Numerical analysis of the thermal relaxation of the dense gas between two parallel plates: the free energy monotonicity for the Enskog equation

This paper numerically investigates the thermal relaxation of dense gas between parallel plates using two versions of the Enskog equation, revealing that the non-equilibrium free energy decreases monotonically over time only when employing a recently proposed modified Enskog factor, unlike the original formulation.

The Gist

The Big Picture: A Crowded Dance Floor

Imagine a very crowded dance floor (a dense gas) trapped between two walls. The walls are kept at a perfect, constant temperature (like a thermostat).

In a normal, empty room, people (gas molecules) move around freely, bumping into each other occasionally, and eventually, everyone settles into a calm, steady rhythm. Physics has a great rulebook for this called the Boltzmann equation. It predicts that the "disorder" (entropy) of the room will always increase, or conversely, the "free energy" (the potential to do work) will always go down until things settle. This is like the Second Law of Thermodynamics: things naturally move toward a state of rest and balance.

But what happens when the room is packed tight?
When the dance floor is so crowded that people are shoulder-to-shoulder, the simple rulebook breaks down. The molecules are no longer just points; they have size, and they can't occupy the same space. This is where the Enskog equation comes in. It's a more complex rulebook designed for crowded gases.

The Problem: Two Versions of the Rulebook

The researchers in this paper are testing two different versions of this "crowded room" rulebook:

1. The Original Rulebook (OEE): This is the classic version proposed by Enskog decades ago. It tries to account for the size of the molecules.
2. The New, Tweaked Rulebook (EESM): This is a newer version proposed by the authors themselves. They made a small, clever adjustment to how the rulebook calculates the likelihood of molecules bumping into each other.

The Goal: They wanted to see which rulebook actually follows the "Law of Relaxation." If you start with a chaotic, uneven crowd, does the "Free Energy" (the measure of how far the system is from perfect calm) go down smoothly and steadily until everyone is settled?

The Experiment: Simulating the Dance

Since they can't build a giant, perfect gas chamber in a lab to test this, they used a supercomputer to simulate it.

• The Setup: They created a virtual gap between two plates.
• The Start: They started with a "Maxwellian" distribution (a standard, calm speed for everyone) but added a little wave to the density—imagine the crowd is slightly thicker on the left and thinner on the right.
• The Process: They let the simulation run, watching how the gas relaxed back to a uniform state.

The Results: One Rulebook Works, The Other Stumbles

Here is the punchline of their discovery:

1. The New Rulebook (EESM) is Perfect:
When they used the new, tweaked version, the "Free Energy" dropped smoothly and steadily every single second. It was like a ball rolling down a perfectly smooth hill. It never went back up. This proves that their new math is consistent with the fundamental laws of thermodynamics. It behaves exactly as nature should.

2. The Original Rulebook (OEE) is Flawed:
When they used the classic, original version, things got weird. The "Free Energy" went down, but then it would jump back up a little bit before going down again. It was like a ball rolling down a hill, hitting a bump, rolling back up a few inches, and then continuing down.

• Why this matters: In physics, if your math predicts that energy spontaneously increases in a closed system without an external push, your math is broken. The original Enskog equation, while useful for many things, fails this specific "thermodynamic sanity check."

The "Ghost" in the Machine: Density Profiles

The researchers also looked at how the density of the crowd changed over time.

• The Surprise: Even though the final state (when everything settled) looked almost identical for both rulebooks, the journey there was different.
• The "New Rulebook" showed the crowd smoothing out in a very specific, logical way.
• The "Original Rulebook" showed the crowd wiggling and shifting in a slightly different, less physically consistent pattern.

The Takeaway: Why Do We Care?

This might sound like a very technical math problem, but it's actually about trust.

As we build smaller and smaller machines (micro-chips, nanobots), the gas inside them behaves like a "dense gas." We need to predict how heat moves and how pressure builds up in these tiny spaces. If we use the "Original Rulebook," our computer simulations might give us results that look okay on the surface but violate the fundamental laws of physics (like the ball rolling back up the hill).

The Conclusion:
The authors have proven that their new, slightly tweaked version of the Enskog equation is the correct tool for the job. It guarantees that the physics works out correctly, ensuring that our simulations of dense gases in tiny systems are reliable and obey the laws of nature.

In short: They found a better map for navigating crowded molecular traffic, ensuring that the "energy" of the system always flows in the right direction.

Technical Summary

1. Problem Statement

The paper investigates the thermal relaxation process of a dense gas confined between two parallel, stationary plates maintained at a constant, uniform temperature ($T_0$). The system starts from a non-equilibrium state characterized by a uniform temperature but a non-uniform density profile (specifically, a sinusoidal variation).

The primary objective is to analyze the time evolution of the Helmholtz free energy ($F$) in this non-equilibrium dense gas system. The study compares two variants of the Enskog equation, which extends the Boltzmann equation to dense gases by accounting for molecular volume and finite-size collision effects:

1. Original Enskog Equation (OEE): Uses the classical Enskog collision factor.
2. Enskog Equation with a Slight Modification (EESM): Uses a modified Enskog factor proposed in Takata & Takahashi (2025) to ensure thermodynamic consistency.

The central question is whether the free energy of the system decreases monotonically over time, as required by the Second Law of Thermodynamics (specifically the H-theorem for systems in contact with a heat bath).

2. Methodology

A. Mathematical Formulation

• Governing Equation: The gas behavior is modeled using the Enskog equation for hard-sphere molecules. The collision integral includes the Enskog factor ($g$), which corrects the collision frequency based on local density and molecular size.
• Two Models for $g$:
  • OEE: $g(X_1, Y_1) = 2S(\frac{4\pi\sigma^3}{3m}\rho(\frac{X_1+Y_1}{2}))$, where $S$ is derived from the Carnahan-Starling equation of state.
  • EESM: $g(X_1, Y_1) = S(R(X_1)) + S(R(Y_1))$, where $R(X_1)$ is a non-local integral of the density. This modification was theoretically proven to satisfy the H-theorem.
• Boundary Conditions: Diffuse reflection at the plates (molecules re-emerge with a Maxwellian distribution at $T_0$).
• Free Energy Definition: The study utilizes an extended definition of free energy ($F$) for non-equilibrium states:
  $$F = RT_0 H + E$$
  Where $H = H^{(k)} + H^{(c)}$ is the generalized H-function (kinetic part + collisional/correlation part), and $E$ is the total energy. For the EESM, it is theoretically proven that $F$ decreases monotonically ($dF/dt \leq 0$).

B. Numerical Approach

• Discretization: The authors employ a hybrid numerical method:
  • Spatial Derivatives: Finite-difference method (upwind schemes).
  • Collision Integral: Fast Fourier Spectral Method (FFSM). This allows for high accuracy in handling the complex collision operator of the Enskog equation.
• Time Integration: A second-order extrapolation scheme (Adams-Bashforth) is used for the collision term to achieve second-order accuracy in time.
• Dimensionless Parameters: The problem is characterized by four dimensionless parameters:
  • $\hat{\sigma} = \sigma/L$ (Molecular size relative to gap width).
  • $\eta_0$ (Reference volume fraction).
  • $\lambda$ (Wavelength of initial density perturbation).
  • $w$ (Amplitude of initial density perturbation).
• Mass Conservation: Since the numerical scheme is not strictly conservative, a normalization step is applied at each time step to control total mass variations.

3. Key Contributions

1. Numerical Validation of the H-Theorem for EESM: The paper provides the first rigorous numerical evidence that the modified Enskog equation (EESM) satisfies the monotonic decrease of free energy in a bounded domain with diffuse reflection boundaries.
2. Demonstration of OEE Failure: The study numerically demonstrates that the Original Enskog Equation (OEE) fails to guarantee the monotonic decrease of free energy. The free energy in OEE simulations exhibits non-monotonic behavior (overshoots), violating the thermodynamic expectation for a system relaxing to equilibrium.
3. Identification of Non-Kinetic Contributions: The analysis reveals that the monotonicity in EESM relies heavily on the non-kinetic (correlation) part of the free energy ($H^{(c)}$). The "ideal gas" part of the free energy ($F - RT_0 H^{(c)}$) does not decrease monotonically in either model, highlighting the critical role of dense-gas correlations in thermodynamic consistency.
4. Transient vs. Steady-State Differences: The study shows that while OEE and EESM converge to nearly identical final steady-state density profiles, their transient time evolutions (density and temperature profiles) differ significantly, particularly when the initial density perturbation is large.

4. Key Results

• Free Energy Evolution:

  • EESM: The total free energy $F$ decreases strictly monotonically over time, confirming the theoretical H-theorem.
  • OEE: $F$ decreases initially but exhibits non-monotonic behavior (overshoots) before eventually settling. This indicates that OEE does not strictly obey the Second Law of Thermodynamics in this non-equilibrium context.
  • Dependence on Perturbation: The deviation between OEE and EESM becomes more pronounced as the amplitude ($w$) of the initial density variation increases.

• Density and Temperature Profiles:

  • Steady State: Both models predict the same non-uniform density profile at equilibrium (higher density near walls due to excluded volume effects), consistent with literature.
  • Transient Phase: Significant differences are observed in the time evolution of density and temperature. For example, at specific time steps (e.g., $\hat{t} \approx 0.08$), the temperature profiles between OEE and EESM show distinct deviations, even if the final states are similar.

• Sensitivity to Molecular Size: The "overshoot" phenomenon in OEE occurs at consistent time scales relative to the molecular collision time ($t_\sigma^0$), regardless of the specific ratio of molecular size to gap width ($\sigma/L$).

5. Significance

• Thermodynamic Consistency in Kinetic Theory: This work resolves a long-standing issue regarding the thermodynamic consistency of the Original Enskog Equation. It confirms that while OEE is widely used, it lacks the rigorous H-theorem guarantee in bounded, non-equilibrium systems.
• Validation of Modified Models: The results strongly support the adoption of the EESM (Modified Enskog Equation) for numerical simulations of dense gases. EESM ensures that numerical solutions respect fundamental thermodynamic laws (monotonic entropy/free energy production).
• Implications for Micro/Nano-Scale Flows: As microfabrication technology advances, understanding dense gas behavior in sub-micro scales is crucial. This paper provides a validated framework (EESM) for simulating heat transfer and relaxation in these regimes, ensuring that simulations do not produce unphysical results (like free energy increases) that could arise from using the OEE.
• Methodological Advancement: The successful application of the Fast Fourier Spectral Method to the modified Enskog equation demonstrates a robust computational approach for handling complex collision integrals in dense gas dynamics.

In conclusion, the paper establishes that the Enskog equation with the modified factor (EESM) is the thermodynamically consistent choice for simulating dense gas relaxation, whereas the Original Enskog Equation (OEE) can yield physically inconsistent transient behaviors despite converging to the correct steady state.