And so I put off doing any cleaning of the lab space for months. Even when guests came over I would leave all the lab equipment out hoping that no one would notice. It got to a point where I was just dumping stuff that I got onto my lab table when the amazon order arrived.

And that would usually be fine! BUT until a week ago it spiralled into me not being able to do experiments because I didn't have enough space. Also, it would take me much longer than usual to do experiments because I couldn't find things, since there was so much crap lying around. There was one time that I spent an hour trying to find my digital caliper, which is time pissed down the drain. This was the final red line, and so I finally caved in an decided to do some cleaning.

Here are some quotes from family about the state of the lab

'The state of your apartment makes me have a lump in my throat' - sister.

'How do you live like this' - parents.

'We need to get you your own room for your hobbies’ - my gf.

This is what the lab looked like. There is literally no space to put stuff on the table or for me to sit with a chair. The monitor takes up quite a bit of space, the HEPA laminar flow hood also takes up a significant portion of the desk, and thus I have no place to put my microscope. And I use my microscope a lot so it's a shame that it doesn't have a permanent home. Right now its on a shelf on my bookshelf.

Here's what the whole space looks like - ignore my drying laundry

Even the contents of my shelves were messy

Also the 5GBP plastering board that I put everything on looks like it's about to break. You can even see the bending in the middle. And to fix this problem I put two of my bedside tables underneath it with some books to give some structural integrity.

The first step was to get a desk. So I got ordered 2m long sturdy looking one from IKEA. This doesn't arrive till next week, but I still went ahead and started pre-organising stuff.

I have a metal cabinet shelf. I wanted to put a system to label things in these shelved properly. I also wanted to digitise the contents of what I had in the lab so that an LLM would be able to instantly tell me which experiments were possible or not. And so use this opportunity not just to clean but to do also some logging of all the stuff I had. So for your interest, here's a partial tour of what I have so far

I started to collect the easy stuff first, like glassware, and put them all in a single shelf. So I put the following in this shelf

• test tubes

• victorian style sterilisation lamp

• glass petri dishes

• assorted beakers

• glass stirrer

Then I put all of my probes into another shelf

• pressure manometer for biosecurity experiments

• temperature, CO2, and particle concentration monitor for biosecurity experiments

• ammeter / voltmenter / multimeter

• cooking temperature probes

And then I put everything to do with electrical and circuitry in another shelf. Here I keep

• small electronic sensors, like ultrasound sensors

• breadboards

• cables

• transistors, amps, resistors

• power adapters

• other assorted chips like ESP32 dev board, arduino nano, etc

I put the things that can control temperature in another shelf

• magnetic heater and stirrer - very useful for a range of things that I need to do - highly recommend as a piece of equipment

• candles

• matches

• and randomly my jewellers' scale

And then safety stuff like

• nitrile gloves

• safety goggles in another shelf

I am thinking to have another cupboard for live samples, samples and reagants. For example I have chemicals like

• magnesium chloride

• silver nitrate

• dilute hydrogen peroxide

• caustic soda

• agar jelly

• silicone oil

Along with minerals for my curiosity in origins of life

• periodite

• olivine

• obisidan / volcanic glass

Anyways, the lab is a work in progress, I'll keep you guys updated with the full cleanup once my shelves arrive.

But we should allow for the option for science to be done outside of universities. The Independent Science Society believes that this yields high risk-reward discovery, has convincing historical precedent, and increases empirical skills amongst the public. That’s what this whole blog is trying to prove. But don’t take it from me - I’m just a guy messing around at home in my free time. There are already many cool FROs and BBNs doing cool work outside of academia!

And so people outside of academia should be able to at least ask for grant funding.

So that’s why, on behalf of The Independent Science Society, we are undersigning Science Works’ letter on Reforming UKRI Eligibility. The letter is here. It argues that the eligibility rules make it impossible for new orgs to ask for money, and the main ask is as quoted

First, UKRI should remove the de facto ban on supporting new organisations. It should achieve this by offering a forward-looking route through which new organisations can demonstrate research capability and financial viability. Significant financial support from funders that UKRI already trusts should provide such a route. To that end, UKRI should publish a list of trusted funders, including at a minimum those on the existing list of endorsed funders, and should update it regularly.

Second, UKRI should remove the artificial distinction between standard and non-standard eligibility. The requirements for standard eligibility are practically identical to those for the highest band of non-standard eligibility, yet organisations meeting these materially equivalent criteria are treated differently. This distinction blocks organisations with non-standard eligibility from accessing support, even when they have been deemed to have sufficient research capability and appropriate financial governance. If non-standard eligibility must be kept, UKRI’s Executive Committee should mandate all calls to be open by default to organisations with either type of eligibility.

I encourage you to click and check it out. Here is a screenshot of the beginning, and please read it in full.

And on behalf of the Independent Science Society I’ve undersigned it with the cool folks from Renaissance Philanthropy, Amodo Design, the Living Physics Lab, Cultivarium, the London Institute for Mathematical Sciences, and other really cool people!

Maybe one day home scientists like me would be able to ask for funding for UKRI to do cost effective and high risk experiments in our homes. One model would be some pool of cash where we ask trusted home scientists to de-risk weird phenomena that we can easily watch or verify.

A small step for science at home!

The most versatile thing independent scientists can buy are tools that control and measure temperature. Temperature control challenges have appeared in every experimental endeavour I've done, like making a steady temperature gradient for Rayleigh Bernard cells, or keeping the right temperature for biological cultures.

In fact temperature control is probably the most important experimental control to get right regardless of the field of science. But damn, it's such a pain in the ass that even I myself am tempted to bite the bullet and spend a couple grand on incubators so I don't have to go through such a hard time hacking a temperature control system together. I've already installed split condenser AC in my bedroom - maybe I should extend this to the lab space as well... But that would be against the hacking spirit of the Independent Science Society!

Last time I modified a mini fridge as a heat shield from the heatwaves in England.

Now I'm trying to test cheap water baths that you can buy on Amazon for 65 GBP and see if they work. So I bought this one... and trying to check if it worked basically put me down a rabbit hole of finding the potential noise around boiling points and freezing points...

It arrived next day thanks to the wonders of Amazon Prime. I set the temperature to 60C and then tested using two identical kitchen temperature probes. Both showed 62.2C. Excuse the dark image but if you squint you can see the temps in the LCD.

The temperature flickered around a bit on both probes, but despite setting it to 60C, the probes showed the temperature as centered around 62C with a flickering of +- 1C, despite the claimed 60C in the setting. Even after stirring, and moving the temperature probes around in the water this is what I got.

Again after a few minutes of mixing and measuring, to make sure I wasn’t fucking this up with uneven heating, I get 62C, 62.8C

I then was confused... could it be that my temperature probes wrong or was the Chinese water bath wrong? Also, were there mixing effects what I wasn't taking account? Was I measuring in the right place in the water?!

So, inspired by Lord Kelvin, I wanted to test if my temperature probes would work at the boiling point of water. Tap water should boil at 100C - although atmospheric pressure, altitude effects, and mineral effects could push this off. But some rough googling convinced me that such effects weren't going to exceed error 1C off a boiling point of 100C. And so, this was a way to check if the kitchen temperature probes were working correctly.

So I dipped the probes in boiling water to see if they would measure 100C. The first one:

The second one:

And then I tried to see if the probes would match an ice water slurry so I got a bunch of ice, waited a bit for it to melt and tried to see if I could get the probes to show 0C:

Not exactly perfect, and this seems like a fiddlier way to test temperature probes.

But after this exercise I was convinced that the temperature probes were working correctly and that the water bath just had a systemic error of 2C which I could adjust for. That's fine. And then two weeks later I set it again to 60C on the setting and measured it again:

62.8C, 62.5C. So I think it’s fine!

So yeah, it looks like the water bath does what it says, as long as you add +2C and have an error of +- 1C !

Previously I demonstrated Rayleigh Bernard convection cells with Mica powder and oil, where I saw cell-like devisions in liquid when I applied a temperature gradient. This was an example of non-equilibrium , self organising phenomena due to the application of a temperature gradient. I've preciously tried and failed to do other things like the Briggs Rauscher reaction, which is a (safer) example of an oscillating chemical reaction.

With regards to complex phenomena, I am trying to figure out which variables actually matter through both math and physical experiments, which is hard. In the case of convection cells, we have variables like the thickness of the liquid layer, the temperature gradient, the size of the container, amongst other things. And these ideas are something I want to get around to understanding myself in this lab.

Another piece of complex phenomena that caught my eye is Liesegang rings. These are rings that form in a gel according to a spacing that goes like (1 + p)^n, which is a geometric series. You can get them through a bunch of different waysHere is what they look like, images taken from this really cool website on Liesegang ring models.

You can see quite clearly the spacing going in a geometric pattern in this one with magnesium hydroxide.

Why do I find this interesting? When reading books about embryology, like the Triumph of The Embryo, I got interested in how chemical reactions give rise to patterning in some animals. I also find that the theory of patterning in chemical reactions is mathematically rich as well, something which I'm trying to understand more through the lens on non-equilibrium statistical mechanics. The aesthetics of these things are just mesmerising. Even though liesegang rings aren't the mechanism behind animal patterning, you still can't help but marvel at what goes on behind the scenes.

Here's a page out of 'A Field Guide To the Birds of Malaysia'.

And to the best of my knowledge, the mechanisms behind Liesegang rings haven't really been fully resolved yet. Which puts this in prime territory to try and replicate some of this stuff at home.

I had a think about trying to replicate Liesegang rings using Potassium dichromate and silver nitrate, which seems to yield the brightest rings. But when I was shopping for potassium dichromate on sigma aldrich and then realised that it was carcinogenic, with inhalation risk. So not really that appropriate for homelab style stuff...

And then the other example on the Liesegang website was using a gel soak of magnesium hydrochloride with ammonia, but I also found the ammonia quite sketchy since the concentrations looked high.

So then I was looking around with Gpt Sol to try and find some safer versions, along with reagants that were more available. So then I came across a paper by Chopard in 1999 which managed to create Liesgang rings out of agar jelly (safe), magnesium chloride (bath salts) and sodium hydroxide. Sodium hydroxide is called caustic soda and can be bought from stores. It's corrosive but doesn't have fumes and I previously used it trying to make quantum dots at home.

Here are photos of the Liesegang rings from the Chopard 1999 paper

I tried the following recipe to make the rings with NaOH, shown in the image on the right.

The reagents were pretty cheap as well, here is th e complete list with the estimated costs from Claude. Again to stress, most of these available pretty readily out there! I would really like to make chemical network research more accessible to home experimentation,

I got Sol to tell me the concentrations and procedure to replicate the paper.

Here is my first attempt. And spoiler alert, it's a fail and I couldn't get the bands to separate properly for some reason. But I think it's just as important to share failures along with successes in the spirit of being a transparent independent scientist so here goes...

Embarassingly the fiddliest part of all this was trying to get the agar jelly nice and smooth so that the gel wouldn't be cloudy, so that I could see the banding.

I first tried to mix 1.5g of agar powder with 90ml and then put it in my magnetic stirrer and heater, but I only managed to get the agar to a temperature of 60C, which isn't enough to fully dissolve the agar powder - you need to get it to near boiling for it to dissolve properly.

It was much quicker and easier to get the agar to 90C temperature by putting it in a microwave for 40s at 900W. You can get agar from any old food shop or on amazon. I used a basic cooking temperature probe to test that it got to temperature.

Putting it into the microwave for 40s managed to get it up to the right temperature for dissolution - here it's 94C.

I then kept the agar warm on my magnetic stirrer at 60C.

And then after that I prepared 2g of Magnesium Chloride Hexahydrate in 98ml of water and then put it in the agar jelly

Stirring it and getting the bubbles out

And then I put it in some test tubes and let it cool, in tubes of 20g each

I then let the MgCl soaked tubes rest for 90 minutes until it became a gel.

After that I put 1g of NaOH in 50ml of water solution, this is corrosive so be careful

And then i poured the NaOH solution into the gel soak.

And then I did this with 3 test tubes and then left one alone.

I left it for 10 hours and here's what I got. As you can see there is a white precipitate interface, which is promising. But I don't see the band structures that I so desperately want to see!

References

1. B. Chopard, M. Droz, J. Magnin, Z. Rácz, and M. Zrinyi, "Liesegang Patterns: Effect of Dissociation of the Invading Electrolyte," J. Phys. Chem. A 103 (10), 1432–1436 (1999). ACS · preprint arXiv:cond-mat/9809385.

2. Liesegang ring experiments and models, insilico.hu.

3. L. Wolpert, The Triumph of the Embryo (Oxford University Press, 1991).

4. Lim Kim Seng, Lim Kim Chuah, and Yong Ding Li (illus. Dana Gardner), A Field Guide to the Birds of Malaysia, including Sabah and Sarawak (John Beaufoy Publishing). NHBS.

My main method of attack given these new developments in AI has to try and see if there are any conjectures that can be disproven or proven through elementary methods that I am actually able to understand. And then I would go about then parsing the generated results by hand, or with tools like the ones my friend George Smith has made. I am hoping for both results and something of pedagogical value to me. Such records of parsing seem to be what other people are doing, like Terence Tao's recent 'digestions' of AI generated results.

I was slightly emboldened by my progress that I made on a conjecture in random matrices last week (I possibly resolved a conjecture). On that project, I emailed the solution to the authors who then replied saying that the solution looks correct. (More on this later)

The downside of this approach of constructing understandable counterexamples is that my current level and knowledge of math is the limiting factor, and so proofs that I force AIs to generate with methods known to me are likely to be long winded and probably don't have some big insight. Also, it seems less aesthetic than trying to come up with a motivated, constructive theory to resolve something, which is something that I've struggled to get AI to do. If I wanted to solve a problem with AI assistance through theory building, which is how most of current research mathematics seems to be done, then the use of AI doesn't really seem appropriate for someone with my aims. I suspect that I would get lost instantly due to complexity. Also, counterexamples only resolve a conjecture when it's false - there are many conjectures which are likely true and it's unclear how an independent mathematician could contribute a convincing result of something outside of the counter-example type stuff that we see AI doing.

As a companion side project, I am also trying to parse the book on non-equilibrium process by De Groot and Mazur, specifically on chemical reactions. On the experimental side I am still trying to find more non-linear phenomena that you can observe safely at home, like the Briggs Rauscher reaction or Liesegang rings. I failed in my last attempt at Briggs Rauscher which I will write up soon.

Anyways, back to my rough initial thoughts on the paper.

Consider the system of processes that drive our circadian rhythms. They oscillate with a noisy period of twenty-four hours. There seem to be many such biological processes that oscillate (albeit with noise). A biochemical clock is a system in which the concentrations of the chemicals involved oscillate at a near-constant period. Such clocks vary in mechanism. Some involve gene regulation; others require no gene regulation at all and can be assembled purely from biochemicals in a test tube. The paper cites one such protein system [2].

We expect noise in biological environments to affect biochemical oscillators, so it seems reasonable that the oscillations decay in time. The paper uses a measure called the number of coherent oscillations [3] to quantify how long they last. I won't go into too much detail on exactly what this means in this post - the reader can check the references.

The main result of Oberreiter, Seifert and Barato is a conjecture that bounds from below the entropy produced by a biochemical oscillator, where the bound is a function of the number of coherent oscillations. Bounds of these form our cool because it might be able to let us constrain the energetics of some biological processes - an approach that Nick Lane talks about in his book The Vital Question [4]. In the early chapters he also makes a nod to the dissipative structures work by Prigogine.

Here is a screenshot of the conjecture:

To the best of my knowledge, this conjecture remains unproven. Right now I am attempting to get GPT Sol to solve it - it has made a counter example and I am trying to check it rigorously. I am emboldened somewhat by my last submission of a counterexample in random matrix theory, which the original authors said "looks correct" in an email eight days after I sent it to them. (I am not really sure what to do with it, though; it feels like publishing such a thing through formal channels would be a waste of time.)

The preliminary counterexample that Sol came up with is here. GPT Sol first reframes the problem in terms of the eigenvalues of the system and then tries to massage a network to fulfil the inequality without changing the entropy.

This counterexample looks correct at first glance. I'm still trying to digest it by going through the proof manually. On the surface it looks like Sol constructs a bilayer network, one of which is in equilibrium, and one which isn't, such that the slow eigenvalues can be massaged without moving the entropy that much.

Right now, if this example was correct, it would indeed be pretty contrived. It would be cool to find other such counterexamples with the aim of finding a biochemical clock that is below this bound.

In other news, I am finding learning statistical mechanics this way to be a lot more fun than trying to parse textbooks. I have four textbooks right now but I've been struggling to find a mental 'hook' to get this stuff into my brain! We might be able to motivate younger students more by trying to mass-find low hanging conjectures for them to chew on, or to find higher quality digestions of.

Other books I am trying to read

• Kondepudi, D. and Prigogine, I. (2015). Modern Thermodynamics: From Heat Engines to Dissipative Structures, 2nd ed. Wiley. ISBN 978-1-118-37181-7. [Wiley]

• de Groot, S. R. and Mazur, P. (1984). Non-Equilibrium Thermodynamics. Dover (reprint of North-Holland, 1962). ISBN 978-0-486-64741-8. [Dover]

• Livi, R. and Politi, P. (2017). Nonequilibrium Statistical Physics: A Modern Perspective. Cambridge University Press. ISBN 978-1-107-04954-3. [CUP]

References

[1] Oberreiter, L., Seifert, U. and Barato, A. C. (2022). Universal minimal cost of coherent biochemical oscillations. Physical Review E 106, 014106. arXiv:2112.01607. https://arxiv.org/abs/2112.01607

[2] Nakajima, M., Imai, K., Ito, H., Nishiwaki, T., Murayama, Y., Iwasaki, H., Oyama, T. and Kondo, T. (2005). Reconstitution of circadian oscillation of cyanobacterial KaiC phosphorylation in vitro. Science 308, 414–415. https://doi.org/10.1126/science.1108451

[3] Barkai, N. and Leibler, S. (2000). Circadian clocks limited by noise. Nature 403, 267–268. https://doi.org/10.1038/35002258

[4] Lane, N. (2015). The Vital Question: Energy, Evolution, and the Origins of Complex Life. Profile Books / W. W. Norton. ISBN 978-1-78125-036-5.

So I asked GPT Sol Ultra to hunt for some conjectures with resolutions that someone with my background could understand and check myself. Disclosure - I’ve studied theoretical physics up to Master’s level at Cambridge, and the math that I read usually has a statistical physics flavour. I also finished my fourth year with a score of 98/100 (though it was covid and so exams were easier).

One result that it found was a disproof of a conjecture on the limiting behaviour of a matrix model, with a fairly simple proof. The original conjecture is conjecture 2 from the following paper:

Masoud Khalkhali, Nathan Pagliaroli, Andrei Parfeni, and Brayden Smith, “Bootstrapping the critical behavior of multi-matrix models,” Journal of High Energy Physics 02 (2025), 158. DOI · arXiv:2409.07565

The resolution is here, which resolves the conjecture falsely. The full proof is below as well at the bottom of this post. The part which made me feel something was the fact that this conjecture used math that was basic enough for me to grasp it, and I have found the subject matter (field theory, random matrix theory) interesting previously.

I am spending some time to try and check if I got it right. My friend George Smith works on tools for math education and I’m trying to use his generated problems to gain a better intuition of what GPT Sol actually did.

I am well aware of being wrong, so I have emailed the authors of this paper to check to see if it makes sense. Formalising it feels to heavy as a tool right now, so I’ll stick to algebra tools. I will disclose in a later post if I have found that I have made mistakes.

Here are some notes on what I’m looking for when I try to make AI assisted proofs

• Find a problem that I can understand.

  • This feels really important, what’s the point of doing research math if you don’t get the joy of trying to understand something new?

    • If you choose problems that you understand you will be more likely to somewhat be invested in the solution and also be more likely to take it in a useful place.

    • The concepts you learn are more likely to mentally hook to something you care about which might make some cool second order effects.

    • If you choose a problem you understand you will feel good when its solved, which I rate highly.

    • A lot of results in maths aren’t applicable to real life, the search space is huge, just chill.

    • Somehow I feel more drawn to math that has some empiricism in scope, or a model of the world that is not obvious.

• If you understand more math you are more likely to have an expanded universe of problems that feel significant (and can hopefully solve with AI assistance)

  • This means that it is still worth trying to understand math.

  • It feels more likely to me that one part of the future of math is trying to digest results made by AI and trying to see if they have wider significance.

  • I still think people want to understand things, I still find understanding things fun and knowing how things work is fun. I don’t think it matters if something is already known, there are infinitely many presentations of knowledge that can be valuable in different ways.

  • I still think there is scope for a much needed, much better thought out way of communicating and storing such results.

Acknowledgements

Thanks to George Smith for the discussions. All mistakes are mine.

Proof

For transparency, the full chat transcript is here.

A lot of my time is spent buying equipment for experiments, a uniquely painful task that all indie lab owners have to do. This is hard and tedious. I also spend too much time trying to optimise prices. Equipment buying is particularly tricky for electrical components required in experiments, like different resistors, transistors, and custom parts that you wouldn’t be able to get from a set. Sometimes I need to select forty different individual components from a website catalog, without error, to do one experiment.

Reducing friction of this process is likely to increase the productivity of independent scientists greatly.

One solution is to try automate online shopping for experiment equipment. Electrical component manufacturers don’t have APIs where I can send order requests. Nor does Amazon or online stores in general. To get around this, I'm using the beta version of Claude Code Desktop's browser feature, which allows Claude Code to interact with a browser in-app, just as if it was a person clicking around the website.

One experiment on my shelf that I've avoided doing because of the pain of getting all the components is "A simple, sensitive circuit to measure Boltzmann's constant from Johnson's noise," Eur. J. Phys. 31 (2010) 675–679 by J. C. Rodríguez-Luna and J. de Urquijo.

So I decided to just give this paper to Claude Code Desktop and let it rip, and asked it put all the equipment I need in the shopping cart.

In short, Claude Code operates the browser autonomously and does the following:

• It read the paper

• Made a list of stuff that I needed (I am not sure if this is correct though, I haven't pulled the trigger), with recommendations for the more expensive items like an oscilloscope

• Adapted it to a breadboard version that I wanted

• Then searched for the items on the online store of Farnell's (an electrical component retailer)

• Made an xlsx matching the store's bulk upload format

• Found the codes of the things I needed and added it to the xlsx

• Uploaded the xlsx

• Then added everything to cart

Some cool things

• It replaced the outdated components (since the paper is old) with the modern version

• It seemed to optimise cost

• Made a widget list so I could follow what's happening

There were two dangerous mistakes

• It mistakenly added qty = 5000 for two components, making the total shop 500GBP

• It mistakenly added parts that had to be ordered from the US which added cost. These were replaceable by parts on the website from the UK (after I prompted it to do so)

And a few annoying things

• It’s slow

• I had to prompt it quite a lot, needed handholding

• I had to delete things manually from the basket if it admitted it made a mistake

• Responses near the end on some questions I had on being able to use audio interfaces to measure voltage variance was just too confusing.

Overall I'm happy with it because even with the mistakes it still cut down a lot of time. Here are some screenshots of what went on (next time I’ll do a screen capture). Claude Code is operating the browser autonomously.

Here it uploads the bulk requirements…

Here its checking if the parts exist….

This is what was in the basket…

The final result we want is a list of non-trivial, phenomena-rich, and novel experiments that might be possible to do in a home setting with some effort.

If we do this we hope to find interesting technical barriers to physical experimentation that can be reduced. We also hope that we can train ourselves to become better empiricists in a world that is becoming increasingly uncertain. Another useful side effect is home empiricism being a channel for theorists to start experimenting, so that the theory-experiment gap is bridged, and that problem selection by theorists is more aligned.

Overall though, it's just fun!

We hope that the experiments below have at least one of the following properties:

• historical significance to humanity, so that they're worth replicating even if they've been done a thousand times

• interesting phenomena that still aren't fully fleshed out

• scope for improvement in experimental technique, such that people could do them at home

• the weird property of being 'table-toppy', as written about in On Reduction to the Table Top, and have a chance of being 'scaled down'

The below is a list of some experiments that I myself I've been thinking about to contribute to the list. So inevitably it is slightly tilted to my interests. Nothing here is original: neither the Society nor I take any credit for the ideas (although I am responsible for what I've written here), and the ideas here are a brainstorm from many ongoing discussions with cool people.

Physics

• Reconstructing Galileo's classic experiments and seeing if they can be done in any reasonable time in a classroom session

  • The classic isochronism experiment of a ball moving down a ramp and a spiral

  • Observing the craters on the Moon in the style of the old telescopes made by the Dutch during the 17th century

  • Dropping two things in a vacuum tube and seeing if they fall at the same rate

    • Corollary — make a much cheaper vacuum tube than the ones you can buy, and then see when the air resistance approximations become legit

• A rehash of the mysteries around the Crookes radiometer

  • A replication of the Crooke's radiometer in a full vacuum

  • I still think this hasn't been fully resolved — is there a more cohesive explanation?

• Foucault's pendulum

  • Currently, a Foucault's pendulum needs to be 67m long for people to see the effect. Is a desktop Foucault's pendulum possible? If not, why not?

• Investigating the boiling point of water under different conditions

  • Can you consistently boil water at the same temperature? (See Hasok Chang's Inventing Temperature)

  • Recreating Lord Kelvin's early experiments in trying to fix the boiling point of water

• More macro-scale demonstrations of thermal physics

  • For example, a really easy way to measure the temperature change of a hot and a cold liquid meeting and then coming to equilibrium

  • A gas piston which shows the Carnot cycle more clearly

• Macro-scale demonstrations of emergent organised phenomena

  • Replicating Rayleigh-Bénard convection cell experiments and the critical temperature at which they occur — predicting the number of cells that form under different conditions

  • A replication of the original Couette flow experiment

    • How do we make a Taylor-Couette flow cell much more cheaply?

  • Whirlpools from flux

• A macro-scale demonstration of non-equilibrium statistical mechanics, like the Onsager effect

  • Welding together a thermocouple to make a current from temperature differences

  • Making an artificially noisy system and then using it to test physics that is impacted by noise / thermal effects

• A stricter replication of Brown's original observations of Brownian motion

• Electrical conductivity in salt water — how does it change as we vary the concentration?

Astronomy

• Observing craters on the Moon like Galileo did

• Amateur astrophotography is already a huge field, but cameras are expensive - is there a way to collect astronomical data that is a lot cheaper?

Chemistry

• Can we find safer oscillating chemical reactions to study at home? Some ideas might be

  • Belousov-Zhabotinsky reactions without the annoying bromine part

  • Briggs-Rauscher with lower concentrations of hydrogen peroxide, since high-concentration hydrogen peroxide is hazardous

• What would be a minimal, non-trivial chemical reaction network to try and measure?

Biology and Ecology

• Beginner-ish safe genetic engineering tasks, like

  • Staining tobacco plant leaves using genetic engineering

  • Making yeast fluorescent using genetic engineering

• Measuring photosynthesis rates from CO2 concentrations with a home bioreactor

• Controlling the environment of algae through light pulses

• Some experiments on closed communities and ecologies

• Physical mechanisms of cell splitting (L-form bacteria)

• Something to demonstrate the Lotka-Volterra equations, but with soil microbiomes / two competing organisms / data found by ecologists

• A thermophoresis experiment to show the organic clumping that could've happened in an alkaline vent

Meta

• A big database of experiment videos, so that procedures are much clearer

• A big database of the things above

• A way to systematically trawl the web for experiments that fit the scaling criteria in On Reduction to the Table Top

Theory

• Extending Dyson's theory of life models to make them non-equilibrium

• Making statistical physics much more explainable, since the terminology and concepts still are really really hard

• A Lean formalisation of the recent, very difficult proofs of Boltzmann's equation by Yu Deng et al.

• Tools that somehow capture the tacit knowledge required to use simulation

Experimental methods

• A repeatable and cheap way to control temperature at home in hot environments / heatwave flats

• Safety guidelines for those doing chemistry at home

• Safer purification methods for getting species out of soil

Biosecurity

• I previously made a big list of biosecurity projects in A Hundred Project Ideas in Biosecurity, it would be cool if anyone wanted to pick these up again

If you had a list of experiments you wanted to add, feel free to reach out on the society discord. Also feel free to reach out if there is something here you'd like to dig into! I’m trying to get better at describing experiments, so if you try any of these, I’d love to know what information was missing from my description that would have helped you.

Acknowledgements

I would like to thank David Jordan, David Pfau, Arina Berezovsky, Ben Trettel, and Isabelle Zane for discussions about many things related here. All mistakes are mine, views are my own.

The ‘good science’ part means finding or replicating natural phenomena that we don’t currently know how to model. This is the job of all scientists, regardless of whether they are in labs or houses. And my current approach to this is just talking to people (which I need to do more of), reading papers, and writing.

But the at home part requires extra steps.

First, I need to figure out if the essential parts of the phenomena can be captured on a kitchen desk. This means that I need to think carefully about if the essential parts scale down appropriately when you reduce the size by making a toy model. By scale down, I mean this in all aspects of the word. I was recently made aware of the mathematical theory of scale physics and dimensional analysis. But by scale down, I also mean reducing models to low N, reducing the volume of a environment, keeping stuff constant, and in general just removing big-ness in all aspects.

For certain problems like the biology of microbes or electron behaviour in materials, this is not a problem because the thing being studied is small to begin with, and you don’t need to reduce it. If I wanted to study the movement of genetically engineered E. Coli, you could engineer it, grab a microscope and record it.

For problems in ecology, it might possible because I could build a biome in a jar, put in some food, and put in some microbes. But I’d need to think carefully about the relative number of the microbes versus the abundance of food. If I wanted to make a toy ecology with two microbes in a jar, then I would need to think about which phenomena in two species systems would reasonably scale up to an ecology with a large number of species, or if any interesting effects could get washed out.

And for something like fluid turbulence, it’s not obvious if turbulence that occurs when you shoot air at a toy plane scales up to a real plane. And then for really complex phenomena, like the climate of the whole earth, this feels like an impossible problem.

The second part of the problem is doing it cheaply.

For example, if you wanted to genetically engineer E-Coli, you would need to store and order plasmids, order a bunch of PCR machines, do sequencing, and more. Which would rack up in costs. But the benefit is that I’d be studying something without having to scale it down. Similar with condensed matter physics - I could try to characterise the band structure of some materials which also is expensive.

Conversely, whilst a toy plane and home made fluid tunnel is cheaper, the translation to the original scaled up science of big planes is less obvious.

One way to get around the expensiveness problem is through ‘replacements’. For example, I’m learning about thermophoresis in Nick Lane’s ‘The Vital Question’. Lane argues that the pores in alkaline vents, along with temperature gradients, cause a clumping of organic material that is necessary for early life. His lab has done demonstrations of such things, which seem fairly expensive to do. I wonder if there are any really cheap replacements like sand and fluorescent material that still captures the essential physics. Sure, it wouldn’t be a faithful replication, but there might be some phenomena their that might be inspiration for something interesting.

So there seems to be a set of systematic criteria which would quantify the types of interesting problems I could work on at home, where perhaps I should rank each problem based on the criteria as follows:

• Is the original phenomena I am trying to study currently uncertain in the scientific community?

• Does the phenomena scale down neatly to table top size?

• Is it cheap to do, and if not, are there replacements that can be done that still capture the essential physics?

Acknowledgements

Thanks to Ben Trettel for making me aware of the field scale physics and modelling! All mistakes and views are mine.

References

Lane, N. (2015). The Vital Question: Why is Life the Way it Is? Profile Books (UK) / W. W. Norton (US). — on thermophoresis and thermal gradients in alkaline hydrothermal vents concentrating organics.

One point of an experiment is to make natural phenomena happen repeatably.

Another point is to test models of the world, by seeing what happens when the experimenter changes a variable. In this case, it is important to only change one thing, and keep constant everything else that could effect the experiment. This allows you to be more confident the thing you are varying is actually causing the effect that you want to measure.

It's not possible to control everything though. What makes things harder is that some factors have varying importance. Some things are less important to control, like the physical location of where you do the experiment. And some things are more important to control, like the humidity of your biological samples.

And so it seems to be the job of the scientist to decide what exactly needs to be controlled. But this is hard because a-priori there needs to be some science done to figure out what actually matters. Also, some systems have a really large sensitivity to initial conditions, and may be chaotic. This is a philosophical problem that I want to write more about.

But anyways. If you are dealing with biology experiments, one thing that is definitely important to control is temperature, because biochemical processes might change as temperature varies. Which means that the experiment would become invalid otherwise.

But figuring out a way to keep the temperature constant is hard. Especially for independent scientists that are working out of their rooms and have less access to controlled environments. This is a real engineering challenge. But given the heatwave in the UK, it’s important to solve it. Here is an attempt to do so.

I'm trying to do experiments with algae in the bioreactor that David Jordan me, so I needed a way to keep the temperature constant. Whilst the bioreactor could do some degree of temperature regulation, it failed above 28C because it was just too hot.

So last week, I got this mini fridge (Subcold Pro 15L Mini Fridge) and tried to figure out a way to use it to ‘keep the temperature of the environment’ constant, using the fridge as an insulating and cooling layer.

Making sure it fits:

The obvious solution is to stick just stick bioreactor in a fridge. But the bioreactor has wires and requires power. Soto do that I needed to drill some holes in the side. I am not a DIY guy so I asked claude to recommend me a drill, and appropriate drill bit such that I could make a 10mm clean hole safely without cracking the plastic of the fridge. I found a corded drill from Screwfix to be the cheapest, along with a conical shaped drill bit for variable hole size.

Here is me drilling:

Hole came out pretty clean!

Also, the fridge only does a ambient minus 18C cooling, which means that if its 30C then it will cool to 12C, but if its 24C it will cool to 6C. So to get it to constant temperature I used an Inkbird, which is basically an automated plug that only turns on the fridge once it gets above a set temperature point, which you can set. Useful. So I put the ink-bird temperature probe in the fridge, attached the fridge plug to the ink-bird.

This solution worked pretty well, the system fluctuates between 23C and 24.5C. This fluctuation is fine because the bioreactor has its own feedback loop. The main thing I really wanted was a heat shield so that I could do experiments in the summer even if my living room is at 30C.

Parts list

• Subcold Pro 15L Mini Fridge — the enclosure (already owned). Thermoelectric; runs ~18 °C below ambient. product page

• Subcold 12 V AC replacement adapter — mains power lead for the fridge (bought 2026-07-25, £14.99). Must be the 12 V version, not the 9 V Classic one. Subcold cables

• Inkbird ITC-308 — dual-stage plug-in temperature controller with two switched sockets (cooling + heating); holds a fixed setpoint (algae target ~24 °C). UK-plug version. ~£30–35.

• Small heater (optional, for the ITC-308 heating socket) — e.g. reptile heat mat or low-watt tube heater, so the setpoint still holds when the room is cooler than target.

And whilst entropy itself may not be measurable with a thermometer or ruler, it is still a useful concept since we can calculate derived quantities from it that are directly measurable.

I've written previously quite vaguely about 'life as entropy'. This was an idea motivated by Schrödinger (1944)² through the concept of negentropy. Through negentropy, life seems to maintain order by feeding on the energy around it, and reducing its local disorder. But up until now, I've been confused about what this actually means in detail.

And so one way I'm trying to understand this is through models. One approach might be to make toy models and then figure out a mathematically consistent way to define entropy in those systems. Then, you could simulate the model and have a better idea of how the entropy evolves.

One such toy model that is useful to get an intuition is Dyson's toy model of a cell. Dyson's toy model of a cell is a Markov chain that settles into one of three equilibrium states, two of which are the 'life' and 'death' states. But if we care about the vague concept of 'life as entropy', then we should be able to make a definition of entropy to apply to the Markov chain.

But the concept of entropy as originally defined by Clausius was a function of work and temperature

And so it is unclear how to relate the two ideas together. Is there a different way to define entropy in the context of Markov chains that gives you the same result, consistent with physical ideas?

Well, one way to attack this is by looking at the work of Boltzmann, who quantified the relationship between entropy and the number of possible states in a system.

Suppose you take a system and then observe a system's state variables, like temperature, pressure and volume, using various instruments. Presumably, there are a number of different configurations of the physical system that are compatible with it being in that state. And if entropy is related to the concept of order and disorder, then presumably entropy would be higher if there were more possible states.

Here is a vague example. If you have a gas of identical molecules at absolute zero, fixed in position, with each at zero velocity at zero temperature, you have fewer possibilities for what the system could be for the gas to be in that state. Whereas if we had a heated gas at a hot temperature, there are many configurations that a state might be in as a result.

Temperature is higher, entropy is higher. Is it any coincidence that the number of possible states increases too?

It turns out there is a way to make these ideas rigorous. Boltzmann showed that entropy can be written as a function of the number of states that a system can be in, if we fixed what the macro state variables were. That means if we could count all the states that were possible in a system at fixed temperature, pressure and volume, then we could calculate the entropy.

This is written in the legendary equation below, which is also surprisingly simple — that the entropy of a system is the logarithm of the number of states, multiplied by a fixed constant called the Boltzmann constant.

Whilst the result is simple it's completely non-obvious. Boltzmann creates a model of gas where each particle only can be in a set of finite velocities and positions and then tries to take the limit. The proof is not easy and mathematicians are still figuring out how to properly rigourise the work of Boltzmann.

But anyway!

To make this example even more concrete, let's try to compute the entropy of a basic system, given we know what the energy (macrostate) is. Consider Curie's model of a magnet. In this model, our physical space consists of a finite number of electrons, which can either be spin up or spin down. In models of magnets, energy (or work) is done when the spins are misaligned with the direction of the magnetic field that the metal is placed in.

And so this suggests that if we had a collection of atoms, the energy is a function of how many atoms are aligned with spin up vs spin down.

Ok, so suppose we had a very, very small system of 5 atoms. And suppose that we measured the energy as E = 1. First, we need to figure out how many atoms need to be spin up and spin down to have this configuration. The obvious answer is three spin up and two spin down because

We could also trivially solve this using the constraint that

n↑+n↓=5, so

And the configurations we could have are the following. Each configuration has three up spins and two down spins, but you can arrange this in 10 ways. The diagram below lists all of the possibilities.

This is 5 choose 3 combinations, which is 10.

Those ten configurations are the entire content of the macrostate "E = 1".

This means that Boltzmann would label the entropy of the system as

or log₂ 10 = 3.32 bits. And so in this case we can see the entropy as a function of energy. If we plot out entropy as a function of the energies we get this graph:

So now we've learnt how to compute entropy from the total number of microstates that a system can have. Ok, so how do we count the entropy of a Markov chain process like in Dyson's cell model?

In Dyson’s toy model that we have N sites, and each site could be in one of three states — empty, active, or inactive. Given some rules, in my last post I showed that this converges to an equilibrium. The equilibrium values are a complicated function of fixed points, but they do exist, and the probability of being empty, inactive or active does converge to a fixed value.

This equilibrium point allows us to define the entropy of a Markov chain through a counting argument. Say we had an equilibrium case where we had a 1/2 chance that a state is empty, a 1/4 chance that a state is active, and a 1/4 chance that a state is inactive.

This means if we had 8 sites, then we would have 4 that were empty, 2 that were active, and 2 that were inactive. So to measure the entropy, we now need to count all of the configurations of this happening. This is shown below, with all of the arrangements.

Counting them is the multinomial coefficient — choose which 4 of the 8 sites are empty, then which 2 of the remaining 4 are active, and the last 2 are inactive. This can be calculated by taking the factorial of the number of spaces there are, and dividing it by the factorials of the number in each distinct state. So in this case, since we have 4 empty, 2 active and 2 inactive, we divide by 96.

And so the entropy of this is the logarithm of the number of states multiplied by the Boltzmann constant.

The next natural question is how the entropy evolves in the system. It turns out we can make some cool general statements about entropy based on the topology of the graph, which I'll explain in a later post, and also write down the conditions for which entropy increases.

Thanks to David Pfau for discussions and help, all mistakes are mine.

References

¹ R. Clausius, Ueber verschiedene für die Anwendung bequeme Formen der Hauptgleichungen der mechanischen Wärmetheorie, Annalen der Physik 201(7), 353–400 (1865).

² E. Schrödinger, What is Life? The Physical Aspect of the Living Cell, Cambridge University Press (1944), ch. 6.

The Independent Science Society had a dinner last night. As part of figuring out how to do excellent science ourselves, we discussed the philosophy of writing science well, which Arina and I motivated in this essay.

We discuss the philosophy of writing because the world is confusing. We have LLMs, more science, more text, and more questions about what counts as good empiricism. And so how are we meant to use language to deal with these issues? There are also ethical questions about what it means to write science safely, which is something I was personally curious about going into the dinner.

To structure the dinner, we printed a list of question prompts along with the essay before hand, and then placed columns of text at each person's table. And then Arina moderated the discussion in a 'single thread format' where only one person spoke at a time, after some initial prompt. Each person got a notepad to take notes. And credit to Arina, people found that this format worked very well. We also had Chatham house rules so people could speak freely without being attributed to their comments.

Here are some pictures of the event below. A full write up to follow!

Here were the question prompts. Often times the discussions did stray away from these prompts though, but that’s fine.

Why?

• What is the point of writing science?

  • What is the job of the scientist?

  • What is good science writing?

    • Should we care at all about writing good science - whatever that is?

    • If so how frequently do we need to write?

  • Is there a point in writing science ourselves given LLMs exist now?

    • Is there a way to get past the human vs LLM competition and still be marginally useful?

  • Is the human act of writing science still worthwhile?

  • Do we expect the purpose of scientific writing today to change?

    • Will scientific writing soon become purely a pedagogical act?

    • If AI provides all the answers then shouldn't we be optimising for human understanding, like Terence Tao's recent digestions on the Jacobian conjecture counter examples.

• How many people need to read a piece of science for it to become common knowledge?

  • Does it matter who these people are?

  • Does the buck stop at experiment replication?

Credibility

• Should we be optimising for legitimacy or the appearance of credibility in any way? And if so why, and to whom?

  • Will scientific experts matter any more given LLMs?

• Are there any credibility structures that still remain that get in the way of communicating the science we do?

Form and Medium

• Is the written word optimal for communicating science?

  • Is the traditional scientific paper optimal for communicating science?

  • Does the written word have enough information density?

  • If not, why not videos / audio / pictures?

    • Or why not everything at once?

    • In practical terms how do we host the optimal form of communication on servers?

• What are the optimal ways for feedback and replication?

Polished-ness And Thought Privacy

• How much of our thinking process should we expose to the public?

• How much can authors change their mind without damaging credibility?

• How good does something have to be before it is written?

• Should government funded scientists write more and more often?

• How do we trade off the time spent writing versus actually doing the thing?

Clarity And Detail

• Clarity and detail are opposing things - how do we trade them off?

• What does it mean for writing to be clear?

  • How do we write clearly?

  • How can we tell if an expression of idea becomes too complex?

  • How much exactness do we need for someone to be able to replicate our experiment to a reasonable level of sameness?

  • What about style?

• Does scientific writing need to be clear given AI?

• Who are we trying to be clear for?

• How do we trade off clarity for time?

• Is clarity audience dependent?

Ethics

• How do we think about writing science that could be used for nefarious purposes?

  • A question in particular for the bio writers out there!

But here's something I noticed when doing this many times. When I need to remote into a cluster, the first part is to log in. I need to first login to the University VPN, which means using some outdated software to get the VPN set up from my computer. And then there is the authentication part, where the VPN software makes you jump through multiple authentication and 2FA processes to log in. And then when the 2FA window finally comes, I need to dig out my phone from somewhere in my room to get the code. Probably on average I find that the VPN + ssh + 2FA process takes 30 seconds every time I need to do something, which generally feels like too much.

It's not too much work overall, but there are a lot of intermediate steps and riff-raff.

I call this clunkiness. Something is clunky when there are many intermediate steps required to do one task. You know those really annoying shower controls in fancy hotels when you need to adjust two separate knobs for the pressure and temperature separately? That's clunky.

It’s clunky to do work in University compute clusters.

A bit of clunkiness in using something is usually fine. But if the clunkiness is even slightly more than a bit, the effects can be disastrous. If there are too many intermediate steps to get started to work on something, I become less motivated to work on it, forget about it, and in the worst case scenario just drop the endeavour entirely. Which means that an entire stream of work can be lost for small and trivial reasons. This happens with a lot of things! For example, it's really clunky to descale my coffee machine, and so I never end up doing it, and so it always breaks.

The friction in using these clusters has been enough to actually stop me from doing work that I would've done otherwise. If I was a full-time PhD student I would probably muscle through each awkward step to login. But I'm not, I work on this independently and am short of time. I am also mentally tired since I do this after work. And these things make it such that access to the compute needs to be instant and no-faff for me to justify getting started. This is something independent scientists try to optimise, which I wrote about in one of my older posts.

In terms of background, many individuals did research whilst they were in university, but had to give it up as they started their jobs. There were also individuals who worked as engineers in top tech companies, science adjacent companies, or start-up founders, who already had high level experience in project management and research. Some had already submitted papers! Most tried to do research in their free time out of pure passion, and chose day jobs in industry to support financial and lifestyle goals.

Source: What I Learned from Speaking to 500 Independent Scientists

There is also a layer of friction between doing the computations and then analysing the logs that comp chemistry programs spit out. If you want to visualise orbitals or look at the outputs of your simulations, it's really awkward to run Python whilst being in the cluster, and the cluster has no GUI display. The way I used to do this was by doing SCP and copying the log files back and forth. Efficient I know... I used to use MacFUSE with sshfs but it turned out to be really unreliable as well, and quite slow. I wrote about the procedure a few years ago.

If you're like me and a beginner on working on remote HPC clusters with Slurm for computational physics / chemistry, one thing that I naively used to do was

- write python scripts, input files and logic on my local computer, using IDEs and tools on my laptop, and then

- ssh / rsync them to the remote cluster, and then

- do the computation, and then

- ssh or rsync them back.

But this is kind of annoying for small projects when you just start out. Especially when you end up writing a bunch of boiler plate code to sync files back and forth - leading to more and more lines of code. And guess what - I spent too much time on that code!

*So instead, I realised that I can use macFUSE with sshfs to mount the remote directory via to my local, so that I can just treat it like a directory as any other. There are some fiddly bits like changing your security on permissions on mac M1, but I found the tutorial pretty easy to follow.*

Source: MacOS sshfs

Most computational chemistry programs output a ridiculous text log file without a proper API. Something that would save the death by a thousand cuts nature of this problem is a python wrapper or parser.

Source: Tools to parse log files easier

And then you have all of the other annoying things when you work on something in a remote cluster. For example, you need to port over custom scripts and hotkeys that you have locally. You might have latency when copying files. It's harder to organise a remote shell and your local stuff all at once. It truly is a nightmare.

And then there is the biggest blocker for most people - they don't have access to a university compute cluster. I'm lucky enough to actually be working with a university that lets me use theirs, but most people don't have this opportunity.

One idea to get over the difficulties of doing computational chemistry on a cluster is to do everything locally. There are two problems with doing this though. The first problem is that installing and configuring comp chemistry software is really hard and complicated. And then the second problem is the obvious one - often times locally there isn't enough compute available on a computer to do tasks in any reasonable time.

But with Claude Code, it's become quite easy to install things! A few weeks ago I asked Claude to install OpenMolcas on my Mac and run some tests and those seemed to pass. And as for compute, I also realised that there are many small tasks that can actually be run on my machine:

For example, I can do a basic SCF optimisation for a small diamide that I'm working with (19 atoms), and it seems to work.

The results look to be the same.

Same input, same geometry, one core each.

Across all 144 Stage-2 runs on the cluster: min 187 s, median 190 s, mean 191 s, max 212 s. Against the Mac's 82 s, the median cluster run is 2.31× slower.

The Mac does this with LINALG=Internal (the bundled reference BLAS), OpenMP off, MPI off — the slowest linear algebra available to it.

Here's what they look like.

Suppose that Bob, in 50 A.D, observes that his oranges fall to the ground every time he cuts them from a branch of his orange tree. He then grows another orange tree to see what happens when he cuts them, and lo and behold, they also fall to the ground. Bob thinks for a while and makes a theory that ALL oranges, once cut from trees, fall to the ground.

Now, Bob tells Alice of his findings through the village publisher. Alice tries to verify this theory by growing her own orange tree. She cuts a branch. And lo and behold, the orange falls to the ground.

Has Alice successfully replicated Bob’s experiment, even though she used a different orange tree? Even if she possibly grew it with different soil? What if Alice’s orange fruit fell down at a different speed, or were a different species?

Is Bob’s theory more plausible as a result of Alice’s replication?

I dunno…

Furthermore, suppose Catherine from the other side of the village starts waxing lyrical about all sour fruits falling to the floor. And then Donald decides to grow a lemon tree and finds that his fruits also fall to the floor when he cuts them. Has Donald validated Catherines general theory of citrus fruits?

Is the general theory of sour fruit falling thingies more valid?!!

I also dunno…

Scientists do experiments. From the result of these experiments, they make general statements about the world called theories. One way to check whether these theories hold any weight is to replicate their experiments.

But what is the goal of replication?

Supposedly the goal of a replication it is to build confidence that a theory actually describes the world to some degree of (rough) precision.

But if you take this as the goal, then it’s actually unclear how strict a replication needs to be to still be useful to accomplish this goal. A true replication of an experiment would be to do something with exactly the same conditions, but this is impossible, because different labs are in different locations and times.

And so if we can give up on the impossible goal of doing exact replications, how do we define the boundaries of what experiments count as valid replications of a theory? It feels like the possibilities are wide. On one hand, you could try replicate everything about an experiment except for its time and place. And on the other hand you could just do what Alice did and try your best to capture the essential points.

It’s unclear what is the best way!

In this blog I’ve done experiments related to theories or phenomena that I’ve suspected to be true.

Some of this has been stuff in the physical world that I can see or measure myself. For example, while I haven’t tried to replicate strict papers on UV killing yeast, I’ve done experiments that demonstrate the same effect. Does this count as a valid piece of science?

I’ve also tried to replicate and measure the claimed air cleaning rates of air purifiers. Does this count? It might not count as a classic academic replication, since I probably used different room dimensions than the original experimenters. But is it still useful to the goal of building confidence that air cleaners are useful? Maybe.

Other replication attempts have been more theoretical. Right now I’ve been working with a university to check an improved method for spectroscopy prediction, and this project involves doing calculations with different approaches that should still converge to the same result in the limit. Is this exercise of value at all?

And on the other end I’ve done really CHILL attempts at observing phenomena, like my experiment in heating up oil and mica powder to observe Rayleigh-Benard convection. Okay, some guy on the internet has filmed Benard convention for the thousandth time… what the marginal contribution of that?!

Right now I’m just a random dude doing experiments in my house, recording it and writing about it.

On the surface, it looks unserious, in a conventional sense.

But if you think about it, it’s really unclear what counts as meaningful scientific process in general. I hope that the independent scientists amongst us think about this more clearly and take advantage of it to do unconventional things of value.

In my last piece on Dyson's toy model of a cell, we modelled a cell as a Markov chain where each monomer is one of three possible states - active, inactive, and empty. In that post, ψ(x) is meant to be the probability factor that increases when the number of active monomers increases.

I explained that in Dyson's model, the fraction of active monomers settles in equilibrium points where

where x is the fraction of monomers that are active, and q and n are the (positive) statistical weights of the empty and inactive states respectively, and ψ(x) is the probability factor that determines the likelihood of an empty monomer becoming active.

In this post I wanted to think out loud on some of the math around this equilibrium condition. Dyson claims that ψ(x) should take the form:

It's a little unclear to me why this should be the case. In a different post I'll clarify why this makes sense. But for now, let's take ψ(x) as given. Dyson then rewrites it as

through the substitutions ψ(x) = bˣ and a = q + n, giving

Both parameters are positive. For ψ(x) = bˣ to increase with the active fraction — the autocatalytic assumption — we need b > 1. And a = q + n > 0, since q and n are positive.

And this reduces the problem to figuring out the roots of φ(x) = x — the points where φ crosses the diagonal.

Now, we have some questions.

• Is it always true that we have three equilibrium points?

• Could we have two or one equilibrium points, or zero?

• If x starts near any of these equilibrium points, how can we tell if one is stable or not?

• What are the sets of a and b that correspond to the different numbers of equilibrium points?

• What is the derivative of φ(x) at the equilibrium points?

• Can we bound the size of the derivatives at each equilibrium point?

Let's first say a few things about this function. It's bounded between 0 and 1. It's continuous. And it's differentiable infinitely many times. It's also monotonic.

To get the number of equilibrium points in the theory of life system, we want to find the number of intersections that this function makes with the y = x line. In other words, we want to find the number of roots that satisfies the equation

This is a tricky problem and should not be taken for granted! It's not guaranteed that we have three equilibrium points all the time.

a = 20, b = 400

By changing the parameters, we can squish the fixed points around. For example, we can make the central and right fixed point go quite far to the right:

But by just playing around with the parameters, we find that not only can we get three equilibrium points, we can also get two. If we set the following parameters to a = 9, b = 91, then we get

We can also get one equilibrium point

a = 9, b = 130

Can we get zero equilibrium points? To do this, we would need to show that the following equation has no roots:

We can answer whether this is true with the the intermediate value theorem. The theorem says that if g is continuous on [0, 1] and g(0) and g(1) have opposite signs, then there is at least one c ∈ (0, 1) with g(c) = 0. This is a diagram of this result is:

Apply it to g(x) = φ(x) - x, which looks like

But note that

and also that

and that g is continuous, since φ is.

Since g(0) > 0 and g(1) < 0, the intermediate value theorem guarantees at least one root in (0, 1). So we can never get zero roots — there is always at least one equilibrium point.

How many roots as a function of *a* and *b*?

Indeed, it's possible that φ(x) has between one and three roots. The function φ(x) is a function of a and b. So the next question is what values of a and b correspond to how many equilibrium points there are in the system?

Dyson puts a diagram of how many roots there are based on the values of a and b here.

Region tour

Here is a tour of what the equilibrium points look like as we sweep across values a and b. The plot on the left shows the φ(x) curve intersecting with y, and the plots on the right show where we are for a given value of a and b. Dyson labels the region where we only have an alive state as the Garden of Eden. And similarly we have a region where we only have one dead state.

a = 5

a = 9, b ∈ [76, 91])

a = 16 (wider, b ∈ [174, 1078])

a = 30 (very tall band, b ∈ [396, 1.8×10⁵])

So the next question is - what values of a and b give three equilibrium points? What is the functional form of those black lines here?

But there's a problem with growing stuff at home. The UK is going through multiple heatwaves, so I can’t get the temperature constant for my experiments.

And whilst I have AC in my bedroom, I don't have it in my workbench in my living room. And to do any reasonable experiments with living things, you need to control the temperature.

For algae, I want to keep it at 24C since they are sensitive creatures.

And even though the bioreactor has Peltier elements for heating with feedback, the cooling doesn't work well enough when the ambient temperature in my house is 30C.

So one solution that David suggested is to get a mini fridge, stuff the bioreactor inside, and then wire in the cables through drilling holes through the front side.

So I got this fridge (Subcold Pro 15L Mini Fridge) (not an endorsement)

And here's what it looks like for scale. Gonna go to Screwfix this weekend to get some drills. Stay tuned!

Acknowledgements

Thank you to David Jordan from Living Physics for the suggestion on the fridge and the support.

In my last post I tried to replicate the physical relationships found from Brownian motion (Improved Tracking Of Brownian Milk Globules). I explained that the diffusion relationship tells us how far something is meant to travel in a heat bath as time goes on.

• ⟨Δr²⟩ = mean squared displacement - the squared distance something travels

• d = dimensions — in this case we are approximately in 2 dimensions.

• D = diffusion coefficient

• k_B = Boltzmann constant

• T = temperature — room temperature.

• η = viscosity — water at room temperature, η ≈ 1.0 × 10⁻³ Pa·s (1.0 mPa·s at 20 °C)

• a = globule radius.

The diffusion constant is a function of the radius of the particle in water. But last time I didn't measure the radius of the milk globules that I used. So in this post I'll go through an attempt at doing that. As usual, it's not that straightforward!

So to estimate the radius of the globule I bought something called a hemocytometer which is originally used in medicine to count the density of cells in a sample. The grid spacing in between is fifty micrometres by fifty micrometres (0.05 mm × 0.05 mm, area 0.0025 mm²). Given that estimates of milk globules I read online were from around 2 to 20 micrometers, I thought this would be a decent start.

So once this arrived I diluted milk with one part milk and 10 parts water, and then pipetted it onto the cell counting slide.

Then I put on the cover slip on top.

It's a bit fiddly, but when you put the coverslip on top, the sample of milk spreads out.

Then I used my microscope which I wrote about here. At the first level of magnification you can see the white line spacers that are meant to show increments of 50um each, along with the dark spots that are the globules.

And then magnifying to 40x we can see the size of the globules a bit better. One of the problems is that it's hard to get both the lines and globules focused at the same time, as you can see below in this gif.

Here are some pictures at varying levels of focus:

So to measure the globules I got Claude to write a program that matches the distance between the white lines, to the number of pixels in the image.

So at this scale from grid: 326 px = 50 µm, which means we have 0.153 µm/px. And then using some image recognition in Numpy I could get the diameters of the globules.

So in terms of data I found that the clean measurements gave me a set of:

• N = 36 globules

• median = 3.4 µm, mean = 3.2 µm, std = 0.8 µm

• interquartile range 2.7–3.7 µm, full range ~1.8–5.2 µm

Ok, so that makes sense.

But here are some issues. The size measurement is janky because there are multiple depths, shown in the image below, and I'm not sure how to count these properly. How do I know that there isn’t some depth variation as a result of the thickness of the slide, especially with stuff so small like milk?!

Cell density measurement is a little trickier. There are a lot of globules to count. A lot of the globules are faded and those are really hard to pick up just with image software. I have some ideas on how to fix this but that’s for a different time.

Also, since I don't want to buy a cell counting slide every time I do an experiment, I had to clean the counting slide. To clean it, I just poured mineral water from Tesco over it in the sink. Then I used a glasses lens cleaner and gently dabbed it. The result after cleaning is in the picture below, you can still see some scratches!

In the future I want to measure some other cell sizes, like algae and yeast. Stay tuned!

Rip the goat.