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Q & A: entropy and more

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Most recent answer: 10/22/2007
Q:
1) The first derivative of kinetic energy with respect to velocity is momentum. But what does this mean conceptually? In other words, it is easy (I think) to visualize the rate of energy change with respect to time, i.e., power; or distance, i.e., force. But how do you visualize the rate of change of energy with respect to velocity? 2) Starting from zero-sum of forces, how do you derive conservation of momentum and energy. If you can, then the conservations LAWs must be called THEOREMs. 3) T=1/(dS/dU)=dU/dS, i.e., at higher temperatures, a given change in U produces lesser entropy than lower temperatures. What does this mean conceptually? 4) Although dS/dU is 1/T, however, this ratio (dS/dU) seems too neat to be conveived as 1 over something (T or anything else). It seems to me that dS/dU must have its own intuitive concept. What is it? After all, dS/dU is the rate of entropy production due to transfer of energy -- the only way that entropy is produced. In other words, entropy is the child of energy change. In fact, originally, I thought dS/dU IS the entropy -- but I found otherwise in a thermodynamics book.
- Mehran
Miami
A:
Hi Mehran- This is a big grab-bag of issues, so I'll start with the ones I can handle best,

"3) T=1/(dS/dU)=dU/dS, i.e., at higher temperatures, a given change in U produces lesser entropy than lower temperatures. What does this mean conceptually?
4) Although dS/dU is 1/T, however, this ratio (dS/dU) seems too neat to be conveived as 1 over something (T or anything else). It seems to me that dS/dU must have its own intuitive concept. What is it? After all, dS/dU is the rate of entropy production due to transfer of energy -- the only way that entropy is produced. In other words, entropy is the child of energy change."

The whole direction of how things go is set by the rule that net entropy S increases. It’s as if the universe wanted to make entropy. dS/dU is a measure of how hungry some system is for energy, i.e. how much S payoff it gets for some U input. Energy will flow toward where it makes the most entropy. We simply define 1/T=dS/dU, so that doesn’t ’mean’ anything, it just gives us a new symbol T that’s shorter than writing out the derivative. High 1/T means ’very hungry for energy’ so that’s where the energy flows to maximize net entropy. I.e. heat flows toward low T. That tells you that this T is something like the T of ordinary language.

"2) Starting from zero-sum of forces, how do you derive conservation of momentum and energy. If you can, then the conservations LAWs must be called THEOREMs."

The conservation of momentum is a trivial consequence of the sum of the forces being zero (Newton’s 3rd law), and vice-versa. If you grant that the sum of forces in an isolated system is zero, then conservation of momentum is a theorem. However, that’s not much of a theorem since what you granted was obviously equivalent to the result. A more interesting theorem, due to Emmy Nöther, is that the invariance of the laws of physics under translation in space implies conservation of momentum.

Actually, conservation of momentum is more useful in a broad set of circumstances, particularly when the forces are either not known or are ill-defined. For example, in high-energy particle collisions, the quantum descriptions describe the exchange of "force carriers", but nowhere does anyone ever compute a force. In fact, some of these force carriers (the W+ for example) can transmute particles from one kind to another, or create entirely new ones. The fact that we can apply conservation of momentum still even when we cannot apply Newton’s laws is one of the things we have Emmy Nöther to thank for.

The sum of the forces being zero does NOT imply conservation of energy. Energy conservation flows as via Nöther’s theorem from another symmetry, that the laws of physics don’t change under translation in time.

Energy takes on many different forms -- kinetic energy of motion, rest energy in the masses of particles, potential energies of chemical bonds, energy stored in electromagnetic fields, among others. It took quite some years after Newton wrote his Principia before conservation of energy was realized to be as universally applicable as it is now, since all the ways energy can be converted to other forms took time to discover. We may still have a few more left to discover (we think we’ve got most understood qualitatively, even if some are difficult to predict quantitatively). In any case, Nöther’s theorem relating time invariance to conservation of energy gives us confidence we are on the right track with this one.

"1) The first derivative of kinetic energy with respect to velocity is momentum.
But what does this mean conceptually? In other words, it is easy (I think) to visualize the rate of energy change with respect to time, i.e., power; or distance, i.e., force. But how do you visualize the rate of change of energy with respect to velocity?"

This relationship is the start of the proof of the work-energy theorem (specifically the work-kinetic-energy theorem). This theorem states that the work, which is the integral of force with respect to distance (and in three dimensions, it's a line integral -- you take the dot product of the force everywhere with the displacement) gives the change in the kinetic energy.

Here’s a
from our introductory mechanics course which proves this theorem for a variable force in 1D (the 3D version just has dot products in the integrals). The last step, integrating mass times velocity with respect to velocity gives kinetic energy, and the steps leading up to that involve the use of the chain rule and Newton’s second law. One important thing to note here: what you say about the first derivative of the kinetic energy wrt velocity being the momentum is not true relativistically, and even the idea of force isn’t used much in relativistic calculations (it’s usually much easier to use the conservation laws of energy and momentum).

Mike W. and Tom

(published on 10/22/2007)

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