There is no butterfly effect

In social sciences, the story of butterfly effect is often invoked to explain catastrophic events such as financial crises. What is butterfly effect?

It is said that when a butterfly flips its wings in Brazil, it could generate a storm in the United States. Is it true? Definitely not! The air movement generated by the butterfly is quickly dissipated, not amplified. This is a basic property of any thermodynamic systems. This is also why   weather can be forecast with reasonable degree of accuracy.

If there is no butterfly effect in real weather, how the theory of butterfly effect was developed? In earlier days, researchers could not solve the equations that represent weather phenomena. So they modified the original equations into simpler equations. For these simplified equations, small changes in initial conditions could lead to big changes in solutions. This is the so called butterfly effects. This type of work were used to explain the difficulty of weather forecasting. With the improvement of mathematical methods and computing power over time, meteorologist can now forecast weather with the original equations  at much higher accuracy than the traditional methods based on personal experiences. 

If there is no butterfly effect in weather, why this term becomes so popular in social sciences? Essentially, this is to cover up the impacts of many policies by attributing catastrophic consequences to random events.  

From thermodynamic theory, we know that it takes large amount of effort to generate large amount of impacts. Take the 2007, 2008 financial crisis for example. It is the consequence of many long term policies and activities. First, Federal Reverse kept the interest rate low for a long time to encourage borrowing. Second, banks originated many innovative mortgages, such as adjustable rate mortgages, which  generate high incomes for themselves but are destined to default when the mortgage rate become adjusted, usually after five years . Third, the market of mortgage backed securities transfer credit risk from informed insiders to uninformed outsiders. Fourth, credit rating agencies certify low quality mortgage bonds as AAA bonds to increase their marketability. Fifth, legislations force employers and employees to fork large amount of money into pension systems, creating huge demand for financial securities, especially fixed income securities. Sixth, when large scale defaults occur or are going to occur, regulatory authorities step in to bail them out. In 1998, when Long Term Capital Management, a hedge fund, was going to default on its contracts, Federal Reserve organized a rescue to unwind its positions smoothly. The help of the regulatory system encouraged financial institutions to engage in ever more leveraged positions. There are many other highly concerted activities needed to generate a financial crisis of such a scale. Of course, the purpose of  these activities is not to generate a financial crisis, but to generate high income. In 2006, one year before Lehman Brothers fail, it generated highest profit in its history. But the coordinated effort to make highest profit for the financial systems make the larger human society very unstable. 

Now let us return to the modelling issue. Atmosphere, as a system involve many particles, is a thermodynamic system. Thermodynamic equations are partial differential equations. In earlier days, people could not solve these equations. So researchers simplify the partial differential equations into ordinary differential equations, which exhibit butterfly effect. But when we can already solve the original equations, should we still cling on the results obtained from watered down systems?

Why partial differential equations are essential in describing any systems involving many particles? This is because interaction and diffusion  are a basic property for any system with many particles, such as human societies. From another perspective, when we understand our society, we not only need to look at the time dimension, but also need to look at the wealth dimension, which influence our behaviors tremendously. So we need at least two variables, time and wealth, to describe our society. The resulting equation will include two variables, which are partial differential equations.