# Quantum Mechanics, NASA interview, Materials Science to start the year

A layman’s guide to Quantum Mechanics:

https://arstechnica.com/science/2021/01/the-curious-observers-guide-to-quantum-mechanics/

A good interview:

https://www.quantamagazine.org/the-nasa-engineer-whos-a-mathematician-at-heart-20210119/

Materials work on the interface of biology and physics: https://physicsworld.com/a/perovskite-sensor-sees-more-like-the-human-eye/

It’d be good to add a little bit about the math of the pandemic, but I’ll leave that out for now.

# End of Year / Missing posts

I think I lost my November post (possibly didn’t save and keep many browser windows open, so finding it’s tough), so here’s a little change of pace:

https://www.discovermagazine.com/the-sciences/just-how-dark-were-the-dark-ages?utm_source=pocket-newtab

Also to a good 2021 ahead!

Cool things I found relating to condensed matter physics (one focus topic of my PhD — though none of these links relate that closely except for the transition temperature one) recently:

Water

https://phys.org/news/2019-11-critical-transition-liquid.html (related article)

https://phys.org/news/2020-02-everywhereand-weirder.html

https://phys.org/news/2020-10-tetrahedra-uniqueness.html

Other material strangeness

https://phys.org/news/2020-10-room-temperature-superconducting-material.html

https://phys.org/news/2020-10-stacking-graphene-rare-magnetism.html

# A PhD Thesis

Someone close to me finished their thesis in a separate discipline in the last 1.5 years. It’s well-lauded (news came out recently about it), so I thought I would share. It is in a discipline, with which I am not that familiar, so I cannot comment that much on the content, though I had some exposure to related fields nearly 15 years ago, so it is not totally unrelated either, so it is interesting. I have not finished reading through, but I am interesting in the stochastic approach portrayed for debris transport.

# Another quick chapter

Work is still busy, but I still come across a number of articles, which could be of interest to the audience of this blog:
https://phys.org/news/2020-03-mathematicians-theory-real-world-randomness.html

And for those interested in studying datascience, you could have a day and a half free access to datacamp:
https://www.datacamp.com/freeweek

# A few data science learning links

Well, it’s been busy this past month and I haven’t had time for derivations, but I have nonetheless come across some interesting articles concerning my interests relating to this blog (in this case, just data science). I hope it’s helpful:

https://towardsdatascience.com/nvidia-gave-me-a-15k-data-science-workstation-heres-what-i-did-with-it-70cfb069fc35
https://towardsdatascience.com/4-free-maths-courses-to-do-in-quarantine-and-level-up-your-data-science-skills-f815daca56f7
https://towardsdatascience.com/mathematics-for-data-science-e53939ee8306

# COVID-19 and What’s With Reporting about Exponential Increases in Cases?

“Flattening the curve” has become a popular expression nowadays, referring to slowing the spread of the new Corona virus (for a reference, https://www.livescience.com/coronavirus-flatten-the-curve.html).  In contrast to a “flattened-curve” (South Korea), there are plots of exponential growth (most other countries and South Korea in the early stages of the disease):

https://www.vox.com/policy-and-politics/2020/3/13/21178289/confirmed-coronavirus-cases-us-countries-italy-iran-singapore-hong-kong

First, off, why do many plots start at the day when 100 cases were reached?  Before there are a lot of cases, statistically, the spread of the disease can be noisy; that is, say quite a few of those infected early on are socially distant, then the disease might only be transmitted over there few interactions  (there’s the separate issue of testing for the disease, but that can be potentially gotten into at a later point).  Also, the slow incubation period/ slow time to show symptoms could cause noisiness in the plot because in the early stages, people weren’t tested until there was a reason to.  Anyhow, many of the plots in the lower right-hand corner (you are look at each country individually) confirm that the early trend is not as clearly linear as after a many cases have shown up: https://gisanddata.maps.arcgis.com/apps/opsdashboard/index.html#/bda7594740fd40299423467b48e9ecf6 (which I got from here: http://www.cidrap.umn.edu/covid-19/maps-visuals). Also note that the slow incubation period is the reason why social distancing efforts take weeks to be noticed.

To explain the exponential beginnings, we can look at a number of models used to describe the spread of the disease:

https://en.wikipedia.org/wiki/Compartmental_models_in_epidemiology

(Or for the commonly referred to SEIR model: https://sites.me.ucsb.edu/~moehlis/APC514/tutorials/tutorial_seasonal/node4.html )

Let’s look at the early stages (and also assume that once infected, you cannot be infected again, though in the early stages, we can ignore this relation).  Taking  $I$ to be the number of infected and $H$ to be the number of healthy individuals, it is assumed  that the rate of infection is proportional to both these values

$\frac{d I}{d t} = k H I = k I (N - I)$

where the total population is either healthy or infected $N = H + I$ .   Note that even if it is not true that every individual has the same number of contacts, statistically-speaking, the relation often holds.  The solution to the differential equation is the sigmoidal function(for reference: https://www.reddit.com/r/dataisbeautiful/comments/fohr58/oc_the_technical_problems_of_fitting_a_logistic/)

$I(t) = \frac{kN}{k+\exp(kNt-t_0)}= N \frac{\exp(kNt-t_0)}{\exp(kNt-t_0)+1/k}$

For small values ($t << t_0 + \ln (1/k)/kN$ or $I(t=0) = N \frac{\exp(-t_0)}{\exp(-t_0)+1/k} << N$), this curve is exponential:

$I(t) \approx k N \exp(kNt-t_0)$

Another way of seeing this is

$\frac{d I}{d t} \approx k I N$

when $I << N$ and the solution of that equation is a exponential

$I(t) \approx k N \exp(kNt-t_0)$

the same as the above!

As an exercise, you can plot the approximate and exact solutions and see how they differ (when they are the same and when they differ significantly).