believe, is that the Web is danger place for diagnosing yourself.

Nonetheless, the tone of the Time's article is slightly sensationalist.  I do believe that 50% is better than nothing. But I do believe that writing about a "false positive rates of as low as one in 100,000" is misleading. Particularly when the article does not report that the final confidence of the diagnosis is 50%.

Articles like this, IMHO, are prone to lead to cyberchondria, and it would be a pity for White and Horvitz to achieve exactly the opposite result they (supposedly) had in mind when writing their original piece.

Below, a few books I read on the topics. From the most technical  (a great tutorial on Bayesian statistics) to medicine ("The patient will see you now", on how health is being disrupted by data analysis), and the always enjoyable "Numbers Behind Numb3rs".

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In reality, errors compensate each others.  You might overestimate one variable, but will underestimate the next one. Unless you are biased, the error will grow like a drunken wanderer.

Say we want to estimate the number $N$ of something. Number of candies eaten by children in the world. Or piano tuners in Chicago. Or whatever.

To estimate $N$, we multiply estimated values $e_i$ of the factors which contribute to $N$, whose real (unknown) values is $a_i$. For estimating the candies, we might have the number of people in the world, fraction of children, sugar-producing crops and so on.

$N = a_1 \cdot a_2 \cdot ... \approx e_1 \cdot e_2 ...$

In the end, will compute our estimate by multiplying all $e_i$.

Now, let's say that you are really bad in estimating, and you never get the right value. All $e_i$ are wrong by a factor 2 –sometimes your estimate $e_i$ is the double, sometimes is one half of the actual value $a_i$.

Now you do what any good engineer would have done before the advent of pocket calculator when had to multiply numbers –you sum logarithms:

$log(N) = \sum_i log(a_i) \approx \sum_i log(e_i)$

($\sum$ means "sum".) But we said your estimates are

$e_i = a_i \cdot random(2, 0.5)$

Or

$log(e_i) = log(a_i) + random(+1, -1)$

Approximating $log(2)$ to $1$.

This allows us to separate the errors from the estimates and write

$log(N) \approx \sum log(a_i) + \sum random(+1, -1)$

$log(N) \approx log(N) + log(\sigma_{final})$

where $\sigma_{final}$ is the error you'll get at the end of the estimate.

The logarithm of the final error, $\sigma_{final}$ , actually diffuses quite slowly. Like drunken wanderers who can only walk on a line, will make one step in one direction, than two steps in the opposite direction, and so on. After $S$ steps, 70% of those drunken wanderers are on average no more than $\sqrt{S}$ steps away from their starting point.

This means that 70% of the times the log of your estimation error is not bigger than $\sqrt{S}*log(\sigma)$, where $\sigma$ is the average (estimated) error factor, which we initially assumed to be 2. Or

$\sigma_{final} = \sigma^{\sqrt{S}}$

With $S$ number of assumptions you made, $\sigma$ the average error for each factor, $\sigma_{final}$ the final estimation error.

The post How big can an error be when we estimate something? appeared first on Visualab.

Such computer would represent the number $1/3$ as:

$1/3 = 0.333333$

That means the number is rounded down by about one millionth. But it might get much worse. Have a look at the following operation:

$a = 1 + 1/30,000 = 1.00003[3333...]$
$b = a - 1$

It leads to $b = 3\cdot 10^{-5}$, which means the computer produced a rounding error of about 10%! Now you should feel sorry you did not write:

$a = 1 - 1$
$b = a + 1/30,000$

which would have given $b = 3.33333\cdot 10^{-5}$.

Normally, this is not a big deal. But in probability computation it is. Take the way the disjunction (union) of probabilities is computed. Given $N$ events whose probability is $p_i$. The probability that at least one of these events happens is:

[1] $P(p_1 \lor p_2 \lor ...\lor p_N) = 1 - (1 - p_1)\cdot(1 - p_2)...\cdot(1 -p_N)$

(the symbols $A \lor B$ means $A$ or $B$ is true, or both). It is immediately clear that if all $p_i$ are very small, and will be rounded like the 1/3 above, the error can become soon big enough to spoil your computation.

And this is actually a pitty! Because:

[2] $(1 - p_1)\cdot(1 - p_2)...\cdot(1 -p_N) = 1 - \sum_i p_i + \sum_{i\neq j} p_i\cdot p_j - \sum_{i\neq j \neq k} p_i\cdot p_j\cdot p_k ...$

where $\sum_i$ means "sum" for all values of $i$, in this case 1 to N, $\sum_{i\neq j}$ all values of $i$ and $j$ not equal between themselves, etc.

Equation [2] tells us that if we solve [1] algebraically, subtract 1 and only after substitute the values in the various $p_i$, the error would be smaller.

Let's make an extreme example. If we had two events with $p_1 = p_2 = 10^{-6}$:

[3] $P(p_1 \lor p_2) = 1 - (1-10^-6)^2 = 2\cdot 10^{-6}-10^{-12} = 1.99999\cdot 10^{-6}$

Imagine you have a "3-digit decimal computer" (3 digits plus the position of the point and the power of 10). If you solve equation [3] using the formula in [2], even with such lousy computer the result would be a decent $1.99\cdot 10^{-6}$. This because both $2\cdot 10^{-6}$ and $10^{-12}$ do not need more than 3 digits. On the contrary, a naive computation, where the products are computed through equation [1] substituting the value of the variables, would give:

[1'] $1 - 0.999\cdot 0.999 = 1.99\cdot 10^{-3}$

A terribly wrong result. The short script below, in Python, solve the equation algebraically:

def probabilities_disjunction(probabilities: list):
    n = len(probabilities)
    # tree is a list of strings whose length is len(probabilities) with all possible 0/1 without the 0s.
    # e.g. for three probabilities: 001, 010, 011, 100, 101, 110, 111 
    tree = []
    for i in range(1, 2**n):
        tree.append(str(bin(i))[2:].zfill(n))
    v = []
    for p in probabilities:
        v.append([1., -p])
    prod = 0
    # multiply all probabilities*(-1) according to tree
    for branch in tree:
        inter_prod = 1.
        for i, leaf in enumerate(branch):
            inter_prod *= v[i][int(leaf)]
        prod += inter_prod
    return -prod

It is clearly non optimised, and the number of computations grows exponentially with the number of probabilities.

Interested in the topic? Below two books on Python (I read the High Performance Python, which IMO should be read by anyone writing scientific code) plus two on the subject (which I admit I haven't read).

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When it's hot, there is high uncertainty about position and speed of the molecules composing the system. The more temperature goes down, the more precise your knowledge about position and speed will be.

Because von Neumann says that entropy is the missing information between a macroscopic description and a microscopic one, you have the answer –entropy goes down when temperature goes down. At 0 kelvin, the macroscopic description gives you exactly the same information than the microscopic one, therefore the entropy is zero....

If heat is released from a system, will entropy increase?

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The Magnum: As Simple as Possible

Giacomo, in his "Cremeria Castiglione", explains to me how he is able to make one of the best gelatos in Bologna, and possibly in Italy. This means, for Italians at least, one of the best ice creams in the world. "Making gelato is like baking pizza," he says. "You wouldn't try to sell a pizza hours after it's baked. The same goes for the gelato –it must be eaten fresh. Only when I realize I’m running out of a flavour do I take all the necessary ingredients, pasteurize them at high temperature, blend them at low temperature and sell it immediately after".

Italians love gelato. On hot summer afternoons, cities fill up with families strolling around, each member with a gelato in hand. I have always been part of this collective passion: when I was a teenager, my friends and I preferred to meet outside gelaterias and indulge in huge ice cream cones, rather than get drunk in bars

In a country where the gelato is sold soon after preparation, and where teenagers prefer it to alcoholic beverages, you may wonder how packaged ice creams can possibly sell at all. But they do, and account for 30 percent of the Italian ice cream market thanks to heavy marketing and widespread distribution. Of the 30 percent, half is in the hands of Anglo-Dutch multinational Unilever, under the brand Algida.

For decades, Algida's strongest seller was the Cornetto, an imitation of the artisanal ice cream cone, which was launched in 1959, the same year Fellini produced La Dolce Vita. Thirty years later, the country was miles away from the economic boom described by Fellini, but the Cornetto hung around and was joined by a considerable number of competitors.

The 1980s  were a time of consumerist excess, with brands offering cherry, amaretto, chocolate and biscuit all in the same ice cream. It was in this environment that Unilever, in 1989, launched Magnum, the simplest ice cream bar ever –vanilla with a chocolate coating.

As an apocryphal quotation of Albert Einstein goes, "things should be as simple as possible, but not simpler"[1]. The Magnum was simple, but not straightforward. Most ice creams had vanilla filling, but only few of them had good quality vanilla, and not one that was covered by thick, good, real chocolate.  To produce good quality coating, Unilever asked Belgian Callebaut to develop a chocolate that could go down to -40 degrees without breaking, something did not exist before.

The Magnum stood out from the crowd because it was simple, yet sophisticated. According to Unilever, it was already in 1992 "Europe’s most popular chocolate ice cream bar"[2]

Simplicity, though, did not last for long with the Magnum. With time, the Magnum evolved from the original ice cream to an ecosystem of elaborated ice creams: Almond, Mint, Caramel and Nuts, Yogurt; bigger and smaller Magnums; even Magnums without sticks. The original Magnum, in this new fauna of Magnum Ice Creams, was renamed Magnum Classic.

The Magnum Syndrome

When this process of differentiation started, and the promise of simplicity was broken, I got upset. When Unilever came out with Moments, small ice creams stuffed with caramel and hazelnut, I decided the company had reached the limit, and prophesied Magnum's fall from greatness to dust. In conversations, whenever dealing with something unnecessarily complex, I would refer to what I called the Magnum Syndrome: "Things start nice and simple, but with time they accumulate complexity. This is when they lose their strength, like in the Magnum's case: it is not the delicacy it used to be, there’s too much noise around."

I could find plenty that had fallen victim to the Magnum Syndrome. One of these was, in my opinion, elementary particle physics, my field of research. CERN, the European Laboratory for Nuclear Physics in Geneva where I started working in 1994, was setting up the Large Hadron Collider. For the first time, thousands of physicists, engineers, software developers were working together on a single experiment, and most scientists only had a very limited view of the experiment they were working on. Scientific articles were signed by hundreds of people, and a few names were still attributed authorship long after they had left the institution. This was a far cry from the physics of the pioneers I studied –and idealised– at university. There, a few great minds created theories and ran experiments in the isolation of their university offices and labs. Everyone involved in an experiment knew everything about it, and in case of a discovery there was never more than three people accounting for the idea –possibly because the Nobel can at most be awarded to three people for the same discovery. In those good old days, particle physics skyrocketed. During my research times in the 1990s, although we still have thousands of scientists working on huge investments, particle physics practically grounded to a halt, with no major discoveries made or in sight for the 2000s.

My explanation was that physics had lost its original "hippy", relaxed atmosphere. Its simplicity. At CERN, there were fewer people strolling around the campus, smoking, wearing shorts and sandals, and more people in suits talking loudly into mobile phones. The way CERN's management reacted to the "invention" of the World Wide Web by Tim Berners-Lee, was a clear sign of a bureaucratic hierarchy paralysing new initiatives. Not only did they not show any interest in Berners-Lee's project during the R&D phase[3], but even after the Web exploded into something undeniably huge[4].

Perhaps as a result of the dire state of information technology at CERN, I became highly pessimistic about computer science in general. In the 1970s, a few programmers had been able to create huge amounts of "free software", isolated in their dormitory room. Now, software development was becoming too complex to be managed by a single developer, and in the future only big corporations would have been able to produce high quality software. According to friends working on 20-year old software used by big banks, the complexity of the code –multiple intricate layers written by generations of programmers– was so high they couldn’t understand how banks could possibly run without major problems.

My small world of particle physics and software seemed irretrievably doomed and as far as I was concerned the real world was not doing much better. A few self-appointed "developed countries" took on the task of saving the world using harsh economic policies imposed to the "developing countries". But these policies appeared, at best, useless. According to Nobel prize Joseph Stiglitz, they were actually worsening the situation. Financial crashes in Asia, Brazil and, finally, Argentina –with people starving to death because of the financial default– did not paint a happy picture for the future of the planet.

In sum, everything was a huge... complex mess.

When Cherry Guevara was launched, together with other terrible Magnum flavours like the John Lemon, the Wood Choc, and the Jami Hendrix, I considered them the four Horsemen of the Apocalypse. The Magnum ecosystem will collapse soon, I was thinking while biting into my Classic. And global capitalism will surely follow.

Taming Complexity

The Network Revolution

In this way, we can draw a graphical representation of the increase in complexity of the Magnum system over time. From the simple "star" at the beginning of the 1990s with one central ice cream and four peripheral ones, to the intricate network arrived at post 2000.

• [1] Like most Einstein's quotations, the quotation is indeed apocryphal. The original one is: "It can scarcely be denied that the supreme goal of all theories is to make the irreducible basic elements as simple and as few as possible without having to surrender the adequate representation of a single datum of experience". Given that Einstein's writing style rarely follows the precepts of the apocryphal one, I believe the use of the latter is justified.
• [2] Floris A. Maljers, "Inside Unilever: The Evolving Transnational Company", Harvard Business Review, September 1992
• [3] Berners-Lee, T., Fischetti, M., & Foreword By-Dertouzos, M. L. (2000). Weaving the Web: The original design and ultimate destiny of the World Wide Web by its inventor. Harper Information.
• [4] The funniest proposal I heard was that CERN should have charged a tiny amount of money –say one millionth of dollar– whenever a Web page (any page, not just CERN's pages) was visited by someone surfing the web. Interestingly, the University of Minnesota applied this business model to Gopher, an hypertext protocol which challenged the Web at its birth. Soon after the announcement, "industry dropped Gopher like a hot potato",  as Tim Berners-Lee said.
• [5] Clarke, C. (2012). "The science of ice cream". Royal Society of Chemistry.

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The horizontal distance D of a projectile with speed V, launched at an angle $\theta$ is:

$D = V^2 \cdot sin(2\theta) / g$

Which gives, for a lion running at 54km/sec, i.e. V=15 m/s, jumping at an angle of 45 degrees, and assuming $g \approx 10m/s^2$:

$D = 22.5 m$

Considering that running at maximum speed and jumping at 45 degrees is very hard, even for a lion, this is an over-estimation. But jumping at 15 degrees would allow the lion to fly over a distance D:

$D \approx 11 m = 36ft$

Below some books on the subject. I only read a few pages from each of them:

Guggisberg, in his "Simba: The life of a lion" reports a speed of 48-59kph, confirmed by Schaller in his "The Serengeti Lion: A Study of Predator-Prey Relations" –the reference book on the African Lion since its publication in 1972.

Previous estimates are given by Howell, in his "Speed in animals, their specialization for running and leaping" (1944): he reports lions' speed at 80kph, which was considered too high by Schaller.

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