Saturday, November 28, 2009

The future of the universe

All of the complexity and structure we see in the world around us will ultimately be degraded and eradicated.

Consider first the fate of gravitationally bound systems, such as galaxies and galaxy clusters. Let us first recall some concepts from Newtonian gravity. Because gravitation is an attractive force, the potential energy U of a system of gravitating objects is negative; the kinetic energy K of the system, the energy of motion of the objects, is positive; and the total energy is E = U + K. A gravitating system is said to be bound if the total energy E is negative. In other words, if the magnitude of the potential energy exceeds the magnitude of the kinetic energy, the system is bound. The more negative the total energy, the more tightly bound the system is.

Each gravitating system has associated with it an escape velocity, which is the speed a constituent object must attain if the distance between it and the other objects in the system is to become unbounded. A bound system is such that the average velocity of the objects in the system is less than the escape velocity. However, after the objects in the system have interacted for a period of time, there will be a distribution of velocities, and some will exceed the escape velocity. These objects will thence depart the system, never to return. This evaporation of objects from the system will remove positive kinetic energy from the system, hence the total energy of the system will become more negative, making the system more tightly bound. Nevertheless, the system will continue to evaporate.

Contemporary spiral galaxies are bright and vibrant star cities, evolving through multiple generations of star formation, and possessing an ecological structure in which material is cycled between the population of stars, and the gas and dust of the interstellar medium. Eventually, however, star formation will cease, all the stars in a galaxy will expend their nuclear fuel, and galaxies will be populated by black holes and the dead cinders of stars. These cold, dark galaxies will then evaporate, as Iain Nicolson explains:

"Although close encounters between stars are extremely rare, given sufficient time, many encounters between dead stars will take place. In each encounter, one star will gain energy and the other will lose energy. Even without any encounters of this kind, an orbiting star will gradually lose energy by radiating gravitational waves and so, very slowly, will migrate closer to the centre of its galaxy. Close encounters will accelerate this process. Over extremely long periods of time, most dead stars will evaporate from their host galaxies and the remainder will coalesce into gigantic 'galactic' black holes at their centres. A similar process is likely to happen to clusters and superclusters of galaxies, with dead galaxies merging at their centres to form 'supergalactic' black holes, and others being ejected into intercluster space." (The End of the Universe, 1998 Yearbook of Astronomy, pp220-232).

Smaller bound systems, such as molecules and atoms will also evaporate, but the reason for this is quite subtle. As John C.Baez explains, any system in thermal equilibrium will minimise its so-called free energy, the amount of energy which is available to perform work. The free energy can be defined to be E - TS, where E is the total (internal) energy, T is the temperature of the system, and S is the entropy of the system. The restriction to internal energy here simply means that one ignores the potential energy a system might possess in an external field, and one ignores any bulk energy of motion; internal energy includes the internal potential energy of the system, and its internal kinetic energy. The entropy S of a system can be seen in this context as the amount of unusable energy in the system, per unit of system temperature; hence, multiply the entropy by the temperature, and one obtains the total amount of unusable energy in the system. Subtract the amount of unusable energy from the total energy, and one obtains the free energy.

The free energy of a system E - TS can clearly be reduced either by reducing E, or by increasing TS. As Baez points out, an ionised gas (a so-called plasma) has more energy than a gas made from atoms or molecules of the same substance. When those atoms or molecules form, electromagnetic radiation is released, decreasing the total energy of the matter in the system. However, the atoms or molecules have less entropy than the ionised system. At high temperatures, the free energy is minimised by the high entropy plasma state. However, at lower temperatures, the free energy of a matter system can be minimised by reducing the total internal energy of the system. (Note, however, that although the atomic or molecular state is a lower entropy state for the matter, the total entropy still increases because of the entropy of the electromagnetic radiation released when the atomic or molecular state is formed).

This assumes, however, that the system occupies a fixed volume. If the volume available to the constituents of the system is constantly increasing, as is the case in an expanding universe, then the maximum available entropy S of the system will be constantly increasing, and eventually, even at very low temperatures, the free energy of a gas will be minimised in the ionised, plasma state, the state which maximizes the entropy of the system.

Black holes, of course, are also capable of evaporating into radiation, but only do so if their temperature is lower than that of their surroundings. Crucially, some theorists currently argue that the presence of the dark energy, responsible for the acceleratory expansion of the universe, equips the universe with a minimum temperature. The temperature of a black hole is inversely proportional to its size, hence if sufficiently large black holes form from the merger of smaller black holes (and they would have to be as large as the currently observable universe), then such black holes would never evaporate.

Thus, (neglecting some questions over the fate of protons) the future of the universe is a future in which all galaxies, stars, planets, complex molecules and atoms eventually evaporate, and all that remains will be gravitational radiation, electromagnetic radiation, black holes, and isolated elementary particles.

Wednesday, November 25, 2009

Lewis Hamilton and instability

The Eurofighter-Typhoon jet aircraft is so unstable, that it cannot be controlled by a pilot alone, and requires the intervention of electronic control systems to prevent it from stalling in flight. Hold that thought in your head as you read the description of Lewis Hamilton's raw driving ability, which McLaren director of engineering, Paddy Lowe, gave to Autosport's Mark Hughes in 2008:

"He's tremendously good at controlling a car in oversteer. We saw that from the first moment he got in our car. We saw the data, and on every entry we could see there was a massive correction on the steering, and our normal drivers would have been bitching like hell that the car was undriveable, yet he didn't even pass comment. So with a driver like that, you're better equipped to push the boundaries to new levels. Speaking generically of that characteristic, a lot of the performance limit of a car is set by stability; if you can't hang on to it, you will have to introduce understeer in that zone. But if you have a driver better able to deal with oversteer in those zones that induce it, then you will have a less-understeery car elsewhere and therefore more total grip over the lap. The great drivers over the years - Senna, Schumacher, Mansell - have all had that ability. Like for like compared to other drivers, they want more front end."

There are two particular concepts in Lowe's analysis which need to be distinguished: corner entry oversteer, and entry instability. To understand Hamilton's unique capabilities, we therefore need to briefly introduce some definitions from stability theory.

If a car (or aircraft) is initially in an equilibrium state, and there is a transitory control input (or external disturbance), a stable vehicle will return towards its initial equilibrium state of its own accord, whilst an unstable vehicle, in the absence of any further control inputs, will diverge even further from the initial state. To be precise, the first condition is sometimes called static stability, and the latter condition is called static instability. Whether a vehicle is stable or not can be speed dependent. For example, a bicycle is stable at higher speeds, but is unstable at low speed, requiring continuous corrective inputs from the rider to remain vertical.

In the case of an F1 car, an initial steering input induces an initial slip-angle in the front tyres, which induces an initial direction change (a rotation about the vertical axis, called a yaw motion). If an F1 car is statically stable, the car will then return towards a state of zero yaw. If an F1 car is statically unstable, an initial steering input would not just induce an initial slip-angle and change of direction, but an ever greater change of direction (in the absence of corrective action from the driver), giving the vehicle a tendency to spin on entry to every corner. In particular, if an initial steering input provokes the car into oversteer, then that oversteer will increase the initial direction-change. Hence, the driver must supply opposite-lock steering corrections to reduce the direction-change. Oversteer and instability are therefore related. To be precise, turn-in oversteer is a statically unstable handling characteristic, albeit one which Lewis Hamilton is clearly capable of dealing with.

There is a further nuance here, however, because even statically stable vehicles can be either dynamically stable or dynamically unstable. After an initial input, the attitude of a dynamically stable vehicle will oscillate with simple harmonic motion of decreasing amplitude about the initial attitude. In contrast, in the case of a dynamically unstable vehicle, whilst its attitude will at first return towards the initial state, it will then oscillate with increasing amplitude about that initial attitude, leading to a loss of control (in the absence of corrective inputs). These two behavioural characteristics are also sometimes dubbed positive stability, and relaxed stability, respectively. The Eurofighter Typhoon possesses dynamic instability (relaxed stability).

If an F1 car was statically stable, but dynamically stable on turn-in, an intial steering input would create an initial direction-change, and the direction-change would then oscillate with decreasing amplitude. If, however, an F1 car was dynamically unstable on entry to a corner, then the direction-change would oscillate with increasing amplitude, (in the absence of corrective action), giving the vehicle a tendency to spin.

Rear-end instability on corner entry is reportedly the handling characteristic which Jenson Button struggles most to deal with, but as his Brawn team-mate, Rubens Barrichello, demonstrated this year, it is a characteristic which different drivers can cope with to different degrees. Perhaps, then, the type of instability exhibited at times by the Brawn in the second half of the 2009 season, was merely the dynamic instability of a statically stable car.

Judging from Paddy Lowe's remarks, one can speculate that not only is Lewis Hamilton able to cope with such dynamic instability on corner entry, but to a degree unique amongst his peers, he is able to supply the corrective inputs necessary to prevent a statically unstable car from spinning on corner entry.

Monday, November 23, 2009

Why did Jenson Button leave Brawn?

On Friday, Eddie Jordan announced to the world that Michael Schumacher will be making a comeback with the Mercedes Formula 1 team. Generally speaking, Eddie Jordan is a reliable source of motorsport information in the same sense that Gillian McKeith is a qualified authority on diet and nutrition. In this case, however, Jordan's prediction makes a lot of sense, for Michael is clearly directionless without the opiate of Formula 1, and Mercedes have conspired to lose their World Champion driver, Jenson Button, to erstwhile partners McLaren.

And here's an interesting thing: Jordan claims that Mercedes's attempts to woo Michael "started with a meeting between Michael, Ross Brawn and Daimler chief executive officer Dieter Zetsche at the Abu Dhabi Grand Prix." This claim was corroborated on Sunday by Willi Weber, latterly Schumacher's manager, who said he was "sure that Schumacher had had talks with Dr Dieter Zetsche, head of Mercedes-Benz and Norbert Haug, who runs the company’s motor-sport division, at the Abu Dhabi Grand Prix."

At Abu Dhabi? Reports that Button could be lured to McLaren had surfaced in the week after he secured the World Championship in Brazil, but still it seemed that an agreement between Button and Brawn/Mercedes was a mere formality. As Ross Brawn commented at the time, "We are working with Jenson to find a balance between what we can afford and what he feels is fair for his status and what he can contribute in the future...You are never 100% but I would say 99% [certain it will happen]."

The possibility of Button switching to McLaren was interpreted as a mutually convenient negotiating ploy: it let Brawn/Mercedes know that Button had another option, and it let McLaren candidate Kimi Raikkonen know that McLaren too had other options. Mercedes motorsport boss Norbert Haug, for one, was rather dismissive of the possibility that Jenson Button and Lewis Hamilton could end up in the same team: "I do understand that people in England are dreaming of an English team with two world champions in the cockpits... However, dreams don't always come true."

By Abu Dhabi, however, Haug and Mercedes were apparently considering the loss of Button as a serious prospect, and began exploring the Schumacher option before Button put pen to paper with McLaren.

Both Mercedes and McLaren now claim that money was not an issue, that Mercedes offered Button the £8 million a year salary he was seeking, and that McLaren ultimately granted Jenson a deal worth less than Mercedes were offering. This, however, is not the point. Whilst Mercedes's final offer matched Button's salary requirements, the initial deal which Brawn/Mercedes offered to Jenson was only £4 million. This constituted little advance on the reduced salary which Jenson had voluntarily accepted to keep the team afloat when Honda pulled out at the end of 2008, and Jenson probably perceived this as something of a slight. By the time that Jenson was escorted on a tour of the Sir Norman Foster-designed McLaren Technology Centre (MTC) at Woking, the damage may already have been done.

To understand the effect this may have had on Jenson, it's worth recalling the words of Ron Dennis, speaking to Nigel Roebuck in late 2001 (Autosport, December 20-27, p23) before the opening of the MTC:

We were looking for perfection, so we didn't want [MTC] to look out over any buildings. When people are working there, all they'll see out of the windows is fields and trees...I believe that good technical resources attract the best people like a magnet...OK, they want money - and money becomes like a rate card...You measure yourself in financial terms - yes, it affects your lifestyle, but primarily it's a reflection of how good or bad you are. The best people get the most money - that should just be common sense. And once you've satisfied that desire, you've got to give them the best facilities.

So perhaps, then, the picture is as follows: Brawn/Mercedes made an offer which undervalued Button's services, at which point Jenson's management team made contact with McLaren to develop some negotiating leverage; in response, Brawn/Mercedes tried to cover the possibility of losing their new World Champion by developing an interest in Michael Schumacher; already offended by the comparatively low nature of the salary on offer, Button possibly became aware of Mercedes's apparent attempt to seduce Schumacher out of retirement, and decided to take the McLaren offer seriously; Jenson's eyes were then opened by the yin and the yang of the McLaren Technology Centre, and he reciprocated the interest of his new suitor, irrespective of salary.

Wednesday, November 18, 2009

Can Jenson beat Lewis?

It seems to be a week for unusual and inexplicable combinations. Erstwhile Toyota F1 driver Jarno Trulli revealed at the weekend that he was seriously considering an eventual move to NASCAR, the American stock-car racing series. On the face of it, this would be as appropriate as Brian Sewell playing centre-forward for Caledonian Thistle.

Jarno, of course, was so indignant at being hung out to dry by Adrian Sutil in the Brazilian Grand Prix, that he turned up at the next race in Abu Dhabi with a portfolio of photographs and video evidence to prosecute his case. Sadly, however, the driving tactics in NASCAR hardly constitute the Queensberry Rules either...

This is completely overshadowed, however, by Jenson Button's apparently counter-intuitive decision to join Lewis Hamilton at McLaren next year. Many pundits have advised Jenson against this because McLaren appears to be very much Lewis's team, and many have predicted that Lewis would blow Jenson away if the two were partnered in the same car.

There is, however, at least one factor in Jenson's favour. Next year's cars will have narrower front tyres, and the presence of larger fuel tanks will shift the centre of mass, and therefore the centre of aerodynamic pressure, further towards the rear of the car. This is potentially very much to the favour of Jenson, and to the detriment of Lewis. A centre of pressure further towards the rear potentially alleviates the possibility of rear instability under braking, a handling trait which Jenson struggles to deal with. Furthermore, Lewis notoriously favours a car with a strong front-end, and the narrowing of the front tyres and the shift in the centre of aerodynamic pressure will both contribute towards making next year's cars more liable to understeer. Perhaps Jenson, then, fancies his chances against Lewis...

The primary solecism of quantum theory

Last night, the BBC's popular science series, Horizon, once again featured the popular comedian Alan Davies, in a programme entitled How long is a piece of string? Like virtually every other popular science book or programme, it is also reiterated the following interpretation of quantum theory:

A particle can be in two different places, A & B, at the same time.

This is a claim repeated not just by many science journalists and popularisers, but also by many working physicists, yet it is completely wrong. Instead, the weirdness of quantum theory arises from the fact that either-or statements of the following form,

Either particle x is at position A or particle x is at position B,

can be true, even though quantum theory doesn't represent either of the constituent disjuncts to be true:

Particle x is at position A.

Particle x is at position B.

Thus, a disjunction can be true in quantum theory without either of the disjuncts being true. In technical terms, the logic of quantum theory is said to be non-distributive. This is the problem which any successful interpretation of quantum theory must deal with, and, in particular, this is why one of the possible approaches is the so-called hidden variables interpretation, which claims that quantum theory provides an incomplete specification of the actual state of a physical system.

For an excellent summary of the issues involved, one could do worse than this excellent review from Oxford philosopher of physics, David Wallace: The quantum measurement problem: state of play.

Paradoxes, however, are good press, and physicists are trying to sell their products to an increasingly ill-educated public, so don't expect the solecisms to abate.

Monday, November 16, 2009

Shell V-power and journalism

Being not disinterested in the world of motoring journalism, I was intrigued the other day to find what, at first sight, appeared to be a competition for aspiring journalists, jointly organised by Shell and Auto Express magazine.*

Sounds good, I thought.

But how, exactly, would the applicants demonstrate their journalistic flair and investigative powers to the competition judges? Well, here's how:

Please now share your thoughts in a written statement of 75 words or fewer in response to the question: "What do you think about Shell V-Power and the difference it makes to you and your car?"

What I like about this is the brazen assumption that there really is no difference at all between a journalist and a PR or press officer. Personally speaking, I was hoping to sell my soul after 5 years of fruitless struggle in journalism, not cave-in at the very outset.

Still, have to be in it to win it, I suppose, so here goes:

As the latest premium brand of fuel from the type of company which Robert Mugabe refuses to deal with on ethical grounds, Shell V-power is designed to fool affluent and the poorly-informed Western motorists, that a higher octane 'quality' will improve power, irrespective of an engine's compression ratio. Manufactured from the pituitary glands of Ugandan infants, and blended with the tears of Filipino child prostitutes, Shell V-power is now able to scrub CO2 from the atmosphere, and is increasingly used as a cure for syphilis and pancreatic cancer.

I look forward to my trip to Shell Global Solutions in Thornton, Cheshire.

*Many thanks to Patrick's Motorsports Ramblings for this link.

Saturday, November 14, 2009

Supersymmetry and the Large Hadron Collider

Like a parent cautiously returning to a firework which failed to launch at the first attempt, scientists at CERN are about to switch the Large Hadron Collider (LHC) on again.

Anil Ananthaswamy duly provides a decent summary in New Scientist of the prospects for the LHC finding evidence of supersymmetric particles as well as the Higgs boson.

The basic idea of supersymmetry is that the two types of elementary particles with which we are familiar, bosons and fermions, are actually just different states of single particle types. In this respect, it is postulated that each type of boson has a fermionic partner, and each type of fermion has a bosonic partner. Supersymmetry therefore predicts the existence of numerous particles which have not hitherto been detected. For example, the photon, (a boson) has a hypothetical supersymmetric partner called the photino (a fermion).

The particle ontology of supersymmetry is then twinned with a cosmological explanation of why bosons and fermions are observed as distinct particles in terrestrial laboratories. It is proposed that the symmetry between bosons and fermions was respected at the higher energy levels found in the early universe, but as the universe expanded and energy levels dropped, supersymmetric symmetry breaking took place, with the consequence that the bosons and fermions with which we are familiar interact very rarely with their supersymmetric partners. These weakly-interacting supersymmetric particles then provide a nice candidate to explain the existence of dark matter in astronomy and cosmology.

Mathematically, supersymmetry also entails an interesting modification to the definition of what an elementary particle is. The latter is intimately related to the local space-time symmetry group, the group of symmetries possessed by every small patch of space-time, irrespective of how those patches are sewn into a global space-time. Without supersymmetry, the local space-time symmetry group of our universe appears to be a subgroup of the Poincare group. Wigner established that each type of elementary particle corresponds to an irreducible unitary Hilbert space representation of this subgroup of the Poincare group (with a few technical qualifications concerning so-called covering groups).

However, if it transpires that our universe is a supersymmetric universe, then the definition of an elementary particle has a straightforward generalisation. The local space-time symmetry group becomes (a subgroup of) the super-Poincare group, and the set of possible supersymmetric elementary particles is then defined by the irreducible unitary Hilbert space representations of (a subgroup of) the super-Poincare group. Each such representation decomposes into a direct sum of unitary irreducible representations of the Poincare group, and the members of such a supersymmetric 'multiplet' are said to be super-partners of each other.

Mathematicians interested in symmetry, and philosophers interested in the mereological concept of elementarity, should therefore share a common interest in the data physicists are set to harvest from the LHC's detectors.

Monday, November 09, 2009

The latitude and longitude of F1

It has been observed on more than one occasion over the years that Bernie Ecclestone could save everyone in Formula One a lot of time and money by the simple expedient of holding each Grand Prix in the same location, and merely changing the scenery every couple of weeks.

There has, of course, also been a trend in recent years for the centre of gravity of the championship calendar to become increasing Oriental. Thus, if one were to hold each Grand Prix in the same location, where would be the most appropriate place in which to hold it? Probably not somewhere in Europe. To be scientific about this issue, then, let us propose instead that we take the average latitude and longitude of all the race tracks on the 2010 Formula One calendar.

Assuming that the British Grand Prix will be held at Silverstone, the 2010 calendar consists of the following geographical locations:

 Race Latitude Longitude Bahrain 26.0325 50.510556 Australia -37.849722 144.968333 Malaysia 2.760556 101.7375 China 31.338889 121.219722 Spain 41.57 2.261111 Monaco 43.734722 7.420556 Turkey 40.951667 29.405 Canada 45.505833 -73.526667 Europe 39.458889 -0.331667 Britain 52.071 -1.016 Germany 49.327778 8.565833 Hungary 47.578889 19.248611 Belgium 50.437222 5.971389 Italy 45.620556 9.289444 Singapore 1.291403 103.864147 Japan 34.843056 136.540556 S. Korea 34.733333 126.416667 Abu Dhabi 24.467222 54.603056 Brazil -23.703611 -46.699722

The average latitude and longitude of the 2010 Formula 1 calendar is therefore 28.95632537, 42.12886447. These coordinates transpire to be a desert region in Saudi Arabia called Al Haiyaniya, (pictured in the satellite image above).

The perfect location, then, for the entire championship.

Saturday, November 07, 2009

The brain of a racing driver

Is there a basic neuro-physiological difference between the brains of racing drivers and the brain of a normal person? Is a racing driver's preference for understeer or oversteer determined by neuro-physiological differences?

The techniques for answering both of these questions are already available, and have been applied by University College London (UCL) to study the structure and activity of the brains of taxi drivers.

An initial UCL study, published in 2000, used magnetic resonance imaging to discern that the size of the posterior hippocampus was enlarged in a sample of 16 London taxi drivers, compared to a control sample of 50 people. It also found a positive correlation between the length of time a taxi driver had spent in the job, and the volume of the right hippocampus. The hippocampus is associated with navigational abilities, (as well as the establishment of long-term memories), hence it was hypothesized that the posterior hippocampus actually grows in response to the navigational demands placed upon it.

A second study, publicised in 2008, examined the activity of a taxi driver's brain in real-time, using functional magnetic resonance imaging (fMRI), as the drivers navigated their way through the streets of a computer simulation:

The hippocampus was only active when the taxi drivers initially planned their route, or if they had to completely change their destination during the course of the journey.

The scientists saw activity in a different brain region when the drivers came across an unexpected situation - for example, a blocked-off junction.

Another part of the brain helped taxi drivers to track how close they were to the endpoint of their journey; like a metal detector, its activity increased when they were closer to their goal.

There is no reason why a similar pair of studies could not be conducted on racing drivers. Very sophisticated driving simulators now exist, including ones which can simulate the kinaesthetic sensations of driving a racing car. The latter point is particularly important, because a racing driver can sense the limit of car's adhesion, not merely by hand-eye coordination, but also by the sensations associated with changes of linear momentum and angular momentum. The first mechanism is a sensory feedback loop involving the visual cortex and motor cortex, whilst the latter is likely to be a feedback loop operating between the somatosensory cortex and the motor cortex.

Is any part of the brain in a racing driver enlarged compared to the background population? Do drivers who prefer oversteer have a larger somatosensory cortex than drivers who prefer understeer? Or are the differences purely psychological, rather than neuro-physiological?

I propose that we introduce UCL's fMRI to McLaren's driving simulator, and ask Mr Hamilton and Mr Raikkonen if they'd like to take part in a small medical study...

Tuesday, November 03, 2009

Fluting

Driving a Jaquar is the closest one can get to driving a neoclassical temple on the open road.

It is rarely emphasised, however, that the iconic design of the Jaquar owes an awful lot to the styling of the bonnet. To be precise, it is the presence of the fluting in the bonnet, and the blending of that fluting into the curves of the protruding headlights, which give the Jaquar its Palladian aesthetic.
Such fluting is rarely seen on modern roadcars, and racing car designers are no longer noted for their stylistic flourishes, but, pleasingly, there is extensive use of it on the engine shroud of Adrian Newey's first Formula 1 car, the March 881.