The Quality of Oscillations in Overdamped Networks

Abstract: The second law of thermodynamics implies that no macroscopic system may oscillate indefinitely without consuming energy. The question of the number of possible oscillations and the coherent quality of these oscillations remain unanswered. This paper proves the upper-bounds on the number and quality of such oscillations when the system in question is homogeneously driven and has a discrete network of states. In a closed system, the maximum number of oscillations is bounded by the number of states in the network. In open systems, the size of the network bounds the quality factor of oscillation. This work also explores how the quality factor of macrostate oscillations, such as would be observed in chemical reactions, are bounded by the smallest equivalent loop of the network, not the size of the entire system. The consequences of this limit are explored in the context of chemical clocks and limit cycles.