Actually everyone so far who has given a definition for the second law of thermodynamics has done so accurately and correctly. They appear in different froms depending on historical context and subject to the mathematical representation available at the time. The most common and accurate from used today is the last one given by musik_man. (Btw, there is a dash over the d is dQ, because the integral is path specific)
The laws of thermodynamics actually do extermely poorly for living beings and systems. The main reason is that thermodynamics assumes the the system is in thermal equilibrium. Most living objects are constanly changing mass, generating heat, and doing stuff (for a lack what a better everyday activity). The law as never meant be used on living things.
In particular, as to what I know, it makes no statement about what is possible or impossble in evolution.
As for the statement
'radiometric dating is a statistical science...'
This is correct. Basically one quantifies the amount of radioactive material left by it's half life -- the average time it takes to decay from one form to another, emmiting a radioactive partical in the process. The process is quantum mechanical in nature thus is basically a random process.
'that assumes in every measurement that no physical process has distributed the individual atomic nuclea in some 'non-random' manner...'
This is also correct. In order to take a measurement, we have to assume some starting point for initial quantities of radioactive material. I will also talk about the basic case. In practice, there maybe some corrections that can be made to get more accuracy. This is how it works.
We know that radioactive carbon-14 is produced in nature at a constant rate, and a very small amount of this stuff is in everything -- food, air, desolved in water. For living organisms, because we eat, drink and breath and grow, the amount of carbon-14 in us is about same as that in the environment.
Once we die, we stop taking in carbon. If the body is preserved somehow, the carbon slowly over thousands of years decays into nitrogen. So in the simplest case by looking at the ratio of carbon to nitrogen, we can tell when the fossil was preserved. We know this is relatively accurate, because we can test this against stuff that we do have dates for, and using other techniques
On rereading the statment, it think that it sounds slightly odd. The words 'distributed the individual atomic nuclea in a 'non-random' manner' are really out of place. In my opinion, it should have read something like 'that assumes in every measurement that there is no physical process which biases the initial distribution of radioactive atomic species'.
'...radiogenic dating neglects the history of the individual atoms before they congealed to solid rocks. all of the processes that give rise to elements generate them at temperatures that exceed their ionization temperatures by many magnitudes. before these elements became bound up in rocks, they existed as plasma in interstellar space....'
Basically he is saying that chemistry in the process of rock formation may affect radioactive decay rates (halflives)
I need to read the rest of the article to tell you if this is plausible or not. i.e. to be sure that it is not just competely wrong because of some schoolboy error. It's not the history of the individual atoms that is important, it is basically the constancy or at least predictibility of the environmental state pertaining to the radioactive element used for dating that is important.
This is because for radioactive dating , it really does not matter how each individual atom behaves, or even the chemistry it undergoes. This is because chemistry has to do with electrons, whereas radioactive dating depends on neutrons decaying into protons, which is basically determined by quantum mechanical barriers of the quarks. It should have nothing to do with the chemistry.
https://en.wikipedia.org/wiki/Beta_particle#.CE.B2.E2.88.92_decay_.28electron_emission.29But like i say, I need to read the complete article and find out if he is talking about special cases where it might (for some reason that I don't know). I will also need to cross referece his claims to other articles in the area.
I think you might find this page interesting, there is loads of science to plough though -- I've only scanned through but believe that it is accurate. However, i think the most interesting bits are in the last couple of pages.
https://www.asa3.org/aSA/resources/Wiens.html
Topic: pianistimo-fading out?