🔬 condensed matter

Link between Continuous and Discrete Descriptions of Noise in Nonlinear Resistive Electrical Components

This paper demonstrates that thermodynamic consistency in modeling noise for nonlinear resistive components necessitates the Hänggi-Klimontovich prescription in both continuous stochastic and discrete Markovian frameworks, leading to a generalized Johnson-Nyquist relation that reduces to the classical form only at low voltages.

Original authors: Lucas Désoppi, Bertrand Reulet

Published 2026-07-03

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Lucas Désoppi, Bertrand Reulet

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

Imagine you are trying to understand how electricity flows through a special, tricky kind of wire. This isn't a normal wire; it's a nonlinear resistor. Think of a normal wire like a smooth, straight slide where the speed of your slide depends only on how steep it is. A nonlinear resistor is more like a slide with bumps, curves, and changing friction; the way electricity moves through it changes depending on how much "push" (voltage) you give it.

The scientists in this paper, Lucas and Bertrand, are trying to figure out how to mathematically describe the noise in this system. In the world of electricity, "noise" isn't a sound; it's the tiny, random jiggling of electrons caused by heat. It's like a crowd of people in a hallway: even if no one is trying to move in a specific direction, they are constantly bumping into each other and jostling around.

The paper tackles a big problem: How do we describe this jiggling accurately without breaking the laws of physics?

The Two Ways to Look at the Jiggling

The authors compare two different ways of modeling this noise:

The "Smooth" Way (Continuous Model): Imagine the jiggling is a constant, smooth stream of tiny bumps. Mathematically, this is described by a "Stochastic Differential Equation" (SDE). It's like describing a river's flow as a smooth, continuous line.
The "Step" Way (Discrete Model): Imagine the jiggling happens in distinct, tiny jumps. Electrons don't flow smoothly; they hop from one spot to another. This is described by a "Master Equation." It's like counting individual steps in a staircase rather than looking at a ramp.

The Problem: The "Brillouin Paradox"

Here is where things get tricky. If you use the wrong math to describe these jiggles, you accidentally create a perpetual motion machine.

The paper mentions something called the Brillouin Paradox. Imagine a particle floating in a warm fluid. If your math is wrong, it might predict that the random bumps from the fluid molecules would push the particle to drift in one specific direction forever, even though the fluid is just sitting there. This would mean you could extract free energy (work) from a single heat source, which violates the Second Law of Thermodynamics (you can't get something for nothing).

The authors found that for a long time, scientists used two common math rules (called Itô and Stratonovich) to handle the "Smooth" model. But when they applied these rules to nonlinear resistors, they hit a dead end: the math either forced the resistor to be a simple, boring linear one, or it predicted that the resistor would generate free energy (the paradox).

The Solution: The "H-K" Prescription

The paper's main discovery is that there is a third way to do the math, called the Hänggi-Klimontovich (H-K) prescription.

Think of the math rules as different ways to decide when to measure the speed of a car while it's accelerating:

Itô: You measure the speed at the start of the second.
Stratonovich: You measure the speed at the middle of the second.
H-K: You measure the speed at the end of the second.

The authors prove that for a nonlinear resistor to obey the laws of thermodynamics (and not create free energy), you must use the H-K rule (measuring at the end).

When you use this specific rule, a beautiful new relationship appears. It's a modern version of the famous Johnson-Nyquist relation (which links noise to temperature and resistance).

Old Rule: Noise depends on a fixed "conductance" (how easy it is to flow).
New Rule (for nonlinear stuff): The noise depends on the ratio of the average current to the voltage.

It's like saying: "The amount of jiggling isn't just about how slippery the slide is; it's about how fast the people are actually sliding right now."

Connecting the Smooth and the Step

The second part of the paper connects the "Smooth" model to the "Step" model.

In the "Step" model, the electrons hop in discrete jumps. The authors showed that if you make the jumps very small (so small they look like a smooth flow), the math naturally leads you to the H-K rule again.
This confirms that the H-K rule isn't just a mathematical trick; it's the only way to make the "step-by-step" reality of electrons match up with the "smooth" math we use for big circuits, without breaking physics.

The Catch: Low Voltage Only

There is one important limit. The authors found that this perfect "New Rule" for noise only works perfectly when the voltage is low.

Low Voltage: The electrons are moving slowly, and the "smooth" approximation works well. The noise follows the new rule perfectly.
High Voltage: The electrons are moving fast and jumping wildly. The "smooth" model starts to break down, and the simple relationship between current and noise gets complicated again.

Summary in a Nutshell

The paper solves a puzzle about how to mathematically describe the random jiggling of electricity in tricky, nonlinear wires.

If you use the old math rules, you accidentally invent a machine that creates free energy (which is impossible).
To fix this, you must use a specific math rule called Hänggi-Klimontovich.
This rule reveals a new, more accurate way to calculate noise in these wires, linking it to how much current is actually flowing.
This new understanding works perfectly when the electrical "push" is gentle (low voltage), ensuring our models of electricity don't violate the fundamental laws of the universe.

Technical Summary: Link between Continuous and Discrete Descriptions of Noise in Nonlinear Resistive Electrical Components

Problem Statement
The modeling of noise in classical, nonlinear resistive electrical components faces a fundamental conflict between theoretical descriptions and thermodynamic consistency. Two primary models exist: the Johnson-Nyquist model, valid for linear resistors using Gaussian white noise, and the shot noise model, which describes discrete charge fluctuations via Markovian master equations. While the shot noise model accommodates nonlinearity, applying Gaussian white noise to nonlinear resistors via Stochastic Differential Equations (SDEs) has historically led to unphysical predictions, most notably the Brillouin paradox. This paradox implies that a nonlinear resistor coupled to a single thermal bath could generate a non-zero average voltage (or drift), effectively acting as a Maxwell's demon and violating the second law of thermodynamics. The core problem addressed is identifying the specific mathematical prescriptions required to reconcile nonlinear resistive behavior with thermodynamic equilibrium in both continuous (Gaussian) and discrete (shot noise) frameworks.

Methodology
The authors employ a dual-method approach to analyze noise in a resistive element parallel to a capacitor $C$ , where the charge $Q_t$ and voltage $U_t$ evolve under thermal fluctuations.

Continuous Gaussian Noise Model:
The system is modeled using an SDE of the form $C dU_t = -I(U_t)dt - J(U_t)dW_t$ , where $I(U_t)$ is the deterministic current-voltage characteristic and $J(U_t)$ represents multiplicative noise dependent on the bias voltage. To define the SDE mathematically, the authors introduce a discretization parameter $\alpha \in [0, 1]$ , which determines the evaluation point of the noise term within a time step. This parameter encompasses the standard Itô ( $\alpha=0$ ) and Stratonovich ( $\alpha=1/2$ ) prescriptions, as well as the Hänggi-Klimontovich (H-K) prescription ( $\alpha=1$ ). The authors derive the associated Fokker-Planck equation and impose the condition that the steady-state probability distribution must satisfy the Maxwell-Boltzmann statistic ( $P_{eq} \propto \exp(-CU^2/2k_B T)$ ).
Discrete Shot Noise Model:
The system is modeled as a Markov jump process where voltage takes discrete values $u_n = n v$ (with $v = q_e/C$ ). The evolution is governed by a master equation involving jump rates $\lambda$ between charge states. A causality parameter $\tilde{\alpha} \in [0, 1]$ is introduced to define whether the jump rates depend on the initial state, final state, or an intermediate state. The authors expand this master equation to the second order in the voltage step $v$ to derive a continuous Fokker-Planck limit, establishing the relationship between the discrete parameter $\tilde{\alpha}$ and the continuous prescription parameter $\alpha$ .

Key Contributions and Results

Necessity of the H-K Prescription in Continuous Models:
By enforcing thermodynamic equilibrium (vanishing probability current and Maxwell-Boltzmann statistics), the authors demonstrate that for a nonlinear resistor ( $I(u)$ is non-linear) to be compatible with Gaussian white noise, the discretization parameter must be $\alpha = 1$ . This corresponds to the Hänggi-Klimontovich (H-K) prescription.
- Under any other prescription ( $\alpha \neq 1$ ), the only solution satisfying equilibrium is a linear resistor with constant conductance, rendering the model incapable of describing nonlinearity without violating thermodynamics.
- With $\alpha = 1$ , the model yields a generalized Johnson-Nyquist relation: $J(u) = \sqrt{2k_B T I(u)/u}$ . This relation replaces the standard conductance $G$ with the ratio of mean current to voltage ( $I(u)/u$ ).
- The authors show that failing to use the H-K prescription (or failing to satisfy the generalized relation) leads to $\langle U_t \rangle_{eq} \neq 0$ , manifesting the Brillouin paradox where work could be extracted from a single thermal bath. A mechanical analog of a particle with velocity-dependent damping confirms that only the H-K prescription prevents unphysical drift.
Connection between Discrete and Continuous Descriptions:
In the discrete shot noise model, the authors find that avoiding the Brillouin paradox requires the causality parameter $\tilde{\alpha} = 1/2$ .
- Taking the continuous limit of the master equation reveals a direct correspondence: $\alpha = 2\tilde{\alpha}$ .
- Consequently, the discrete choice of $\tilde{\alpha} = 1/2$ (which ensures thermodynamic consistency in the jump process) maps exactly to the continuous H-K prescription ( $\alpha = 1$ ).
- However, a critical distinction is noted: while the H-K prescription is mandatory for the continuous Gaussian model to exist for any voltage, the generalized Johnson-Nyquist relation derived from the discrete shot noise model is recovered only at low voltages ( $|q_e u| \ll k_B T$ ). At higher voltages, the discrete nature of the charge prevents the simple recovery of the generalized relation found in the continuous limit.

Significance and Scope
The paper establishes that thermodynamics imposes strict constraints on the modeling of noise in nonlinear resistors. It proves that a nonlinear resistor can be described by Gaussian noise, but only if the stochastic differential equation utilizes the Hänggi-Klimontovich prescription and the noise spectral density follows the generalized Johnson-Nyquist relation. Furthermore, it bridges the gap between discrete and continuous theories, showing that the specific causality choice required to avoid the Brillouin paradox in discrete jump processes naturally converges to the H-K prescription in the continuous limit.

The authors remain modest regarding the scope of these findings. They explicitly state that the results apply to classical fluctuations in a single nonlinear resistor coupled to a single degree of freedom. The work does not address quantum effects (relevant at low temperatures) or the retroactive effects arising from complex electromagnetic environments (impedances), noting that extending these results to circuits with multiple nonlinear elements and environmental feedback is left for future work. The paper does not propose new experimental setups but rather clarifies the theoretical foundations required to avoid unphysical predictions in existing models.