In classical mechanics you want to understand the motion of all constituents of a system in detail. The trajectory of each ‘particle’ can be calculated from the forces between them and initial positions and velocities. In statistical mechanics you try to work out what can still be said about a system even though – or…

# Category: Physics

I am thinking out loud about well-known mathematical physics. Re-learning an re-viving the basics again and again.

## Integrating The Delta Function (Again and Again) – Penrose Version

I quoted Nobel prize winner Paul Dirac’s book, now I will quote this year’s physics Nobel prize winner Roger Penrose. In his book The Road to Reality Penrose discusses not-so-well-behaved functions like the Delta Function: They belong in the category of Hyperfunctions. A Hyperfunction is the difference of two complex functions: Each of the complex…

## The RSA Algorithm

You want this: Encrypt a message to somebody else – using information that is publicly available. Somebody else should then be able to decrypt the message, using only information they have; nobody else should be able to read this information. The public key cryptography algorithm RSA does achieve this. This article is my way of…

## Integrating the Delta Function (Again) – Dirac Version

The Delta Function is, roughly speaking, shaped like an infinitely tall and infinitely thin needle. It’s discovery – or invention – is commonly attributed to Paul Dirac[*]. Dirac needed a function like this to work with integrals that are common on quantum mechanics, a generalization of a matrix that has 1’s in the diagonal and…

## Delta Function Haiku

I have proved that a Lorentzian bell curve becomes the Dirac Delta Function in the limit. Now I want to look at another representation of the Delta Function. As this is a shorter proof, a haiku will do. ~ Infinite numbers of oscillations added. Need to damp them down Symmetrically attach an exponential for each…

## The Improper Function and the Poetry of Proofs

Later the Delta Function was named after their founder. Dirac himself called it an improper function. This time, the poem is not from repurposed snippets of his prose. These are just my own words to describe a proof: ~ In the limit the Lorentzian becomes the improper function. In the limit of tiny epsilons it…

## Heat Conduction Cheat Sheet

I am dumping some equations here I need now and then! The sections about 3-dimensional temperature waves summarize what is described at length in the second part of this post. Temperature waves are interesting for simulating yearly and daily oscillations in the temperature below the surface of the earth or near wall/floor of our ice/water…

## Can the Efficiency Be Greater Than One?

This is one of the perennial top search terms for this blog. Anticlimactic answer: Yes, because input and output are determined also by economics, not only by physics. Often readers search for the efficiency of a refrigerator. Its efficiency, the ratio of output and input energies, is greater than 1 because the ambient energy is…

## Consequences of the Second Law of Thermodynamics

Why a Carnot process using a Van der Waals gas – or other fluid with uncommon equation of state – also runs at Carnot’s efficiency. Textbooks often refer to an ideal gas when introducing Carnot’s cycle – it’s easy to calculate heat energies and work in this case. Perhaps this might imply that not only must the…

## The Heat Source Paradox

It is not a paradox – it is a straight-forward relation between a heat pump system’s key data: The lower a heat pump’s performance factor is, the smaller the source can be built. I would not write this post, hadn’t I found a version of this statement with a positive twist used in an advert!…

## Entropy and Dimensions (Following Landau and Lifshitz)

Some time ago I wrote about volumes of spheres in multi-dimensional phase space – as needed in integrals in statistical mechanics. The post was primarily about the curious fact that the ‘bulk of the volume’ of such spheres is contained in a thin shell beneath their hyperspherical surfaces. The trick to calculate something reasonable is…

## The Collector Size Paradox

Collector harvest does not change much if we only use half the collector. Perhaps counter-intuitive but explained by the characteristics of heat exchangers connected in series: Especially if one of the heat exchangers (in the tank or collector) is already much superior to the other one, it does not help to optimize the good one further.

## 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…

## 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…

## 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…

## You Never Know

… when obscure knowledge comes in handy! You can dismantle an old gutter without efforts, and without any special tools: Just by gently setting it into twisted motion, effectively applying ~1Hz torsion waves that would lead to fatigue break within a few minutes. I knew my stint in steel research in the 1990s would finally…

## 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…

## Mr. Bubble Was Confused. A Cliffhanger.

This year we experienced a record-breaking January in Austria – the coldest since 30 years. Our heat pump system produced 14m3 of ice in the underground tank. The volume of ice is measured by Mr. Bubble, the winner of The Ultimate Level Sensor Casting Show run by the Chief Engineer last year: The classic, analog…

## Ice Storage Hierarchy of Needs

Data Kraken – the tentacled tangled pieces of software for data analysis – has a secret theoretical sibling, an older one: Before we built our heat source from a cellar, I developed numerical simulations of the future heat pump system. Today this simulation tool comprises e.g. a model of our control system, real-live weather data,…

## On Photovoltaic Generators and Scattering Cross Sections

Subtitle: Dimensional Analysis again. Our photovoltaic generator has about 5 kW rated ‘peak’ power – 18 panels with 265W each. Peak output power is obtained under so-called standard testing condition – 1 kWp (kilo Watt peak) is equivalent to: a panel temperature of 25°C (as efficiency depends on temperature) an incident angle of sunlight relative to…

## 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…

## And Now for Something Completely Different: Rotation Heat Pump!

Heat pumps for space heating are all very similar: Refrigerant evaporates, pressure is increased by a scroll compressor, refrigerant condenses, pressure is reduced in an expansion value. *yawn* The question is: Can a compression heat pump be built in a completely different way? Austrian start-up ECOP did it: They invented the so-called Rotation Heat Pump….

## Re-Visiting Carnot’s Theorem

The proof by contradiction used in physics textbooks is one of those arguments that appear surprising, then self-evident, then deceptive in its simplicity. You – or maybe only: I – cannot resist turning it over and over in your head again, viewing it from different angles. tl;dr: I just wanted to introduce the time-honored tradition…

## Alien Energy

I am sure it protects us not only from lightning but also from alien attacks and EMP guns … So I wrote about our lightning protection, installed together with our photovoltaic generator. Now our PV generator is operational for 11 months and we have encountered one alien attack, albeit by beneficial aliens. The Sunny Baseline…

## 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…

## Temperature Waves and Geothermal Energy

Nearly all of renewable energy exploited today is, in a sense, solar energy. Photovoltaic cells convert solar radiation into electricity, solar thermal collectors heat hot water. Plants need solar power for photosynthesis, for ‘creating biomass’. The motion of water and air is influenced by the forces caused by the earth’s rotation, but by temperature gradients…

## Peter von Rittinger’s Steam Pump (AKA: The First Heat Pump)

Peter von Rittinger’s biography reads like a Victorian novel, and his invention was a text-book example of innovation triggered by scarcity. Born 1811, he was poor and became an orphan early. Yet he was able to study mathematics and physics as his secondary education had been financed by the Piarist Order. He also studied law…

## Heat Pump System Data: Three Seasons 2012 – 2015

We have updated the documentation of monthly and seasonal measurement data – now including also the full season September 2014 to August 2015. The overall Seasonal Performance Factor was 4,4 – despite the slightly lower numbers in February and March, when was the solar collector was off during the Ice Storage Challenge. Edit: I have…

## Lest We Forget the Pioneer: Ottokar Tumlirz and His Early Demo of the Coriolis Effect

Two years ago I wrote an article about The Myth of the Toilet Flush, comparing the angular rotation caused by the earth’s rotation to the typical rotation in experiments with garden hoses that make it easy to observe the Coriolis effect. There are several orders of magnitude in difference, and the effect can only be…

## An Efficiency Greater Than 1?

No, my next project is not building a Perpetuum Mobile. Sometimes I mull upon definitions of performance indicators. It seems straight-forward that the efficiency of a wood log or oil burner is smaller than 1 – if combustion is not perfect you will never be able to turn the caloric value into heat, due to…