# Does classical simple harmonic motion violate thermodynamics?

If SHM allows for motion to occur forever, we can consider it perpetual motion, does this imply that the second law of thermodynamics is violated? Or does the presence of an external force act on the system? However, there should be still a loss of energy, causing, for example, a pendulum to stop moving eventually.

## 4 Answers

However, there should be still a loss of energy, causing, for example, a pendulum to stop moving eventually.

The pendulum will stop moving eventually, if there is loss of energy (like friction). Hence real simple harmonic oscillators always damp out.

Idealized SHOs will continue in motion forever, but there is no prohibition against objects moving forever. For example, two objects can remain orbiting each other forever (neglective gravitational waves). The second law prohibits a perpetual motion of a different kind:

A perpetual motion machine of the second kind is a machine that spontaneously converts thermal energy into mechanical work. When the thermal energy is equivalent to the work done, this does not violate the law of conservation of energy. However, it does violate the more subtle second law of thermodynamics (see also entropy). The signature of a perpetual motion machine of the second kind is that there is only one heat reservoir involved, which is being spontaneously cooled without involving a transfer of heat to a cooler reservoir. This conversion of heat into useful work, without any side effect, is impossible, according to the second law of thermodynamics.

• I assume, this is a form of 'perpetual motion', is this not the case? Having motion that goes forever is not 'perpetual motion'? May 29, 2022 at 14:06
• @user1007028 it's perpetual motion, but perpetual motion is not prohibited by the second law. May 29, 2022 at 14:07
• Is there a law that stops this 'perpetual motion'? It's simply that we were taught that 'any device that undergoes perpetual motion breaks the rules of the first or second law, does it violate the first, or is it that in reality we cannot make a machine without violating either? May 29, 2022 at 16:28
• @user1007028 there is no law that stops 'perpetual motion' and if someone taught you that, they are wrong. The thing that is impossible is a machine that produces work indefinitely, since that violates energy conservation. Simple motion without producing work is not impossible. May 29, 2022 at 23:21
• Even if there is no friction the pendulum will emit radiation right? But the thermal energy (related to random motion) that is taken out of the pendulum shouldn't affect its motion right? It would just make it colder. Jul 4, 2022 at 0:14

No, a simple harmonic oscillator does not violate any of the laws of thermodynamics. However, it does represent an idealized system with no dissipation that cannot be exactly realized in Nature.

First law. The total internal energy of an isolated system is conserved (assuming there is no work done on the system and no heat transfer).

This law is not violated since energy is conserved in simple harmonic motion (and we are assuming there is no work done or flow of heat).

Second law. The entropy of an isolated system never decreases.

The entropy of an oscillator undergoing simple harmonic motion does not decrease. The oscillator is in a single microstate, specified by the initial position and momentum. Therefore the entropy is always $$0$$ (the log of the number of microstates, since $$\log 1 = 0$$).

Third law. The entropy of a system approaches a constant value as the temperature approaches zero.

The harmonic oscillator's ground state is unique, therefore the entropy at zero temperature is $$0$$.

• I was taught that any 'perpetual motion' machine must break these laws, is it simply because it is idealized, and to make any device in practice must break one of the laws? May 29, 2022 at 16:30
• @user1007028 Essentially, yes. In more detail, what is forbidden by the laws of thermodynamics is a machine that returns to its original state after having performed nonzero work on its environment -- in other words, you can't have a machine that runs forever without any energy input and from which you can extract useful work. If you attach the oscillator to a system so that it does work, then the work done will extract energy from the oscillator, and the oscillator will lose energy and (if enough energy is extracted) come to a stop. May 29, 2022 at 16:44
• I see, so perpetual motion itself does not break any laws, only perpetual motion machines that do work, but it is generally not possible in reality except in idealized systems or large orbits (even then the classical view is not realistic), May 29, 2022 at 17:07
• @user1007028 Yes, I agree with your conclusion. It's only possible for idealized systems. Even objects orbiting each other due to gravity will lose power to gravitational wave emission and inspiral. May 29, 2022 at 20:50

Every harmonic oscillator is doomed to halt eventually. The force driving the oscillation, be it gravity, electromagnetism, or whatever force, involves radiation. Gravitational waves (gravitons) or EM radiation (photons) will take energy away. Two black holes orbiting will send out a considerable part of their energy into the universe which we can even detect here on Earth. Only in a stable, non radiating state, like of an electron around a proton, no radiation follows. But this ain't an oscillator, of course. The only true oscillator is the virtual field of particles. But then again, that's virtual...

• This answer misses a central point of the question: The part about the second law of thermodynamics. Also I am not sure what "The only true oscillator is the virtual field of particles." is even supposed to mean. May 29, 2022 at 16:57
• A perfect periodically process doesnt exist precisy because o the fact its an non-reversible process. You can see which direction time goes by looking at it. The quantum vacuum doesnt have this feature. Its composed of perfect oscillators. May 29, 2022 at 16:59

Thermodynamics applies to macroscopic systems, i.e., systems large enough to allow negligible fluctuations around the average values.

Moreover, although some concepts of thermodynamics can be applied to systems made by a small number of degrees of freedom, the fundamental concept of thermodynamic equilibrium requires some efficient mechanism to have a mixing dynamics. Integrable systems like one or more harmonic oscillators are not suitable for a thermodynamic description.

• yeah, using these laws makes more sense with control volumes and heat engines etc, I was confused between perpetual motion and perpetual motion machines the latter would generally be macroscopic. May 29, 2022 at 17:09