# Tinkering, Science, and (Not) Sharing It

I stumbled upon this research paper called PVC polyhedra:

We describe how to construct a dodecahedron, tetrahedron, cube, and octahedron out of pvc pipes using standard fittings.

In particular, if we take a connector that takes three pipes each at 120 degree angles from the others (this is called a “true wye”) and we take elbows of the appropriate angle, we can make the edges come together below the center at exactly the correct angles.

A pivotal moment: What you consider tinkering is actually research-paper-worthy science. Here are some images from the Chief Engineer’s workbench.

The supporting construction of our heat exchangers are built from standard parts connected at various angles:

The final result can be a cuboid for holding meandering tubes:

… or cascaded prisms with n-gon basis – for holding spirals of flexible tubes:

The implementation of this design is documented here (a German post whose charm would be lost in translation unless I wanted to create Internet Poetry).

But I also started up my time machine – in order to find traces of my polyhedra research in the early 1980s. From photos and drawings of the three-dimensional crystals in mineralogy books I figured out how to draw two-dimensional maps of maximally connected surface areas. I cut out the map, and glued together the remaining free edges. Today I would be made redundant by Origami AI.

I filled several shelves with polyhedra of increasing number of faces, starting with a tetrahedron and culminating with this rhombicosidodecahedron. If I recall correctly, I cheated a bit with this one and created some of the pyramids as completely separate items.

I think this was a rather standard hobby for the typical nerdy child, among things like growing crystals from solutions of toxic chemicals, building a makeshift rotatable telescope tripod from scraps, or verifying the laws of optics using prisms and lenses from ancient dismantled devices.

The actually interesting thing is that this photo is the only trace of any of these hobbies. In many years after creating this stuff – and destroying it again – I never thought about documenting it. Until today. It seems we weren’t into sharing these days.

# Heat Transport: What I Wrote So Far.

Don’t worry, The Subversive Elkement will publish the usual silly summer posting soon! Now am just tying up loose ends.

In the next months I will keep writing about heat transport: Detailed simulations versus maverick’s rules of thumb, numerical solutions versus insights from the few things you can solve analytically, and applications to our heat pump system.

So I checked what I have already written – and I discovered a series which does not show up as such in various lists, tags, categories:

[2014-12-14] Cistern-Based Heat Pump – Research Done in 1993 in Iowa. Pioneering work, but the authors dismissed a solar collector for economic reasons. They used a steady-state estimate of the heat flow from ground to the tank, and did not test the setup in winter.

[2015-01-28] More Ice? Exploring Spacetime of Climate and Weather. A simplified simulation based on historical weather data – only using daily averages. Focus: Estimate of the maximum volume of ice per season, demonstration of yearly variations. As explained later (2017) in more detail I had to use information from detailed simulations though – to calculate the energy harvested by the collector correctly in such a simple model.

[2015-04-01] Ice Storage Challenge: High Score! Our heat pump created an ice cube of about 15m3 because we had turned the collector off. About 10m3 of water remained unfrozen, most likely when / because the ice cube touched ground. Some qualitative discussions of heat transport phenomena involved and of relevant thermal parameters.

[2016-01-07] How Does It Work? (The Heat Pump System, That Is) Our system, in a slide-show of operating statuses throughput a typical year. For each period typical temperatures are given and the ‘typical’ direction of heat flow.

[2016-01-22] Temperature Waves and Geothermal Energy. ‘Geothermal’ energy used by heat pumps is mainly stored solar energy. A simple model: The temperature at the surface of the earth varies sinusoidally throughout the year – this the boundary condition for the heat equation. This differential equation links the temporal change of temperature to its spatial variation. I solve the equation, show some figures, and check how results compare to the thermal diffusivity of ground obtained from measurements.

[2016-03-01] Rowboats, Laser Pulses, and Heat Energy (Boring Title: Dimensional Analysis). Re-visiting heat transport and heat diffusion length, this time only analyzing dimensional relationships. By looking at the heat equation (without the need to solve it) a characteristic length can be calculated: ‘How far does heat get in a certain time?’

[2017-02-05] Earth, Air, Water, and Ice. Data analysis of the heating season 2014/15 (when we turned off the solar/air collector to simulate a harsher winter) – and an attempt to show energy storages, heat exchangers, and heat flows in one schematic. From the net energy ‘in the tank’ the contribution of ground can be calculated.

[2017-02-22] Ice Storage Hierarchy of Needs. Continued from the previous post – bird’s eye view: How much energy comes from which sources, and which input parameters are critical? I try to answer when and if simple energy accounting makes sense in comparison to detailed simulations.

[2017-05-02] Simulating Peak Ice. I compare measurements of the level in the tank with simulations of the evolution of the volume of ice. Simulations (1-minute intervals) comprise a model of the control logic, the varying performance factor of the heat pump, heat transport in ground, energy balances for the hot and cold tanks, and the heat exchangers connected in series.

(Adding the following after having published this post. However, there is no guarantee I will update this post forever ;-))

[2017-08-17] Simulations: Levels of Consciousness. Bird’s Eye View: How does simulating heat transport fit into my big picture of simulating the heat pump system or buildings or heating systems in general? I consider it part of the ‘physics’ layer of a hierarchy of levels.

Planned episodes? Later this year (2017) or next year I might write about the error made when considering simplified geometry – like modeling a linear 1D flow when the actualy symmetry is e.g. spherical.

# Spheres in a Space with Trillions of Dimensions

I don’t venture into speculative science writing – this is just about classical statistical mechanics; actually about a special mathematical aspect. It was one of the things I found particularly intriguing in my first encounters with statistical mechanics and thermodynamics a long time ago – a curious feature of volumes.

I was mulling upon how to ‘briefly motivate’ the calculation below in a comprehensible way, a task I might have failed at years ago already, when I tried to use illustrations and metaphors (Here and here). When introducing the ‘kinetic theory’ in thermodynamics often the pressure of an ideal gas is calculated first, by considering averages over momenta transferred from particles hitting the wall of a container. This is rather easy to understand but still sort of an intermediate view – between phenomenological thermodynamics that does not explain the microscopic origin of properties like energy, and ‘true’ statistical mechanics. The latter makes use of a phase space with with dimensions the number of particles. One cubic meter of gas contains ~1025 molecules. Each possible state of the system is depicted as a point in so-called phase space: A point in this abstract space represents one possible system state. For each (point-like) particle 6 numbers are added to a gigantic vector – 3 for its position and 3 for its momentum (mass times velocity), so the space has ~6 x 1025 dimensions. Thermodynamic properties are averages taken over the state of one system watched for a long time or over a lot of ‘comparable’ systems starting from different initial conditions. At the heart of statistical mechanics are distributions functions that describe how a set of systems described by such gigantic vectors evolves. This function is like a density of an incompressible fluid in hydrodynamics. I resorted to using the metaphor of a jelly in hyperspace before.

Taking averages means to multiply the ‘mechanical’ property by the density function and integrate it over the space where these functions live. The volume of interest is a  generalized N-ball defined as the volume within a generalized sphere. A ‘sphere’ is the surface of all points in a certain distance (‘radius’ R) from an origin

$x_1^2 + x_2^2 + ... + x_ {N}^2 = R^2$

($x_n$ being the co-ordinates in phase space and assuming that all co-ordinates of the origin are zero). Why a sphere? Because states are ordered or defined by energy, and larger energy means a greater ‘radius’ in phase space. It’s all about rounded surfaces enclosing each other. The simplest example for this is the ellipse of the phase diagram of the harmonic oscillator – more energy means a larger amplitude and a larger maximum velocity.

And here is finally the curious fact I actually want to talk about: Nearly all the volume of an N-ball with so many dimensions is concentrated in an extremely thin shell beneath its surface. Then an integral over a thin shell can be extended over the full volume of the sphere without adding much, while making integration simpler.

This can be seen immediately from plotting the volume of a sphere over radius: The volume of an N-ball is always equal to some numerical factor, times the radius to the power of the number of dimensions. In three dimensions the volume is the traditional, honest volume proportional to r3, in two dimensions the ‘ball’ is a circle, and its ‘volume’ is its area. In a realistic thermodynamic system, the volume is then proportional to rN with a very large N.

The power function rN turn more and more into an L-shaped function with increasing exponent N. The volume increases enormously just by adding a small additional layer to the ball. In order to compare the function for different exponents, both ‘radius’ and ‘volume’ are shown in relation to the respective maximum value, R and RN.

The interesting layer ‘with all the volume’ is certainly much smaller than the radius R, but of course it must not be too small to contain something. How thick the substantial shell has to be can be found by investigating the volume in more detail – using a ‘trick’ that is needed often in statistical mechanics: Taylor expanding in the exponent.

A function can be replaced by its tangent if it is sufficiently ‘straight’ at this point. Mathematically it means: If dx is added to the argument x, then the function at the new target is f(x + dx), which can be approximated by f(x) + [the slope df/dx] * dx. The next – higher-order term would be proportional to the curvature, the second derivation – then the function is replaced by a 2nd order polynomial. Joseph Nebus has recently published a more comprehensible and detailed post about how this works.

So the first terms of this so-called Taylor expansion are:

$f(x + dx) = f(x) + dx{\frac{df}{dx}} + {\frac{dx^2}{2}}{\frac{d^2f}{dx^2}} + ...$

If dx is small higher-order terms can be neglected.

In the curious case of the ball in hyperspace we are interested in the ‘remaining volume’ V(r – dr). This should be small compared to V(r) = arN (a being the uninteresting constant numerical factor) after we remove a layer of thickness dr with the substantial ‘bulk of the volume’.

However, trying to expand the volume V(r – dr) = a(r – dr)N, we get:

$V(r - dr) = V(r) - adrNr^{N-1} + a{\frac{dr^2}{2}}N(N-1)r^{N-2} + ...$
$= ar^N(1 - N{\frac{dr}{r}} + {\frac{N(N-1)}{2}}({\frac{dr}{r}})^2) + ...$

But this is not exactly what we want: It is finally not an expansion, a polynomial, in (the small) ratio of dr/r, but in Ndr/r, and N is enormous.

So here’s the trick: 1) Apply the definition of the natural logarithm ln:

$V(r - dr) = ae^{N\ln(r - dr)} = ae^{N\ln(r(1 - {\frac{dr}{r}}))}$
$= ae^{N(\ln(r) + ln(1 - {\frac{dr}{r}}))}$
$= ar^Ne^{\ln(1 - {\frac{dr}{r}}))} = V(r)e^{N(\ln(1 - {\frac{dr}{r}}))}$

2) Spot a function that can be safely expanded in the exponent: The natural logarithm of 1 plus something small, dr/r. So we can expand near 1: The derivative of ln(x) is 1/x (thus equal to 1/1 near x=1) and ln(1) = 0. So ln(1 – x) is about -x for small x:

$V(r - dr) = V(r)e^{N(0 - 1{\frac{dr}{r})}} \simeq V(r)e^{-N{\frac{dr}{r}}}$

3) Re-arrange fractions …

$V(r - dr) = V(r)e^{-\frac{dr}{(\frac{r}{N})}}$

This is now the remaining volume, after the thin layer dr has been removed. It is small in comparison with V(r) if the exponential function is small, thus if ${\frac{dr}{(\frac{r}{N})}}$ is large or if:

$dr \gg \frac{r}{N}$

Summarizing: The volume of the N-dimensional hyperball is contained mainly in a shell dr below the surface if the following inequalities hold:

${\frac{r}{N}} \ll dr \ll r$

The second one is needed to state that the shell is thin – and allow for expansion in the exponent, the first one is needed to make the shell thick enough so that it contains something.

This might help to ‘visualize’ a closely related non-intuitive fact about large numbers, like eN: If you multiply such a number by a factor ‘it does not get that much bigger’ in a sense – even if the factor is itself a large number:

Assuming N is about 1025  then its natural logarithm is about 58 and…

$Ne^N = e^{\ln(N)+N} = e^{58+10^{25}}$

… 58 can be neglected compared to N itself. So a multiplicative factor becomes something to be neglected in a sum!

I used a plain number – base e – deliberately as I am obsessed with units. ‘r’ in phase space would be associated with a unit incorporating lots of lengths and momenta. Note that I use the term ‘dimensions’ in two slightly different, but related ways here: One is the mathematical dimension of (an abstract) space, the other is about cross-checking the physical units in case a ‘number’ is something that can be measured – like meters. The co-ordinate  numbers in the vector refer to measurable physical quantities. Applying the definition of the logarithm just to rN would result in dimensionless number N side-by-side with something that has dimensions of a logarithm of the unit.

Using r – a number with dimensions of length – as base, it has to be expressed as a plain number, a multiple of the unit length $R_0$ (like ‘1 meter’). So comparing the original volume of the ball $a{(\frac{r}{R_0})}^N$ to one a factor of N bigger …

$aN{(\frac{r}{R_0})}^N = ae^{\ln{(N)} + N\ln{(\frac{r}{R_0})}}$

… then ln(N) can be neglected as long as $\frac{r}{R_0}$ is not extreeeemely tiny. Using the same argument as for base e above, we are on the safe side (and can neglect factors) if r is of about the same order of magnitude as the ‘unit length’ $R_0$. The argument about negligible factors is an argument about plain numbers – and those ‘don’t exist’ in the real world as one could always decide to measure the ‘radius’ in a units of, say, 10-30 ‘meters’, which would make the original absolute number small and thus the additional factor non-negligible. One might save the argument by saying that we would always use units that sort of match the typical dimensions (size) of a system.

Saying everything in another way: If the volume of a hyperball ~rN is multiplied by a factor, this corresponds to multiplying the radius r by a factor very, very close to 1 – the Nth root of the factor for the volume. Only because the number of dimensions is so large, the volume is increased so much by such a small increase in radius.

As the ‘bulk of the volume’ is contained in a thin shell, the total volume is about the product of the surface area and the thickness of the shell dr. The N-ball is bounded by a ‘sphere’ with one dimension less than the ball. Increasing the volume by a factor means that the surface area and/or the thickness have to be increased by factors so that the product of these factors yield the volume increase factor. dr scales with r, and does thus not change much – the two inequalities derived above do still hold. Most of the volume factor ‘goes into’ the factor for increasing the surface. ‘The surface becomes the volume’.

This was long-winded. My excuse: Also Richard Feynman took great pleasure in explaining the same phenomenon in different ways. In his lectures you can hear him speak to himself when he says something along the lines of: Now let’s see if we really understood this – let’s try to derive it in another way…

And above all, he says (in a lecture that is more about math than about physics)

Now you may ask, “What is mathematics doing in a physics lecture?” We have several possible excuses: first, of course, mathematics is an important tool, but that would only excuse us for giving the formula in two minutes. On the other hand, in theoretical physics we discover that all our laws can be written in mathematical form; and that this has a certain simplicity and beauty about it. So, ultimately, in order to understand nature it may be necessary to have a deeper understanding of mathematical relationships. But the real reason is that the subject is enjoyable, and although we humans cut nature up in different ways, and we have different courses in different departments, such compartmentalization is really artificial, and we should take our intellectual pleasures where we find them.

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Further reading / sources: Any theoretical physics textbook on classical thermodynamics / statistical mechanics. I am just re-reading mine.

# Ploughing Through Theoretical Physics Textbooks Is Therapeutic

And finally science confirms it, in a sense.

Again and again, I’ve harped on this pet theory of mine – on this blog and elsewhere on the web: At the peak of my immersion in the so-called corporate world, as a super-busy bonus miles-collecting consultant, I turned to the only solace: Getting up (even) earlier, and starting to re-read all my old mathematics and physics textbooks and lecture notes.

The effect was two-fold: It made me more detached, perhaps more Stoic when facing the seemingly urgent challenges of the accelerated world. Maybe it already prepared me for a long and gradual withdrawal from that biosphere. But surprisingly, I felt it also made my work results (even ;-)) better: I clearly remember compiling documentation I wrote after setting up some security infrastructure with a client. Writing precise documentation was again more like casting scientific research results into stone, carefully picking each term and trying to be as succinct as possible.

As anybody else I enjoy reading about psychological research that confirms my biases one-datapoint-based research – and here it finally is. Thanks to Professor Gary for sharing it. Science says that Corporate-Speak Makes You Stupid. Haven’t we – Dilbert fans – always felt that this has to be true?

… I’ve met otherwise intelligent people, after working with management consultant, are convinced that infinitely-malleable concepts like “disruptive innovation,” “business ecosystem,” and “collaborative culture” have objective value.

In my post In Praise of Textbooks with Tons of Formulas I focused on possible positive explanations, like speeding up your rational System 2 ((c) Daniel Kahneman) – by getting accustomed to mathematics again. By training yourself to recognize patterns and to think out of the box when trying to find the clever twist to solve a physics problem. Re-reading this, I cringe though: Thinking out of the box has entered the corporate vocabulary already. Disclaimer: I am talking about ways to pick a mathematical approach, by drawing on other, slightly related problems intuitively – in the way Kahneman explains the so-called intuition of experts as pattern recognition.

But perhaps the explanation is really as simple as that we just need to shield ourselves from negative effects of certain ecosystems and cultures that are particularly intrusive and mind-bending. So this is my advice to physics and math graduates: Do not rely on your infamous analytical skills forever. First, using that phrase in a job application sounds like phony hollow BS (as unfortunately any self-advertising of social skills does). Second, these skills are real, but they will decay exponentially if you don’t hone them.

# Simulating Peak Ice

This year ice in the tank was finally melted between March 5 to March 10 – as ‘visual inspection’ showed. Level sensor Mr. Bubble was confused during the melting phase; thus it was an interesting exercise to compare simulations to measurements.

Simulations use the measured ambient temperature and solar radiation as an input, data points are taken every minute. Air temperature determines the heating energy needed by the house: Simulated heat load is increasing linearly until a maximum ‘cut off’ temperature.

The control logic of the real controller (UVR1611 / UVR16x2) is mirrored in the simulation: The controller’s heating curve determines the set temperature for the heating water, and it switches the virtual 3-way valves: Diverting heating water either to the hygienic storage or the buffer tank for space heating, and including the collector in the brine circuit if air temperature is high enough compared to brine temperature. In the brine circuit, three heat exchangers are connected in series: Three temperatures at different points are determined self-consistently from three equations that use underground tank temperature, air temperature, and the heat pump evaporator’s power as input parameters.

The hydraulic schematic for reference, as displayed in the controller’s visualization (See this article for details on operations.)

The Coefficient of Performance of the heat pump, its heating power, and its electrical input power are determined by heating water temperature and brine temperature – from polynomial fit curves to vendors’ data sheet.

So for every minute, the temperatures of tanks – hot and cold – and the volume of ice can be calculated from energy balances. The heating circuits and tap water consume energy, the heat pump delivers energy. The heat exchanger in the tank releases energy or harvests energy, and the collector exchanges energy with the environment. The heat flow between tank and ground is calculated by numerically solving the Heat Equation, using the nearly constant temperature in about 10 meters depth as a boundary condition.

For validating the simulation and for fine-tuning input parameters – like the thermal properties of ground or the building – I cross-check calculated versus measured daily / monthly energies and average temperatures.

Measurements for this winter show the artificial oscillations during the melting phase because Mr. Bubble faces the cliff of ice:

Simulations show growing of ice and the evolution of the tank temperature in agreement with measurements. The melting of ice is in line with observations. The ‘plateau’ shows the oscillations that Mr. Bubble notices, but the true amplitude is smaller:

Simulated peak ice is about 0,7m3 greater than the measured value. This can be explained by my neglecting temperature gradients within water or ice in the tank:

When there is only a bit of ice yet (small peak in December), tank temperature is underestimated: In reality, the density anomaly of water causes a zone of 4°C at the bottom, below the ice.

When the ice block is more massive (end of January), I overestimate brine temperature as ice has less than 0°C, at least intermittently when the heat pump is turned on. Thus the temperature difference between ambient air and brine is underestimated, and so is the simulated energy harvested from the collector – and more energy needs to be provided by freezing water.

However, a difference in volume of less than 10% is uncritical for system’s sizing, especially if you err on the size of caution. Temperature gradients in ice and convection in water should be less critical if heat exchanger tubes traverse the volume of tank evenly – our prime design principle.

I have got questions about the efficiency of immersed heat exchangers in the tank – will heat transfer deteriorate if the layer of ice becomes too thick? No, according also to this very detailed research report on simulations of ‘ice storage heat pump systems’ (p.5). We grow so-called ‘ice on coil’ which is compared to flat-plate heat exchangers:

… for the coil, the total heat transfer (UA), accounting for the growing ice surface, shows only a small decrease with growing ice thickness. The heat transfer resistance of the growing ice layer is partially compensated by the increased heat transfer area around the coil. In the case of the flat plate, on the contrary, also the UA-value decreases rapidly with growing ice thickness.

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For system’s configuration data see the last chapter of this documentation.

# Learning General Relativity

Math blogger Joseph Nebus does another A – Z series of posts, explaining technical terms in mathematics. He asked readers for their favorite pick of things to be covered in this series, and I came up with General Covariance. Which he laid out in this post – in his signature style, using neither equations nor pop-science images like deformed rubber mattresses – but ‘just words’. As so often, he manages to explain things really well!

Actually, I asked for that term as I am in the middle of yet another physics (re-)learning project – in the spirit of my ventures into QFT a while back.

Since a while I have now tried (on this blog) to cover only the physics related to something I have both education in and hands-on experience with. Re General Relativity I have neither: My PhD was in applied condensed-matter physics – lasers, superconductors, optics – and this article by physicist Chad Orzel about What Math Do You Need For Physics? covers well what sort of math you need in that case. Quote:

I moved into the lab, and was concerned more with technical details of vacuum pumps and lasers and electronic circuits and computer data acquisition and analysis.

So I cannot find the remotest way to justify why I would need General Relativity on a daily basis – insider jokes about very peculiarly torus-shaped underground water/ice tanks for heat pumps aside.

My motivation is what I described in this post of mine: Math-heavy physics is – for me, that means a statistical sample of 1 – the best way of brazing myself for any type of tech / IT / engineering work. This positive effect is not even directly related to math/physics aspects of that work.

But I also noticed ‘on the internet’ that there is a community of science and math enthusiasts, who indulge in self-studying theoretical physics seriously as a hobby. Often these are physics majors who ended up in very different industry sectors or in management / ‘non-tech’ jobs and who want to reconnect with what they once learned.

For those fellow learners I’d like to publish links to my favorite learning resources.

There seem to be two ways to start a course or book on GR, and sometimes authors toggle between both modes. You can start from the ‘tangible’ physics of our flat space (spacetime) plus special relativity and then gradually ‘add a bit of curvature’ and related concepts. In this way the introduction sounds familiar, and less daunting. Or you could try to introduce the mathematical concepts at a most rigorous abstract level, and return to the actual physics of our 4D spacetime and matter as late as possible.

The latter makes a lot of sense as you better unlearn some things you took for granted about vector and tensor calculus in flat space. A vector must no longer be visualized as an arrow that can be moved around carelessly in space, and one must be very careful in visualizing what transforming coordinates really means.

For motivation or as an ‘upper level pop-sci intro’…

Richard Feynman’s lecture on curved space might be a very good primer. Feynman explains what curved space and curved spacetime actually mean. Yes, he is using that infamous beetle on a balloon, but he also gives some numbers obtained by back-of-the-envelope calculations that explain important concepts.

For learning about the mathematical foundations …

I cannot praise these Lectures given at the Heraeus International Winter School Gravity and Light 2015 enough. Award-winning lecturer Frederic P. Schuller goes to great lengths to introduce concepts carefully and precisely. His goal is to make all implicit assumptions explicit and avoid allusions to misguided ‘intuitions’ one might got have used to when working with vector analysis, tensors, gradients, derivatives etc. in our tangible 3D world – covered by what he calls ‘undergraduate analysis’. Only in lecture 9 the first connection is made back to Newtonian gravity. Then, back to math only for some more lectures, until finally our 4D spacetime is discussed in lecture 13.

Schuller mentions in passing that Einstein himself struggled with the advanced math of his own theory, e.g. in the sense of not yet distinguishing clearly between the mathematical structure that represents the real world (a topological manifold) and the multi-dimensional chart we project our world onto when using an atlas. It is interesting to pair these lectures with this paper on the history and philosophy of general relativity – a link Joseph Nebus has pointed to in his post on covariance.

Learning physics or math from videos you need to be much more disciplined than with plowing through textbooks – in the sense that you absolutely have to do every single step in a derivation on your own. It is easy to delude oneself that you understood something by following a derivation passively, without calculating anything yourself. So what makes these lectures so useful is that tutorial sessions have been recorded as well: Tutorial sheets and videos can be found here.
(Edit: The Youtube channel of the event has not all the recordings of the tutorial sessions, only this conference website has. It seems the former domain does not work any more, but the content is perserved at gravity-and-light.herokuapp.com)

You also find brief notes for these lectures here.

For a ‘physics-only’ introduction …

… I picked a classical, ‘legendary’ resource: Landau and Lifshitz give an introduction to General Relativity in the last third of the second volume in their Course of Theoretical Physics, The Classical Theory of Fields. Landau and Lifshitz’s text is terse, perhaps similar in style to Dirac’s classical introduction to quantum mechanics. No humor, but sublime and elegant.

Landau and Lifshitz don’t need manifolds nor tangent bundles, and they use the 3D curvature tensor of space a lot in addition to the metric tensor of 4D spacetime. They introduce concepts of differences in space and time right from the start, plus the notion of simultaneity. Mathematicians might be shocked by a somewhat handwaving, ‘typical physicist’s’ way to deal with differentials, the way vectors on different points in space are related, etc. – neglecting (at first sight, explore every footnote in detail!) the tower of mathematical structures you actually need to do this precisely.

But I would regard Lev Landau sort of a Richard Feynman of The East, so it takes his genius not make any silly mistakes by taking the seemingly intuitive notions too literally. And I recommend this book only when combined with a most rigorous introduction.

For additional reading and ‘bridging the gap’…

I recommend Sean Carroll’s  Lecture Notes on General Relativity from 1997 (precursor of his textbook), together with his short No-Nonsense Introduction to GR as a summary. Carroll switches between more intuitive physics and very formal math. He keeps his conversational tone – well known to readers of his popular physics books – which makes his lecture notes a pleasure to read.

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So this was a long-winded way to present just a bunch of links. This post should also serve as sort of an excuse that I haven’t been really active on social media or followed up closely on other blogs recently. It seems in winter I am secluding myself from the world in order to catch up on theoretical physics.

# Rowboats, Laser Pulses, and Heat Energy (Boring Title: Dimensional Analysis)

Dimensional analysis means to understand the essentials of a phenomenon in physics and to calculate characteristic numbers – without solving the underlying, often complex, differential equation. The theory of fluid dynamics is full of interesting dimensionless numbers –  Reynolds Number is perhaps most famous.

In the previous post on temperature waves I solved the Heat Equation for a very simple case, in order to answer the question How far does solar energy get into ground in a year? Reason: I have been working on simulations of our heat pump system since a few years. This also involves heat transport between the water/ice tank and ground. If you set out to simulate a complex phenomenon you have to make lots of assumptions about materials’ parameters, and you have to simplify the system and equations you use for modelling the real world. You need a way of cross-checking if your results sound plausible in terms of orders of magnitude. So my goal has been to find yet another method to confirm assumptions I have made about the thermal properties of ground elsewhere.

Before I am going to revisit heat transport, I’ll try to explain what dimensional analysis is – using the best example I’ve ever seen. I borrow it from theoretical physicist – and awesome lecturer – David Tong:

How does the speed of a rowing boat depend in the number of rowers?

References: Tong’s lecture called Dynamics and Relativity (Chapter 3), This is the original paper from 1971 Tong quotes: Rowing: A similarity analysis.

The boat experiences a force of friction in water. As for a car impeded by the friction of the surrounding air, the force of friction depends on velocity.

Force is the change of momentum, momentum is proportional to mass times velocity. Every small ‘parcel’ of water carries a momentum proportional to speed – so force should at least be proportional to one factor of v. But these parcel move at a speed v, so the faster they move the more momentum is exchanged with the boat; so there has to be a second factor of v, and force is proportional to the square of the speed of the boat.

The larger the cross-section of the submerged part of the boat, A, the higher is the number of collisions between parcels of water and the boat, so putting it together:

$F \sim v^{2}A$

Rowers need to put in power to compensate for friction. Power is energy per time, and Energy is force times distance. Since distance over time is velocity, thus power is also force times velocity.

So there is one more factor of v to be included in power:

$P \sim v^{3}A$

For the same reason wind power harvested by wind turbines is proportional to the third power of wind speed.

A boat does not sink because downward gravity and upward buoyancy just compensate each other; buoyancy is the weight of the volume of water displaced. The heavier the load, the more water needs to be displaced. The submerged volume of the boat V is proportional to the weight of the rowers, and thus to their number N if the mass of the boat itself is negligible:

$V \sim N$

The volume of something scales with the third power of its linear dimensions – think of a cube or a sphere; so the surface area scales with the square of the length, and the cross-section A scales with V – and thus with N:

$A \sim N^{\frac{2}{3}}$

Each rower contributes the same share to the total rowing power, so:

$P \sim N$

Inserting for A in the first expression for P:

$P \sim v^{3} N^{\frac{2}{3}}$

Eliminating P as it has been shown to be proportional to N:

$N \sim v^{3} N^{\frac{2}{3}}$
$v^{3} \sim N^{\frac{1}{3}}$
$v \sim N^{\frac{1}{9}}$

… which is in good agreement with measurements according to Tong.

Heat Transport and Characteristic Lengths

In the last post I’ve calculated characteristic lengths, describing how heat is slowly dissipated in ground: 1) The wavelength of the damped oscillation and 2) the run-out length of the enveloping exponential function.

Both are proportional to the square root of a simple number:

$l \sim \sqrt{D \tau}$

… the factor of proportionality being ‘small’ on a logarithmic scale, like π or 2 or their inverse. τ is the period, and D was a number expressing how well the material carries away heat energy.

There is another ‘simple’ scenario that also results in a length scale described by
$\sqrt{D \tau}$ times a small number: If you deposit a confined ‘lump of heat’, a ‘point heat’ it will peter out and the average width of the lump after some time τ is about this length as well.

Using very short laser pulse to heat solid material is very close to depositing ‘point heat’. Decades ago I worked with pulsed excimer lasers, used for ablation (‘shooting off) material from ceramic targets.This type of lasers is used in eye surgery today:

Heat is deposited in nanosecond pulses, and the run-out length of the heat peak in the material is about $\sqrt{D \tau}$ with tau being equal to the very short laser’s pulse length of several nanoseconds. As the pulse duration is short, the penetration depth is short as well, and tissue is ‘cut’ precisely without heating much of the underlying material.

So this type of $\sqrt{D \tau}$ length is not just a result of a calculation for a specific scenario, but it rather seems to encompass important characteristics of heat conduction as such.

The unit of D is area over time, m2/s. If you accept the heat equation as a starting point, analysing the dimensions involved by counting x and t you see that D has to contain two powers of x and one of t. Half of applied physics and engineering is about getting units right.

But I pretend I don’t even know the heat equation and ‘visualize’ heat transport in this way: ‘Something’ – like heat energy – is concentrated in space and closely peters out. The spreading out is faster, the more concentrated it is. A thin needle-like peak quickly becomes a rounded hill, and then is flattened gradually. Concentration in space means curvature. The smaller the space occupied by the lump of heat is, the smaller its radius, the higher its curvature as curvature is the inverse of the radius of a tangential circular path.

I want to relate curvature to the change with time. Change in time has to be measured in units including the inverse of time, curvature is the inverse of space. Equating those, you have to come with something including the square of spatial dimension and one temporal dimension – something like D [m2/s].

How to get a characteristic length from this? D has to be multiplied by a characteristic time, and then we need to take a the square root. So we need to put in some characteristic time, that’s a property of the specific system investigated and not of the equation – like the yearly period or the laser pulse. And the resulting length is exactly that $l \sim \sqrt{D \tau}$ that shows up in any of of the solutions for specific scenarios.

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The characteristic width of the spreading lump of heat is visible in the so-called Green’s functions. These functions described a system’s response to a ‘source’ which resemble a needle-like peak in time. In this case it is a Gaussian functions with a ‘width’ $\sim \sqrt{D \tau}$. See e.g. equation (14) on PDF-page 14 of these lecture notes.