Simple Models of Complex Systems

The range of phenomena addressed by physicists in trying to understand nature is very wide. In length scales, it varies from the Planck scale ( about 10 meters), to the size of the universe ( ∼ 10 meters). In time scales from 10 seconds to billions of years. The enormity of this range is somewhat mind-boggling. While the problems of understanding ultra-short and very-large length scales are certainly interesting and intriguing problems, it is good to remember that there are many interesting and fascinating phenomena at human length scales also, where our understanding today is still rather rudimentary. Here I will like to discuss some examples of such problems, taken the general area of statistical physics, as that is the subject I am most familiar with. One of the main thrusts of physics research in the last century has been to understand what are the fundamental constituents of matter, and interactions between them. However, knowing these does not directly help explain the world around us. Because, most of the time, one deals with phenomena involving a large number of particles. The mathematical problem of determining the dynamics of n interacting bodies is usually quite intractable, for n > 2. For example, in Newton’s classical theory of gravitation, one can analytically calculate the motion of a planet around the sun. However, the three-body problem, of three massive bodies moving under each other’s gravitational field (say the sun, the earth, and Jupiter) cannot be solved exactly. There are well-known examples of systems with few degrees of freedom that undergo deterministic evolution, e.g. the logistic map, or the Lorentz model, that chaos, i.e. a sensitive dependence on initial conditions, making long time predition of evolution impossible in practice. Thus, for such systems, knowledge of basic laws of interaction does not lead to effective predictability of future. System with many more than 2 or 3 degrees of freedom are usually chaotic, but in a way , a more chaotic evolution actually makes the system more predictable, if we agree to give only a probabilistic description of the system. Then, by what is called the law of large numbers, the average properties of the system can be determined quite accurately. This is the domain of statistical physics, where we try to determine the properties of a system consisting of a large number of interacting parts, with the interaction between parts given. The most interesting feature of systems with many degrees of freedom is captured in the aphorism: the whole is bigger than the sum of its parts. For example, from atoms and molecules, at larger scales of aggregation we go to chemistry and chemical reactions, to structure and function of large molecules like proteins, and then to cells, animals, and society. At each new level of organization, we find it useful to introduce new concepts, like life, love and culture. While the whole is more than its parts, it has to be understood in terms of the parts. The first problem with many degrees of freedom that was understood was the equilibrium properties of a box of a gas, made up of many molecules. The lessons learned there gave rise to the subject called statistical physics. It is reasonable to expect that the techniques developed there would also be useful in the more complicated cases mentioned above.