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Chinese stocks lost 30% this year, has China’s lost decade begun?

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I predicted dire times for China six years ago, when Xi Jinping amended the constitution to make himself leader for life, in charge of the government, the party, the military, and the banks. Emperor, I called him, here. It now seems the collapse has begun, or at least stagnation. Chinese history is cyclic. Good times of peace and plenty give rise to a supreme emperor whose excesses bring war and famine, or at least stagnation. The cycle repeats every 50 to 100 years. Since Nixon opened China in 1973, the country has seen 50 years of prosperity and spectacular growth, but the growth has stopped and may be in decline. The stock market (Shanghai Shenzen 300) peaked in 2021 and has declined 50% from there. It’s down 30% for the last 12 months to levels seen in December 2010. US growth seemed slower than China’s but it’s been more steady. The main US stock market, the S+P 500, has more than tripled since 2010, up 24.5% this year.

Five years of the Shanghai 300 index with hardly any change. There has hardly been change in 15 years. One could argue that the lost decade is here and on-going. .

Each year Chairman Xi’s behaves more dictatorial. Last year he arrested his predecessor, Hu Jintao in front of the Communist party. He now tracks all his citizens actions by way of face recognition and phone software, and gives demerits for wrong thinking and wrong behaviors. You lose merits by buying western cars or visiting western internet sites. Taking money abroad is generally illegal. Needless to say, such behavior causes people to want to take money abroad, just in case. Last week, Xi proposed a limit on video game playing and clamped down on banks, demanding low interest rates. This is bad for the gaming corporations and teenagers, and banks, but so far there are no protests as there is no war.

Kissinger said that war was likely, though. Xi is building the navy at a fast pace, adding fast surface ships, nuclear submarines, aircraft carriers, and new attack airplanes. They’ve added hypersonic missiles too, and added listening stations and bases. There’s now a naval base in Djibouti, at the entrance to the Red Sea, where they oversee (or promote?) Iran’s attacks on Western shipping. Then there are the new Chinese Islands that were built to take oil and fishing rights, and to provide yet more military bases on key trade routes. These could easily be a trigger for war, but so far just one military interaction in the region. Last month, the Chinese and Philippines navy clashed over fishing!

In the Gulf of Finland last Month, a Chinese ship, New New Polarbear, destroyed the offshore cables and gas pipes between Finland and Estonia, in protest of Finland’s entry into NATO. It’s belligerent but not war. Undersea cables are not covered by the UN charter, law of the sea. Then there is the evidence that COVID-19 was the result of Chinese bioweapon development, and the Chinese spy ballon that was sent over the US. We maintain at peace, but an unsettled sort of peace — is it a preface to war? Wars don’t have to be big war against the west or Taiwan, more likely is Vietnam, IMHO.

China’s negative population growth means that property values will drop along with product consumption. Kids buy stuff; old folks don’t.

News from China is increasingly unreliable so it’s hard to tell what’s going on. There were claims of a coupe, but perhaps it was fake news. Reporters and spies have been arrested or shot so there is no window on anyone who knows. There are claims of high unemployment, and COVID deaths, and claims of a movement to “lie flat” and stop working. Perhaps that was behind the ban on excessive gaming. Who knows? Xi claims that China is self sufficient in food production, but record food shipments from the US to China suggest otherwise.

Major businesspeople have disappeared, often to reappear as changed men or women. Most recently, Jimmy Lai, the Hong Kong clothing magnate, was indicted for sedition by tweets. Perhaps he just wanted to fire workers, or pay down debt, or move abroad (his daughter is). Many businesses exist just to make jobs, it seems. Not all of these businesses are efficient, or profitable. Some exist to violate US patents or steal technology, particularly military technology. I suspect that China’s hot new car company, BYD, is a money-losing, job factory, behind Tesla in every open market. Some 91 public firms have delisted over the last two years, effectively vanishing from oversight. Are they gone, or still operating as employment zombies. Will BYD join them? If China manages to avoid war, I have to expect stagnation, a “lost decade” or two, as in Japan saw from 1990 to 2010, as they unwound their unprofitable businesses.

A sign suggesting that a Chinese lost decade has begun is that China’s is seeing deflation, a negative inflation rate of -0.2%/year according to the world bank. It seems people want to hold money, and don’t want Chinese products, services, or investment. Japan saw this and tried a mix of regulation and negative interest rates to revive the interest, basically paying people to borrow in hopes they spend.

In Japan, the main cause of their deflation seems to have been an excess of borrowing against overvalued and unoccupied real estate. The borrowed money was used to support unprofitable businesses to buy more real estate. This seems to be happening in China too. As in Japan, China originally needed new lots of new apartments when they opened up and people started moving to the cities. The first apartments increased in value greatly so people built more. But now they have about 100% oversupply: one unoccupied or half-built apartment for every one occupied, with many mortgaged to the hilt against other overvalued apartments and flailing businesses.

Chinese Dept, personal and corporate match Japan’s at the start of the lost decade(s). Personal debt is at 150% of GDP, corporate debt is at65% of GDP, all propped up by real estate.

As in Japan 30 years ago, China’s corporate + personal debt is now about two times their GDP. Japan tried to stop the deflation and collapse by increased lending, and wasteful infrastructure projects. People in the know sent the borrowed money abroad confident that they would repay less when they repaid. We are already seeing this; low interest loans, money flowing abroad and a profusion of fast trains, unused roads, and unused bridges. I suspect most fast trains don’t pay off, as planes are faster and cheaper. These investments are just postponing the collapse. China is also seeing a birth dearth, 1.1 children per woman. This means that within a generation there will be half as many new workers and families to use the trains, or occupy the apartments. As the country ages, retirees will need more services with fewer people to provide them. China’s culture promotes abortion. China’s working population will decline for the next 30 years at least.

Japan came through all this without war, somewhat poorer, but unified and modern. It helped that Japan was a democracy, unified in culture, with an open press and good leaders (Abe). There was no collapse, as such, but 20 years of stagnation. China is a dictatorship, with a disunited culture, and a closed press. I think it will get through this, but it will have a much rougher time.

Robert Buxbaum January 9, 2024. China isn’t alone in facing collapse and/or lost decades. Germany is in a similar state, especially since the start of the Ukraine war. It’s a democracy like Japan, and pacifist for now.

A series solution to the fussy suitor/ secretary problem

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One way to look at dating and other life choices is to consider them as decision-time problems. Imagine, for example that have a number of candidates for a job, and all can be expected to say yes. You want a recipe that maximizes your chance to pick the best. This might apply to a fabulously wealthy individual picking a secretary or a husband (Mr Right) in a situation where there are 50 male choices. We’ll assume that you have the ability to recognize who is better than whom, but that your pool has enough ego that you can’t go back to anyone once you’ve rejected the person.

Under the above restrictions, I mentioned in this previous post that you maximize your chance of finding Mr Right by dating without intent to marry 36.8% of the fellows. After that, you marry the first fellow who is better than any of the previous. My previous post had a link to a solution using Riemann integrals, but I will now show how to do it with more prosaic math — a series. One reason for doing this by series is that it allows you to modify your strategy for a situation where you can not be guaranteed a yes, or where you’re OK with number 2, but you don’t like the high odds of the other method, 36.8%, that you’ll marry no one.

I present this, not only for the math interest, but because the above recipe is sometimes presented as good advice for real-life dating, e.g. in a recent Washington Post article. With the series solution, you’re in a position to modify the method for more realistic dating, and for another related situation, options cashing. Let’s assume you have stock options in a volatile stock company, if the options are good for 10 years, how do you pick when to cash in. This problem is similar to the fussy suitor, but the penalty for second best is small.

The solution to all of these problems is to pick a stopping point between the research phase and the decision phase. We will assume you can’t un-cash in an option, or continue dating after marriage. We will optimize for this fractional stopping point between phases, a point we will call x. This is the fraction of guys dated without intent of marriage, or the fraction of years you develop your formula before you look to cash in.

Let’s consider various ways you might find Mr Right given some fractional value X. One way this might work, perhaps the most likely way you’ll find Mr. Right, is if the #2 person is in the first, rejected group, and Mr. Right is in the group after the cut off, x. We’ll call chance of of finding Mr Right through this arrangement C1, where

C1 = x (1-x) = x – x2.

We could used derivatives to solve for the optimum value of x, but there are other ways of finding Mr Right. What if Guy #3 is in the first group and both Guys 1 and 2 are in the second group, and Guy #1 is earlier in the second line-up. You’d still marry Mr Right. We’ll call the chance of finding Mr Right this way C2. The odds of this are

C2 = x (1-x)2/2

= x/2 – x2 + x3/2

There is also a C3 and a C4 etc. Your C3 chance of Mr Right occurs when guy number 4 is in the first group, while #1, 2, and 3 are in the latter group, but guy number one is the first.

C3 = x (1-x)3/4 = x/4 – 3x2/4 + 3x3/4 – x4/4.

I could try to sum the series, but lets say I decide to truncate here. I’ll ignore C4, C5 etc, and I’ll further throw out any term bigger than x^2. Adding all smaller terms together, I get ∑C = C, where

C ~ 1.75 x – 2.75 x2.

To find the optimal x, take the derivative and set it to zero:

dC/dx = 0 ~ 1.75 -5.5 x

x ~ 1.75/5.5 = 31.8%.

That’s not an optimal answer, but it’s close. Based on this, C1 = 21.4%, C2 = 14.8%, C3 =10.2%, and C4= 7.0% C5= 4.8%Your chance of finding Mr Right using this stopping point is at least 33.4%. This may not be ideal, but you’re clearly going to very close to it.

The nice thing about this solution is that it makes it easy to modify your model. Let’s say you decide to add a negative value to not ever getting married. That’s easily done using the series method. Let’s say you choose to optimize your chance for either Mr 1 or 2 on the chance that both will be pretty similar and one of them may say no. You can modify your model for that too. You can also use series methods for the possibility that the house you seek is not at the last exit in Brooklyn. For the dating cases, you will find that it makes sense to stop your test-dating earlier, for the parking problem, you’l find that it’s Ok to wait til you’re less than 1 mile away before you settle on a spot. I’ll talk more about this latter, but wanted to note that the popular press seems overly impressed by math that they don’t understand, and that they have a willingness to accept assumptions that bear only the flimsiest relationship to relaity.

Robert Buxbaum, January 20, 2020

The Brexit, Trump, Johnson anti-crash

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Before Brexit, I opined, against all respectable economists, that a vote for Bexit would not sink the British economy. Switzerland, I argued, was outside the EU, and their economy was doing fine. Similarly, Norway, Iceland, and Israel — all were outside the EU and showed no obvious signs of riots, food shortages, or any of the other disasters predicted for an exited Britain. Pollsters were sure that Britain would vote “No” but, as it happened, they voted yes. The experts despaired, but the London stock market surged. It’s up 250% since the Brexit vote.

Lodon stock market prices from January 2016 through the Brexit vote, August 2016, to the Boris Johnson election, August 2019. The price has risen by more than 250%.

A very similar thing happened with the election of Trump and of Boris Johnson. In 2016 virtually every news paper supported Ms Clinton, and every respectable economic expert predicted financial disaster if he should, somehow win. As with Brexit, the experts were calmed by polls showing that Trump would, almost certainly lose. He won, and as with Brexit, the stock market took off. Today, after a correction that I over-worried about, the S+P index remains up 35% from when Trump was elected. As of today, it’s 2872, not far from the historic high of 3049. Better yet, unemployment is down to record levels, especially for black and hispanic workers, and employment is way up, We’ve added about 1% of adult workers to the US workforce, since 2017, see Federal Reserve chart below.

Returning to Britain, the economic establishment have been predicting food shortages, job losses and a strong stock market correction unless Brexit was re-voted and rejected. Instead, the ruling Conservative party elected Boris Johnson to prime-minister, “no deal” Brexiter. The stock market responded with a tremendous single day leap. See above

Ratio of Civilian Employment to US Population. Since Trump’s election, we’ve added about 1% of the working age US population to the ranks of the employed.

You’d think the experts would show embarrassment for their string of errors. Perhaps they would save some face by saying they were blinded by prejudice, or that their models had a minor flaw that they’ve now corrected, but they have not said anything of the sort. Paul Krugman of the New York Times, for example, had predicted a recession that would last as long as Trump did, and has kept up his predictions. He’s claimed a bone rattling stock crash continuously for nearly three years now, predicting historic unemployment. He has been rewarded with being wrong every week, but he’s also increased the readership of the New York Times. So perhaps he’s doing his job.

I credit our low un-employment rate to Trump’s tariffs and to immigration control. When you make imports expensive, folks tend to make more at home. Similarly, with immigration, when you keep out illegal workers, folks hire more legal ones. I suspect the same forces are working in Britain. Immigration is a good thing, but I think you want to bring in hard-working, skilled, honest folks to the extent possible. I’m happy to have fruit pickers, but would like to avoid drug runners and revolutionaries, even if they have problems at home.

I still see no immediate stock collapse, by the way. One reason is P/E analysis, in particular Schiller’s P/E analysis (he won a Nobel prize for this). Normal P/E analysis compares the profitability of companies to their price and to the bond rate. The inverse of the P/E is called the earnings yield. As of today, it’s 4.7%. This is to say, every dollar worth of the average S+P 500 stock generates 4.7¢ in profits. Not great, but it’s a lot better than the 10-year bond return, today about 1.5%.

The Schiller P/E is an improved version of this classic analysis. It compares stock prices to each company’s historic profitability, inflation adjusted for 10 years. Schiller showed that this historic data is a better measure of profitability than this year’s profitability. As of today, the Schiller P/E is 29.5, suggesting an average corporate profitability of 3.5%. This is still higher than the ten-year bond rate. The difference between them is 2%, and that is about the historic norm. Meanwhile, in the EU, interest rates are negative. The ten year in Germany is -0.7%. This suggests to me that folks are desperate to avoid German bank vaults, and German stocks. From my perspective, Trump, Johnson, and the Fed seem to be doing much better jobs than the EU bankers and pendents.

Robert E. Buxbaum, August 16, 2019.

Less than 1 year to the crash

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Stock market crashes happen for a reason, and generally the reason is that owning stock is seen as less profitable than owning bonds, gold, guns, or hundred-dollar bills stuffed into one’s mattress. For this essay, I thought I might explain the reasoning behind the alarm bells that virtually every economist has been sounding. For the last year and a half they’ve been sure a severe correction is imminent. The reason has to do with price and predictions of profitability.

Let’s begin with Nobel Laureate economist, Paul Krugman of the New York Times. He has been predicting severe job losses, and a permanent stock collapse since Trump’s election in November 2016. Virtually every week he announces that the end is near, and every month the economy looked better. A lesser man would give up, but he has not. Why? Mostly it’s his hatred of all things Trumpian: Krugman can not accept that Trump could avoid destroying the economy, and con not imagine that any investor would see things otherwise.

Apparently some folks felt otherwise, and caused unemployment to drop and the market to rise. but then, in September 2017, Krugman’s dire predictions were echoed by Robert Schiller, 2013 Nobel winner, and author of a textbook the majority of schools use to teach market analysis. Robert Schiller, has argued that valuations are extremely expensive. “This stock market bears striking similarities to that of 1929. “The market is about as highly priced as it was in 1929,” “In 1929 from the peak to the bottom, it was 80 percent down. And the market really wasn’t much higher than it is now in terms of my CAPE [cyclically adjusted price-to-earnings] ratio. So, you give pause when you notice that.”

What Schiller is referring to is his particular version of the price to earnings ratio, the price of the average stock share divided by the amount of the average earnings per share. Schiller’s CAPE version uses the ten-year, inflation-averaged earnings, rather than today’s earnings, and finds the ratio is high, as the graph below shows. When he made these comments, this ratio was 25, nearly as high as the 1929 peak. The ratio is now higher, 32.74, higher than it stood on “Black Tuesday.” Why this number is important is that the profitability of a stock-share is merely the inverse of the Price/ Earnings ratio. The current ratio, 32.74 suggests that the average dollar’s worth of shares will return about 3.05% (1/32.74 = 3.05%). By comparison, one could buy a five-year treasury bond and get 2.96%. That’s hardly less, and federal bonds are totally safe. More alarming yet, the Federal Reserve has indicated that it will continue to raise interest rates at planned rate of 1%/year for at least the next year. At some point, people will decide bonds are the far better bargain, and will exit stocks en-mass. And then it’s crash-city, or so the theory goes.

The Schiller Price to Earnings ratio as of July 27, 2018. It suggests a crash is past due.

The Schiller Price to Earnings ratio as of July 27, 2018. It suggests a crash is past due.

Shown above is a historical plot of Schiller’s particular version of the price to earnings ratio based on the S+P 500 index, with data going back to 1880. It’s argued that his version using a ten-year, trailing average of corporate profits, is better than the non-adjusted, one year P/E ratio: the version you find in the newspapers. In the newspaper version, the peaks don’t show up until just after the crash because company profits tend to spike along with prices. In this version, profits can’t exactly spike, and  stock crashes show up as valuation peaks. The crash is seen as a consequence to high values of the Schiller P/E.  In terms of CAPE, we are at a more dangerous spot than in 1929. We are more exuberant than in 2008, or when Alan Greenspan warned of irrational exuberance. Schiller: “you give pause when you notice that.”

Schiller Price to earnings ratios are a good predictor of future stock prices. We are past the end of this chart, suggesting a significant loss of stock value ahead.

Schiller Price to earnings ratio plotted versus 20 year stock return. The higher the Schiller P/E, the lower the return. We are past the end of this chart suggesting we should expect a significant loss of capital value.

Stock pull-backs are sometimes gradual, as in 1968 through 1982, but more often the pullback is sudden, a crash. People typically expect a stock return in excess of bonds of 2% or so. They sometimes accept less, and sometimes demand more. Schiller calls the cause “animal spirits.” The fear is that investors will suddenly go back to the historical norm and demand of stocks 2% more return than the 3.05% they get from bonds. If they’d suddenly demand a 5.05% return on stocks to balance, the stock prices would fall by 40%. If the crash happened now, it would take a 40% drop in stock prices to raise the earnings ratio to 5.05%. But if they wait a year, until after the Fed raised the interest rate to 3.5%, we’d expect a greater pull-back 50% or so, a major crash. As early as last year, Schiller has advised moving out of US stock into foreign stocks, particularly European, noting that the US market was  the most expensive in the world. I don’t agree that Europe is a safe haven, but agree that a crash is likely given current return rates, snd the treasury plan to raise interests by 1% over the next year.

Schiller claims that the reason the recession has not hit so far is that people trust Trump. I would not have expected a comment like that from a Yale economist, especially given the constant carping from the TV news. Still Schiller may be on to something. The stock market went up dramatically after the Trump election. There are some advantages to a narcissist president. It also seems Trump’s tariffs are helping to provide jobs, as I predicted. In this quarter, the GDP rose at an impressive 4.1% rate. Gains came even where you’d expect otherwise. US soybean exports rose by 9600% despite a boycott from China. If the economy keeps going like this it might be as much as a year before the correction. A likely scenario is that the Fed raises interest rates, growth slows to 2.5% or less, and with bond interest rates at 3.5% people will get out of stocks in a big way. My expectation is that China will suffer too, and with it Europe. With luck, the Fed will then lower interest rates to 2%, or so. In my opinion interest rates should matches the inflation rate, more or less. I don’t know why the Federal Reserve does not do this, but instead swings its interest rates from very high to low, now aiming for a far excess of inflation rate. I suspect it’s mistake, one that we will pay for soon.

Robert Buxbaum, July 29, 2018. My only other stock analysis post was on bitcoin, In December 2017 I thought it had gone about as far as it would go. Shortly there-after bitcoin value crashed. I hope I don’t cause a crash

Bitcoin risks, uses, and bubble

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Bitcoin prices over the last 3 years

Bitcoin prices over the last 3 years

As I write this, the price of a single bitcoin is approximately $11,100 yesterday, up some 2000% in the last 6 months. The rise rate suggests it is a financial bubble. Or maybe it’s not: just a very risky investment suited for inclusion in a regularly balanced portfolio. These are two competing views of bitcoin, and there are two ways to distinguish between them. One is on the basis of technical analysis — does this fast rise look like a bubble (Yes!), and the other is to accept that bitcoin has a fundamental value, one I’ll calculate that below. In either case, the price rise is so fast that it is very difficult to conclude that the rise is not majorly driven by speculation: the belief that someone else will pay more later. The history of many bubbles suggests that all bubbles burst sooner or later, and that everyone holding the item loses when it does. The only winners are the brokers and the last investors who get out just before the burst. The speculator thinks that’s going to be him, while the investor uses rebalancing to get some of benefit and fun, without having to know exactly when to get out.

That bitcoin is a bubble may be seen by comparing the price three years ago. At that point it was $380 and dropping. A year later, it was $360 and rising. One can compare the price rise of the past 2-3 years with that for some famous bubbles and see that bitcoin has risen 30 times approximately, an increase that is on a path to beat them all except, perhaps, the tulip bubble of 1622.

A comparison between Bitcoin prices, and those of tulips, 1929 stocks, and other speculative bubbles; multiple of original price vs year from peak.

A comparison between Bitcoin prices, and those of tulips, 1929 stocks, and other speculative bubbles; multiple of original price vs year from peak.

That its price looks like a bubble is not to deny that bitcoin has a fundamental value. Bitcoin is nearly un-counterfeit-able, and its ownership is nearly untraceable. These are interesting properties that make bitcoin valuable mostly for illegal activity. To calculate the fundamental value of a bitcoin, it is only necessary to know the total value of bitcoin business transactions and the “speed of money.” As a first guess, lets say that all the transactions are illegal and add up to the equivalent of the GDP of Michigan, $400 billion/year. The value of a single bitcoin would be this number divided by the number of bitcoin in circulation, 15,000,000 currently, and by the “speed of money,” the number of business transactions per year per coin. I’ll take this to be 3 per year. It turns out there are 5 bitcoin transactions total per year per coin, but 2/5 of that, I’ll assume, are investment transactions. Based on this, a single bitcoin should be worth about $8890, slightly below its current valuation. The gross speed number, 5/year, includes bitcoin transactions that are investments and never traded for goods, and those actively being used in smuggling, drug-deals, etc.

If the bitcoin trade will grow to $600 billion year in a year with no other change, the price rise of a single coin would surpass that of Dutch tulip bulbs except that more coins are bing minted, and that the speed is increasing. If you assume that coin use will reach $1,600 billion/year, the GDP of Texas in the semi-near future, before the Feds jump in, the fundamental value of a coin should grow no higher than $44,000 or so. There are several problems for bitcoin investors who are betting on this. One is that the Feds are unlikely to tolerate so large an unregulated, illegal economy. Another is that bitcoin transactions are not likely to go totally legal. It is very hard (near impossible) to connect a bitcoin to its owner. This is a plus for someone trying to deal in drugs or trying hide profits from the IRS (or his spouse), but a legal merchant will want the protection of courts of law. For this, he or she needs to demonstrate ownership of the item being traded, and that is not available with bitcoin. The lack of a solid, legitimate business need suggests to me that the FBI will likely sweep in sooner or later, and that the value of a coin will never reach $44,000.

Yet another problem for those wishing to invest in bitcoin is the existence of more bitcoins (undiscovered, or un-mined so far) and the existence of other cryptocurrencies with the same general qualities: Litecoin (LTC), Ethereum (ETH), and Zcash (ZEC) as examples. The existence of these coins increases the divisor one should use when calculating the value of a bitcoin. The total number of bitcoins is capped at 21,000,000, that is 6,000,000 coins more than known today. Assuming more use and more acceptance, the speed (turnovers per year) is likely to increase to four or five, similar to that of other currencies. Let’s assume that the bitcoin will control 1 trillion dollars per year of a $1.6 trillion/year illegal market. One can now calculate the maximum long term target price of a bitcoin by dividing $1 trillion/year by the number of bitcoins, 21,000,000, and by the speed of commercial use, 4.5/year. This suggests a maximum fundamental value of $10,582 per coin. This is just about the current price. Let the investment buyer beware.

For an amusing, though not helpful read into the price: here are Bill Gates, Warren Buffet, Charlie Munger, and Noam Chomsky discussing Bitcoin.

Robert Buxbaum, December 3, 2017.

The Gift of Chaos

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Many, if not most important engineering systems are chaotic to some extent, but as most college programs don’t deal with this behavior, or with this type of math, I thought I might write something on it. It was a big deal among my PhD colleagues some 30 years back as it revolutionized the way we looked at classic problems; it’s fundamental, but it’s now hardly mentioned.

Two of the first freshman engineering homework problems I had turn out to have been chaotic, though I didn’t know it at the time. One of these concerned the cooling of a cup of coffee. As presented, the coffee was in a cup at a uniform temperature of 70°C; the room was at 20°C, and some fanciful data was presented to suggest that the coffee cooled at a rate that was proportional the difference between the (changing) coffee temperature and the fixed room temperature. Based on these assumptions, we predicted exponential cooling with time, something that was (more or less) observed, but not quite in real life. The chaotic part in a real cup of coffee, is that the cup develops currents that move faster and slower. These currents accelerate heat loss, but since they are driven by the temperature differences within the cup they tend to speed up and slow down erratically. They accelerate when the cup is not well stirred, causing new stir, and slow down when it is stirred, and the temperature at any point is seen to rise and fall in an almost rhythmic fashion; that is, chaotically.

While it is impossible to predict what will happen over a short time scale, there are some general patterns. Perhaps the most remarkable of these is self-similarity: if observed over a short time scale (10 seconds or less), the behavior over 10 seconds will look like the behavior over 1 second, and this will look like the behavior over 0.1 second. The only difference being that, the smaller the time-scale, the smaller the up-down variation. You can see the same thing with stock movements, wind speed, cell-phone noise, etc. and the same self-similarity can occur in space so that the shape of clouds tends to be similar at all reasonably small length scales. The maximum average deviation is smaller over smaller time scales, of course, and larger over large time-scales, but not in any obvious way. There is no simple proportionality, but rather a fractional power dependence that results in these chaotic phenomena having fractal dependence on measure scale. Some of this is seen in the global temperature graph below.

Global temperatures measured from the antarctic ice showing stable, cyclic chaos and self-similarity.

Global temperatures measured from the antarctic ice showing stable, cyclic chaos and self-similarity.

Chaos can be stable or unstable, by the way; the cooling of a cup of coffee was stable because the temperature could not exceed 70°C or go below 20°C. Stable chaotic phenomena tend to have fixed period cycles in space or time. The world temperature seems to follow this pattern though there is no obvious reason it should. That is, there is no obvious maximum and minimum temperature for the earth, nor any obvious reason there should be cycles or that they should be 120,000 years long. I’ll probably write more about chaos in later posts, but I should mention that unstable chaos can be quite destructive, and quite hard to prevent. Some form of chaotic local heating seems to have caused battery fires aboard the Dreamliner; similarly, most riots, famines, and financial panics seem to be chaotic. Generally speaking, tight control does not prevent this sort of chaos, by the way; it just changes the period and makes the eruptions that much more violent. As two examples, consider what would happen if we tried to cap a volcano, or provided  clamp-downs on riots in Syria, Egypt or Ancient Rome.

From math, we know some alternate ways to prevent unstable chaos from getting out of hand; one is to lay off, another is to control chaotically (hard to believe, but true).