Monthly Archives: March 2024

Einstein’s theory of diffusion in liquids, and my extension.

In 1905 and 1908, Einstein developed two formulations for the diffusion of a small particle in a liquid. As a side-benefit of the first derivation, he demonstrated the visible existence of molecules, a remarkable piece of work. In the second formulation, he derived the same result using non-equilibrium thermodynamics, something he seems to have developed on the spot. I’ll give a brief version of the second derivation, and will then I’ll show off my own extension. It’s one of my proudest intellectual achievements.

But first a little background to the problem. In 1827, a plant biologist, Robert Brown examined pollen under a microscope and noticed that it moved in a jerky manner. He gave this “Brownian motion” the obvious explanation: that the pollen was alive and swimming. Later, it was observed that the pollen moved faster in acetone. The obvious explanation: pollen doesn’t like acetone, and thus swims faster. But the pollen never stopped, and it was noticed that cigar smoke also swam. Was cigar smoke alive too?

Einstein’s first version of an answer, 1905, was to consider that the liquid was composed of atoms whose energy was a Boltzmann distribution with an average of E= kT in every direction where k is the Boltzmann constant, and k = R/N. That is Boltsman’s constant equals the gas constant, R, divided by Avogadro’s number, N. He was able to show that the many interactions with the molecules should cause the pollen to take a random, jerky walk as seen, and that the velocity should be faster the less viscous the solvent, or the smaller the length-scale of observation. Einstein applied the Stokes drag equation to the solute, the drag force per particle was f = -6πrvη where r is the radius of the solute particle, v is the velocity, and η is the solution viscosity. Using some math, he was able to show that the diffusivity of the solute should be D = kT/6πrη. This is called the Stokes-Einstein equation.

In 1908 a French physicist, Jean Baptiste Perrin confirmed Einstein’s predictions, winning the Nobel prize for his work. I will now show the 1908 Einstein derivation and will hope to get to my extension by the end of this post.

Consider the molar Gibbs free energy of a solvent, water say. The molar concentration of water is x and that of a very dilute solute is y. y<<1. For this nearly pure water, you can show that µ = µ° +RT ln x= µ° +RT ln (1-y) = µ° -RTy.

Now, take a derivative with respect to some linear direction, z. Normally this is considered illegal, since thermodynamic is normally understood to apply to equilibrium systems only. Still Einstein took the derivative, and claimed it was legitimate at nearly equilibrium, pseudo-equilibrium. You can calculate the force on the solvent, the force on the water generated by a concentration gradient, Fw = dµ/dz = -RT dy/dz.

Now the force on each atom of water equals -RT/N dy/dz = -kT dy/dz.

Now, let’s call f the force on each atom of solute. For dilute solutions, this force is far higher than the above, f = -kT/y dy/dz. That is, for a given concentration gradient, dy/dz, the force on each solute atom is higher than on each solvent atom in inverse proportion to the molar concentration.

For small spheres, and low velocities, the flow is laminar and the drag force, f = 6πrvη.

Now calculate the speed of each solute atom. It is proportional to the force on the atom by the same relationship as appeared above: f = 6πrvη or v = f/6πrη. Inserting our equation for f= -kT/y dy/dz, we find that the velocity of the average solute molecule,

v = -kT/6πrηy dy/dz.

Let’s say that the molar concentration of solvent is C, so that, for water, C will equal about 1/18 mols/cc. The atomic concentration of dilute solvent will then equal Cy. We find that the molar flux of material, the diffusive flux equals Cyv, or that

Molar flux (mols/cm2/s) = Cy (-kT/6πrηy dy/dz) = -kTC/6πrη dy/dz -kT/6πrη dCy/dz.

where Cy is the molar concentration of solvent per volume.

Classical engineering comes to a similar equation with a property called diffusivity. Sp that

Molar flux of y (mols y/cm2/s) = -D dCy/dz, and D is an experimentally determined constant. We thus now have a prediction for D:

D = kT/6πrη.

This again is the Stokes Einstein Equation, the same as above but derived with far less math. I was fascinated, but felt sure there was something wrong here. Macroscopic viscosity was not the same as microscopic. I just could not think of a great case where there was much difference until I realized that, in polymer solutions there was a big difference.

Polymer solutions, I reasoned had large viscosities, but a diffusing solute probably didn’t feel the liquid as anywhere near as viscous. The viscometer measured at a larger distance, more similar to that of the polymer coil entanglement length, while a small solute might dart between the polymer chains like a rabbit among trees. I applied an equation for heat transfer in a dispersion that JK Maxwell had derived,

where κeff is the modified effective thermal conductivity (or diffusivity in my case), κl and κp are the thermal conductivity of the liquid and the particles respectively, and φ is the volume fraction of particles. 

To convert this to diffusion, I replaced κl by Dl, and κp by Dp where

Dl = kT/6πrηl

and Dp = kT/6πrη.

In the above ηl is the viscosity of the pure, liquid solvent.

The chair of the department, Don Anderson didn’t believe my equation, but agreed to help test it. A student named Kit Yam ran experiments on a variety of polymer solutions, and it turned out that the equation worked really well down to high polymer concentrations, and high viscosity.

As a simple, first approximation to the above, you can take Dp = 0, since it’s much smaller than Dl and you can take Dl to equal Dl = kT/6πrηl as above. The new, first order approximation is:

D = kT/6πrηl (1 – 3φ/2).

We published in Science. That is I published along with the two colleagues who tested the idea and proved the theory right, or at least useful. The reference is Yam, K., Anderson, D., Buxbaum, R. E., Science 240 (1988) p. 330 ff. “Diffusion of Small Solutes in Polymer-Containing Solutions”. This result is one of my proudest achievements.

R.E. Buxbaum, March 20, 2024

Defending against deadly attacks on Jews.

There have been many attacks on Jewish schools, homes , and markets. The press likes to blame white supremicists. But in the US, Islamicists and “Black Hebrews” have been the more regular assailants. Along with them are equal opportunity killers — those who kill, for no obvious reason. I note that mostly attackers don’t wear body armor, suggesting that a small revolver is the best choice for defense. The police come, but never in time.

The Monsey, NY, 2019 attack is fairly typical of a small-scale hate crime, though it was not charged as such. A member of the “Black Hebrew” movement who had attacked Jews in the. past, always released by police, waled into a Channuka celebration in a home in Monsey, NY, pulled a large knife, and stabbed the rabbi and four others before being chased out by folks with chairs. One of those stabbed died from the wounds, and several others spent time in hospital. The attacker, undeterred, drove attack another Jewish establishment, a nearby orthodox shul, and attacked there. It seems he’d committed an anti-Jewish stabbing shortly before this murder, but was released as always before the final, deadly attack. As with most black on Jewish attacks, this was not ruled a hate crime by the police.

Kessler before the attack. The claim is that his flag triggered an accidental attack by Professor Alnaji and his compatriot.

In the US Islamic on Jewish attacks tend to be ruled as accidents or legitimate expressions, and never as hate crimes. In Thousand Oaks California, 2023, Paul Kessler 69 was standing with an Israeli flag (right) when two Islamic activists crossed the street to shout at him. One of them, Professor Loay Abdelfattah Alnaji, hit him fatally on the head with a bull horn. The police ruled it accidental, involuntary manslaughter, despite that it was two on one, deliberate, premeditated, and the assailant kept yelling: “stop killing our children,” even after Kessler was down after being hit. Alnaji is free on bail of $50K. It was not ruled a hate crime.

Poway synagog shooter, Shot four, killed one before gun jammed.

The court reacts quite differently to white on Jewish crimes, ruling these hate crimes and punishing to the full extent of the law. An example, in Poway, CA, 2019, a white man, left, entered the Orthodox, Chabad synagog during services carrying a semi-automatic pistol. He shot and killed the first person he met, then shot the rabbi, entered a side room, and shot two more, an adult and an 8 year old. Then his gun jammed. At that point he left, and called 911. He claimed he hated Jews, Moslems, and President Trump. I note that gun jams are common in stressful situations, but police showing up in time is uncommon. A revolver for personal defense would’ve helped, but they are mostly illegal in California — not that the antigun laws deterred the killer.

Organized attacks are more deadly, and almost impossible to defend against. They tend to be Islamic. The recent attack on a music festival in Israel, for example. An air – land assault with machine guns by an armed group civilians (and UN workers!) that left 1500 dead, and 250 captured. Most of the victims were unarmed, but some were armed. They were over-run, and killed. It is very hard to defend against multiple assailants with training and the advantage of surprise.

A smaller-scale versions of these military stile Islamic attacks have play out regularly around the world. For example, Mumbai, 2019, two Islamic activists entered an orthodox Jewish hostel and school, and barricaded themselves in. Over the course of three days, they killed the rabbi and his wife, and five of their children. It was part of a wider program of well-planned attacks on Jews and Jewish businesses in India. The two perpetrators were eventually killed by the police, but the support network escaped justice. These are the folks who planned the attack, and armed the two; IMHO they are as guilty as the murderers.

The shooter who attacked the Hyper Kasher kosher store in Paris. He was trained, but worked alone, and wears no bulletproof vest. First he shot the person nearest to him and those behind the counter — anyone who might reasonably stop him. He then closed the metal grate around the store, started talking and killing for 4 hours. A well timed shot or two could have taken him out.

In Paris, as a similar Islamic general attack on Jews and businesses included the killing of 12 at the humor magazine “Charlie Hebdot” A trained Islamic activist entered a kosher market, “hypercasher” with two Kalashnikov AK47s provided by the same network who armed the Charlie Hebdot killers. Ownership of most guns is illegal in France, but that makes for easy targets. On entering, he immediately killed the person next to him and shot the two people behind the counter (one died). He then asked that the store be sealed by its steel gratings so he could keep on killing in peace. Secure in the market, the attacker then asked if he should kill someone else. When every shouted no, he laughed and killed the person. The killer talked and killed for the next 4 hours while the police gathered outside and watched. One unarmed customer tried to attack him, but was killed in the process, and jeered at besides — jeers seem to be common. Eventually, the French police killed the attacker and rescued those still alive. As with the Indian attack, the support network escaped or were found non-guilty. If someone had a pistol, maybe the killing would have ended quicker.

White supmemicist, right killed 11 in Pittsburgh. Survivor, center picture will testify. From the NY Post.

In Pittsburgh, PA, 2018, a “White supremicist” entered the “tree of life synagogue” with four semi-automatic pistols (three of them Glocks). He killed 11, going from room to room, sometimes talking to people. One survivor hid under the sink for hours, unable to reach his phone in deadly fear that it would ring and expose him. Eventually the killer just left, and as he did, someone with a gun shot after him, missing. Clearly, this fellow had that gun all along but was afraid to draw it, or could not find it. I’m glad he missed, by the way. If he’d hit the guy as he left, the shooter would have gone to jail. According to US law, you can’t shoot a fleeing attacker. My lesson is that you want a gun that’s small enough to hide well and draw easily, and you want to practice enough to be comfortable using it.

Another deadly attack from “Black Hebrews”, this time organized, military stile. In Jersey City, 2019, two “Black Hebrews” attacked the patrons of an orthodox, Kosher market, starting to shoot from the street, from 50 feet away. Once they were sure that no one inside was armed, they entered and killed three individuals who were doing their best to hide. The recent Gaza attacks used this military style, too. They attacked from a distance first to drive folks into hiding, then set the buildings afire or shot cowering individuals point blank. it’s very hard to defend against this sort of attack, especially if you are unarmed, but even if you are armed and trained.

Enhanced photo from the shooting at the Jersey City Kosher market. This is a rare example of military tactics being used. Two attackers of the “Black Hebrews” started shooting from outside the store, and only entered later to finish up.

The majority of other deadly attacks are by “Islamic youths” against older Jews. The youths will enter a house, threaten, kill, and leave. In one case the victim (a professor) was beheaded on the main street. He’d shown cartoons to his class that suggested that Islam is not peaceful. As with beatings that go with “Palestine Independence” rallies, these attacks are not considered “hate crimes,” but teen violence or political expression.

Hate crimes or not, they mostly target Jews, and they seem to be religiously motivated. Typically, it’s only one or two assailants attacking a chosen, visibly orthodox individual or place. Killing is mostly in close quarters over a relatively long period, often jeering the dead. So far, none appear to use a bulletproof vest. The police do not come on time, ever.

From the above, I suggest a stubby revolver for its concealment and reliability. Carrying a gun is not a good idea if you have children in the house, or if you spend a lot of time in schools, even though these are among the locations that need defending most. You need permission to carry in large venues, including big stores, synagogues and churches, as well as most clubs.

J. Edgar Hoover’s 1939, 32 caliber, “Pocket perfect,” Detective.

A gun suggestion is a “detective special” revolver like the S+W 642 “airweight, 14.6 ounces. It’s about half of the weight of a standard Glock, and shoots five bullets of 38 caliber. A step smaller are 32 caliber revolvers as were carried by J. Edgar Hoover. Smaller yet, are 22LR and/or 22WMR, revolvers like the S+W 351C or 351 PD, and all the NAA mini revolvers, 6 to 11 oz. They are easy to carry, non-obvious, and more reliable than a semi. Five to seven bullets can be enough. Robert Kennedy was killed with a 22lr. Semi-automatic pistols are good for the range, but they need to be racked, and tend to jam in tense situations.

I suggest a revolver that takes different loads. You can practice with cheaper ammo, and carry it loaded with more expensive. Especially with semis, make sure you can draw fast and shoot accurately without jamming.

Robert Buxbaum, March 10, 2024. A common claim in the press is that guns should be banned as in Europe, or highly regulated as in New York, New Jersey and California. I disagree. Europe has a very high rate of violent crime, including quite a few deadly attacks on jews.