In the context of the gravitational two-body problem, when two objects orbit each other, the system as a whole does not violate the second law of thermodynamics, which states that the entropy of an isolated system tends to increase with time.
The increase of entropy in a system is typically associated with a transition from a more ordered or lower entropy state to a more disordered or higher entropy state. However, when it comes to the gravitational two-body problem, the increase in entropy is not apparent at first glance because the motion of the objects appears to be highly ordered.
However, it's important to note that the entropy of a system is not solely determined by the macroscopic motion of its constituents but also by the microscopic details. In the case of a two-body system, the entropy increase can be understood by considering the statistical distribution of possible microstates associated with the system.
When two objects initially separate from each other and eventually start orbiting, their initial state typically has a lower entropy due to the high degree of order or correlation in their motion. However, as time progresses, the system explores a larger number of microstates consistent with the overall macroscopic properties of the system, such as the conservation of energy and angular momentum.
In the process of orbiting each other, the objects continuously exchange energy and angular momentum, leading to a statistical distribution of possible states that is more consistent with a higher entropy. The system explores a larger phase space, encompassing various orbital configurations, velocities, and positions.
Therefore, while the macroscopic motion of the objects may seem ordered and repetitive, the underlying microscopic dynamics and the statistical distribution of possible states contribute to an overall increase in entropy over time. The arrow of time is preserved, and the second law of thermodynamics is not violated in the gravitational two-body problem, even though it may not be immediately obvious.