To lobby or to imitate?

In the official Thursday press conference for this weekend's Malaysian Grand Prix, Julien Febreau of L'Equipe asked the five drivers present for their opinion on Red Bull's flexible front wings: "I have nothing interesting to say, unfortunately," replied Nico Rosberg; "Not much to say," remarked Sebastien Buemi; "No, nothing to say," answered Jarno Trulli; "No," said Felipe Massa flatly.

Jenson Button, however, was a little more loquacious: "I know a few people that I have spoken to say it flexes more than what they expect is correct." Elsewhere, Button's McLaren team-mate Lewis Hamilton was quoted as saying, "When I asked what the rules are about how much the wing is allowed to flex they said 20mm. If I you show you a picture that is probably not what is happening."

In an exasperated response, Christian Horner protested that the Red Bull is simply running with greater rake than the McLaren. Which would constitute a rather disappointing return on all that investment in Fluid-Structure Interaction (FSI), one would have thought.

No, Red Bull have clearly stolen a march on the opposition here. This first became apparent at the Hungarian Grand Prix last August, yet McLaren have so far displayed no inclination to imitate the technology. Rather, their strategy has been to lobby the governing body, often via the media, in an attempt to limit the extent to which Red Bull can utilise their advantage. This strategy appeared to be successful last year, with the introduction of more stringent front-wing load-tests at Spa, and a change in the permissible articulation of the under-body plank at Monza. Despite this, Red Bull still appear to be deriving a significant performance benefit from their investment in FSI, so McLaren and the other teams have a choice: lobby or imitate.

One presumes that if McLaren were intending to introduce similar technology in the near future, then their drivers would not be making statements challenging the very legality of such developments. But why would McLaren not be seeking to imitate Red Bull? It may be that a flexible front-wing would simply not work well with the other elements of their package; alternatively, given McLaren's relatively recent troubled relationship with the governing body, they might still be reluctant to develop a technology of dubious legality. Or, more ominously, perhaps McLaren have taken a close look at this, and concluded that they're at least two years behind Red Bull in terms of developing the relevant simulation or carbon-fibre capability. If such is the case, it would certainly explain why McLaren have decided that the best strategy is to continue with the lobbying...

How Red Bull simulate front-wing aeroelasticity

When it first became apparent in 2010 that Red Bull were using aeroelasticity to generate extra front-wing downforce, a number of eagle-eyed observers pointed out that the Milton Keynes based team had established a relationship with MSC Software in 2009, precisely to develop simulation software capable of representing this type of Fluid-Structure Interaction:

Using MSC's latest MD (multi-discipline) software versions of Nastran and Adams, we already combine mechanism and deformable finite element simulations. We also increasingly use aerodynamic output directly from CFD analysis to generate more accurate loads for the structural simulations. There are rule restrictions to limit this, but multi-physics coupling of these effects allows us to legally enhance the performance of deformable components, for example to optimise down-force and drag characteristics for flexible wing components.

It's clear that Red Bull's front wings are still flexing under load this year, so let's dig a little bit deeper to try and understand what has been achieved here, and what the other teams need to do to respond. Careful scrutiny of MSC's website reveals a detailed explanation of exactly how their software works in this respect:

The objective of modeling fluids in a structural analysis is to account for the influence of fluid pressures on the structure and for improved accuracy in structural response prediction. Structures are generally modeled using Lagrangian scheme where material is tied to a finite element mesh. On the other hand, fluids are solved with Eulerian scheme with material being independent of the mesh, but instead flowing through the mesh. Dual schemes are required because of the way structures and fluids behave.

When fluids and structures need to be modeled in a single analysis, the challenge is running these different schemes in a single run. This is accomplished through an automatic coupling algorithm, where two meshes – one for structure and another for fluid, exist. A coupling surface is created between these two domains which acts as a boundary to the flow of material in Eulerian mesh, while enabling transfer of the stresses to the Lagrangian structural mesh causing it to deform.

So we can hypothesise that Red Bull are representing the solid interior of their front-wing with a Lagrangian simulation, and coupling it to a Eulerian hydrocode to represent the airflow around it. Some further explanation of these terms is perhaps in order.

Both Lagrangian and Eulerian computer simulations divide a continuous domain, occupied by a solid or fluid, into a discrete mesh of cells. The corners of the cells are called the nodes of the mesh. In a Lagrangian simulation, the mesh moves with the motion of the solid or fluid, whereas in a Eulerian simulation, the mesh is fixed in space, and the solid or fluid moves through the mesh. A computer simulation also divides the continuous flow of time into a sequence of discrete time-steps, using the final data from one time-step as the initial data for the next.

If Red Bull are using a Lagrangian simulation to represent the solid front-wing, then they are using a mesh which moves with the wing as it deforms. At the beginning of each time-step in such a simulation, one has the coordinates of each node, the velocity of each node, and the stress and strain associated with each cell. The strain represents the relative displacement, or stretch, which the points inside the solid body are subjected to under the influence of external loads. A solid body responds to strain by generating internal restoring forces, called stresses. The stress and the strain each have isotropic components, and when the isotropic components are subtracted, what remains are essentially the shear stresses and shear strains. These are dubbed the deviatoric stress and deviatoric strain.

The task of a Lagrangian simulation is to go from the nodal positions and velocities, and the stresses and strains inside each cell, at the beginning of each time-step, to the positions, velocities, stresses and strains at the end of the time-step. In the case of Red Bull's front-wing, the aerodynamic pressures at the beginning of each time-step will simply be part of the boundary data.

The basic method for solving this problem is as follows: Use the stresses to calculate the forces on the nodes, thence the acceleration of the nodes, and from this update the velocities of the nodes; use the velocities of the nodes to update the positions of the nodes; calculate the rate-of-strain from the velocities, infer the rate-of-stress from the rate-of-strain, and update the stress and the strain. (In reality there's a lot of shuttling back-and-forth with half time-steps and the like to minimise numerical errors, but these are the basic ideas).

The method of updating the stress is quite involved. Given the nodal velocities, one can simply take the gradient of the velocity field, (and symmetrise it), to obtain the rate-of-strain. (In the case of a racing-car front-wing, all the strain will be elastic, so there is no need to worry about plastic strain). The rate-of-strain can be divided into the deviatoric rate-of-strain and the isotropic rate-of-strain. The rate of deviatoric stress can then be calculated using the shear modulus of the material, and the rate of isotropic stress can be calculated using the bulk modulus. (In the case of Red Bull's front-wing, one can expect the bulk modulus and shear modulus to vary with position across the wing). The deviatoric and isotropic stresses can then be updated.

With the Lagrangian time-step for the solid front-wing deformation completed, the boundary of the deformed configuration can be fed as initial data to the next time-step in the Eulerian hydrocode representing the airflow over the front-wing. (Although, once again, one presumes there is a more sophisticated shuttling back-and-forth between the Lagrangian and Eulerian codes to minimise errors). The Eulerian code will calculate new pressure forces on the boundary of the front-wing, and the cycle will begin all over again.

Of course, realising the desired front-wing performance not only requires the development of this type of simulation technology, but also an understanding of how to implement the requisite elasticity via the orientation of the carbon-fibre plies. Nevertheless, it remains a surprise that so many F1 teams rely on off-the-shelf simulation software rather than developing their own.

KERS, DRS and Button's short-cut

At the beginning of 2010, Bernie Ecclestone mischievously suggested that Formula One tracks should incorporate short-cuts. "It would stop people getting stuck behind others and be good for TV," explained Bernie. At the time, Jenson Button didn't seem too keen on the idea, remarking "I suppose if you are the only person who knows about [the shortcut] then it is a great idea...Bernie has a lot of very, very good ideas, and that is not one of the best."

By lap 11 of today's Australian Grand Prix, however, it seemed that Bernie's proposal must somehow have lodged deeply in Jenson's subconscious. After persistent and hitherto unsuccessful attempts to overtake Felipe Massa's obstructive Ferrari, Jenson had got alongside the Brazilian going into the fast switchback of turns 11/12. Taking the outside line into the left-hander, Jenson had the inside for turn 12, but with both cars momentarily heading for the eye-of-the-needle at high speed, Jenson bailed-out via a short-cut, and rejoined ahead of Massa.

At this point, one expected Jenson to immediately cede the place back, for the overtaking move had clearly not been completed on the racetrack. The McLaren driver, however, appeared to be in no mood to slow down and let Felipe re-pass. Perhaps sensing a means of aggravating Jenson's predicament, Massa immediately lifted off en-route to turn 13, defending briefly from Alonso, before ceding the place to his team-mate into turn 14.

The punishment meted out to Alonso for a similar pass on Kubica at Silverstone last year established a clear and recent precedent here, yet McLaren seemed intent on asking race control for an opinion rather than taking the initiative themselves. The short window of opportunity before Massa pitted a couple of laps later was squandered, and Button's race was ruined by the drive-though penalty served upon him.

McLaren team principal Martin Whitmarsh later claimed that he hadn't seen the incident at the time, a statement which reminded one of Mr Alex Ferguson's habitual claim not to have seen a foul committed within the penalty box by one of his own players.

Jenson did, of course, later pass Felipe on the track, and herein lies an interesting detail concerning the joint use of KERS and the newly instigated Drag Reduction System (DRS).

First, we need to go back to the early stages of the race. The DRS system cannot be used in the first couple of laps, but on lap 3 Button and Massa were going at it hammer and tongs, and coming out of the final corner Button duly deployed his DRS. Jenson, however, had already fully discharged his KERS during the course of the lap, and the Ferrari merely triggered its own KERS to cancel-out any potential gain by the McLaren.

At some stage over the next lap or so, Button was heard to enquire over the radio how the Ferrari could be so fast on the straight. It would be interesting to know what information the team fed back to Jenson, for by lap 7 he was saving half of his KERS charge for the exit of the final corner. Jenson was now able to deploy both the DRS and the remaining KERS charge simultaneously, yet still this wasn't sufficient to facilitate an overtaking opportunity.

When Jenson finally overtook Massa starting lap 48, the Ferrari was on worn tyres, about to pit, and this enabled him to stay closer to Felipe through the final corner. In addition, on this occasion Jenson had almost an entire lap's worth of KERS available. The McLaren duly deployed the DRS, and expended the entire KERS charge between the exit of the final corner and the timing beam, at which point a new lap's worth of KERS became available, which Jenson tapped into to complete the manoeuvre.

So, assuming that Massa used his KERS in response, and with the caveat that the DRS zone in Melbourne was perhaps not located in the ideal place, the limited conclusion we can deduce is the following:

KERS + DRS + healthy tyres > KERS + worn tyres

A History of Modern Witchcraft: Part 2

I swallowed another clutch of paracetamol, and washed them down with a mouthful of sweet tea. This was the fifth consecutive day I'd developed a piercing head-ache by noon, and now my jaw ached like I'd been involved in a brawl with Jeff Bridges. Idly, I wondered if a brain-tumour the size of a golf-ball might be developing in some lobe or other, flagrantly disregarding my ongoing pension contributions.

Marina was strolling with cat-like insolence around the lab, picking up various pieces of kit, peering intently at their mute exteriors, and replacing them without comment. I watched her out of the corner of my eye, and tried to gauge the size of her bust. Deconvolving the volumetric displacement of a woman's breasts from the bra-dependent contour and uplift was increasingly a black art, and despite the amount of cleavage available as observational evidence, I was forced to conclude that theory was under-determined by data on this occasion.

"Do you not miss the daylight down here, Dr Bones?" asked Marina, staring down the muzzle of an ancient golf-leaf electroscope. She was still using my formal appellation; keeping an emotional distance, then.

"We have interior lighting, Marina. Admittedly, we miss out on some of that bracing UV radiation the Outer-Worlders receive, but this is more than compensated for by the gamma rays."

The lab-door opened, and Dr Burgher strode confidently across the threshold to greet our eminent VIP. "Ah, Miss Petrovium, pleased to meet you," announced Dr Burgher, his sense of brisk self-importance sadly undermined, literally at every step, by his leather shoes squeaking their way across the PVC floor. "I'm the Chief Project Scientist here, as Brod' has no doubt explained to you."

Marina's cobalt-blue eyes sparkled as she met Dr Burgher's outstretched hand. Most women at the base appeared to find Dr Burgher more attractive than a bar magnet in a box of iron filings, and as 'Miss Petrovium' flicked her eyes up and down with studied coyness, I concluded that the IAEA's Head of Paranuclear Phenomena also lacked diplomatic immunity to such attractions.

The analgesics were by now surfing down my capillaries, and things were beginning to feel delightfully fuzzy. As Burgher briefed Marina in the background, I disengaged, and felt that familiar sensation of being in mental free-fall. Inchoate ideas floated past, just out of grasp; I knew there was something I needed to be thinking about, a problem which needed resolving, a question which required an answer, but as soon as I directed my attention to any of the passing notions, they dissolved in a noxious cloud of confusion and receded from conscious awareness.

"...which we're currently unable to explain with any known physics," concluded Dr Burgher with the melodramatic flourish he'd been carefully honing over the past fortnight, ever since we'd discovered the shaft descending from the fifth chamber.

The Book of Universes

"Universes are big at the moment." John D.Barrow.

This is a beautifully written, highly accessible account of the theoretical universes defined by modern cosmology. As Barrow comments, cosmology "is not just an exercise in describing our universe as completely and accurately as possible. It seeks to place that description in a wider context of possibilities than the actual."

Barrow takes a chronological approach, explaining the circumstances in which the various universes were proposed, and the nature of the people who devised them. The biographical information is fascinating, and Barrow's explanation of each universe is absolutely crystal-clear. This is popular science of the highest quality.

A History of Modern Witchcraft: Part 1

It was in the long, torpid Summer of '06 that I first met Marina, and stumbled upon the discovery which would change everything forever. She stepped off a Venetian gondola in a cherry-pattern dress, clasping her IAEA accreditation in one hand, and holding her hat down with the other, as a cool Adriatic breeze blew her golden tresses across one cheek.

I greeted her with a curt nod and a perfunctory handshake, and escorted her to the waiting limousine, which was purring with satisfaction at its own mechanical harmony.

We sat in silence as the stone facades of green-shuttered apartments scrolled past, the townscape blending into scattered farmhouses and walled orchards, then dwindling into olive trees and scrub as the limo followed a dishevelled road climbing reluctantly into the hills.

Torn between the need to commence conversation with small-talk, but wary of sounding superficial, we sat, bound together in an introspective impasse.

Turning off the mountain road onto a gravel track, she glanced in my direction. "Shouldn't you be blindfolding me at this stage, Dr Bones?"

"Please," I retorted, "call me Broderick. And a blindfold won't be necessary. For the moment at least..."

A couple of miles along the track, we passed through a dense cypress grove, their boughs interlocking to form an arboreal arch, through which we emerged into a steep-sided gorge. A shallow, sparkling river swept along the rocky base of the ravine, the track winding along a narrow ledge beside it.

"You will, however, be needing this," I said, handing her a thermoluminescent dosemeter. "Keep it on you at all times. This facility doesn't officially exist, but we're still compelled by Euratom recommendations to keep a record of the radiation dose to all visitors."

Rounding a tight bend, the gully opened up slightly to reveal a large, perfectly oval pool of limpid water, from which the river channel descended. "What is this, Broderick," enquired my fellow passenger, from whom I was already detecting a distinctly acerbic air, "a geography field trip?"

"Not exactly, Marina. Watch carefully."

Our chauffeur circumnavigated the limb of the pool, and brought the limo to a stop at the end of the track, facing a blank expanse of sheer rock. He flipped a switch on the dashboard, and with a jolt, and a brief flurry of dust, the rock-face began descending with a hydraulic hum into the floor of the canyon.

"It's a good job we're not in the Pacific, Dr Bones, or I'd swear that we've just arrived on Tracy Island. Or perhaps you'll be escorting me into the caldera of an extinct volcano instead?"

"Extinct volcanoes are passé, Marina. Besides, they tend to attract various nosey scientists from the US Geological Survey, which we actively discourage."

The rock-face had by now fully retracted, disclosing a dark tunnel hewn into the naked granite. Our chauffeur switched his lights on full beam, revealing two lines of reflective studs, converging to a point in the subterranean distance. With only the briefest of pauses, we plunged at full pelt into the darkness.

To be continued...

Motorsport Magazine subscription

There are many things the internet can do. It provides, for example, an unprecedented array of specialised, up-to-the-minute information and opinion for the curious mind. In terms of immediate access to timely information, the motorsport enthusiast has never been better served.

There are, however, things which the internet has yet to do. It rarely, for example, provides a coherent explanation and interpretation of events stretching over a period of time; it rarely tells a good story, or puts events into context. In contrast, all these things have traditionally been done by printed periodicals. So, motorsport magazines should have nothing to fear from the internet, right? As long as such magazines concentrate on the things they do best, there shouldn't be a problem.

And yet, over the past decade, many motorsport periodicals seem to have done the very opposite, and wilfully made themselves look more like the internet. Word counts have plummeted, and news stories have descended into fragments, snippets, and sidebars. There is now no magazine which provides the motorsport enthusiast with the quality of information provided by Motoring News in the 1980s and early 1990s, and F1 Racing Magazine, in particular, appears to be targeted at those with a single-digit mental age.

This seems almost like a form of business suicide, for why would anyone pay for a magazine which simply provides a less up-to-date, and sometimes inferior version of the information one can get for free on the internet?

One also gets the feeling that many modern journalists consider it unnecessary to describe, in words, events which the reader will already have seen replayed, ad nauseam, on television. This is an enormous mistake, for part of the art and craft of a great writer or journalist is to re-cast known events in an original light, evoking humour or poignancy, drawing attention to subtle details and hidden patterns.

There are, however, some surviving outposts of quality: Autosport's Mark Hughes continues to be beacon from the world of Motoring News journalistic values, and the monthly publication Motorsport Magazine, provides the type of story-telling capability which has evaporated from other periodicals. Thus it was that I recently took out a subscription to Motorsport Magazine.

Indycar designer Ben Bowlby comments that the February issue of the magazine was "stunning. There was so much good content I read it for about seven hours straight." In fact, I almost had the same experience when I received the April issue last weekend.

I am, of course, desperately simplifying the issues here, for ultimately even magazines will have to seek an electronic means of distribution, and this will not be without difficulty, given the expectation that information content on the internet should be free of charge. Nevertheless, there remains some hope that a business model based upon quality of information, story-telling capability, explanation, interpretation, and the ability to paint a picture in words, will survive the transformation.

The implications of Fukushima

The ongoing nuclear emergency in Japan highlights a design flaw in Boiling Water Reactors (BWRs). Most Light Water Reactors (LWRs), such as the BWRs, use zirconium alloy cladding to encase the nuclear fuel, and prevent fission products mixing with the cooling water. There are basically three requirements for the cladding in an LWR: (i) it must have a low thermal neutron absorption cross-section; (ii) it must be resistant to corrosion in water; and (iii) it must have a melt/ignition temperature greater than the operating temperature of the reactor. The temperature is designed to be as high as possible to maximise the thermal efficiency of the heat engine which generates electricity from a nuclear reactor.

Now, what Fukushima has revealed is that when a complete power failure stops the flow of cooling water in a BWR, the water turns to steam, and oxidises the zirconium fuel cladding, releasing the hydrogen component of the H₂O coolant. The hydrogen is then able to escape from the reactor, whence it almost inevitably causes a detonation or deflagration in the presence of atmospheric oxygen. These are the explosions suffered by Fukushima 1 on Saturday and Fukushima 3 on Monday.

Whilst these hydrogen explosions have not damaged the reactor pressure vessels themselves, it is unwise to suggest, as Malcolm Grimston, from the Energy Policy and Management Group at Imperial College, did after Saturday's detonation, that "The explosion... wasn't a terribly important event." On the contrary, such explosions are clearly highly disruptive and destructive to the efforts being made to feed cooling water into the reactor vessels and prevent a meltdown. Reports today suggest that the explosion in Fukushima 3 has damaged four of the five available pumps, and all three reactors now seem to have suffered at least a partial meltdown.

Are there alternatives to using zirconium cladding? Well, in the Magnox reactors used in the UK for some decades, the nuclear fuel was clad in magnesium oxide, which has a low neutron absorption cross-section, and which by virtue of already being an oxide, is incapable of undergoing an oxidation reaction with the carbon dioxide cooling gas used in these reactor designs. Unfortunately, magnesium oxide corrodes in water, so cannot be used in light water reactors.

There is, however, another alternative: stainless steel cladding was used in the UK's Advanced Gas-Cooled Reactors (AGRs), because the latter operate at higher temperatures for greater thermal efficiency, and magnesium oxide would become soft and potentially flammable in such circumstances. Stainless steel cladding would be resistant to oxidation in the presence of steam, but possesses a larger neutron absorption cross-section. To compensate for this the nuclear fuel would need to be subjected to further enrichment, which costs money, power and time, possibly making the whole thing uneconomical.

The nuclear industry might well suggest that the Japanese crisis has been triggered by unique tectonic circumstances irrelevant in Europe, that only a sizeable earthquake and tsunami could cause the simultaneous joint failure of the national grid and back-up diesel generators. Disturbingly, however, one can also imagine a terrorist attack which achieves exactly the same result.

Tom Paulin on the 2011 F1 grid

I leiked the McLaren. It spoke to me of an audacious and extravagant spirit. The sidepods made me think of Joyce at his finest, a willingness to find stark beauty in bleak alphabetic form. It has a wonderful anarchy to it. It's operatic, lyrical, and poetic; I mean it's complete nonsense, but none the worse for it.

The Ferrari, however, I saw as insipid and cliched in comparison, as if the team had asked themselves a series of questions so dull that they'd fallen into a narcoleptic coma before completing the design.

The Lotus-Renault is simply delightful and extraordinary, like it's been hewn from a block of polished obsidian. At times, I felt like it was some sort of blind, dark, repressed Freudian urge, which had been hauled out of the deep unconscious by a psychoanalytical trawler, and thrust naked into physical form.

The Red Bull is a very confident design. It's subtle, marvellous, brilliant and powerful. Its curves have a restless rhythm to them. It feels no need to be showy or flashy, or self-consciously radical, yet it exudes a quiet sense of sublime perfection.

The Hispania is provincial and amateurish. It's pure sawdust and glue. In fact I don't think I've ever seen anything so rotten, degenerate, and downright awful in my entire leif.

Systems Engineering and Technology Insertion

Manufacturing constitutes about 12% of the UK economy compared to 25% of the German economy. There's no need to lose any sleep over this, however, because with the assistance of the East Midlands Development Agency, the UK now has a Systems Engineering Innovation Centre:

The Systems Engineering Innovation Centre at Loughborough focuses on systems engineering aspects which provide a framework for the integration of people, processes, tools and technology in order to improve the management of risk, product configuration and technology insertion for the development of innovative products.

Systems Engineering is an attempt to uplift engineering from a concrete, practical activity, into the information economy. It is predicated on the notion that because complex engineering projects are really 'systems of systems', it is necessary to create a separate discipline, with its own group of practitioners, which specialises in the abstract, top-down analysis of engineering projects in terms of requirements, capabilities and stakeholders, abstracted from concrete engineering issues.

One thing Systems Engineering has been very successful at is codifying the blindingly obvious, and transforming it into self-important acronyms and obscure management-theory language.

Consider technology insertion, one of the phrases used above to define the remit of the UK's Systems Engineering Innovation Centre. What could this possibly mean, given that engineering is all about technology anyway? What else could an engineering project consist of? Literary criticism perhaps? Fortunately, the University of Bath School of Management are at hand to provide a definition:

Technology insertion is achieved through managing supply networks whose business plans and supply interfaces are aligned to a system-level capability to insert technological upgrades into existing complex platforms.

So it's about buying new components to upgrade an existing system? Thank goodness we've been able to codify that. David Kirkpatrick of the Royal United Services Institute is able to offer further insight into this difficult concept:

Both military and industrial opinion has increasingly supported a policy of 'technology insertion' (TI) by which ageing weapon systems can be rejuvenated by successive injections of advanced technology to enhance system effectiveness and/or to reduce operating and support costs. The term 'technology insertion' can be used to describe minor improvements, in which one subsystem is replaced by a more modern version with essentially the same functionality, and also to describe major upgrades (such as the mid-life upgrades of some warship and combat aircraft) which transform the weapon system and its military capabilities. In either case the TI must be carefully designed to achieve satisfactory spatial, mechanical, electrical and electronic interfaces with the original weapon system.

Note that we've now got an acronym, TI, so this must be an important and profound idea. We've also got a proposed list of the requirements for successful TI: "it must achieve satisfactory spatial, mechanical, electrical and electronic interfaces with the original...system."

Golly. This means that a new component must actually function when it's fitted to the existing system. In particular, it must achieve a satisfactory spatial interface. I think that means it has to fit. If it's bigger than the space available for it, then it's no good brother.

But why does this list of requirements only speak of mechanical, electrical and electronic interfaces? Surely if we're going to explicitly codify the obvious, then we need to be exhaustive? Hence, there's the thermodynamic interface to think about; if the new component generates more heat than the old one, then you could be in trouble brother. And let's not forget the vibratory and shock interface; the new component might not be able to withstand the thumps and bangs the old one was capable of withstanding. And what about the radiation interface? Can the new component tolerate the electromagnetic interference from the rest of the system, and will it generate some interference of its own?

It's a good job that Systems Engineers are available to explain this type of thing, and the East Midlands Development Agency certainly aren't wasting tax-payers' money in this respect.

Ricci flow and the entropy of space

The entropy of the universe appears to be increasing, in accordance with the Second Law of thermodynamics, and this is commonly taken to imply that the early universe resided in a state of extremely low entropy. However, given that the geometry of space in the early universe was highly uniform, and given that matter at this time resided in thermal equilibrium, there is a problem. If the universe began in what appeared to be a state of equilibrium (i.e., a maximum entropy), then why did it not remain in such a state?

The common answer proffered is to propose that there is some form of gravitational entropy, and this type of entropy is maximised by clustering, not by homogeneity. There are two primary observations which seem to support this argument: (i) a homogeneous distribution of matter is clearly unstable under gravitational attraction, and over time matter clusters into stars and galaxies; (ii) the entropy of a black hole, the most concentrated form of gravitational collapse conceivable, is huge.

Philosopher of physics David Wallace has already dispelled the complacent notion that the matter in the early universe was in a state of equilibrium. As David explains, the rate of expansion of the universe displaced the matter in the early universe from equilibrium, leaving most of it stranded in the form of hydrogen and helium. The formation of stars increases entropy because it permits such lighter elements to undergo nuclear fusion. Black holes, argues Wallace, are a special case. In addition, let us not forget that all gravitationally bound systems eventually evaporate, which is hardly consistent with clustering itself being the driving force behind the Second Law.

But what about the other half of that complacent assumption concerning the early universe? Does a homogeneous geometry of constant curvature really possess low gravitational entropy? Well, here's a possible reason for thinking that it doesn't: the Ricci flow:



The expression on the left here is the rate-of-change of the metric tensor, whilst the expression on the right is a negative multiple of the Ricci tensor. The Ricci flow is an equation used by mathematicians for, (amongst other things), evolving the geometry on a 3-dimensional manifold. There are two crucial facts about this equation: (i) it evolves a geometry of varying curvature into one of constant curvature; and (ii) it's a kind of diffusion equation.

Now, the point about diffusion equations is that they increase entropy. In the field of radiation transport, for example, whilst the radiative transfer equation is the relevant equation from non-equilibrium statistical mechanics, it is fiendishly difficult to solve, and a diffusion equation is used as a macroscopic approximation instead.

So, if the evolution of an initially random 3-dimensional geometry towards constant curvature, is represented by a diffusion equation, and if diffusion equations represent physical processes which increase entropy, doesn't this suggest that a geometry of constant curvature possesses greater geometrical entropy than one of varying curvature? Could the Ricci flow ultimately be cast as a macroscopic approximation to a random walk in the space of underlying quantum geometries?

Which is all very well, but how do we reconcile this with the fact that black holes possess a huge entropy? Well, recall Roger Penrose's long-held proposal that gravitational entropy will be associated in some way with Weyl curvature. Black holes possess zero Ricci curvature and non-zero Weyl curvature, whilst the big bang possesses the opposite combination of zero Weyl curvature and non-zero Ricci curvature. If we restrict attention to 3-dimensional geometries of zero Weyl curvature, we may be able to argue that it is actually those with the highest degree of symmetry, those of constant curvature, which possess the highest entropy. Nevertheless, it may be that gravitational entropy is increased by clustering, not because it reduces geometrical homogeneity or symmetry, but purely because it increases the Weyl curvature.