Passive two-plateau relaxation from Tricomi confluent hypergeometric kernels

This paper introduces a non-fractional, passive framework based on Tricomi confluent hypergeometric functions to model anomalous relaxation with two-plateau dispersive laws, proving its physical realizability and demonstrating its superior accuracy in fitting broadband dielectric and electrochemical data compared to classical models.

The Gist

Imagine you are trying to describe how a sponge soaks up water, or how a battery slowly loses its charge over years. In the world of physics and engineering, these processes are called relaxation.

For a long time, scientists used simple models to describe this, like saying "the sponge dries out at a steady, predictable rate." But real life is messy. Real sponges, tissues, and batteries don't dry out or charge in a straight line. They have "long memories." A sponge might dry fast at first, then very slowly for a long time. A battery might age in complex ways that simple math can't capture.

This paper introduces a new, smarter way to model these messy, "long-memory" behaviors without getting lost in complicated, unmanageable math.

Here is the breakdown of their idea, using some everyday analogies:

1. The Problem: The "One-Size-Fits-All" Trap

Imagine you are trying to tune a radio.

• The Old Way (Fractional Models): Scientists used to use "fractional calculus" (math with weird, non-whole-number exponents) to describe these messy signals. It's like having a radio dial that can stop at any point, even between the numbers. It's incredibly flexible and fits the noise perfectly. But, it's hard to build a physical radio that works that way, and it's hard to simulate on a computer because the math is "non-local" (it needs to remember everything that happened since the beginning of time).
• The Goal: We need a model that is as flexible as the fractional one but behaves like a standard, buildable electronic circuit (like a resistor and a capacitor) that a computer can easily simulate.

2. The Solution: The "Tricomi" Magic Ingredient

The authors found a special mathematical function called the Tricomi confluent hypergeometric function.

• The Analogy: Think of this function as a super-elastic rubber band.
  • Standard rubber bands (simple models) stretch and snap back in a predictable, exponential way.
  • This "Tricomi rubber band" is weird. It can stretch slowly at first, then speed up, then slow down again. It can mimic almost any shape of "memory" you throw at it.
• The Catch: On its own, this rubber band is too wild. If you pull it too hard (at very low frequencies), it might stretch infinitely, which doesn't make sense in the real world (nothing has infinite resistance).

3. The Trick: The "Möbius Normalization" (The Safety Valve)

To fix the "infinite stretch" problem, the authors put the Tricomi rubber band inside a smart safety valve (a mathematical transformation called a Möbius map).

• The Analogy: Imagine a water pipe with a pressure gauge. If the pressure gets too high, the valve automatically limits the flow to a safe maximum. If the pressure drops too low, it limits the flow to a safe minimum.
• The Result: This creates a "Two-Plateau" system.
  • Low Frequency (Slow time): The system settles at a specific, stable value (like a fully charged battery).
  • High Frequency (Fast time): It settles at a different, stable value (like a battery that can't react instantly).
  • The Middle: In between, it flows smoothly and messily, just like real life, but it never breaks the laws of physics (it stays "passive," meaning it doesn't create energy out of thin air).

4. Why This is a Big Deal

The authors proved three amazing things about this new model:

1. It's Physically Real: Because of the math they used, they can prove this model represents a real, buildable circuit. It won't violate the laws of thermodynamics.
2. It's Tunable: You can adjust two "knobs" (parameters $a$ and $b$) to change how the system behaves.
   • One knob controls how it behaves at slow speeds (low frequency).
   • The other knob controls how it behaves at fast speeds (high frequency).
   • Old models usually forced the slow and fast behaviors to be linked. This model lets you tune them independently, giving it much more flexibility.
3. It's Computer-Friendly: They showed how to turn this complex, continuous rubber band into a stack of simple, standard electronic components (resistors and capacitors). This means engineers can put this model into standard circuit simulators without needing supercomputers.

5. Testing it in the Real World

They tested this new model on two very different things:

• Human Tissues (The "Sponge" Test): They tried to model how electricity moves through heart muscle, fat, and kidney tissue.
  • Result: The new model fit the data much better than the old "Cole-Cole" models. It could see subtle details in the data that the old models missed, almost like upgrading from a standard-definition TV to 4K.
• Batteries (The "Aging" Test): They looked at how lithium-ion batteries change as they get old.
  • Result: The model didn't just fit the numbers; it told a story. It showed that as batteries age, their internal "dissipation" (energy loss) shifts toward slower, heavier processes. It's like watching a runner slow down not just because they are tired, but because their shoes have changed weight.

Summary

Think of this paper as inventing a new type of Lego brick.

• Old bricks (Debye models) were simple but couldn't build complex shapes.
• Fractional models were like liquid clay—super flexible but hard to stack into a stable tower.
• This Tricomi brick is the best of both worlds: it's flexible enough to build complex, messy shapes (like real biological tissues or aging batteries), but it's solid enough to stack into a stable, predictable tower that engineers can actually use.

It gives scientists a powerful new tool to understand the "long memories" of the world around us, from our bodies to our phones.

Technical Summary

1. Problem Statement

Anomalous relaxation phenomena, characterized by non-exponential decay and broad distributions of relaxation times, are ubiquitous in disordered solids, soft matter, biological tissues, and electrochemical interfaces.

• Limitations of Current Models:
  • Single-time constant models (Debye): Fail to capture the dispersive behavior of heterogeneous materials.
  • Fractional-order models (e.g., Fractional Cole-Cole, Havriliak-Negami): While efficient at capturing power-law tails with few parameters, they suffer from intrinsic non-locality in time. This makes long-time time-domain simulations computationally expensive and complicates integration into standard circuit or state-space solvers. Furthermore, fractional parameters can lead to spectral ambiguity and degeneracy in practical identification.
• The Gap: There is a need for a non-fractional, passive framework that:
  1. Captures broad dispersive transitions and power-law regimes.
  2. Enforces physically meaningful low-frequency (DC) and high-frequency plateaus.
  3. Guarantees passivity, causality, and complete monotonicity.
  4. Admits a constructive finite-dimensional realization (rational approximation/state-space) for simulation and circuit implementation.

2. Methodology

The authors propose a novel modeling framework based on the Tricomi confluent hypergeometric function, denoted as $U(a, b, z)$.

A. Mathematical Construction

1. Core Function: The model utilizes the integral representation of the Tricomi function $U(a, b, z)$, which acts as a Laplace transform of a non-negative kernel $g_{a,b}(t)$. This ensures the underlying memory kernel is physically admissible (non-negative and completely monotone).
2. Bounded Normalization: To address the divergence of the bare Tricomi function at low frequencies, the authors introduce a bounded Möbius normalization:
   $$\Phi(z) = \frac{z}{1+z}$$
   The normalized mapping is defined as $W_U(s) = \Phi(U(a, b, s\tau))$. This transformation enforces:
   • A unit low-frequency plateau ($W_U \to 1$ as $s \to 0$).
   • A zero high-frequency plateau ($W_U \to 0$ as $s \to \infty$).
3. Impedance Element: The final impedance model $Z_U(s)$ is anchored between prescribed resistive limits $R_0$ (low-frequency) and $R_\infty$ (high-frequency):
   $$Z_U(s) = R_\infty + (R_0 - R_\infty) W_U(s)$$

B. Theoretical Properties

• Asymptotic Behavior: The model exhibits two independent algebraic regimes controlled by parameters $a$ and $b$:
  • Low-frequency: Approaches the DC plateau with an exponent $b-1$.
  • High-frequency: Decays from the high-frequency plateau with an exponent $a$.
  • This allows for asymmetric two-plateau responses, unlike symmetric models like Cole-Cole.
• Passivity and Stieltjes Representation: For the parameter range $a \in (0, 1)$ and $b \in (1, 2)$, the authors prove that the normalized mapping admits a Stieltjes representation with a non-negative spectral density. This mathematically guarantees:
  • Complete monotonicity of the time-domain impulse response.
  • Passivity (Positive Realness).
  • Causality.
• Subcases: The framework naturally contains the Debye model ($a=1, b=2$) and the Cole-Cole model ($b=a+1$) as exact subfamilies.

C. Discretization and Realization

To enable practical simulation, the authors derive a Gauss–Stieltjes discretization:

• The continuous spectral integral is approximated using Gauss quadrature on the log-rate axis.
• This yields a Foster-type rational approximation (sum of first-order terms) with positive poles and residues.
• The result is a first-order state-space realization that is strictly passive and compatible with standard circuit solvers.

3. Key Contributions

1. New Passive Framework: Introduction of a non-fractional, bounded two-plateau relaxation model based on Tricomi functions that guarantees passivity and complete monotonicity.
2. Asymmetric Flexibility: The ability to independently tune low-frequency ($b-1$) and high-frequency ($a$) power-law exponents, extending beyond the symmetric constraints of Cole-Cole.
3. Constructive Realization: A rigorous derivation of passive, finite-dimensional rational approximations (state-space models) from the continuous special-function kernel, bridging the gap between spectral theory and circuit implementation.
4. Theoretical Proof: Proof of the Stieltjes property and non-negative spectral density for the normalized mapping, ensuring physical consistency.

4. Results and Validation

The framework was validated against experimental data in two distinct domains:

A. Biological Tissue Dielectric Spectroscopy

• Data: Broadband dielectric data (Gabriel dataset) for heart muscle, breast fat, white matter, and kidney.
• Performance: Multi-block Tricomi mixtures ($N=2$ to $4$) achieved systematically lower complex-domain errors (RMSE reduced by ~38–63%) compared to classical Cole-Cole baselines.
• Modal Analysis: The model successfully resolved distinct relaxation modes. Bootstrap analysis confirmed that additional blocks were physically supported in highly dispersive tissues (kidney, heart) but identified overfitting in low-loss tissues (breast fat), demonstrating the model's ability to distinguish structured dispersion from noise.

B. Battery Electrochemical Impedance Spectroscopy (EIS)

• Data: Lithium-ion battery impedance spectra across different aging cycles.
• Findings: The model captured the evolution of aging by resolving the redistribution of dissipative dynamics.
  • Aging Effect: The intermediate-frequency (MF) mode shifted systematically toward lower frequencies (slower time scales), and the high-frequency (HF) mode broadened and merged with the MF mode.
  • Interpretation: This revealed a progressive slowing of interfacial and mesoscopic transport processes, a level of detail difficult to extract from global fractional fits.

C. Numerical Convergence

• The Gauss–Stieltjes rational approximations showed monotonic convergence to the continuous response as the number of terms ($M$) increased.
• Even in "long-tail" regimes (small $a$), high-order approximations ($M=45$) achieved indistinguishable accuracy from the continuous model, validating the method for finite-dimensional simulation.

5. Significance and Impact

This work provides a constructive alternative to fractional-order models for anomalous relaxation. Its significance lies in the unique combination of four features rarely found together in a single model class:

1. Bounded Two-Plateau Behavior: Physically realistic limits at DC and high frequencies.
2. Certified Passivity: Guaranteed by the Stieltjes representation and non-negative spectral density.
3. Explicit Spectral Structure: Clear interpretation of low/high-frequency exponents and modal distributions.
4. Constructive Rational Realization: Direct conversion to state-space/circuit models without complex fractional calculus solvers.

The framework is particularly valuable for applications requiring interpretable modal structure (e.g., identifying aging mechanisms in batteries) and efficient simulation (e.g., circuit co-simulation), offering a middle ground between the simplicity of fractional phenomenology and the complexity of unconstrained multi-pole fitting.