Continuous vs. discrete behavior

Tom,

What you describe is correct, however, like with my guitar string, I will argue that I can describe the movement of the individual ball falling and hitting each nail with equations that are continuous, even if for that I have to know all the information concerning each nail, its exact position and elasticity, etc.. (Just for the sake of the argument, I am not Marquis de Laplace!)

Statistics, however, rely on the basic counting of events and, therefore, introduce a discrete dimension to our modeling of the world. There is no wonder that Quantum Mechanics introduces discrete behavior because it is derived from statistics. What is amazing is that we seem to live in a discrete universe that obeys continuous laws in the macro-scale. Every time I store a file in my thumb drive I remember how absurd the tunnel effect is from the classical mechanics point of view, yet it works!

Ramon

I think that an expert more versed in mathematics can provide a better reply or expand this one.

I am not sure whether Neha refers to the non-linear and discontinuous behavior of most natural systems or to the discrete behavior of nature in Quantum Mechanics. In any case, here is my reply:

Neha,

Nature is very complex, and to succeed in understanding and modeling it we have to decompose the problems into smaller chunks that we can analyze independently. Most mathematical models, at least those learnt at the beginning of our scientific training, are analytical or continuous and do describe with sufficient accuracy many real situations, like the trajectory of a ball or the propagation of a wave. These are easier to solve and an obvious starting point for the beginner. In general,  linear systems are only sufficiently linear in a limited range of operation. Most real systems are non-linear, like the progressive spring of a motorbike or the circuits in a radio receiver. Mathematical methods exist to "linearize" the behavior or divide the range of interest in segments that can be considered sufficiently linear. Other systems exhibit what seems to be a discrete behavior, like an AC thermostat that snaps to open the circuit to stop the AC at one temperature, then closes the circuit to start AC at another one. These systems are described by considering their different "states" and often continuous equations exist within each state until the conditions are reunited that make the system jump to a different one. The transition between states can often be also analyzed as a continuous process.

There are some systems in nature that seem to exhibit certain discrete behaviors, for  example the modes of vibration of a guitar string or the membrane of a drum. The string can only support waves that have nodes at the fixed points of the string, i e. integer multiples of the fundamental wave. A drum membrane supports radial and angular vibration modes that should fit in its circumference. Yet, I will argue that even the vibration modes of these systems have some continuity. The fix points of the string, for example, have some dimension and the string is not stopped at a single point, but its movement restricted over an albeit small but not zero length. This gives rise to vibrations that do not have an harmonic relation with the fundamental, or inharmonic modes, which happen to be what gives character to real instruments and makes them so difficult to synthesize.

We find the only behavior that I would qualify as really discontinuous in the realm of the atomic particles or Quantum Mechanics. These particles can only jump between specific energy levels while emitting specific frequencies related to the height of such jumps. Their behavior is so contradictory with our everyday experience that raises philosophical questions about the way in which we perceive the Universe (Cosmology).

I do not know if I have replied to the question but I will end by encouraging you to pursue the study of mathematics, or any scientific branch of your liking, to be able to make further inroads into the understanding and description of nature that interests you so much.

Stay curious,

    Ramon