Nonlinear dynamic elastic moduli from equilibrium stress fluctuations

This paper derives transient-time correlation function expressions for both linear and nonlinear dynamic elastic moduli using equilibrium stress fluctuations and DOLLS/SLLOD equations, thereby enabling the computation of anharmonic viscoelastic responses from equilibrium molecular dynamics simulations without explicit nonequilibrium deformation protocols.

The Gist

Imagine you have a giant, invisible trampoline made of billions of tiny springs and balls (atoms) bouncing around. You want to know how this trampoline reacts when you push or pull it. Does it snap back instantly? Does it wobble? Does it get squishy or stiff depending on how hard you push?

In the world of physics, these reactions are called elastic and viscoelastic moduli. Usually, to measure them, scientists have to physically stretch or squeeze the material in a computer simulation and watch what happens. This is like trying to figure out how a car engine works by actually driving it into a wall over and over again. It works, but it's messy, expensive, and hard to control.

This paper introduces a clever new way to figure out these reactions without ever actually pushing the material.

The "Time Travel" Trick

The authors (Garbuzov and Beltukov) found a mathematical shortcut. They realized that if you just watch the material sitting still at room temperature (in equilibrium), the tiny, random jiggles and fluctuations of the atoms contain all the secret information you need.

Think of it like this: If you stand in a crowded room and watch people bumping into each other randomly, you can actually predict how the crowd would react if someone suddenly started shoving them. You don't need to start the shoving to know the answer; the random bumps already hold the blueprint.

The Problem They Solved

Scientists already knew how to use these "random bumps" to predict:

1. Static reactions: How the material feels when you push it and hold it still.
2. Simple, linear reactions: How it feels when you push it gently and quickly.

But there was a huge gap. No one knew how to use the random bumps to predict complex, changing reactions. What happens if you push the material, then pull it, then push it harder, all in a rhythm? This is called nonlinear dynamic response. It's like trying to predict how a rubber band behaves if you stretch it, let it snap back, and then stretch it again while it's still vibrating. Until now, there was no formula to calculate this just by watching the material sit still.

The Solution: A New Recipe

The authors derived a new "recipe" (a mathematical formula) that acts like a translator.

• The Ingredients: They look at the stress (the internal pressure) and the Born-Kinetic terms (a fancy way of describing the combined energy of the atoms' positions and their speeds).
• The Process: They calculate how these ingredients correlate with each other over time. It's like listening to the rhythm of the random bumps.
• The Result: They get a formula that tells you exactly how the material will react to any complex, time-changing push or pull, just by analyzing the data from a calm, undisturbed simulation.

Why It Matters (According to the Paper)

The paper claims this is a major upgrade because:

1. It's Safer and Cheaper: You don't need to run expensive, difficult "deformation" simulations where you physically stretch the material. You just run a standard simulation of the material sitting still.
2. It's More Accurate: When you try to stretch materials very slightly in a simulation, the signal is often weak and noisy (like trying to hear a whisper in a storm). By using the "random bumps" method, you get a clearer picture without the noise.
3. It Unifies Everything: Their formula is a "master key." If you turn the knobs to zero frequency, it becomes the old static formula. If you turn off the complex parts, it becomes the old linear formula. But it also unlocks the door to the complex, nonlinear world that was previously locked.

The Bottom Line

This paper gives scientists a new tool to predict how materials behave under complex, changing forces. Instead of "breaking" the material in a computer to see how it reacts, they can now just "listen" to the material's natural, random vibrations to predict its future behavior. It turns a chaotic, noisy room of bouncing atoms into a clear instruction manual for how the material will respond to the world.

Technical Summary

Technical Summary: Nonlinear Dynamic Elastic Moduli from Equilibrium Stress Fluctuations

Problem Statement
While fluctuation formulas for computing elastic and viscoelastic moduli from equilibrium molecular dynamics (MD) simulations are well-established for quasi-static (zero-frequency) linear and nonlinear cases, as well as for linear dynamic (time-dependent) moduli, a significant theoretical gap remains. Specifically, no fluctuation formula exists for nonlinear time-dependent moduli. These moduli are required to describe the anharmonic viscoelastic response of materials subjected to small but finite time-dependent strains. Existing methods for obtaining these properties typically rely on explicit nonequilibrium deformation protocols, which can be computationally expensive or difficult to control. This work aims to derive the missing fluctuation expressions for these nonlinear dynamic moduli.

Methodology
The authors derive the required expressions starting from the microscopic equations of motion for a system of interacting particles.

1. Equations of Motion: The derivation utilizes the DOLLS/SLLOD equations of motion. The authors restrict their analysis to irrotational deformations (where the velocity gradient tensor $L$ is symmetric), a condition under which the DOLLS and SLLOD formulations coincide. This allows the use of a Hamiltonian framework augmented by a coupling term between particle momenta and the streaming velocity field.
2. Pulled-Back Variables: To handle the time-dependence of the strain, the authors employ a "pulled-back" representation of phase-space variables (coordinates $\vec{R}_a$ and momenta $\vec{P}_a$). This transformation maps the current deformed state back to a reference configuration, ensuring that the variables remain continuous even when a strain is suddenly applied.
3. Nonequilibrium Distribution: The authors construct the nonequilibrium probability distribution function by evolving the initial canonical equilibrium distribution. They express the nonequilibrium average of the stress tensor as an average over the equilibrium ensemble, weighted by the exponential of the energy change due to phase point evolution ($\Delta H_\Gamma$).
4. Perturbation Expansion: The stress tensor and the energy change are expanded in a power series with respect to the Lagrangian finite strain tensor $E$. The authors separate the contributions into linear and nonlinear terms, identifying "Born–kinetic" tensors ($C^{BK}$ and $N^{BK}$) which represent instantaneous elastic constants derived from the potential and kinetic energy terms.
5. Correlation Function Derivation: By substituting these expansions into the expression for the nonequilibrium stress average and retaining terms up to second order in strain, the authors derive explicit formulas for the dynamic moduli as time-dependent correlation functions involving the stress tensor and the Born–kinetic terms.

Key Contributions and Results
The primary result of the paper is the derivation of closed-form fluctuation expressions for both linear and nonlinear dynamic elastic moduli:

• Linear Dynamic Moduli ($C_{ijkl}$): The authors recover the known linear viscoelastic fluctuation formula (Eq. 45), which expresses the modulus as a combination of the equilibrium average of the Born–kinetic tensor and time-correlation functions of the stress tensor.
• Nonlinear Dynamic Moduli ($N_{ijklmn}$): The central contribution is the derivation of the fluctuation formula for the sixth-order nonlinear dynamic modulus tensor (Eq. 46). This expression involves:
  • The equilibrium average of the nonlinear Born–kinetic tensor ($N^{BK}$).
  • Cross-correlations between the stress tensor and the linear Born–kinetic tensor.
  • Higher-order correlations involving products of stress tensors and the difference between the Born–kinetic tensor at different times.
  • Terms scaling with $\beta^2$ (inverse temperature squared) involving triple stress correlations.

The derived tensors satisfy specific symmetry properties: the Born–kinetic tensors possess both minor and major symmetries, whereas the full time-dependent dynamic moduli satisfy only minor symmetries (and a specific permutation symmetry for the nonlinear tensor).

Significance and Claims
The paper claims that these derived formulas constitute a natural extension of previous work, unifying separate results for quasi-static linear, quasi-static nonlinear, and linear dynamic moduli into a single coherent framework.

The authors emphasize the practical utility of these results:

• Equilibrium-Only Computation: The formulas allow for the computation of the full time-dependent anharmonic response of materials entirely from equilibrium MD simulations. This eliminates the need for explicit nonequilibrium deformation runs.
• Applicability: This approach is particularly advantageous for systems where nonequilibrium protocols are costly, difficult to control, or suffer from poor signal-to-noise ratios at small strains.
• Accessibility: The required correlation functions involve only equilibrium averages of stress, Born–kinetic terms, and their time-correlations, all of which are readily accessible in standard equilibrium simulations.

The authors note that while the current work is restricted to irrotational flows, the formalism provides a foundation for future extensions to rotational flows and higher-order moduli, as well as applications to amorphous solids like polymers and biological materials. They also identify the numerical implementation and validation against direct nonequilibrium simulations as a necessary next step.