# Entropy of environment of pendulum?

I remember reading a statement along the lines of:

Suppose our system is a simple pendulum. Then the entropy change in it is overall zero because the system is periodic. However, the entropy of the environment does change.

I don't know physics, but from the little I know about entropy it feels almost everything in the sentence is ill-defined.

Can someone explain what's going on here?

• This is kind of out of context, so it is hard to tell what they are referring to. Are they assuming that a wound spring is driving the pendulum so as to overcome frictional and air drag losses? – Chet Miller Mar 1 '16 at 13:44
• If a system is periodic entropy can not increase because the state of the system at $t+\Delta t$ is identical to that at $t$ for all $t$, where $\Delta t$ is the period. – tfb May 11 '17 at 23:27

Entropy from a system is always increasing (or in some theorical ways, is conserved) this is because the system wants to reach thermal equilibrium with its environnement. No gain of energy, no lost, just transformation. The more entropy is developped against environement, the less its overall energy is usable. To be usable, energy must be transfomable. So everything here is relative to its closest environement. Sorry for my english

I think this is a bad interpretation of the meaning of entropy. The energy on an oscillator loops between potential and kinetic, and while the system keeps oscillating, entropy keeps steady. The only way for entropy to increase is if it exchanges energy with 'the environment'. Let me explain.

If you think of entropy as 'energy dispersal', an apple suspended 1m over the ground has less entropy than an apple that has fallen.

When it falls, the potential energy polarized in both extremes (the apple and the earth) disperses across both bodies. Entropy grows, and you can say that essentially, the entropy 'of the environment' has grown. But as said, what has grown is not the entropy of the apple, but the entropy of both bodies together, that is, the entropy of the whole closed system.

Now, make the same experience, but before, dig a hole across the whole planet just below the apple. Now, let it fall.

If there is no friction, then the apple will oscillate across the planet forever. Entropy on the apple and 'on the environment' (strictly, the planet) will never grow.

But if there's friction due to the air on the tunnel, then, both bodies will exchange energy and eventually the apple will stop oscillating at the center of the planet. The entropy of the planet (the environment) will grow, as well as the entropy of the apple. Because the energy of the apple has dispersed 'into the environment'.

Entropy is originally a thermodynamic assessment, but if needed on another type of system, perhaps 'energy dispersal' is an adequate notion, since each system type has a different type of entropy, depending on the energy manifestations it depends on.

In such case, there is a second possibility: assume that entropy is related only to one energy type: for example, kinetic. In such case, a pendulum with no air friction disperses its full energy at each period (kinetic entropy increases) on 'the environment'. But potential entropy decreases. And so on.

Of course, this is an incomplete, reductionist appreciation of entropy (like assuming the entropy of a gas has decreased with a temperature decrement without considering a volume increase). But that is what the paragraph above seems to point to.

I think that what is meant in the sentence is that a pendulum considered "theoretically" is not a thermodynamic system. In this theoretical consideration it is a periodic system with no damping, so by no way we are getting close to taking into account any entropy variation.

Would you consider the pendulum with fluid friction then you will imply in the description of the motion some energy loss that involve equations where entropy can be created. On a mechanical point of view the entropy variation is hidden in the equation of motion when you add the fluid damping contribution :

$${\bf f}_{damp} = -\lambda {\bf v}$$

To justify this equation, you need to consider more complex fluid dynamics equation where you will be able to describe entropy creation on a microscopical scale when studying the flow of air around the pendulum.

Maybe the meaning of the sentence is that systems with no entropy creation can only be theoretical systems. By the same way in thermodynamics, the 0 Kelvin state is a theoretical state and can never be reached.