Monthly Archives: April 2015

No need to conserve energy

Earth day, energy conservation stamp from the 1970s

Energy conservation stamp from the early 70s

I’m reminded that one of the major ideas of Earth Day, energy conservation, is completely unnecessary: Energy is always conserved. It’s entropy that needs to be conserved.

The entropy of the universe increases for any process that occurs, for any process that you can make occur, and for any part of any process. While some parts of processes are very efficient in themselves, they are always entropy generators when considered on a global scale. Entropy is the arrow of time: if entropy ever goes backward, time has reversed.

A thought I’ve had on how do you might conserve entropy: grow trees and use them for building materials, or convert them to gasoline, or just burn them for power. Under ideal conditions, photosynthesis is about 30% efficient at converting photon-energy to glucose. (photons + CO2 + water –> glucose + O2). This would be nearly same energy conversion efficiency as solar cells if not for the energy the plant uses to live. But solar cells have inefficiency issues of their own, and as a result the land use per power is about the same. And it’s a lot easier to grow a tree and dispose of forest waste than it is to make a solar cell and dispose of used coated glass and broken electric components. Just some Earth Day thoughts from Robert E. Buxbaum. April 24, 2015

Zombie invasion model for surviving plagues

Imagine a highly infectious, people-borne plague for which there is no immunization or ready cure, e.g. leprosy or small pox in the 1800s, or bubonic plague in the 1500s assuming that the carrier was fleas on people (there is a good argument that people-fleas were the carrier, not rat-fleas). We’ll call these plagues zombie invasions to highlight understanding that there is no way to cure these diseases or protect from them aside from quarantining the infected or killing them. Classical leprosy was treated by quarantine.

I propose to model the progress of these plagues to know how to survive one, if it should arise. I will follow a recent paper out of Cornell that highlighted a fact, perhaps forgotten in the 21 century, that population density makes a tremendous difference in the rate of plague-spread. In medieval Europe plagues spread fastest in the cities because a city dweller interacted with far more people per day. I’ll attempt to simplify the mathematics of that paper without losing any of the key insights. As often happens when I try this, I’ve found a new insight.

Assume that the density of zombies per square mile is Z, and the density of susceptible people is S in the same units, susceptible population per square mile. We define a bite transmission likelihood, ß so that dS/dt = -ßSZ. The total rate of susceptibles becoming zombies is proportional to the product of the density of zombies and of susceptibles. Assume, for now, that the plague moves fast enough that we can ignore natural death, immunity, or the birth rate of new susceptibles. I’ll relax this assumption at the end of the essay.

The rate of zombie increase will be less than the rate of susceptible population decrease because some zombies will be killed or rounded up. Classically, zombies are killed by shot-gun fire to the head, by flame-throwers, or removed to leper colonies. However zombies are removed, the process requires people. We can say that, dR/dt = kSZ where R is the density per square mile of removed zombies, and k is the rate factor for killing or quarantining them. From the above, dZ/dt = (ß-k) SZ.

We now have three, non-linear, indefinite differential equations. As a first step to solving them, we set the derivates to zero and calculate the end result of the plague: what happens at t –> ∞. Using just equation 1 and setting dS/dt= 0 we see that, since ß≠0, the end result is SZ =0. Thus, there are only two possible end-outcomes: either S=0 and we’ve all become zombies or Z=0, and all the zombies are all dead or rounded up. Zombie plagues can never end in mixed live-and-let-live situations. Worse yet, rounded up zombies are dangerous.

If you start with a small fraction of infected people Z0/S0 <<1, the equations above suggest that the outcome depends entirely on k/ß. If zombies are killed/ rounded up faster than they infect/bite, all is well. Otherwise, all is zombies. A situation like this is shown in the diagram below for a population of 200 and k/ß = .6

FIG. 1. Example dynamics for progress of a normal disease and a zombie apocalypse for an initial population of 199 unin- fected and 1 infected. The S, Z, and R populations are shown in (blue, red, black respectively, with solid lines for the zombie apocalypse, and lighter lines for the normal plague. t= tNß where N is the total popula- tion. For both models the k/ß = 0.6 to show similar evolutions. In the SZR case, the S population disap- pears, while the SIR is self limiting, and only a fraction of the population becomes infected.

Fig. 1, Dynamics of a normal plague (light lines) and a zombie apocalypse (dark) for 199 uninfected and 1 infected. The S and R populations are shown in blue and black respectively. Zombie and infected populations, Z and I , are shown in red; k/ß = 0.6 and τ = tNß. With zombies, the S population disappears. With normal infection, the infected die and some S survive.

Sorry to say, things get worse for higher initial ratios,  Z0/S0 >> 0. For these cases, you can kill zombies faster than they infect you, and the last susceptible person will still be infected before the last zombie is killed. To analyze this, we create a new parameter P = Z + (1 – k/ß)S and note that dP/dt = 0 for all S and Z; the path of possible outcomes will always be along a path of constant P. We already know that, for any zombies to survive, S = 0. We now use algebra to show that the final concentration of zombies will be Z = Z0 + (1-k/ß)S0. Free zombies survive so long as the following ratio is non zero: Z0/S0 + 1- k/ß. If Z0/S0 = 1, a situation that could arise if a small army of zombies breaks out of quarantine, you’ll need a high kill ratio, k/ß > 2 or the zombies take over. It’s seen to be harder to stop a zombie outbreak than to stop the original plague. This is a strong motivation to kill any infected people you’ve rounded up, a moral dilemma that appears some plague literature.

Figure 1, from the Cornell paper, gives a sense of the time necessary to reach the final state of S=0 or Z=0. For k/ß of .6, we see that it takes is a dimensionless time τ of 25 or to reach this final, steady state of all zombies. Here, τ= t Nß and N is the total population; it takes more real time to reach τ= 25 if N is high than if N is low. We find that the best course in a zombie invasion is to head for the country hoping to find a place where N is vanishingly low, or (better yet) where Z0 is zero. This was the main conclusion of the Cornell paper.

Figure 1 also shows the progress of a more normal disease, one where a significant fraction of the infected die on their own or develop a natural immunity and recover. As before, S is the density of the susceptible, R is the density of the removed + recovered, but here I is the density of those Infected by non-zombie disease. The time-scales are the same, but the outcome is different. As before, τ = 25 but now the infected are entirely killed off or isolated, I =0 though ß > k. Some non-infected, susceptible individuals survive as well.

From this observation, I now add a new conclusion, not from the Cornell paper. It seems clear that more immune people will be in the cities. I’ve also noted that τ = 25 will be reached faster in the cities, where N is large, than in the country where N is small. I conclude that, while you will be worse off in the city at the beginning of a plague, you’re likely better off there at the end. You may need to get through an intermediate zombie zone, and you will want to get the infected to bury their own, but my new insight is that you’ll want to return to the city at the end of the plague and look for the immune remnant. This is a typical zombie story-line; it should be the winning strategy if a plague strikes too. Good luck.

Robert Buxbaum, April 21, 2015. While everything I presented above was done with differential calculus, the original paper showed a more-complete, stochastic solution. I’ve noted before that difference calculus is better. Stochastic calculus shows that, if you start with only one or two zombies, there is still a chance to survive even if ß/k is high and there is no immunity. You’ve just got to kill all the zombies early on (gun ownership can help). Here’s my statistical way to look at this. James Sethna, lead author of the Cornell paper, was one of the brightest of my Princeton PhD chums.

Winning the peace at Appomattox

George A. Custer with captured confederate prisoner. Custer was a man of action but not of cruelty.

George A. Custer with a captured confederate prisoner. Custer was a man of action, but not of cruelty.

It is often forgotten that the aim of generalship is not winning a war, but winning a stable peace. In that sense, most generals and most diplomats are failures; their victories benefit only the undertaker; their peace-treaties only provide time to reload. That was the case with the Mexican civil war but not the US civil war. The choices and surrender at Appomattox, 150 years ago lead to a genuine, stable peace. It’s worthwhile, therefore to consider the how that was done here and not in Mexico, perhaps as a lesson for the future.

I begin, near the end of the war with a much-maligned general, George A Custer on April 8, 1865; this  is the day the 13th Amendment passed, four days after Lincoln walked through a defeated, smoldering Richmond, the capital of the south. The war would end soon, but would the result be peace, or chaos. George A. Custer had graduated at the very bottom of his class at West Point, the position known as goat. As is not atypical with goats, he was not particularly suited to following orders during peacetime, but was supremely suited to war and action. Custer liked to attack first and think later, but he was also a man of peace; he become the youngest Union Brevet General in US history. On April 8, with Lee at Appomattox Court House (that’s the name of the town), Custer led a small group of men to attack a nearby town, Appomattox Station, a rail depot three miles to the southwest. There he captured, without a fight, three, rail cars full of desperately needed arms, ammunition and supplies that had been sent to Lee’s army from Lynchburg.

While leaving the station, Custer’s men ran into the artillery unit of Confederate Brig. Gen. Reuben Walker, and attacked (of course) eventually capturing 25 artillery pieces, nearly 1,000 prisoners and all of their supplies. It took several attacks to win, but the results were worth it. Custer took the cannon and his troops, and positioned them on Lee’s likely escape route, on the Richmond-Lynchburg Stage Road south of Appomattox Courthouse. Lee was now nearly trapped, but didn’t know it yet.

Paining in honor of the 45th regiment colored troops: Afro-American soldier stands with flag before a bust of Washington.

Painting in honor of the 45th regiment colored troops: Afro-American soldier stands, with flag, before a bust of Washington and a depiction of battle (Fair Oaks? Petersburg?)

On the same day, April 8, General Grant sent Lee a proposition to surrender. Lee responded that he was not interested in that, but would like to meet at the McLean House at 10:00 A.M. April 9 to discuss “restoration of peace.” Grant replied that he didn’t have that power but agreed to meet Lee, none the less.

In the meantime, Lee prepared his forces to clear the Stage Road so his forces could escape south-west, to Appomattox station and home. Grant, no newcomer to war, ordered two corps (XXIV and V) under the commands of Maj. Gen. John Gibbon and Bvt. Maj. Gen. Charles Griffin to march all night to the west and north. These corps included 5000 Afro-American troops, mostly in the 45th and 116th U.S. Colored Troop Brigades. On the morning of April 9, Lee attacked to the south and managed to capture the forward pickets defending the Richmond-Lynchburg Stage Road. But when he reached the rise of the hill, he saw his escape was blocked. His 45,000 troops were surrounded by 113,000, better-armed Union soldiers, cannon and cavalry. It was then suggested that Lee disband his troops for an extended guerrilla war, an option he refused as it would lead to murdering bands roaming the county, and would make peace nearly impossible. Instead Lee rode off to discuss surrender to Grant. 

Map of the troop arrangements April 9, 1865. Checkmate. Lee's forces, x or + are out numbered, out gunned and surrounded. The end.

Map of the troop arrangements April 9, 1865. Checkmate. Lee’s forces, x or +, are out numbered, out gunned, and surrounded.

Lee’s surrender, finalized that afternoon, was penned by Grant’s aide-de camp, Lt Col’, Ely S. Parker, a Seneca Indian who had an engineering degree and studied law. The kindness to Indians may have suggested similar kindness to the surrendering Confederates. Parker eventually rose to the rank of general, and then to head of the Bureau of Indian Affairs. The terms of surrender too, were chosen to be unusually generous. Grant did not take the confederates soldiers captive, but instead allowed them to return home, relatively unmolested. Also, he allowed the officers to keep their swords and personal weapons. Kind acts like these may have eased reconstruction. Custer had demanded unconditional surrender from General Beauregard on April 9, something he probably imagined U.S. (Unconditional Surrender) Grant would have wanted; he was over-ruled by his commanding officer, Phillip Sheridan, who probably knew Grant better. 

After the war, most of the confederates swore loyalty to the US. Lee did what he could to promote reconciliation; he supported civil rights and reconstruction, and became president of Washington and Lee College. Some confederate generals and 2500 soldiers headed south to Mexico to join the French/Austrian forces of Emperor, Maximilian I, engaged in a civil war of his own. Maximilian, only 34 years old and a highly decorated Austrian officer, had little local support. He was captured and executed, June 19, 1867. Mexico then descended into chaos: a Pyrrhic victory, and a model to avoid.

Surrender at Appomattox; with Grant are Philip H. Sheridan, Orville E. Babcock, Horace Potter, Edward O.C. Ord, Seth Williams, Theodore S. Bowers, Ely S. Parker and George A. Custer. With Lee is Charles Marshall, his military secretary.

Surrender at Appomattox; with Grant are Philip H. Sheridan, Orville E. Babcock, Horace Potter, Edward O.C. Ord, Seth Williams, Theodore S. Bowers, Ely S. Parker and George A. Custer. With Lee is Charles Marshall, his military secretary. After the signing, most of the furnishings were purchased by Union officers as souvenirs. Lee shook Parker’s hand and said, I’m glad to see a real American here.” Parker replied, “We are all Americans.”

What did Mexico do wrong? For one, in order to win a peace, they failed to get the other side to agree to the peace, with clear documentation about what it is that’s been agreed to (That’s why Parker’s role is so important). Instead of killing Maximilian, they should have had him sign some sort of document and retire him to a farm or college where he could support the peace. In order to win a peace, it’s important to leave a stable country, with stable borders and a strong military, one that can govern itself fairly and well. A stable peace generally involves recognition of your government by other nations, and that too requires not killing your defeated enemy wholesale.

Robert E. Buxbaum, April 7-12, 2015. My sense is that the conditions for building a lasting peace get far too little attention in the study of war and history. I should mention that the 45th were mostly escaped slave volunteers. The 116th were ex-slaves that the Union purchased from Kentucky slave-owners at the beginning of the war to fight for the Union cause. This was thought to be a good emollient for peace, and may have helped keep Kentucky on the Union side. I should note too, that Lee freed his slaves in 1862, near the beginning of the war, a time when Grant still owned some. I’ve noted that men who choose beards tend to show a surprising republican (or communist) generosity. As Lincoln said, “Do I not defeat my enemy when I make him a friend?” For more thoughts on Lincoln’s Gettysburg Address, see here.

For All Fools Day, April 1, 2015

On this April Fools day I’m reminded of:

Democratic Senator Thomas Hart Benton, of Missouri. Before being elected to the US senate, he had been the lawyer for another Democrat, Andrew Jackson (before he was president). Like most prominent Americans of their day both men liked to settle arguments through by use of gunnery at close range. After Andrew Jackson participated in a duel with Benton’s brother, Benton himself challenged Jackson to a duel (I try to avoid this with my lawyers).

Seantor Thomas Hart Benton

Senator Thomas Hart Benton; his feud with A. Jackson, ended by common hatred of the Federal Reserve Bank

When asked about it later, Benton said, “Yes, sir, I knew him, sir; General Jackson was a very great man, sir. I shot him, sir. Afterward he was of great use to me, sir, in my battle with the United States Bank.” Ain’t that America. You can be shooting enemies one day, best of buddies the next.

In world politics and life this doesn’t happen as regularly as perhaps it should, but it happens. A few years after WWII we were allies with Germany and Italy, but enemies with the USSR. After WWI we were allies with Japan, until we weren’t; now we are again. Some day ISIS and Iran will be friends (won’t Mr A Bomb be sad).

I should also mention, and recommend the most amusing work of philosophy you are ever likely to encounter: St Erasmus of Rotterdam’s panegyric, “In Praise of Folly”. The thesis, if I may summarize: that most people like a fool and some folly; so does God. That most people don’t like long, boring lectures; neither does God. Etc. Good folly is likely to hurt you in a way you don’t mind (e.g. music, drinking, chasing women, reading long boring books if you like them); bad folly hurts others, e.g. when priest lives royally off of charity.

You might imagine that Erasmus would be executed for this but it seems he was not. It seems to me that Erasmus did something even more remarkable and made himself into a saint by claiming that God would not mind if people prayed to him for all good things. Apparently they started to, and got answered. A cheerful answer from a world of fools.

Robert E. Buxbaum, April 1, 2015.