Why is potential energy of a system lowered?

Question

In both physics and chemistry I have often heard my teachers informally say. Xyz process happens since it leads to lower potential energy of the system. So the lowering in potential energy of a system seems to be quite a fundamental law of physics with how many processes it seems to explain and how vast it's reach is, quite like the second law of thermodynamics. However, unlike the latter it has no formalisation in the form of a name or subtopic dedicated to the law. Why is this?

Answer 1

In both physics and chemistry I have often heard my teachers informally say. Xyz process happens since it leads to lower potential energy of the system.

...

Why is this?

The negative gradient of the potential energy gives the direction of the conservative forces on the system. Therefore, the conservative forces on the system try to force the system towards a state of lower potential energy.

Of course, if there are only conservative forces, the system will just move back and force from high to low potential and back again (like a perfect pendulum).

In real life there are also non-conservative forces (e.g., air resistance or friction), which cause dissipation of the total energy of the system. This causes a system to move to a state of lowest total energy, which often has zero kinetic energy and the lowest possible potential energy.

For example, a simple harmonic oscillator (ideal spring) potential is given by $$ U(x) = \frac{1}{2}kx^2, $$ which is minimized when $x=0$.

The conservative force on the oscillator is: $$ F = -\frac{dU}{dx} = -kx. $$ If we add in a non-conservative damping force like ($-b\dot x$), then we see that a damped spring obeys the equation of motion: $$ m\ddot x = -k x - b \dot{x}. $$ An underdamped solution to the above equation looks like $$ x(t) = Ae^{-bt/2}\cos\left[\left(\sqrt{\frac{k}{m}-\frac{b^2}{4}}\right)t\right], $$ which, as time goes to infinity, goes to $$ x(t) \to 0, $$ which is the minimum of the potential energy.

Answer 2

In short, it is not a general principle, and is not applicable in all circumstances. It follows from the second law at zero temperature. Mathematically, equilibrium of an isolated system is characterized by a state of maximum entropy $S$. If you allow energy fluctuations by putting into contact with a thermostat of temperature $T$ it needs to be amended.

The new law is that equilibrium is reached at the minimum if free energy $F=U-TS$ with $U$ internal energy. Setting $T=0$ gives you energy minimization. Notice that conversely, at infinite temperature, you essentially recover the case of a closed system, so you just need to maximise $S$ again.

You do not need to be rigorously at absolute zero to apply the principle, you just need to be at temperatures low enough compared to a characteristic scale. This is the case at room temperature for most macroscopic systems like pendulums. Indeed, the typical energy is of 1J, so the corresponding temperature is $10^{25}K$, so room temperature ($300K$) is essentially absolute zero.

Mathematically, this means that you need to modify Newton's law: $$ m\ddot x = -\nabla V $$ which conserves energy $H = \frac12\dot x^2+V$, to: $$ m\ddot x+\gamma\dot x = -\nabla V $$ with an extra damping factor to model the energy exchange with the bath. If temperature is high enough (typically if your system is small or you are interested in very precise measurements), you need to add thermal fluctuations as well: $$ m\ddot x+\gamma\dot x = -\nabla V+\sqrt{2\gamma k_BT}\eta $$ with $\eta$ a gaussian white noise. It corresponds to the entropic term in free energy. As a sanity check, it vanishes at small temperature.

Finally, on a technical note, even in chemistry, you do not minimise energy. Since you are typically at fixed pressure rather than volume, you need to replace energy by enthalpy, but the same reasoning applies.

Answer 3

The tendency of systems to go to states of lower potential energy is a correct and wide-reaching observation, but it is often stated rather loosely, which leads to the question. It does not need to be raised to the status of a law, because it is not quite general and in the cases where it does apply, it is owing to the second law of thermodynamics.

First let's see why the observation is not general. Consider an isolated system. This is one which has no interactions with anything else, and, in particular, cannot exchange energy with anything else. For such a system the total energy is constant. Therefore if any part of the system goes to a state of lower potential energy, then either that part gains kinetic energy or some other part gains energy of whatever form (or both). A good example is a mass on a spring in the absence of friction: it just keeps oscillating forever. Of course that is an idealization since there is always some friction (or friction-like force of some kind) in practice. But there are examples in astronomy where motions like this last for billions of years (think of a planet on an elliptical orbit).

Next, consider the common case which we usually have in mind. This is a system whose main dynamics are simple: say, a pendulum or a ball rolling in a bowl, but it also has (perhaps small but non-zero) an interaction with some other system of many small parts. For example, this could be a solid or gas made of made atoms, or else just electromagnetic radiation with many possible frequencies and directions of travel. In this case entropy comes into play. What happens is that when the system moves towards lower potential energy, it gains kinetic energy, but typically it does not gain quite enough to compensate for the loss of potential energy, because a little energy is transferred to the other things around it. If that transferred energy goes to many small motions (e.g. vibrations) of small parts then there is an increase in entropy. This makes it impossible for the energy to come back to the system again. So the system is now at lower potential energy but without enough kinetic energy to bring it back to its first location.

This process continues, and the result is that the system eventually arrives at a minimum of potential energy. The reason it arrives there and does not come away again is essentially a combination of the laws of thermodynamics: energy conservation (the first law) combined with non-decrease of entropy (the second law).

When we study thermodynamics more fully, and also in the study of chemistry, it is important to consider other cases, such as a system which is in good thermal contact with a large heat bath. Such a system will exchange heat with the bath, in such a way as to keep its temperature $T$ constant. In this case there is a quantity called Helmholtz function, or free energy, given by $$ F = U - T S $$ where $U$ is the internal energy of the system, and $S$ the entropy. Now one finds that the system's state goes to the one which minimizes $F$ rather than $U$. This is a bit like potential energy, but not quite the same. When the system is at constant pressure another function called enthalpy comes into play.

Answer 4

Since the force for a conservative potential is given by $F=-\nabla V$, the forces will tend to do work on a body by lowering the potential energy. By the work-energy theorem we know that work acts by converting potential energy into kinetic energy by putting the body into motion.

There are, of course, many examples where this motion goes past some equilibrium point and then the potential energy starts to rise again only to eventually fall, as in the case of a ball rolling down then back up a ramp. But at all times the forces are acting in a fashion that will in principle lower the potential energy: I.e. gravity will always push the ball to the lower part of the hill even if it has enough kinetic energy that it ends up rolling back up the other side again.