The idea that systems will achieve the lowest energy state they can because they are more "stable" is clear enough. My question is, what causes this tendency? I've researched the question and been unable to find a clear answer, so I was hoping someone could explain what's going on behind the scenes here (or if nothing is and it's just an observed law that isn't explainable).
Answer
Large systems with many degrees of freedom (e.g. a ball consisting of many molecules) tend to settle into low energy states. This is a direct consequence of two fundamental laws, the first and second laws of thermodynamics: energy conservation and entropy increase.
A system with many degrees of freedom can be in many different microscopic states (think about a ball for which each molecular position and vibration etc is specified). Each such feasible micro state is equally likely. However, what we typically observe is not a micro state, but a coarse-grained description (the position of the ball) corresponding to incredibly many micro state. Certain macro states correspond to far fewer micro states than other macro states. As nature has no preference for any of these micro states, the latter macro states are far more likely to occur. The evolution to ever more likely macro states (until the most likely macro state, the equilibrium state, is reached) is called the second law of thermodynamics.
The decrease of potential energy is the consequence of the first (energy conservation) and second (evolution to more likely macro states) law of thermodynamics. As macro states with a lot of energy stored in heat (our ball with random thermal motion of its molecules) contain many more micro states and are therefore much more likely, energy tends to get transferred from potential energy to thermal energy. This is observed as a tendency towards a decrease in potential energy.
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