Overview: Coherent vs random motion

Prerequisites

In the motions we have considered so far in this class — a ball thrown upward, a box pushed along a floor, a bird poking a hole in a tree — the motions have been coherent. That is, we can treat the motion as if all parts of the object move together in the same way, with a single or only a few forces being responsible for the motion of the object.

It isn't only macroscopic objects that can be treated this way. When two atoms in a gas collide, we can treat each atom as moving coherently (i.e. we can ignore the internal motions of the electrons). Newton's laws help us calculate the accelerations and velocities associated with such coherent motion. In other words, we can predict the future position and velocity of an object based on its current motion. There are still additional interesting things to say about coherent motion: defining momentum and learning about the conservation of momentum, a universal law that is as absolute as the conservation of energy.

However, predicting the motion of objects becomes harder if the object interacts with many other objects. For that situation, we need to build a different mathematical model than one based on Newton's laws and differential equations.

Atoms and molecules in a gas collide very frequently with each other, rapidly changing direction. Physicists working on chaos theory show that when there are multiple collisions even a very small uncertainty about the current position and velocity makes it impossible to predict position and velocity of an object accurately. To see for yourself how hard it is to predict the motion after even two collisions. Imagine dropping one soccer ball on top of another that is sitting on the floor. It is very hard to get the top ball to bounce off the bottom ball twice!

Fortunately, in cases of many interacting objects such as gases or fluids we can still make a number of very useful predictions about what will happen — on average — even after many collisions!  We refer to this description of motion in response to many interactions of comparable strength as random motion (sometimes also known as incoherent or thermalmotion)

Random motion is everywhere: As you know from your earlier science classes, all matter is made up of atoms and molecules. These atoms and molecules all are in constant motion, a motion that is a reflection of the object's temperature. Atoms and molecules in solids vibrate pretty much around a fixed location, but atoms and molecules in liquids are in continual and unconstrained motion, wandering from place to place, and atoms in typical gases move long distances (compared to their sizes) before colliding with other atoms and molecules and changing their direction. Newton's laws have too many variables to handle this situation, even if we knew all the molecular positions and velocities with high accuracy (which we don't).

So we'll develop a different mathematical approach. Since there are lots Newtonian forces in all directions experienced by molecules in matter, the result is that they move in unpredictable directions. We can describe all these kinds of motion of an object's atoms and molecules through a random motion framework.

Though Newton's laws and momentum conservation are no longer enough to predict the trajectories of objects in random motion, they help us to understand important phenomena such as random walk, diffusion, and entropy. Let's look at one very important example here:  When a group of objects undergoes random motion, for every object moving to the right you will find one object there is one that is moving to the left.  That implies that the total momentum of a group of objects averages out to zero. For example, the average momentum of all atoms in a water filled balloon is zero. Conservation of momentum then tells us that the average momentum of the fluid in the water balloon will stay zero. In other words the water balloon stays at rest even though the atoms inside it wiggle around and each atom and water molecule changes its momentum in each collision. If we throw the water balloon we change the total momentum of the whole group of atoms, giving the atoms some coherent motion in addition to their random motion.

The critical property that differentiates coherent from random motion is that when the parts of an object are in coherent motion, the object carries a net momentum. When the parts of an object are in random motion, the sum of all the internal momentum cancels out.

Since a lot of what happens at the microscopic level in biological and chemical systems depends heavily on the properties of the random motion happening in a gas or fluid, we will study how what we know of the properties of motion leads to an understanding of the properties of random motion. This will lead us to important phenomena such as random walk, diffusion, an understanding of the ideal gas law and temperature, and entropy.

In this section, we'll consider the mathematical models for both coherent and random motion.

• Linear momentum — We introduce the concept of momentum and the Impulse-Momentum theorem, an important way of seeing the mechanism of Newton's 2nd law. We then discover a fundamental principle of the universe: the conservation of momentum.
• The role of randomness: Biological implications — We then make the transition to a new way of modeling motion mathematically: randomness.
• Diffusion and random walks — We develop the mathematical treatment of random motion and the emergent principles of diffusion and Fick's law.
• The ideal gas law — We finish the chapter by showing how our understanding of linear momentum applied in a situation of random motion and emergence, leads to an understanding of pressure, temperature, and the ideal gas law.

Joe Redish & Wolfgang Losert 10/20/12

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