Information-theoretic transitions in oscillator circuits: A Landau ferromagnetic analogy

Oscillator circuits, as non-chaotic autonomous systems, undergo a supercritical Hopf bifurcation at a bifurcation point, where the output undergoes a qualitative change from 0 V (i.e., a stable fixed point) to a sinusoidal wave (i.e., a limit cycle). Oscillator circuits generate periodic waves (order) from noise (disorder), but the theory of supercritical Hopf bifurcations does not inherently include noise. Information theory, on the other hand, can describe unpredictable states such as noise. By applying information theory, we can account for the presence of noise and gain insight into the physical essence of the system. To date, very few studies have discussed the oscillation of oscillator circuits (i.e., supercritical Hopf bifurcations) from an information-theoretic perspective. Therefore, the objective of this study is to analyze changes in the oscillator’s output signal from an information-theoretic perspective and to demonstrate that oscillation constitutes an informationtheoretic transition (ITT). In the information-theoretic analysis, we modeled the probability density function based on the predictable output signals of the oscillator circuit (0 V, sinusoidal wave, and square wave) and calculated the differential entropy. At the bifurcation point, as the output signal transitioned from a 0 V convergence to a sinusoidal wave, the differential entropy exhibited a sharp change; furthermore, we confirmed spontaneous symmetry breaking in the output signal. We define this sudden change in differential entropy, accompanied by spontaneous symmetry breaking at the bifurcation point, as an ITT. Noting that ITTs closely resemble second-order phase transitions, we conducted a comparison based on an analogy with magnetic phase transitions and confirmed a strong correspondence. Furthermore, we theoretically demonstrated that the amplitude is an exponential function of the control parameter, and that an entropy-like quantity and an information capacity exist. Since the difference in differential entropy between the zero-voltage state and the square wave state is 1 bit, we discussed the possibility of verifying Landauer’s principle in a non-equilibrium transient system. Based on an analogy with time crystals, we further proposed the concept of a classical temporal atom as a new characterization of the oscillator circuit. Not limited to oscillator circuits, real-world physical systems often contain noise; since ITTs account for the contribution of noise, they may potentially interpolate supercritical Hopf bifurcations that do not inherently contain noise.