STIMULATE blogs

Statistical mechanics, complex systems and econophysics

Dale — Tue, 21 Jul 2020 09:15:28 +0000

My field of research is Lattice Quantum Chromodynamics, in which the dynamics of quarks and gluons are simulated. These particles make up nucleons, protons and neutrons, which in combination with electrons form atoms. This simulation uses techniques from the field of statistical mechanics, which is used to describe the behaviour of systems with many components, to obtain quantitative results such as the masses of said nucleons, through the connection between the averaging over a statistical ensemble and the integration of Feynman's path integrals.

Another area where techniques of statistical mechanics can be applied is in economics, which is of course highly relevant to our everyday lives. As well as governing our job opportunities, the wellbeing of the economy can statistically determine to some extent our own health and wellbeing as a populace. It also shapes the economic and social politics which may itself have effects that last for decades or more. It is therefore of premium importance to monitor and model the economy and to try to avoid or abate economic crashes. In this blog entry, I will explore the topic of financial crashes that underlines some connections between economics, statistical mechanics and complex systems.

First, a disclaimer: I am in no way claiming any expertise in economics or indeed in complex systems. I felt that the fundamentals of Lattice QCD were explained so well in Floriano's blog that it was pointless to try to reexplain the basics, and through research for the blog, I decided to have a bit of fun by diving headfirst into the wonderful world of econophysics. It is worth noting that the arguments put forth in this blog entry are from the "fractal school" of the market analysis of finance, pioneered by Mandelbrot. This is an alternative to other more traditional schools of thought on the subject like the efficient market hypothesis, an approximation that all prices reflect all available information; financial bubbles cannot be explained within this approximation.

Complex system analysis studies the behaviour of a system with many components that interact with each other. These interactions may be nonlinear in nature, making the system difficult to model using more conventional techniques.
In an economic model, we may distinguish between endogenous and exogenous variables. Exogenous variables are factors whose origin lies externally to an economic model, and are then imposed upon the model. Earthquakes for example may be described as exogenous. On the other hand, an endogenous variable is innate to a model, and its value is determined by the model. A topical example: we may decide to model our economy with the current Covid-19 epidemic as an exogenous factor. This is necessarily a simplification of the complex interplay between human society and the biosphere from which plagues emerge. We may also enrich our model by accounting for the endogenous factors of urbanisation and globalisation etc. that hasten its spread. If we found a link between the origin of a particular epidemic and, for example, animal agriculture, we could call it an endogenous factor of the current system of agricultural production.

Speculative bubbles, as D. Sornette of the University of California explains in his book "Why Stock Markets Crash", are created by investors who expect an unrealistic future return on present investment. The bubble grows due to the positive feedback effects endogenous in the system, such as 'herding' to buy stocks during bubble growth and to sell during a crash. These behaviours do not require any conscious coordination of the individual investors but are instead encouraged by the self-organising system. This sort of behaviour drives the system away from a stable state to an unstable state, where a crash may be triggered by small, local exogenous effects. The bubble may alternatively end by a continuous regime change, i.e. a crash is not necessarily inevitable. Decelerating market devaluations, so-called "anti-bubbles", can also be described by the same "log-periodically decorated" power law as found for accelerating bubbles. Financial crashes have been shown to belong to a different distribution to small-scale market fluctuations, giving us some distinction in scale which can be converted into predictive power.

In contrast, some systems are impossible to predict outright. Earthquakes may be an example of this. The philosophy behind this is that earthquake events display a self-similarity in scale, meaning that large earthquakes are only the continuation of small earthquakes. As the nucleation is identical, the catastrophic large earthquakes are not a priori distinguishable from small earthquakes. One can rather forecast the probability of an earthquake happening within a given relatively large period of time.

According to this theory, a financial crash is characterised by a critical point, in this case in the time domain. As the system evolves in time ever closer to this point, the probability of a crash increases, as the system becomes more unstable with respect to small perturbations. At the critical point, a phase transition occurs, corresponding to a crash. The phases in this case are a "sell" regime, where supply outweighs demand, and a "buy" regime, where demand outweighs supply. Power laws govern largely the behaviour of the variables of the system, such as price, volatility and crash hazard rate, around the critical point. This power-law evolution, however, is difficult to detect in practice due to the inherent noise of the system. It is therefore beneficial to look for other characteristic indicators of an impending financial crash.

There are, fortunately, other indications of a system moving closer to the critical point. As mentioned earlier, bubbles can be characterised by a "log-periodic decorated" power law. This means that the basic power law is 'decorated' by a 'log-periodic' oscillation, with a frequency that increases as the system approaches the critical point. This decoration can provide another indication of an impending crash which is more discernable over a background of noise.

As explained in D. Sornette's book, this is a general feature of 'models of cooperative behaviours resulting from imitation between agents organised within a hierarchical structure'. Here, the hierarchical structure refers to the presence of actors who operate with differing sizes of investment. Take, for instance, the difference in scale between an individual investor and an investment bank. The origin of this log-periodicity could emerge from the impact of analysts that seek to restore the price to its fundamental value, which, in combination with trend-following strategies e.g. positive feedback or herding, overshoots the target with increasing frequency as the price accelerates. More formally, log-periodicity is a general feature of systems with discrete scale invariance, meaning a discrete scaling factor of self-similarity i.e. the magnifying factor from one level to the next. For example, log-periodic decorated power laws are also observed in self-similar models of the Ising type familiar to all those who have taken courses in statistical mechanics.

Discrete self-similarity in investment model

The renormalisation group approach forms a major part in statistical mechanics, as well as in quantum field theory. It is based upon the change of system variables with scale, and is used to probe the critical phenomena of a system. Using as an example a spin system on a lattice, we can change the scale of observation of the system by 'blocking', meaning that we take the average of a block of adjacent lattice sites, and treat this average as a new site on a sparser lattice. The state of the system after blocking therefore depends upon the state of the system before blocking.
The critical point of this spin system is given by a certain critical length scale. The correlation length of a system is the characteristic length at which the spins are correlated i.e. point in the same direction. Depending upon how the correlation length compares to the critical length, the system will evolve in different ways due to blocking.

We have three cases:
1) L < L_c: With blocking the system is driven away from criticality, becoming more disordered.

2) L > L_c: With blocking the system is driven away from criticality, becoming more ordered.

3) L = L_c: With blocking the system remains at the critical length scale, remaining at the same level of order. This regime of criticality shows self-similarity, meaning it is scale-invariant for continuous changes in scale. The structure of the lattice correlations is fractal.

For all values apart from L = L_c, the 'renormalisation group flow' drives the system away from the critical value, with a speed of seperation from the critical point characterised by the critical exponents for different quantities. Therefore, modelling the renormalisation flow allows us to quantify the critical exponents that characterise the critical point.

It turns out that the log-periodic decorated power law is equivalent to the projection onto the real plane of the observable of the real order parameter (L - L_c) taken to the power of a complex exponent. This corresponds to a change in magnitude and a rotation around the complex plane as the magnitude of the order parameter decreases i.e. as we get closer to the critical point. The oscillations in the real observable are manifestations of projections from rotation in the complex plane. This rotation stems from the exponent, leading to log-periodic oscillations in the observable, meaning that there is a regular oscillation in the logarithm of the observable. Discrete scale-invariance is a property of complex critical exponents just as continuous scale-invariance at the critical point corresponds to a real critical exponents.

In this blog post, I have talked about some methods of potentially predicting financial crashes through the use of methods from physics. Within the above theory, some markets are intrinsically instable due to endogenous factors, which are practically impossible to change within a system due to the self-organistion of the system. The instability may therefore be solvable through either control of exogenous factors, i.e. through governmental policy, or through complete reorganisation of the financial system from the outside. Through physical modelling, we may also analyse different systems to find the optimally stable system into which to transition.

Inspiration and further reading

Sornette, D. (2017). Why stock markets crash: Critical events in complex financial systems. Princeton, NJ: Princeton University Press.

Khamzin, A. A., Nigmatullin, R. R., Popov, I. I. and Zhelifonov, M. P. (2012). Log-periodic oscillations in the specific heat behaviour for self-similar Ising type spin systems. Journal of Physics: Conference Series, 394, 012008. doi:10.1088/1742-6596/394/1/012008

Yalamova, R. and Mckelvey, B. (2011). Explaining What Leads Up to Stock Market Crashes: A Phase Transition Model and Scalability Dynamics. Journal of Behavioral Finance, 12(3), 169-182. doi:10.1080/15427560.2011.602484

On collective motion and flocking

Simeoni — Mon, 04 May 2020 09:04:39 +0000

(Stefano Ghezzi - Unsplash.com)

Have you ever had the chance to admire big flocks of birds flying in the sky? Even though it is made up of hundreds of animals, the flock acts almost as a single entity, swirling and stretching, constantly changing its form in a way that is really appealing to the observing eye.

This particular kind of collective motion that creates really appealing and hypnotizing patterns can be found everywhere in Nature: from schools of fish to human crowds, passing through bacterial colonies, cellular tissues, swarms of bugs, group of chemically activated particles, et cetera.

Lance Anderson - Unsplash.com

The phenomenon, being at the crossing of different topics, has been studied extensively by biologists and physicists, and has led to some neat insights on how large groups behave and self organize, even though there is still a lot to uncover.
I've decided to take the occasion of writing this blog post to deepen a bit my knowledge on the subject and to try to artificially reproduce the appealing visuals in the pictures above on the small screen of my laptop, with the help of some good ol' programming.

But first, a tiny bit of introduction.

With the name Active Matter, one points to all those systems made up of a big number of constituent elements that possess the ability to convert chemical energy into kinetic energy (think of it as being able to do jogging after having had a big meal). Furthermore, active systems show the onset of complex structures due to the collective motion of the particles, qualitatively different from the motion of the individual constituents. This particular characteristic goes under the name of swarming or flocking and it is an emergent behavior that arises from a simple set of rules followed by the individuals in reaction to the behavior of the neighbors and the ambient. In most cases, there is not a leader that coordinates the group, and the structure simply arises by the interactions of the individuals.

This is the kind of behavior I wanted to try to reproduce, and more specifically I targeted fish schooling (just because, why not). First, I needed a model, so I focused on a famous one, the Vicsek_model. Physicists like this model a lot because it is quite simple and general, and yet it manages to describe universal properties for these kind of phenomena.

It consists of point-like self-propelled particles that evolve at constant speed in a two (or eventually three) dimensional environment. At every time step, the particles' orientation is dictated by the orientation of all the other neighbors that are within a set distance: a fish will always try to align to its neighbors so that the whole school keeps swimming in the same direction.

The second ingredient is random noise: since this kind of mechanism is conditioned by a big amount of different factors (think for example to signal misinterpretations, stimuli from external ambient, etc.) a random factor is put to affect the direction of the individuals.

Next, the individuals take a step in the new direction. And then the procedure starts all over again. These simple rules are already sufficient to have an interesting dynamic at hand: it can be seen that with large noise or low density of individuals, particles are not aligned and they move with no apparent order. On the other hand, at low noise and large density, particles are globally aligned and move in the same direction. And the transition between the two states is not as smooth as one would think, since when going from one state to the other some interesting patterns appear in the group.

In order to simulate our school, we also need two other ingredients: a repulsion rule for when the fishes get to close and an attraction rule to keep them together (this last term represents the social tendency of the individuals to stick together). In the end, we get a totally believable simulation.

Then, just to have some more fun, the repulsion dynamic can be extended to include some reefs in the simulation, and to have a look at a simple prey-predator dynamic one can introduce sharks as well!

Following, a video of the results that can be obtained with the simple dynamic rules described above:

Lastly, some interesting videos you can check:

Additional resources:

The deadlock in climate policy and the science of extreme event attribution

Goedert — Mon, 09 Dec 2019 04:19:51 +0000

Climate. It is among the most discussed topics of our days: from international policy making to public demonstrations, media outlets, and even elevator small talk. There is no lack of controversy and strong opinions when climate change is on the table, but that is not exactly surprising. When concrete actions must be taken, responsibility must be assigned; and it seems to be human nature that such discussions must be hard and tiring. Naturally, even more so when people's livelihoods are affected. We may never finish discussing to what extent human action has affected our planet’s climate system, what actions need to be taken and by whom. As importantly, we could spend days debating why our governments seem stuck in the drafting of effective measures to mitigate and fight back climate change (or even recognize it, in some cases). Though the divisive issues of responsibility and effective actions are essential, they are not the focus of this humble essay. Not exactly at least; I would like to take the topic closer to home.

The science being developed over climate systems is amazing! There is a staggering amount of physical phenomena and chemical reactions interacting over an immense range of spatial and temporal scales. There is so much diversity in the people involved as well: from the finest theoretical physicists and mathematicians, to the most ingineous meteorologists, spatial scientists and computer scientists, employing the most powerful computers ever built. But this is also not my motivation today.

My objective today is to raise the debate on a sensitive topic in-between: the responsibility we hold as researchers in both the social and political debate of the ever more pressing debate that are climate actions. In particular, the challenges we need to tackle towards the development of effective policies.

Good quality and solid research is essential to inform and guide effective actions, be they local, national or global in scale. On the other hand, we need to constantly reach to our communities so that people can take their positions on the matter with well educated opinions. This is not only a fundamental social responsibility, but also essential to actually drive governments and civil institutions to take necessary policies and measures. The scientific community has been working ever harder on both ends of this debate.

We have seen, all types of communities cry out for action and rise up in protest against the inadequacy of the policies made by our leaders. This is probably the issue that most united our young generation, across geographic and social barriers. This green wave has been reshaping the practices of companies and creating new markets, as people tend more and more to prioritize products that are environmentally friendly. And yet, this is far from enough.

Below, we see the probability distributions for anomalies in the summer mean temperatures, projected over the next decades for different scenarios. The vertical line marks the average anomaly recorded during the famous 2003 European heat wave. Even if the most ambitions measures had been taken, we still would need to get used to such extreme conditions.

Not only heatwaves, other extreme weather conditions and natural disasters such as floods and snow storms are becoming more frequent and intense. After facing such a disaster, someone will always raise the question “was this event influenced by climate change?”. Let us break it down to its core assumptions:

How do we characterize an event as extreme? (e.g. location, duration, likelihood, social impact...)
What does “influenced by”? (how are the above metrics conditioned of affected?)
How do we characterize climate change? (comparison between past, present and future)

These three points summarize the core of the research field that is called Extreme Event Attribution, which searches for robust causal and probabilistic relations between these extreme events and the changes between current climate conditions, its recorded past and our future predictions. These relations are instrumental in defining policies for prevention, damage control and in the payment of reparations.

Only by understanding to which extent human actions affect the variability and frequency of extreme events, can we realistically assign responsibility and take concrete steps towards breaking the deadlock that has plagued our previous attempts at achieving a global compromise towards the protection of our climate and our environment.

Symmetry and Physics: how to explain to my grandma what I do.

Todaro — Tue, 10 Sep 2019 12:00:15 +0000

15th of August. National holiday in Italy. All my family meets together to celebrate. Break between the first and the second course. Frozen ready meals - my standard menu when I'm away - seem like a distant memory. It's a beautiful day, the birds sing in the trees, the sun is bright in the sky, everything is perfect when it happens:

"But, what do you do exactly?"

It's my grandma. She's looking at me with a skeptical face, like I was a door -to-door salesman that wants to fool her. The tone of the question leaves no doubts, this time a vague answer won't be enough. My usually escape strategy in these situations is using the tested formula "I am a researcher in Physics. Theoretical Physics'' - like "My name is Bond. James Bond'', but a bit more nerdy - that, after a moment of solemn silence, is followed by "I work with computers'', just to give a more concrete image of my job. At that point my grandma usually nods making clear she still has no idea of what I do. But this time is different, this time she demands a satisfactory and detailed explanation and I know there is no way to avoid it. I'm screwed.

I know, you are probably thinking I'm too melodramatic and you are probably right. Some of you maybe don't even get what the matter is. The fact is that is extremely hard to explain to people who are not in the scientific sector what a researcher actually does at his job. Just to be clear, I am not saying that it's because we are smarter or better than others or because our work is tougher than others; if you are a specialist in your own field, it's clear that people that never saw those things in their life won't be able to have a clear and specific idea of what you do. Whatever the field is.

What I'm saying is that, for instance, if you say you study law, people have in their mind an image of what your job will consist of - "You want to become a lawer'' - even if this won't be actually true and you are probably going to do something completely different. But people have an understanding of what you deal with, or at least they believe they have one. Engineering? You build bridges. Biology? You spend your days outdoors studying plants and animals. Chemistry? In a lab wearing a white coat, surrounded by bottles of all shapes and dimensions.

Of course, they are stereotypes, generic frames, but you have a frame, at least! Instead, when I said I was studying Theoretical Physics, lots of people do not know what to think. Some of them ask me if I want to become a mad scientist. Others say "You surely are a genius or a sociopath or both'' (Thank you, Sheldon Cooper!). My uncle, for instance, is convinced that I am a sort of inventor and every time he meets me, he asks me to realize his last revolutionary idea that obviously will make us rich. My other grandma, instead, relates the word physicist to the famous picture of Darwin with the long white beard, that she saw somewhere, and every time we meet and I haven't shaven in 3 months, she thinks I'm climbing the career ladder. And I could go on.

This is exactly what I mean! ^{_{(Source: www.nowyourmeme.com)}}

Thus I noticed from my personal experience that science, and Physics in particular - although maybe this is my personal perception and Mathematicians wouldn't agree - are seen as a sort of foggy and cryptic mystery that surrounds also people involved in this sector.

This is the reason why all the efforts that go in the direction of getting people close to science, have an invaluable impact and must be encouraged. Efforts like outreach events, for instance the European Researcher's Night that I recommend to everyone who wants to see 100% original researchers exposing their research activities in a funny and interactive way (last year there was this robotic hand you could control just by moving your arms! And you could try it. It was amazing!)

And, in my small way, I want to do my part too. So what better way then talking about my work, trying to make it accessible to the lovely, not highly educated, 90-year-old woman that my granny is. So grandma, if you are reading this, even if with some delay, but I'm about to answer your question.

Let's start from the beginning. The main goal of my PhD project is the investigation of the properties of matter, and in particular if the violation of the so-called CP symmetry occurs. What are you saying? The only things you studied at school are sum and product, you don't remember subtraction and division was an advanced topic? Mmmh, ok forget for one moment what I said above. Focus on this word: Symmetry!

Symmetry is a very intuitive concept, that we can experience in every day life: look for instance at yourself at the mirror. Your face is an example of symmetry! If you draw a vertical line following your nose, you can notice that all the elements of your face - your eyes, your ears, your mouth are disposed in the same way on the left and on the right part, as in a mirror image. This is an example of reflection symmetry.

There is lot of symmetry in nature: the astonishing colours of the wings of butterflies are disposed in a perfect symmetric way, and so the structure of lot of plants and vegetables that show an ordered and symmetric pattern. The shape of snowflakes is another example, in particular they show a rotational symmetry. Try googling 'Symmetry and nature' and tons of images will show you what I mean. And probably you will find most of them very nice. This because the human mind is naturally attracted to symmetry: we usually find symmetric shapes as more harmonious and beautiful, and this is the reason why there are lot of examples of symmetry also in the arts.

Symmetry in nature: reflection symmetry of butterfly wings, of left and right parts of human faces, and translational symmetry of a honeycumb.^{_{(Sources: left https://ecstep.com/natural-patterns/, right https://it.depositphotos.com )}}

- No grandma, I don't study snowflakes. Just 5 more minutes, I'm getting there. - This simple concept of symmetry can be defined in a rigorous and mathematical language, that connects symmetries with transformations. An object is symmetric under the action of a particular transformation (or a set of them) if its action leaves the object unchanged. The meaning is still quite easy to visualize.

Take for instance something spherical, like a ball, or an orange and let's say this is our 'object', and then we want to rotate it, so the rotation is our 'transformation'. If you leave the room, I rotate it and then you come back, you cannot actually state if I did it or not, because the rotated orange and the unrotated one are indistinguishable. This is an example of spherical symmetry. But there are many others that play an important role in physics. They usually allow you to extract some non-trivial or fundamental properties of a system like conservation laws or quantization conditions.

Consider that the current understanding we have of the fundamental forces of our world, for instance, is basically grounded in some models that have symmetry as a crucial ingredient. I know, lot of unclear words. But just to stress how important symmetries are, these are the words of P.W. Anderson, who won the Nobel Prize for Physics in 1977:

"it is only slightly overstating the case to say that physics is the study of symmetry."

But ok! Go back to our discussion. Above I mentioned the so called CP symmetry. What is that?

'P' stands for parity, the transformation of space that changes the sign of all the axes. It's like inverting all the directions. Imagine you are playing a videogame and you decide to reconfigure the controller and 'up' and 'down', 'left' and 'right' now are inverted. You are in a sort of 'mirrored' space. - What, Grandma? You never played videogames? Mmmh, let's try something else. Stranger Things? Does the upside down mean anything to you? Ok, don't mind. - I'm just saying that there are physical phenomena that are invariant under parity, that means you cannot distinguish between what occurred in the normal world and what in the 'mirror world'. Others that are not.

'C' instead is a bit more tough to explain. It stands for 'charge conjugation' and consists in switching all the particles with the corresponding anti-particles and changing the sign of the charges (electric charge, color charge and so on). I know, it sounds a bit technical, but maybe there is a way to circumvent this difficulty.

My research project aims to test if something is 'symmetric' under CP or not. There is an important theorem (the so-called CPT theorem) that, in practice, tells us that physical laws seem to be invariant under CPT transformations. So if something does not change if you imagine it in a 'mirror world' (parity P) with particles and anti-particles switched (charge conjugation C), then it must be invariant also under T. But what is T? This is easier to understand.

'T' stands for time-reversal, and it consist in inverting the direction of time. If with parity, we could imagine a 'mirror world', now imagine a world where time flows backward, like when you rewind a tape. As before, there are physical phenomena that are invariant if you imagine them taking place in this world and others that are not.

Asking if something is CP symmetric = asking if something is T symmetric.

Now, I have defined the 'transformation', but what is the 'object'? In my case the 'object' is the neutron, one of the building blocks that constitute the surrounding matter. - Yes Grandma, you already knew this because you read the previous blog. Well done, less work for me! - The missing piece of the puzzle is how I know if neutrons are invariant or not when I 'invert' the time direction. Electromagnetism comes to help. Neutrons, as suggested by the word itself, are 'neutral', namely their total electric charge is equal to zero. But, in principle, it could be possible that the distribution of the charge is a bit unbalanced even if the total amount (positive plus negative) is still zero. Therefore a so-called dipole moment would appear that change its direction under T.

I know, it's a bit hard to visualize. Let's take the same orange we used before. It's smooth and spherical, thus, as we said it's quite symmetrical under rotations. But if I take this other orange, with the small stalk still attached, and I rotate it, you can distinguish the object before and after my modification. If the orange has the stalk is not invariant under rotation. If the neutron has a dipole moment is not invariant under T ( and thus under CP ).

A graphical representation of what happens when i rotate an orange. The rotated one
can be recognized only when something (like the stalk) breaks the rotational invariance.

Schemtatic representation of a neutron under a T transformation. d is the neutron EDM, while mu is the magnetic dipole moment. Notice that here 'upward' and 'downward' is meaningless - there is no table that gives us a reference frame - what makes sense is to compare the directions between d and mu.
(^{_{Source: www.wikiwand.com/en/Neutron_electric_dipole_moment}})

Accordingly to the Standard Model - the best theory we have to describe the fundamental forces present in nature - the neutron shouldn't show such a behavior. Finding a neutron Electric Dipole Moment would mean that we have to add something to our Standard Model, it wouldn't be enough to explain everything anymore.

Of course, I don't deal with real neutrons, but I simulate them, by means of this powerful computer clusters - yes, this kind of simulations are extremely computational demanding - and then I collect data and analyze them, using several different techniques, and facing a lot of different possible issues that could arise - and they usually do arise - in a challenge that often puts a strain on you. Most of this things happens while I am in front of my laptop, so the description "I work with computers'' is technically true. It says nothing, but it's true.

Now, maybe, your idea of what I do is a bit more clear, maybe you can even understand the meaning behind the title of my research project, that is: "Lattice QCD methods (i.e. computer simulations) for nucleon observables (i.e. for wondering if there is a nEDM) probing beyond Standard Model physics".

- Do you finally know what my job is? - I think that at this point my grandma is probably nodding her head. This time it's not so clear if she still has no idea of what I do or not.

Spacecrafts, particles and Lattice QCD

Manigrasso — Sun, 14 Jul 2019 10:21:29 +0000

On March 2 1972, the Pioneer 10 space probe was launched from Cape Canaveral, Florida: direction Jupiter. Once completed its mission, this small spacecraft travelled towards the deep space, being the first artificial object to leave the solar system. Besides the 250 kilograms of scientific instruments, the probe contained also a peculiar piece of art: a message from humanity to alien civilisations, the so-called pioneer plaque. Being to possible to communicate with any language, this gold-anodized aluminium plaque, represented only pictures whose aim was to provide information about the origin of the spacecraft. There were representations of a male and a woman, of the probe itself, a map showing the position of the solar system and well as a picture of the planets composing it. Apart from it, in the top left corner the picture depicting the transition between two states of the hydrogen atom. Why? Because it is by far the most abundant element in the cosmos, the first to be studied by any civilisation. Richard Feynman , one of the most influent theoretical physicists of the 20th century, claimed

If, in some cataclysm, all of scientific knowledge were to be destroyed, and only one sentence passed on to the next generations of creatures, what statement would contain the most information in the fewest words? I believe it is the atomic hypothesis [...] that all things are made of atoms.

Richard P. Feynman, The Feynman Lectures on Physics

The Pioneer Plaque. Picture taken from www.planetary.org

What is the atom made of? How many other particles are there?

All of you have heard about proton, neutron and electron at least once in your life. Indeed, these three particles constitute the atom as a whole: neutrons and protons bound together in a very small region of space (which is equivalent to one meter divided by 10 to the fourteenth power) and the electrons move around it in a region thousand times bigger than the nucleus itself.

Sketch of an atom. Picture taken from www.whoinventedfirst.com

These are probably the most widely known examples of particles, although there are many more and they differentiate essentially in two important categories: composite and elementary particles. While the former, if one looks close enough, contain other particles bounded together inside them, the latter do not hide a substructure. This means that they are not composed of smaller constituents.
There are seventeen elementary particles known up to now, and they have been artistically organised in the table below.

The particles composing the Standard Model of particle physics.
The picture was taken from https://www.symmetrymagazine.org/standard-model/

As you can see from the colours, they are divided into four main families:

Leptons (green): They are six and each of them possesses its own antiparticle. Leptons are subdivided into two subcategories, the first one is composed by electron, tauon and muon, while the second contains the neutrinos “counterparts” of these particles.
Quarks (red): As the leptons, they are six and each of them possesses its own antiparticle. There are two light quarks, up and down, that constitutes a huge percentage of the “ordinary matter”, i.e. they are contained in the atoms composing everything in the world; Then the fancy names come out: the remaining quarks are the strange, charm, top and bottom. The latter is much heavier than up and down, and they bound together giving heavier particles which usually are unstable and decay quickly.
Vector bosons (cyan): These particles are conceptually different from the latter since their role is to “mediate” interactions. What does it mean mediate? Well, let’s consider a simple example of two particles with opposite charges (positive and negative) put close to each other: our common experience says that they will attract each other. But how do they feel the presence of each other? By means of a vector boson, the photon! You can imagine this interaction as an exchange of photons between the two charges and in this sense, they “carry” the interaction.

There are mainly four vector bosons corresponding to three forces: apart for the photon, there are the W and Z bosons which carry the weak interaction and are responsible for radioactive decays, and the gluon which mediates the strong interaction acting on a subatomic scale binding the quarks inside the nuclei.
Last but not least, gravity: as far as we know, is a fundamental force but there not exist yet a theory explaining it from a microscopical point of view, in terms of vector bosons. This is probably one of the biggest open problems in physics.
Scalar bosons (purple): Last but not least, the famous Higgs boson. You have probably head about it, because of the fact that it has been theoretically predicted a long time ago and in 2012 experimental physicist s finally claimed they revealed the first signal supporting its existence. The Higgs boson interacts with many elementary particles, like for instance the electron, and through this interaction, it gives them a mass.

Starting from these elementary constituents, hundreds of different particles can be built.

At this stage you may think the physicist as someone collecting boring data about all these tiny particles in huge tables, to organise them like a stamp collector. Is this the real point of conducting research in particle physics?

The answer is at the same time simple and ambitious: finding a big scheme explaining everything, a definitive theory able to explain every phenomena in nature starting from the microscopical scale, predicting the way particles interacts among themselves, moving towards huge distances, explaining the structure of the Universe, how it evolved starting from the Big Bang and why it keeps expanding. In other words, being a physicist means to try to put together all these phenomena, explaining them according to a common set of rules, the physical laws. Saying it with Feynman words:

Curiosity demands that we ask questions, that we try to put things together and try to understand this multitude of aspects as perhaps resulting from the action of a relatively small number of elemental things and forces acting in an infinite variety of combinations.

Richard P. Feynman, The Feynman Lectures on Physics

Solving the puzzle of particle physics.
The picture was taken from https://physicsforme.com

Standard Model of particle physics

The most promising theory, attempting to give an explanation to all these phenomena is called “Standard model”. It is not a definitive theory, in the sense that it cannot explain every physical phenomenon observed in nature. The Standard Model is composed of different sub-theories, each one of them explaining one fundamental force: the electroweak theory was originally composed of two distinct theories, explaining electromagnetism and weak interaction separately. However, later on, Weinberg, Salam and Glashow discovered that the two kinds of interaction are really interconnected and that they can be explained altogether with a unified theory. For this reason, in 1979 they were awarded the Nobel Prize in Physics. The second theory is Quantum Chromodynamics (QCD) and explains the strong interaction between subatomic particles, how quarks bind together giving the proton or neutron, as well as many other composite particles.

QCD, why so difficult?

Even though QCD, as we know it nowadays, has been formulated in the early 70s, it is still a hot topic in research and many of its properties have to be discovered yet. The reason for this resides in the so-called confinement phenomenon. In order to understand what this means let’s take a step back and consider the case of the better-known electromagnetic interaction.
Consider again the example of the two charges, with opposite signs, set apart in the space. Probably many of you have studied in school that the force between the charges is described by the Coulomb interaction,

Coulomb law.
The picture was taken from https://www.toppr.com/

Which means that if we increase the distance between the bodies, the force decreases.

What if the force increases with the distance?

An example of such a scenario is familiar to all of us: if we put a rubber band around two fingers and try to set them apart, the bigger is the separation the stronger gets the force that obstacles our effort. Of course, the reality is much complicated, but this simple example helps to visualize how the force between quarks composing the proton acts. In particular if, for any reason, a quark and an antiquark are set apart the force that binds them increases with the distance separating the two particles. Moreover, like a rubber band that reached a threshold tension breaks, also the quark pairs may break giving rise to two quark pairs! This actually means that we can never observe a free quark, i.e. a not bounded with an antiquark.

When a quark-antiquark pair is set apart it breaks giving two new pairs.
The picture was taken from Wikipedia

The confinement phenomenon is a very fundamental physical property that affects how quarks bind together giving rise to composite particles, like the proton. It is quite remarkable the fact that, even though the confinement phenomenon is commonly recognized in the physical community, there is no mathematical proof of it yet! This means that, for instance, solving the equations of QCD for obtaining a description of the proton is not possible and other methods need to be employed in order to study these particles.
Beside some approximate theories that allow to approximately describe QCD in some particular situations, there was the need to change perspective in order to deeply understand the consequences of this fascinating theory. In 1974 Wilson proposed a new paradigm: instead of approaching QCD analytically one may try to simulate it. This idea, which was possible due to the increased computational power computers are capable of, was furtherly developed and refined during the years and allowed physicist to obtain high accurate physical predictions about QCD.

If you can't solve it...just simulate!

The basics idea behind these simulations is to rewrite the equations of Quantum Chromodynamics within a finite volume discretizing space and time. For this reason, this field of study is called Lattice QCD.

Discretization of space with a grid of points.
The picture was taken from http://www.physics.adelaide.edu.au

It is interesting to see how the volume that physicist has been able to simulate increased with the time, due to the speed-up and the improvement in memory of computational resources. In order to simulate QCD on large enough volumes, today physicist writes simulation software able to run on the most powerful supercomputers all over the world.

The Summit supercomputer.
The picture was taken from https://www.olcf.ornl.gov/

Operator Splitting Methods

Diab — Fri, 24 May 2019 22:50:14 +0000

Operator splitting is a powerful method for numerical investigation of complex models. The basic idea of the operator splitting methods based on splitting of complex problem into a sequence of simpler tasks, called split sub-problems. The sub operators are usually chosen with regard to different physical process. Then instead of the original problem, a sequence of sub-models is solved, which gives rise to a splitting error. In practice, splitting procedures are associated with different numerical methods for solving the sub-problems, which also causes a certain amount of error.

Complex physical processes are frequently modelled by the systems of linear or non-linear partial differential equations. Due to the complexity of these equations, typically there is no numerical method which can provide a numerical solution that is accurate enough while taking reasonable integrational time. In order to simplify the task, operator splitting procedure has been introduced, which is widely used for solving advection-diffusion-reaction problems and Navier-Stokes equation including modelling turbulence and interfaces.

Different algorithms like Lie-Trotter, Strang and Additive splitting were introduced to solve ODEs and PDEs. One of the simplest problems is the initial value Cauchy problem:

Lie-Trotter Splitting

Lie-Trotter splitting is a first order splitting method which solves two sub-problems sequentially on subintervals where and . The different subproblems are connected via the initial conditions as follows:

where . The approximated split solution at the point is defined as .

Algorithm:

Step 1:
Step 2:
Step 3:
Step 4: if , go to step 1, otherwise stop.

Strang Splitting

One of the most popular and widely used operator splitting method is Strang splitting. By small modification it is possible to make the splitting algorithm second order accurate. The idea is that instead of first solving the first sub-problem for a full-time step, we solve it for a half time step. We then solve the second sub-problem for a full-time step, and then back to the first sub-problem and solve it for a half time step.

where , and the approximated split solution at the point is defined as .

Algorithm:

Step 1:
Step 2:
Step 3:
Step 4:
Step 5: if , go to step 1, otherwise stop.

Additive Splitting

Additive operator splitting method is a first order method and it is based on a simple idea in which we solve the different sub-problems using the same initial condition. The computed split solution of the two sub-problems is added and then the initial condition is subtracted from the sum. In this manner we obtain a splitting method where the different sub-problems have no effect on each other. This motivates the use of the additive splitting, first because the result of an additive operator splitting is independent of the order in which the operators are applied. For methods like Lie-Trotter and Strang splitting, the operators will generally not commute in the non-linear case, which means the result depends on the order in which the operators are applied. Another advantage is that the operators are applied independently, so that they can be computed in parallel.

The additive splitting algorithm is as follows:

What is next?

In nowadays research in the field of mathematical modeling and simulations, the systems of partial differential equations are getting more complicated due to the complexity of the modeled physical, biological and engineering problems. Often, one wishes a model to capture different aspects of a situation, for instance both convective transport and dispersive oscillations on a small scale. These different aspects of the model are then reflected in a partial differential equation by different operators like diffusion or convection which are hard to analyze both theoretically and numerically, especially in the non-linear cases. For this reason, operator splitting methods where developed.

There are many situations in which the phenomena are modeled with more complicated system of equations than those described by PDEs. In circuit simulation, gas networks and power networks, partial differential-algebraic systems arise, but still the operator splitting methods dose not cover all the situations.

For more details about operator splitting methods, their consistency and stability analysis, refer to: Splitting methods for partial differential equations with rough solutions. Analysis and MATLAB programs, Kenneth H. Karlsen, Knut-Andreas Lie and Henrik Risebro.

Machine Learning and Physics

Davide — Tue, 26 Mar 2019 10:13:02 +0000

Machine Learning - A very short introduction

Machine Learning (ML) is a particular branch of a very broad discipline called Artificial Intelligence (AI). Whereas AI tries to solve the fundamental problem of "creating" a sentient or intelligent being from a silicon-based machine, or anything you could define as artificial in general, the problem with ML is somehow simpler.

The purpose of ML is, in fact, creating algorithms that are alternative to the usual way one would program a machine, to solve a problem. In a standard code, the computer runs a series of instructions one after the other, that is to say, it behaves like a blind slave who does exactly what he's told to do. After the execution of the code, the computer will have learnt nothing about the problem it was used to solve and will have no knowledge of the data that were passed to it. Therefore, the next time you will run the code on another set of parameters, your program will go exactly through the same sequence of steps, without there being a solution of continuity to this process.

This is a very simple strategy for solving problems that are structured in a repetitive and simple way. This approach, however, fails when you apply it to something that for your brain would be extremely easy. As a banal example, if I asked you to compute the following sum to you or a seven-year-old kid, you would be able to do it practically instantly

4+5=?

On the other hand, if I were to ask you to compute the following sum, it would most likely take you more time than the previous one, and an awful lot of time more time than it would take a computer.

1935428762456 + 42649955246 = ?

Yet, the practical things you have to do are exactly the same. This is because computers are extremely optimised to perform arithmetical operations such as the one above, and can do that in a minuscule fraction of a second. Our brains are, however, specialised in other types of operations. For instance, here's a picture of a red panda, one of my favourite animals living on this planet.

Image from the Binder Park Zoo website

If I asked you which of the following pictures has a red panda in, you would be able to give me an answer immediately. A computer programmed in the usual way would not be able to give me a sensible answer at all, it could only retrieve a random choice.

Images by Joseph Nebus and Mathias Appel (Flickr) respectively

This happens because if you had no idea what a red panda looked like before you saw the picture above, after having experienced that picture, you now have a knowledge of what that animal looks like. In the second iterations of my showing pictures to you, you had stored this information in your brain, and you were able to use it to answer the question. That is to say, your "state" before and after looking at that first picture and being told what it contained changed.

As said previously, a normal implementation of a program on a computer is not able to do this, it could only scan through the pixel and try to find a previously defined pattern to match. This could cover one particular image, but it won't be enough to try and find a red panda in any image I supply to the program, which is however what your brain would be able to do. We can exploit, however, a similar approach to what your brain is doing, that is to say, we want to try and find a way to let the computer memorise what characteristics a picture must have in order to contain a red panda.

Artificial Neural Networks - A way to mimic the human brain

An approach that can be used to solve such problems on a computer is to use so-called Artificial Neural Networks (ANN). These are structures that were first thought of in the 1950s, and later developed and put to practical use in recent years thanks to the advancements in computational power and memory access. Their structure, at its basis, is a series of layers of so-called neurons. Neurons are simple objects that take many inputs, make a weighted average of them (that is to say, they sum the product of each of the values with a corresponding weight), and then compute a function of this result.

Image from https://www.analyticsvidhya.com/blog/2016/08/evolution-core-concepts-dee...

The overall structure of the network might look like the picture above, where each of the circles is a neuron. In this kind of ANN, I am using 3 inputs (numbers) to compute 2 outputs. One can, therefore, think about passing an image via passing each of its pixels as inputs, and retrieve one output that tells them whether the image contains a red panda or not, or more in general, whether the ANN has found anything in it. Although a real-life ANN that performs this operation needs to be more complicated than that, the core ideas are the same.

The way the ANN structure learns (for instance) to recognise if a particular animal is in a picture is by being provided with a series of pictures, and being told if they contain that animal or not. For each of these pictures, the ANN algorithm will guess whether it contains an animal and will compare its result with the given (right) one. If there is a difference between the two results, the weights of all the layers will be changed, so that the answer the ANN would have given matches the right one. This is repeated for every picture in a sample. By this repetition, the ANN will form a core structure of weigths that contains abstract information on whether the animal is in a or not. The ANN will be able to recognise, thanks to the "trained weights" whether the particular animal is in any of the pictures in the sample we have used to train the ANN. Therefore, when provided with new images, it will give an answer that is based on its experience, just like we would do. Therefore, after having trained the ANN, which basically means having run this process long enough for the weights to be exact for it to give the right answer an acceptable number of times, we can then deploy this ANN and its weight on however many images we want, expecting a good number of outcomes to be exact.

Since AI experts have seen a real possibility of application of ANN for many problems, they have been exploited in many a sector, producing a lot of surprising outcomes. A few examples include

Composition of music https://www.youtube.com/watch?v=HAfLCTRuh7U
Write fiction https://www.theverge.com/2017/12/12/16768582/harry-potter-ai-fanfiction (although the result is by no means good, it's amazing to think a program wrote this)
Play Go https://www.youtube.com/watch?v=vFr3K2DORc8

As well as many other examples. It is straightforward to see a development in many applications starting with the simple problems we discussed above. For instance, self-driving cars are now able to recognise vehicles and any sort of object lying around or moving on the road, thanks to ANN technology. Therefore, we can ask ourselves, as scientists, whether ANN can be used for scientific purposes as well as utilitarian ones.

Scientific uses of Machine Learning

ML, and Neural Networks, in particular, have been used more and more often is a wide range of applications, that deviate from the mere pattern recognition, or image classification. In many a sector of physics, for instance, a rise has been observed in the attempts at using ML applications to automate work usually done by scientists.

The need for automation has increased in the last years due to the increasing dimensionality of data scientists would have to examine. Disciplines like Astronomy, that have to deal with images (in the broader sense of the term) are currently exploiting ML techniques for classifications of images, that are not too dissimilar from the ones discussed in the model above.

Even in our group at the Cyprus Institute, Srijit Paul, a fellow in the programme preceding ours (HPC-LEAP) has collaborated in using machine learning techniques to study phase transitions in an Ising Model. This is a very classical problem that works like a toy model for how magnets work. It is possible to simulate their behaviour by having a grid structure of arrows that can be - for instance - either turned up or down. The proximity of equally facing arrows influences the energy of the system and, at a given temperature, it is possible to simulate how the system would behave in reaching the so-called ground state, that is to say, the state at lowest possible energy, which would be the state one observes most likely in nature, as this has the tendency to prefer objects in their lowest possible energy state. Varying the temperature, it is possible to observe two very different behaviours, that can be identified with a phase at which the material has magnetic properties and a phase at which it does not. Recognising the transition phase can be pretty easily done by computing observables from the ensemble of arrows one has. In their work (https://arxiv.org/pdf/1903.03506.pdf), they, however, show how it is possible to use ML techniques to estimate the temperature at which the transition between the phases happens, the so-called "critical temperature", as well as other quantities.

On a more fundamental scale, some scientists were able to construct a ML algorithm that was able to extrapolate Newton's Law of motion by looking at the motion of a pendulum. This is something that could revolutionise how we think about science, since a machine was able to predict, starting from data and a set of mathematical operations, the link between observation and a fundamental law of nature. (https://www.wired.com/2009/04/newtonai/)

It is fair to predict that the attempts will not stop here and we will see a rise in the number of people trying to use ML to both automatise work and help scientists make fundamental discoveries.

Turing machines, Memcomputing and High Performance Computing

Ramirez — Thu, 28 Feb 2019 13:10:56 +0000

Many of us have seen The Imitation Game, a movie describing the life of Alan Turing, considered the father of theoretical Computer Science. In 1936 Turing wrote an incredible piece of art: a paper titled On Computable Numbers, with an Application to the Entscheidungsproblem.

In his paper he basically proposes a very simple "machine" (which later on became known as the Turing Machine - TM). This TM gives us a description of what such a device can and cannot compute, thereby showing us as well that (if we take the TM as the only "object" capable of calculating things) the number of things we can't compute/calculate is infinitely larger than the number of things we can compute.

But let's forget about that infinite set of things that we can't compute and about the fact that our knowledge is so infinitely small. That's not what concerns us here.

On its own, the Turing Machine has both advantages and limitations, and consequently many alternatives to it have been proposed over the last sixty years. Very recently, a new proposal has emerged: Memcomputing.

Before going over some interesting aspects of Memcomputing, let us dive a little deeper into the topic of Turing Machines (and a couple more concepts, relevant to our discussion here).

Turing Machines

In its (almost) simplest form, the Turing machine can be visualized as follows:

(image taken from here)

and can be understood in the following way: the Tape can have values written on it, let's say 0, 1 or Nothing. The Reader/Writer (the double-headed arrow between the Tape and the Program-box, labeled as "Read/write head") reads the value that is written in one specific cell of the Tape, and reacts to what's written there, by doing for example the following:

1. write a 0 in the cell
2. move (the whole Reader/Writer + the Program-box) one position to the left
3. change your state to Blue

(assuming than in the Program-box there is a label that indicates the "color" feature - for some inexplicable reason).

All sorts of (weird and) interesting things can be done building up from this very basic example/concept of the Turing Machine.

Based on this very basic example, one can generalize and define something called the Universal Turing Machine (UTM), and prove, based on the definition of the UTM, that many of the computers, calculators and complicated things that we use everyday can be considered somehow equivalent to a UTM.

Von Neumann - architecture

The TM is an abstraction, an ideal model of a machine that computes "things". In real life, we need something less abstract. Usually, a UTM is realized in practice through the von Neumann architecture.

(image taken from here)

Fair enough, the picture above describing the von Neumann architecture is again an abstraction, but at least a more down to earth one. (Again, understanding the von Neumann architecture can be easy). To understand the von Neumann architecture, let's focus on its individual components:

Central Processin Unit: like in your phone or laptop, this is the piece of electronics (buried deep within your device) that ultimately takes care of additions, multiplications, etc.
Memory Unit: like in your phone or laptop, this would be the internal memory of your device, your SD card, etc.
INPUT, OUTPUT: irrelevant at the moment.
Control Unit, Arithmetic/Logic Unit: (too) specific parts of the CPU that you don't need to worry about right now.

Disclaimer: even though the von Neumann architecture is a very simplified and naive approach which suffers from many pitfalls, so don't take it too seriously, although it is indeed the original architectural base to all modern computers.

Note, in particular, how the Memory Unit and the Central Processing Unit are separated.

Memcomputing

This is the center of our discussion, with the (U)TM as its base.

Recently, a new paradigm has been proposed, an alternative to the Turing Machine that is not based on the UTM concept and inspired by the functionality of our own brain.

In 2014, Fabio Traversa and Massimiliano Di Ventra published a paper titled Universal Memcomputing Machines. In this paper they write: "Like the brain, memcomputing machines would compute with and in memory without the need of a separate CPU. The memory allows learning and adaptive capabilities, bypassing broken connections and self-organizing the computation into the solution path, much like the brain is able to sustain a certain amount of damage and still operate seamlessly".

Rather than going into the specifics let's try to motivate the idea a bit more, perhaps in simpler terms.

From an architectural point of view (architectural, in the sense of the von Neumann architecture), the authors of the paper propose a new Memcomputing Architecture:

In the same way as the von Neumann architecture implements the Turing Machine, the Memcomputing Architecture implements the Memcomputing Machine. Thus raising the question: what is a Memcomputing Machine?

Think of a bottom-up approach: you can start studying the von Neumann architecture, then go and start thinking of generalizations and come up with Turing Machine (TM)-like devices, and finally go one step further and put "everything" into a single classification/framework and call it Universal Turing Machine.

Analogously, you can take the Memcomputing Architecture, start thinking of generalizations, and eventually come up with something that the authors of the paper call Universal Memcomputing Machine.

There are many details in Traversa & Di Ventra's paper, and the goal here is not to go over all of that, but rather to talk about some of the implications/claims discussed in the paper. To this end, let's divert shortly.

High Performance Computing (HPC)

It's not easy to give a definition of what High Performance Computing is. Also, you might be thinking: "everything was going so well, why muddy the waters by adding HPC to the picture?". Soon you'll see why.

For the sake of simplicity, let's assume that High Performance Computing is the same as Parallel Computing.

Parallel Computing is gathering a bunch of devices (say, your laptop or your phone) and putting them to work all together on a single task. If using one computer is good for predicting the weather next week, two must be better, right? And four even better. And so on.

Let's go back to the von Neumann architecture picture:

If you use a single computer for a huge problem with a lot of data (all that data stored in the Memory Unit), when the Central Processing Unit is computing stuff, it needs to bring data from the Memory Unit to calculate things. If, for the same problem, we use two computers, then each computer will have 1/2 of the total original data, and therefore less work to do.

If we use four computers, then we would have 1/4 of the problem-data for each, hence even less work for an individual computer.

But you might notice that something is off: we need to add more and more computers due to the "bad" design of the von Neumann architecture. This bad design originates (indirectly) all the way from the Turing machine: we have the Tape, which is the memory, and we have the Program-box (CPU) to decide how to "react" based on the data in the Tape.

That is how High Performance Computing comes to "save" the day. (Well, not really; it comes to alleviate some "memory pains")

Memcomputing to the rescue I: intrinsic parallelism ----> replacing HPC??

Traversa & Di Ventra claim multiple things in their proposal of a new way of doing Computer Science.

One of them is intrinsic parallelism. They claim: "the Universal Memcomputing Machine approach to parallel computing can account for a number of processing units working in parallel that can grow also exponentially, if required". That doesn't sound very clear.

Let's try again, first in pictures (image taken from here):

The idea of intrinsic parallelism is that Memcomputers are parallel in a more natural sense than the regular computers we currently know/use. Since the fundamental building-blocks of the Memcomputer are connected in some sort of network, and due to the physical properties of each of those blocks, the way information is processed in a Memcomputer has a radically different dynamics compared to our current computers, allowing for an exponentially different parallelism.

With this, Memcomputing faces the challenges of High Performance Computing from a completely new and fresh perspective.

Memcomputing to the rescue II: solving NP-complete problems in polynomial time!

A problem in Computer Science is a situation that has to be solved by means of an algorithm.

For example, sorting a list of numbers in ascending order. This is a relatively simple task, and there are many algorithms for tackling this problem. Even more, if you go from ordering a list of 10 numbers to a list of 1000 numbers, the problem can still be solved in a decent amount of time.

There are of course more complicated problems. For example, imagine you want to sell bread in your hometown. For personal reasons, you decide you want to distribute it by bike. If you have, say, 4 clients, it's not terribly complicated to figure out the fastest path to deliver bread to all of your clients.

One thing you could do (in order) to solve this bike-traveling salesperson problem, is to calculate the distance between all possible pairs of clients, and with the knowledge of your average speed on a bike, try all possible combinations and calculate all possible total distances/times, (and then decide.)

Now, going from client A to client B might not be the same as going from B to A, because the road might be steep.

How many different paths do you have to calculate to find the shortest?

Well, the factorial function gives you the answer to this question: 4! = 24. So, in the case of 4 clients, 24 different paths. For 6 clients, 720 paths and for 9 clientes: 9! = 362880.

This is crazy. The time it takes to solve this problem (meaning, to compute all possible distances/times and then minimize) increases (insanely) rapidly with the problem size (problem size = number of clients).

Many difficult problems, such as the bread-distribution problem mentioned here, don't have a known way of being solved. We don't know how to solve them in a decent amount of time, and getting the exact correct answer. There are many ways of approximating the solution (or at least a "god enough" one) quickly, but no way of solving quickly such problems.

Traversa & Di Ventra claim that Universal Memcomputing Machines are capable of solving such difficult problems, and they show this by applying their ideas to one hard problem in particular (named the Subset-Sum problem).

Dream or reality?

Up to here, only general descriptions and claims regarding Memcomputing have been given. Want to see some criticism to this newly proposed paradigm? See this blog by Scott Aaronson.

In that article, Scott explains the physical impediments of building a Memcomputing, at least in the way proposed by Traversa & Di Ventra.

For a deeper understanding of Memcomputing, see the website of the project (http://memcpu.com/) and (their) papers listed therein (http://memcpu.com/publications/).