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.

Simulations: Levels of Consciousness

In a recent post I showed these results of simulations for our heat pump system:

I focused on the technical details – this post will be more philosophical.

What is a ‘simulation’ – opposed to simplified calculations of monthly or yearly average temperatures or energies? The latter are provided by tools used by governmental agencies or standardization bodies – allowing for a comparison of different systems.

In a true simulation the time intervals so small that you catch all ‘relevant’ changes of a system. If a heating system is turned on for one hour, then turned off again, he time slot needs to be smaller than one hour. I argued before that calculating meaningful monthly numbers requires to incorporate data that had been obtained before by measurements – or by true simulations.

For our system, the heat flow between ground and the water/ice tank is important. In our simplified sizing tool – which is not a simulation – I use average numbers. I validated them by comparing with measurements: The contribution of ground can be determined indirectly; by tallying all the other energies involved. In the detailed simulation I calculate the temperature in ground as a function of time and of distance from the tank, by solving the Heat Equation numerically. Energy flow is then proportional to the temperature gradient at the walls of the tank. You need to make assumptions about the thermal properties of ground, and a simplified geometry of the tank is considered.

Engineering / applied physics in my opinion is about applying a good-enough-approach in order to solve one specific problem. It’s about knowing your numbers and their limits. It is tempting to get carried away by nerdy physics details, and focus on simulating what you know exactly – forgetting that there are huge error bars because of unknowns.

This is the hierarchy I keep in mind:

On the lowest level is the simulation physics, that is: about modelling how ‘nature’ and system’s components react – to changes in the previous time slot. Temperatures change because energies flows, and energy flows because of temperature differences. The heat pump’s output power depends on heating water temperature and brine temperature. Energy of the building is ‘lost’ to the environment via heat conduction; heat exchangers immersed in tanks deposit energy there or retrieve it. I found that getting the serial connection of heat exchangers right in the model was crucial, and it required a self-consistent calculation for three temperatures at the same point of time, rather than trying to ‘follow round the brine’. I used the information on average brine temperatures obtained by these method to run a simplified version of the simulation using daily averages only – for estimating the maximum volume of ice for two decades.

So this means you need to model your exact hydraulic setup, or at least you need to know which features of your setup are critical and worthy to model in detail. But the same also holds for the second level, the simulation of control logic. I try to mirror production control logic as far as possible: This code determines how pumps and valves will react, depending on the system’s prior status before. Both in real life and in the simulation threshold values and ‘hystereses’ are critical: You start to heat if some temperature falls below X, but you only stop heating if it has risen above X plus some Delta. Typical brine-water heat pumps always provide approximately the same output power, so you control operations time and buffer heating energy. If Delta for heating the hot water buffer tank is too large, the heat pump’s performance will suffer. The Coefficient of Performance of the heat pump decreases with increasing heating water temperature. Changing an innocuous parameter will change results a lot, and the ‘control model’ should be given the same vigilance as the ‘physics model’.

Control units can be tweaked at different levels: ‘Experts’ can change the logic, but end users can change non-critical parameters, such as set point temperatures.We don’t restrict expert access in systems we provide the control unit for. But it make sense to require extra input for the expert level though – to prevent accidental changes.

And here we enter level 3 – users’ behavior. We humans are bad at trying to outsmart the controller.

[Life-form in my home] always sets the controller to ‘Sun’. [little sun icon indicating manually set parameters]. Can’t you program something so that nothing actually changes when you pick ‘Sun’?

With heat pumps utilizing ground or water sources – ‘built’ storage repositories with limited capacity – unexpected and irregular system changes are critical: You have to size your source in advance. You cannot simply order one more lorry load of wood pellets or oil if you ‘run out of fuel’. If the source of ambient energy is depleted, the heat pump finally will refuse to work below a certain source temperature. The heat pump’s rated power has match the heating demands and the size of the source exactly. It also must not be oversized in order to avoid turning on and off the compressor too often.

Thus you need good estimates for peak heat load and yearly energy needs, and models should include extreme weather (‘physics’) but also erratic users’ behaviour. The more modern the building, the more important spikes in hot tap water usage get in relation to space heating. A vendor of wood pellet stoves told me that delivering peak energy for hot water – used in private bathrooms that match spas – is a greater challenge today than delivering space heating energy. Energy certificates of modern buildings take into account huge estimated solar and internal energy gains – calculated according to standards. But the true heating power needed on a certain day will depend on the strategy or automation home owners use when managing their shades.

Typical gas boilers are oversized (in terms of kW rated power) by a factor of 2 or more in Germany, but with heat pumps you need to be more careful. However, this also means that heat pump systems cannot and should not be planned for rare peak demands, such as: 10 overnight guests want to shower in the morning one after the other, on an extremely cold day, or for heating up the building quickly after temperature had been decreased during a leave of absence.

The nerdy answer is that a smart home would know when your vacation ends and start heating up well in advance. Not sure what to do about the showering guests as in this case ‘missing’ power cannot be compensated by more time. Perhaps a gamified approach will work: An app will do something funny / provide incentives and notifications so that people wait for the water to heat up again. But what about planning for renting a part of the house out someday? Maybe a very good AI will predict what your grandchildren are likely to do, based on automated genetics monitoring.

The challenge of simulating human behaviour is ultimately governed by constraints on resources – such as the size of the heat source: Future heating demands and energy usage is unknown but the heat source has to be sized today. If the system is ‘open’ and connected to a ‘grid’ in a convenient way problems seem to go away: You order whatever you need, including energy, any time. The opposite is planning for true self-sufficiency: I once developed a simulation for an off-grid system using photovoltaic generators and wind power – for a mountain shelter. They had to meet tough regulations and hygienic standards like any other restaurant, e.g.: to use ‘industry-grade’ dishwashers needing 10kW of power. In order to provide that by solar power (plus battery) you needed to make an estimate on the number of guests likely to visit … and thus on how many people would go hiking on a specific day … and thus maybe on the weather forecast. I tried to factor in the ‘visiting probability’ based on the current weather.

I think many of these problem can be ‘resolved’ by recognizing that they are first world problems. It takes tremendous efforts – in terms of energy use or systems’ complexity – to obtain 100% availability and to cover all exceptional use cases. You would need the design heat load only for a few days every decade. On most winter days a properly sized heat pump is operating for only 12 hours. The simple, low tech solution would be to accept the very very rare intermittent 18,5°C room temperature mitigated by proper clothing. Accepting a 20-minute delay of your shower solves the hot water issue. An economical analysis can reveal the (most likely very small) trade-off of providing exceptional peak energy by a ‘backup’ electrical heating element – or by using that wood stove that you installed ‘as a backup’ but mostly for ornamental reasons because it is dreadful to fetch the wood logs when it is really cold.

But our ‘modern’ expectations and convenience needs are also reflected in regulations. Contractors are afraid of being sued by malicious clients who (quote) sit next their heat pump and count its operating cycles – to compare the numbers with the ones to be ‘guaranteed. In a weather-challenged region at more than 2.000 meters altitude people need to steam clean dishes and use stainless steel instead of wood – where wooden plates have been used for centuries. I believe that regulators are as prone as anybody else to fall into the nerdy trap described above: You monitor, measure, calculate, and regulate the things in detail that you can measure and because you can measure them – not because these things were top priorities or had the most profound impact.

Still harvesting energy from air - during a record-breaking cold January 2017

Bing Says We Are Weird. I Prove It. Using Search Term Poetry.

Bing has done so repeatedly:

Bing Places asks for this every few months.

In order to learn more about this fundamental confusion I investigated my Bing search terms. [This blog has now entered the phase of traditionally light summer entertainment.]

Rules:

  • Raw material: Search terms shown in Bing Web Master Tools for any of my / our websites.
  • Each line is a search term of a snippet of a search term – snippets must not be edited but truncation of phrases at the beginning or end is allowed.
  • Images: Random pick from the media library of elkement.blog or our German blog (‘Professional Tinkerers – Restless Settlers‘)

_______________________

you never know
my life and my work

is life just about working
please try to give a substantial answer

A substantial answer.

element not found
map random elkement

Random elkement, confused by maps.

so as many of you may know (though few of you may care)
my dedication to science

this can happen if
the internet is always a little bit broken

The internet is always a little broken

google translate repetetive glitch poetry

_______________________

So let’s hear what Google has to say!
Same rules, only for Google Search Console:

_______________________

so called art
solar energy poem

ploughing through
proof of carnot theorem
thermodynamics in a nutshell

Thermodynamics in a nutshell.

mr confused
why him?

Mr Confused

plastic cellar
sublime attic
in three sentences or fewer, explain the difference

Plastic tank - 'ice storage'

Sublime attic.

name the three common sources of heat for heat pumps
magic gyroscope
frozen herbs
mice in oven

Had contained frozen herbs. Isomorphic to folding of ice/water tank pond liner.

shapeshift vs kraken
tinkers construct slimey
you found that planet z should not have seasons

Slimey

the best we can hope for is that
tv is dangerous

what is the main function of the mulling phase?
you connect a packet sniffer to a switch

Connect a sniffer - to your heat pump

how to keep plastic water tank cool in summer
throwing boiling water into freezing air

Freezing air.

just elke
self employed physicist
the force is strong in me

The force is strong in me.

  _______________________

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.

Cistern-Based Water-Source Heat Pump System Design, 1993[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.

Simple simulations of volume of ice[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.

Ice formation during the 'ice storage challenge'[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.

System in September - typical operations conditions[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.

Measured 'wave' and propagation time[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?’

Temperature waves in ground - attenuation length of about 10 meters[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.

Energy storage, heat exchangers, heat flow[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.

Hierarchy of needs - ambient energy in ice months[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.

Simulations of brine and tank temperature and volume of ice, based on system state in 1-minute intervals.(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.

Simulation - levels of consciousnessPlanned 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.

The Future of Small Business?

If I would be asked which technology or ‘innovation’ has had the most profound impact on the way I work I would answer: Working remotely – with clients and systems I hardly ever see.

20 years ago I played with modems, cumbersome dial-in, and Microsoft’s Netmeeting. Few imagined yet, that remote work will once be the new normal. Today I am reading about Industry 4.0, 3D printing, the Internet of Things, and how every traditional company has to compete with Data Krakens like Google and Amazon. Everything will be offered as a service, including heating. One consequence: Formerly independent craftsmen become preferred partners or subcontractors of large companies, of vendors of smart heating solutions. Creative engineering is replaced by calling the Big Vendor’s hotline. Human beings cover the last mile that robots or software cannot deal with – yet.

Any sort of customization, consulting, support, and systems integration might be automated in the long run: Clients will use an online configurator and design their systems, and possibly print them out at home. Perhaps someday our clients will print out their heat exchangers from a blueprint generated on Kraken’s website, instead of using our documentation to build them.

Allowing you to work remotely also allows everybody else in the world to do so, and you might face global competition once the barriers of language and culture have been overcome (by using ubiquitous US culture and ‘business English’). Large IT service providers have actually considered to turn their consulting and support staff into independent contractors and let them compete globally – using an online bidding platform. Well-known Data Krakens match clients and freelancers, and I’ve seen several start-ups that aspire at becoming the next matching Kraken platform for computer / tech support. Clients will simply not find you if you are not on the winning platform. Platform membership becomes as important as having a website or an entry in a business directory.

One seemingly boring and underappreciated point that works enormously in favor of the platforms is bureaucracy: As a small business you have to deal with many rules and provisions, set forth by large entities – governments, big clients, big vendors. Some of those rules are conflicting, and meeting them all in the best possible way does not allow for much creativity. Krakens’ artificial intelligence – and their lawyers and lobbyists – might be able to fend off bureaucracy better than a freelancer. If you want to sell things to clients in different countries you better defer the legally correct setup of the online shop to the Kraken Platform, who deals with the intricacies of ever evolving international tax law – while you become their subcontractor or franchisee. In return, you will dutiful sign the Vendor’s Code of Conduct every year, and follow the logo guidelines when using Kraken’s corporate identity.

In my gloomy post about Everything as a Service I came to the conclusion that we – small businesses who don’t want to grow and become start-ups – aspiring at Krakenhood themselves – will either work as the Kraken’s hired hands, or …

… a lucky few will carve out a small niche and produce or customize bespoke units for clients who value luxurious goods for the sake of uniqueness or who value human imperfection as a fancy extra.

My personal credo is rather a very positive version of this quote minus the cynicism. I am happy as a small business owner. This is just a single data-point, and I don’t have a self-consistent theory on this. But I have Skin in this Game so I share my anecdotes and some of the things I learned.

Years ago I officially declared my retirement from IT Security and global corporations – to plan special heat pump systems for private home owners instead. Today we indeed work on such systems, and the inside joke of doing this remote-only – ‘IT-style’ – has become routine. Clients find us via our blog that is sometimes mistaken for a private fun blog and whose writing feels like that. I have to thank Kraken Google, begrudgingly. A few of my Public Key Infrastructure clients insisted on hiring me again despite my declarations of looming ignorance in all things IT. All this allows for very relaxed, and self-marketing-pressure-free collaborations.

  • I try to stay away, or move farther away from anything strictly organized, standardized, or ‘platform-mediated’. Agreements are made by handshake. I don’t submit any formal applications or replies to Request for Proposals.
  • “If things do not work without a written contract, they don’t work with a contract either.”
  • I hardly listen to business experts, especially if they try to give well-meant, but unsolicited advice. Apply common sense!
  • Unspectacular time-tested personal business relationships beat 15 minutes of fame any time.
  • My work has to speak for itself, and ‘marketing’ has to be a by-product. I cannot compete with companies who employ people full-time for business development.
  • The best thing to protect your inner integrity is to know and to declare what you do not want and what you would never do. Removing the absolute negatives leaves a large area of positive background, and counter the mantra of specific ‘goals’ this approach lets you discover unexpected upsides. This is Nassim Taleb’s Via Negativa – and any career or business advice that speaks to me revolves around that.
  • There is no thing as the True Calling or the One and Only Passion – I like the notion of a Portfolio of Passions. I think you are getting to enjoy what you are learning to be good at – not the other way around.
  • All this is the result of years of experimenting in an ‘hyperspace of options’ – there is no shortcut. I have to live with the objection that I have just been lucky, but I can say that I made many conscious decisions whose ‘goal’ was to increase the number of options rather than to narrow them down (Taleb’s Optionality).

So I will finally quote Nassim Taleb, who nailed as usual – in his Facebook post about The New Artisan:

Anything you do to optimize your work, cut some corners, squeeze more “efficiency” out of it (and out of your life) will eventually make you hate it.

I have bookmarked this link for a while – because sometimes I need to remind myself of all the above.

Taleb states that an Artisan …

1) does things for existential reasons,
2) has some type of “art” in his/her profession, stays away from most aspects of industrialization, combines art and business in some manner (his decision-making is never fully economic),
3) has some soul in his/her work: would not sell something defective or even of compromised quality because what people think of his work matters more than how much he can make out of it,
4) has sacred taboos, things he would not do even if it markedly increased profitability.

… and I cannot agree more. I have lots of Sacred Taboos, and they have served me well.

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.

___________________________________

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

Other People Have Lives – I Have Domains

These are just some boring update notifications from the elkemental Webiverse.

The elkement blog has recently celebrated its fifth anniversary, and the punktwissen blog will turn five in December. Time to celebrate this – with new domain names that says exactly what these sites are – the ‘elkement.blog‘ and the ‘punktwissen.blog‘.

Actually, I wanted to get rid of the ads on both blogs, and with the upgrade came a free domain. WordPress has a detailed cookie policy – and I am showing it dutifully using the respective widget, but they have to defer to their partners when it comes to third-party cookies. I only want to worry about research cookies set by Twitter and Facebook, but not by ad providers, and I am also considering to remove social media sharing buttons and the embedded tweets. (Yes, I am thinking about this!)

On the websites under my control I went full dinosaur, and the server sends only non-interactive HTML pages sent to the client, not requiring any client-side activity. I now got rid of the last half-hearted usage of a session object and the respective cookie, and I have never used any social media buttons or other tracking.

So there are no login data or cookies to protect, but yet I finally migrated all sites to HTTPS.

It is a matter of principle: I of all website owners should use https. Since 15 years I have been planning and building Public Key Infrastructures and troubleshooting X.509 certificates.

But of course I fear Google’s verdict: They have announced long ago to HTTPS is considered a positive ranking by its search engine. Pages not using HTTPS will be tagged as insecure using more and more terrifying icons – e.g. http-only pages with login buttons already display a striked-through padlock in Firefox. In the past years I migrated a lot of PKIs from SHA1 to SHA256 to fight the first wave of Insecure icons.

Finally Let’s Encrypt has started a revolution: Free SSL certificates, based on domain validation only. My hosting provider uses a solution based on Let’s Encrypt – using a reverse proxy that does the actual HTTPS. I only had to re-target all my DNS records to the reverse proxy – it would have been very easy would it not have been for all my already existing URL rewriting and tweaking and redirecting. I also wanted to keep the option of still using HTTP in the future for tests and special scenario (like hosting a revocation list), so I decided on redirecting myself in the application(s) instead of using the offered automated redirect. But a code review and clean-up now and then can never hurt 🙂 For large complex sites the migration to HTTPS is anything but easy.

In case I ever forget which domains and host names I use, I just need to check out this list of Subject Alternative Names again:

(And I have another certificate for the ‘test’ host names that I need for testing the sites themselves and also for testing various redirects ;-))

WordPress.com also uses Let’s Encrypt (Automattic is a sponsor), and the SAN elkement.blog is lumped together with several other blog names, allegedly the ones which needed new certificates at about the same time.

It will be interesting what the consequences for phishing websites will be. Malicious websites will look trusted as being issued certificates automatically, but revoking a certificate might provide another method for invalidating a malicious website.

Anyway, special thanks to the WordPress.com Happiness Engineers and support staff at my hosting provider Puaschitz IT. Despite all the nerdiness displayed on this blog I prefer hosted / ‘shared’ solutions when it comes to my own websites because I totally like it when somebody else has to patch the server and deal with attacks. I am an annoying client – with all kinds of special needs and questions – thanks for the great support! 🙂