🌀 nonlinear sciences

Dissipation and fluctuations of CMOS ring oscillators close to criticality

This paper analyzes CMOS ring oscillators near criticality by deriving a thermodynamically consistent Hopf bifurcation normal form to demonstrate that entropy dissipation decreases with increased stability in the oscillating state, while finite-size fluctuations induce noise-driven oscillations with anomalously short decoherence times that scale sub-linearly with system size.

Original authors: Ashwin Gopal, Massimiliano Esposito, Jan Meibohm

Published 2026-08-26

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Ashwin Gopal, Massimiliano Esposito, Jan Meibohm

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the quiet hum of modern electronics, a fundamental tension exists between order and chaos. On one side lies the need for perfect timing, where billions of transistors must switch in unison to keep computers running. On the other lies the inevitable whisper of thermal noise, the random jiggling of atoms that introduces uncertainty into every electrical signal. This struggle is most visible in the subthreshold regime, a mode of operation where electronic circuits run on voltages so low they approach the natural energy of heat itself. In this fragile zone, the line between a stable, silent state and a rhythmic, oscillating one becomes incredibly thin. Understanding how these systems behave at this edge is not just a matter of engineering efficiency; it touches on deep questions about how energy is spent to create order and how randomness can be harnessed rather than suppressed.

A team of researchers has now mapped this delicate boundary using a specific type of electronic circuit known as a ring oscillator. Imagine a loop of electronic switches, or inverters, connected end-to-end. In a standard setup, these switches are designed to flip back and forth, creating a continuous wave of voltage that acts as a clock signal for a computer. The researchers focused on what happens when the power supplied to this loop is dialed down to a critical point, just low enough that the oscillations are about to die out. By building a model that accounts for the thermodynamic laws governing heat and energy, they discovered that the behavior of these circuits changes in surprising ways as they cross the threshold from silence to sound.

The study began by simulating a ring oscillator made of an odd number of these switches, a configuration necessary to force the circuit into an unstable state where it must oscillate. The team watched what happened as they slowly increased the supply voltage. Below a certain critical value, the voltage in the loop fluctuated randomly, a chaotic mess of small jitters with no clear pattern. But as the voltage crossed a specific threshold, the system underwent a sudden transformation. The random noise coalesced into a steady, rhythmic pulse. This transition is known in physics as a Hopf bifurcation, a point where a stable resting state becomes unstable and gives birth to a stable cycle of motion. The researchers found that once this rhythm is established, the circuit actually becomes more stable against small disturbances, even as it consumes less energy to maintain that rhythm, provided the loop contains more than three switches.

This finding challenges a common intuition that creating order always requires a heavy tax on energy. The team calculated the rate at which the circuit dissipates heat, a measure of how much energy is wasted as the system runs. They found that for rings with more than three inverters, the moment the oscillation starts, the rate of energy dissipation actually drops compared to the non-oscillating state. It is as if the system finds a more efficient way to move once it is in motion. This efficiency is linked directly to the stability of the rhythm; the more stable the oscillation becomes, the less energy it wastes. This relationship, which the authors call a stability-dissipation relation, suggests that nature favors a sweet spot where a system is both robust and frugal.

However, the story becomes more complex when the researchers considered the real-world size of these circuits. In their simulations, they treated the system as infinitely large to see the pure, deterministic rules of the transition. But real circuits are finite, composed of a specific, countable number of electrons. When they introduced these finite-size effects, they found that the boundary between silence and sound blurred. Even when the voltage was set slightly below the critical threshold, the random thermal noise was strong enough to kick the system into motion. The circuit would start to oscillate, not because it was driven hard enough, but because the random jitters of electrons were large enough to push it over the edge.

These noise-induced oscillations are fleeting. Unlike the steady, long-lasting rhythms of a well-tuned clock, these random bursts of activity lose their coherence very quickly. The researchers measured how long it took for the rhythm to fall apart and found that near the critical point, this time is surprisingly short. It scales in a way that is much slower than expected for large systems, meaning that as the circuit gets bigger, the rhythm does not hold together for proportionally longer. Instead, the randomness persists, and the signal remains fuzzy. This rapid loss of coherence is not a flaw in this specific context; it is a feature. Because the signal is so sensitive to random fluctuations and breaks down so quickly, it becomes an excellent source of true randomness.

This insight opens a door for new types of computing hardware. While traditional computers strive to eliminate noise to ensure perfect logic, emerging fields like neuromorphic computing and secure cryptography rely on genuine unpredictability. The researchers suggest that these ring oscillators, operating right at the edge of instability, could serve as highly efficient generators for true random numbers. By tuning the voltage to the critical point, engineers could create devices that produce a stream of unpredictable bits, essential for encryption and secure communication. The study confirms that by understanding the thermodynamics of these transitions, we can turn the chaotic noise of the microscopic world into a useful resource, turning the very instability that threatens to break our circuits into the engine of their randomness.

Technical Summary: Dissipation and Fluctuations of CMOS Ring Oscillators Close to Criticality

Problem Statement
The paper investigates the thermodynamic and dynamical behavior of CMOS-based ring oscillators near the onset of coherent voltage oscillations. While self-sustained oscillations are well-understood in the thermodynamic limit (large system size) via deterministic bifurcation theory, the behavior at the "edge of instability" on the mesoscopic scale remains complex. In this regime, thermal fluctuations are comparable to the order parameter, blurring the distinction between deterministic dynamics and noise. This is particularly relevant for ultra-low-power applications and neuromorphic computing, where circuits operate in the subthreshold regime with supply voltages approaching the thermal voltage. The authors aim to characterize the interplay between deterministic stability, entropy dissipation, and finite-size fluctuations in these systems.

Methodology
The authors employ a thermodynamically consistent model of a ring oscillator composed of $N$ (odd) inverters connected in a loop.

Stochastic Modeling: The system is modeled using a Master equation where electron transitions through MOSFETs are treated as bidirectional Poisson processes. The model incorporates thermal noise via a heat bath at temperature $T$ and accounts for gate-body and drain-source capacitances.
Thermodynamic Limit Analysis: To analyze the onset of oscillations, the authors take the thermodynamic limit ( $\Omega \to \infty$ , where $\Omega$ is the system size parameter related to physical dimensions). They derive a deterministic drift equation and perform a linear stability analysis of the trivial fixed point (zero voltage).
Normal Form Derivation: Near the critical voltage $V^*_{dd}$ , the system undergoes a Hopf bifurcation. The authors derive the normal form for this bifurcation by expanding the dynamics in terms of small amplitude variations and utilizing the cyclic permutation symmetry of the ring. This allows them to isolate the critical Fourier mode governing the transition.
Stochastic Thermodynamics: Using the framework of stochastic thermodynamics, the authors calculate the entropy production rate and the phase-space contraction rate to establish a relationship between dissipation and stability.
Finite-Size Fluctuations: To address mesoscopic effects, the authors perform a system-size expansion of the Master equation to the next-to-leading order, resulting in a Fokker-Planck equation. They derive a "stochastic normal form" to analyze noise-induced oscillations and phase decoherence close to the critical point.

Key Contributions and Results

Hopf Bifurcation and Normal Form:
The study identifies the critical voltage $V^*_{dd}$ at which the trivial fixed point becomes unstable, leading to a Hopf bifurcation. For odd $N$ , the critical mode corresponds to Fourier mode $k^* = (N-1)/2$ . The authors derive explicit expressions for the amplitude and frequency of the resulting limit cycle in the thermodynamic limit, showing that the amplitude scales as $\sqrt{\lambda}$ (where $\lambda$ is the distance from the critical voltage) and the frequency increases with $\lambda$ .
Stability-Dissipation Relation:
A central finding is the derivation of a stability-dissipation relation. The authors demonstrate that for ring oscillators with more than three inverters ( $N > 3$ ), the entropy production rate (dissipation) decreases in the oscillating state compared to the static fixed point. Conversely, the phase-space contraction rate (a measure of local stability) increases. This establishes a linear relationship between the relative change in entropy production ( $\Delta \dot{\sigma}$ ) and the relative change in phase-space contraction ( $\Delta L$ ) near the bifurcation. For $N=3$ , the linear contribution to the change in dissipation vanishes, and higher-order terms lead to an increase in dissipation.
Finite-Size Fluctuations and Noise-Induced Oscillations:
Moving beyond the thermodynamic limit, the authors show that finite-size fluctuations remain significant arbitrarily close to the critical voltage, even as the system size tends to infinity. These fluctuations induce coherent oscillations with finite amplitude even below the critical voltage ( $\lambda \leq 0$ ), a phenomenon absent in deterministic dynamics.
Anomalously Short Decoherence Time:
The paper characterizes the phase decoherence time ( $\tau_c$ ) of these noise-induced oscillations. Far from criticality, $\tau_c$ scales linearly with system size ( $\tau_c \sim \Omega$ ). However, close to the bifurcation, the decoherence time scales sub-linearly ( $\tau_c \sim \Omega^{1/2}$ ). This results in an anomalously short coherence time and a scaled quality factor that vanishes as $\Omega^{-1/2}$ near the transition, indicating that noise-induced oscillations are highly decoherent.

Significance and Claims
The paper claims to provide a unified thermodynamic and dynamical description of ring oscillators near criticality. By linking reduced dissipation to enhanced local stability for $N > 3$ , it offers a theoretical basis for the efficiency of oscillating states in certain regimes.

The authors highlight that the anomalously short decoherence time of noise-induced oscillations near criticality could be advantageous for generating high-rate, statistically independent true random numbers. They suggest this property may be harnessed for probabilistic hardware, such as "Ising machines" or stochastic probabilistic-bit (p-bit) systems.

The authors remain modest regarding the scope of their findings, noting that their analysis is strictly valid for small supply voltages (subthreshold regime, near the thermal voltage). They acknowledge that the permutation symmetry used to simplify the model may not hold or simplify dynamics further away from the critical voltage, and they do not propose specific experimental implementations beyond the theoretical framework.

Technical Summary: Dissipation and Fluctuations of CMOS Ring Oscillators Close to Criticality

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