Extended dynamical density functional theory for nonisothermal binary systems including momentum density

This paper derives a new extended dynamical density functional theory (EDDFT) for nonisothermal binary systems by incorporating momentum and energy densities via the Mori-Zwanzig-Forster projection operator technique, thereby enabling the description of both diffusive and convective dynamics while yielding exact functionals for hard spheres and correctly predicting the speed of sound.

The Gist

Imagine you are trying to predict how a crowd of people moves through a busy train station.

Standard theories (like the original Dynamical Density Functional Theory, or DDFT) are like watching a time-lapse video of the crowd. They are great at telling you where the people are on average and how they slowly drift toward the exits (diffusion). But they fail miserably if a sudden rush happens, if the crowd starts running, or if there's a strong wind blowing them in a specific direction. They assume everyone is just shuffling slowly, ignoring the fact that people have momentum and can crash into each other or get pushed by a gust of air.

This new paper introduces an "Extended" version of that theory (EDDFT) that fixes these problems. It's like upgrading from a time-lapse video to a high-speed, 4K action camera that captures every sprint, every collision, and every gust of wind.

Here is a breakdown of what the authors did, using simple analogies:

1. The Problem: The "Slow Shuffle" vs. The "Sprint"

The old theory was designed for "overdamped" systems—think of people wading through waist-deep water. They move slowly, and if you stop pushing, they stop immediately.

• Real life: But in many situations (like metal alloys cooling down, cement flowing, or even how a virus spreads through air currents), things have inertia. They keep moving even after you stop pushing them. They have momentum.
• The Fix: The authors added Momentum Density to the equation. Now, the theory doesn't just track where the particles are; it tracks how fast they are going and where they are heading. It accounts for the "oomph" behind the movement.

2. The Ingredients: The "Four Dials"

To describe this complex dance of particles, the authors built a model with four main dials (variables) that they track simultaneously:

1. Total Mass Density: How crowded is the area? (Is it a packed station or empty?)
2. Concentration: What is the mix? (Are there more "colloidal particles" like sand, or "solvent" like water?)
3. Momentum Density: How fast is the crowd moving, and in what direction? (The "flow" or wind).
4. Energy Density: How hot is it? (Temperature).

Why this matters: In the old theory, temperature was often assumed to be constant (like a room with perfect AC). But in reality, friction creates heat, and heat changes how things flow. This new theory connects heat, flow, and concentration all in one big equation.

3. The Magic Trick: The "Projection Operator"

How did they get from the chaotic motion of billions of tiny atoms to a clean, usable equation?

• The Analogy: Imagine trying to describe a chaotic mosh pit. You could try to track every single person's footstep (impossible!). Instead, the authors used a mathematical tool called the Mori-Zwanzig-Forster projection.
• The Metaphor: Think of this as a "smart camera filter." It ignores the tiny, fast, random jiggling of individual atoms and "projects" the image down to the big, slow, important patterns (like the overall flow of the crowd). It filters out the noise to reveal the signal.

4. The "Hard Sphere" Recipe

The authors didn't just write abstract math; they cooked up a specific "recipe" for a very common type of particle: Hard Spheres (think of billiard balls or marbles that can't overlap).

• They derived an exact formula for the Entropy (disorder) and Free Energy (potential to do work) of these bouncing balls.
• Why it's cool: This allows scientists to plug this new theory directly into existing computer simulations used for materials science. It's like giving a chef a new, precise recipe that works with the old kitchen equipment.

5. The "Speed of Sound" Test

One of the biggest achievements of this paper is that it correctly predicts the speed of sound.

• The Old Way: Previous theories treated sound waves as if the temperature stayed perfectly constant. But sound waves are actually adiabatic—they compress and heat up the air as they travel. The old theories got the speed wrong because they ignored this heating effect.
• The New Way: Because this new theory tracks Energy (Heat) and Momentum together, it realizes that when a sound wave squeezes particles, they get hotter, which changes how fast the wave moves.
• The Result: The theory now gives the exact right answer for the speed of sound, proving it captures the physics correctly.

6. Real-World Applications

Why should a regular person care? This theory is a Swiss Army knife for engineers and scientists:

• Industrial Mixing: Better models for mixing cement or metal alloys, where heat and flow are critical.
• Airborne Disease: Understanding how viruses travel in air currents (convection) rather than just drifting slowly.
• Battery Tech: Modeling how ions move and heat up inside batteries.
• Glass Formation: Understanding why liquids turn into glass (a "glass transition") when they cool down too fast.

Summary

In short, the authors took a theory that was good at describing "slow, lazy" particle movement and supercharged it. They added momentum (speed/force) and energy (heat) to the mix. They used a clever mathematical filter to simplify the chaos of billions of atoms into a set of equations that can predict how complex fluids flow, heat up, and even carry sound waves correctly. It's a bridge between the microscopic world of atoms and the macroscopic world of flowing liquids and gases.

Technical Summary

Here is a detailed technical summary of the paper "Extended dynamical density functional theory for nonisothermal binary systems including momentum density" by Michael te Vrugt et al.

1. Problem Statement

Standard Dynamical Density Functional Theory (DDFT) is a powerful framework for describing the diffusive dynamics of colloidal suspensions and binary mixtures. However, it has two critical limitations:

1. Lack of Inertia: It fails to describe convective dynamics (fluid flow) because it does not include momentum density as a variable. This makes it unsuitable for systems with significant solvent flow, inertial effects, or rapid changes.
2. Isothermal Assumption: Standard DDFT typically assumes a constant temperature, failing to capture nonisothermal phenomena such as heat transport, thermal convection, and the coupling between temperature gradients and concentration (Ludwig-Soret and Dufour effects).

Existing extensions often address inertia or temperature separately but lack a unified, microscopic derivation that handles both simultaneously for binary mixtures (e.g., colloids in a solvent) while maintaining thermodynamic consistency.

2. Methodology

The authors derive a new Extended Dynamical Density Functional Theory (EDDFT) using the Mori-Zwanzig-Forster Projection Operator Technique (MZFT). This approach projects the microscopic Hamiltonian dynamics of a many-particle system onto a set of slow, relevant macroscopic variables.

Key Steps in the Derivation:

• Microscopic Model: The system is modeled as a binary mixture of $N_c$ colloidal particles and $N_s$ solvent particles with masses $m_c$ and $m_s$, interacting via pair potentials and external fields.
• Relevant Variables: The theory is derived for four specific macroscopic fields:
  1. Total mass density ($\rho_m$)
  2. Local concentration of one species ($c$)
  3. Total momentum density ($\vec{g}$)
  4. Energy density ($\varepsilon$)
• Projection: The MZFT is applied to the Liouville equation. The unknown microscopic phase-space distribution is approximated by a relevant density $\rho(t)$, which is a functional of the average values of the four variables above.
• Thermodynamic Conjugates: The authors define conjugate variables based on the entropy functional $S$ (rather than just free energy), allowing for a natural description of nonisothermal systems. The conjugate to energy density is identified as the inverse temperature $\beta = 1/k_B T$.
• Hard Sphere Approximation: To make the theory practical, the authors derive an explicit form for the entropy functional and the free-energy functional specifically for hard spheres. They utilize Fundamental Measure Theory (FMT) for the excess free energy, generalized to nonisothermal conditions.
• Adiabatic Approximation: The pressure tensor term in the momentum equation is approximated using the functional derivative of the free energy, closing the system of equations.

3. Key Contributions

• Unified Framework: The paper presents the first derivation of EDDFT that simultaneously includes momentum density (inertia/flow) and energy density (temperature/heat) for binary mixtures.
• Exact Entropy Functional: An exact expression for the entropy functional of nonisothermal hard spheres is derived. This allows the use of well-established DFT methods (like FMT) within a dynamic, nonisothermal, inertial context.
• Generalized Navier-Stokes: The resulting equations constitute a set of generalized Navier-Stokes equations for two-component systems. They naturally reduce to standard Navier-Stokes equations for simple liquids and standard DDFT for overdamped, isothermal limits.
• Thermodynamic Consistency: The derivation ensures that the system obeys the H-theorem (entropy production is non-negative), proving the thermodynamic consistency of the approximations made (specifically the Markovian approximation for dissipative terms).

4. Key Results

The derived EDDFT equations (Eqs. 49–52) describe the time evolution of the four fields:

1. Mass Continuity: $\dot{\rho}_m = -\nabla \cdot \vec{g}$
2. Concentration Evolution: Includes convective transport ($\nabla \cdot (c \vec{v})$), diffusive flux driven by concentration gradients, and thermal diffusion (Soret effect).
3. Momentum Evolution (Navier-Stokes-like):
   • Includes the convective term $(\vec{v} \cdot \nabla)\vec{v}$.
   • Includes the divergence of the pressure tensor $\Pi$, which depends on density, concentration, and temperature.
   • Includes viscous dissipation terms derived from the diffusion tensors.
   • Includes external force density.
4. Energy Evolution: Describes the transport of energy via convection, heat conduction, and coupling to concentration gradients (Dufour effect).

Specific Findings:

• Speed of Sound: The authors demonstrate that their EDDFT correctly predicts the speed of sound in liquids. Previous inertial DDFTs failed here because they assumed isothermal conditions for sound waves (which are adiabatic). By including energy density, the theory captures the adiabatic nature of sound propagation.
• Hydrodynamic Limit: In the limit of small wavenumbers and frequencies, the equations reduce to standard hydrodynamic equations for suspensions, recovering the correct transport coefficients (viscosity, thermal conductivity, diffusion).
• Relation to Mode-Coupling Theory (MCT): The paper clarifies the relationship between EDDFT and MCT. While MCT focuses on memory effects in the glass transition, EDDFT provides a complementary framework where memory effects are approximated by Markovian dissipation, valid when relevant variables are slow compared to microscopic dynamics.

5. Significance and Applications

This work significantly extends the applicability of density functional theory to complex industrial and physical scenarios:

• Industrial Applications: It enables the modeling of multiphase flows, metal alloys, and cement flow, where inertia, temperature gradients, and phase separation occur simultaneously.
• Biological Physics: It can describe the airborne transmission of infectious diseases (e.g., COVID-19), where fluid mechanics, temperature gradients, and particle concentration are coupled.
• Soft Matter Physics: It provides a rigorous tool for studying thermal convection, thermodiffusion, and phase transitions in colloidal suspensions.
• Theoretical Advancement: By deriving the correct speed of sound and establishing a clear H-theorem, the paper resolves inconsistencies in previous inertial DDFT approaches and bridges the gap between microscopic statistical mechanics and macroscopic hydrodynamics.

In summary, this paper provides a rigorous, thermodynamically consistent, and practically applicable theoretical framework for simulating complex, nonisothermal, and inertial binary fluid systems.