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Reconsidering Einstein’s Derivation of Diffusion Equation to Provide Solutions Consistent with Nano-Scale and Multi-Scale Physics

This paper proposes a corrected diffusion equation derived by revisiting Einstein's original method to retain higher-order terms, thereby preserving fluid structure information and enabling accurate modeling of mass transport at nano-scales and in multi-scale systems.

Original authors: Gregory Aranovich

Published 2026-07-14

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Gregory Aranovich

Original paper licensed under CC BY 4.0 (https://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 describe how a crowd of people moves through a hallway. For over a century, scientists have used a very smooth, very calm map to predict this movement. This map, based on a famous idea by Albert Einstein, treats the crowd like a flowing river of water. It assumes that if you zoom in close enough, the movement is so tiny and continuous that you can ignore the fact that people are actually distinct individuals taking distinct steps.

But what if that smooth map is actually hiding the most interesting parts of the story?

In this paper, Gregory Aranovich suggests that the classic "smooth river" map is missing a crucial detail: the actual size of the steps people take. He argues that by smoothing everything out, we accidentally "kill" the physics that happens at the tiny, nano-scale level and in complex, multi-scale environments.

The "Smoothie" vs. The "Chunky" Reality

To understand the problem, imagine Einstein's original math as a blender. You put in the real, messy data of how molecules jump around—some jump a little, some jump a lot. Einstein's classic method blended this data until it became a smooth liquid, throwing away the "chunks" (the higher-order terms) because they seemed too small to matter.

The result was the famous diffusion equation, which looks like a gentle, continuous curve. But Aranovich says, "Wait a minute!" He argues that those "chunks" we threw away actually hold the secret to how fluids really work. By blending them away, we lost information about the fluid's structure.

Aranovich went back to Einstein's original blender and decided not to smooth it out. Instead of turning the data into a smooth liquid, he kept it "chunky." He derived a new equation that looks like a finite difference equation. Think of this not as a smooth river, but as a staircase. It acknowledges that molecules don't glide; they hop. They jump from one spot to another, and the size of that jump is the "mean free path" (let's call it $\lambda$ ).

Why the Smooth River Fails

The paper shows that when you use the smooth, classic equation in certain situations, it predicts things that are physically impossible.

Imagine a pipe where gas is flowing from a crowded end to a vacuum (an empty end). The classic equation suggests that as the empty end gets emptier, the flow of gas should shoot up to infinity. That's like saying if you open a door to an empty room, the wind should blow so hard it tears the house apart. The paper demonstrates that this "singularity" is a mathematical error caused by the smoothing process.

In contrast, the new "chunky" equation (the staircase model) handles this vacuum scenario perfectly. It predicts a steady, reasonable flow that makes sense, even when the other end is empty. It fixes the "unphysical" behavior by remembering that molecules have a finite size and take finite steps.

The Secret Life of Molecules

The paper also tackles a big question: Do molecules move randomly, like a drunk person stumbling in a straight line (a "Markovian" process), or is there a pattern?

The authors argue that real fluids are not purely random. They point out that after a molecule collides with another, its next collision isn't totally random. There's a "sweet spot" where the next collision is most likely to happen. It's like playing a game of pinball; the ball doesn't hit the bumpers at completely random spots. It tends to hit areas where the geometry of the machine guides it.

Because of this, the paper suggests that fluid diffusion has a "memory." The next step depends on where the molecule just was. This makes the process "Non-Markovian," meaning the old smooth equation, which assumes total randomness, is fundamentally incomplete.

What This Means for Tiny Worlds

The authors show that their new "chunky" equation is much better at describing the nano-scale world. For example:

Near Surfaces: When fluid gets close to a wall, the density of molecules bounces up and down in waves. The classic smooth equation says the density should just be flat and boring. The new equation allows for these waves, matching what we see in computer simulations of hard disks bouncing between walls.
Adsorption: It can describe how molecules stick to surfaces (adsorption) in a way the old equation cannot, because the old equation forces the density to be constant right at the wall.

The Bottom Line

This paper doesn't claim to have solved every mystery of fluid dynamics. Instead, it suggests that the classic, smooth diffusion equation is a useful approximation for big, slow things, but it fails when we look at the nano-scale or when conditions change rapidly.

By keeping the "steps" (the mean free path) in the math instead of blending them away, the new finite difference equation:

Fixes the impossible predictions (like infinite flow in a vacuum).
Captures the wiggly, wave-like density patterns near surfaces.
Acknowledges that molecules have a structure and a memory between collisions.

In short, the paper argues that to understand the tiny, complex world of nano-physics, we need to stop pretending the world is a smooth river and start treating it like a staircase of hops. The math is still there, but now it's got some "meat" on the bones.

Technical Summary: Reconsidering Einstein's Derivation of Diffusion Equation

Problem Statement
The paper identifies a fundamental limitation in the classical diffusion equation, which serves as the core of modern fluid dynamics and mass transfer models. The author argues that the classical equation fails at the nano-scale and in multi-scale physics because it lacks information regarding the fluid's molecular structure. This deficiency arises from the historical derivation by Einstein, which relies on a mathematical truncation of higher-order terms in a Taylor series expansion. By discarding these terms, the derivation eliminates the mean free path ( $\lambda$ ) as an independent parameter, combining it with velocity ( $V$ ) into a single diffusion coefficient ( $D$ ). Consequently, the classical differential equation cannot capture phenomena dependent on discrete molecular steps, such as local density fluctuations, pair correlations, or adsorption mechanisms near surfaces.

Methodology
The author re-examines Einstein's original integral evolution equation for diffusion in fluids:
$n(x, t+\tau) = \int_{-\infty}^{\infty} W(\xi)n(x+\xi, t)d\xi$
where $n(x,t)$ is the density distribution, $\tau$ is the time step, and $W(\xi)$ is the symmetric probability density of a jump $\xi$ .

Instead of applying the standard Taylor series expansion and truncating higher-order derivatives (the classical approximation), the author performs an exact derivation without truncation. By utilizing the symmetry of $W(\xi)$ and combining expansions for $n(x+\xi, t)$ and $n(x-\xi, t)$ , the author derives a finite difference equation for mass transport:
$n(x, t+\tau) - n(x, t) = \frac{1}{2} \int_{-\infty}^{\infty} W(\xi)[n(x+\xi, t) + n(x-\xi, t) - 2n(x, t)]d\xi$
This equation governs diffusion via the averaging of a discrete Laplacian rather than a continuous second derivative. The paper analyzes this model using:

Separation of variables for infinite space solutions.
Simplistic free path distributions (Dirac delta functions) to illustrate the role of the mean free path $\lambda$ .
Thermodynamic equilibrium limits to compare classical and finite difference predictions for zero flux.
Variable mean free path scenarios to model multi-scale diffusion (e.g., gas diffusing from high pressure to vacuum).

Key Contributions and Results

Derivation of the Finite Difference Diffusion Model: The primary contribution is the exact derivation of a finite difference equation that retains the mean free path ( $\lambda$ ) as a distinct physical parameter. This equation is shown to be consistent with Einstein's original integral equation without the loss of structural information.
Consistency with Nano-Scale Physics:
- The paper demonstrates that the classical steady-state solution ($dn/dx = 0$) implies a constant density, which cannot accommodate adsorption profiles or pair correlation functions.
- In contrast, the finite difference model ( $n(x-\lambda) - n(x+\lambda) = 0$ ) admits multiple solutions, including periodic functions. The author shows that pair correlation functions (describing local density fluctuations) and adsorption density profiles near surfaces can be represented as Fourier series where each term satisfies the finite difference equation.
- New boundary conditions are proposed that explicitly represent adsorption mechanisms, linking surface density to bulk flux via molecular interaction energies.
Resolution of Multi-Scale Anomalies:
- In a scenario involving variable density (e.g., gas flowing from atmospheric pressure to vacuum), the classical approximation predicts an unphysical flux singularity (divergence) as the density ratio approaches zero.
- The finite difference model predicts a finite, physically reasonable flux in the same limit. The model correctly transitions to a "one-way" flux behavior at vacuum conditions where collisions are negligible, avoiding the mathematical singularity inherent in the differential form.
Non-Markovian Nature of Diffusion: The paper argues that real fluid diffusion is Non-Markovian. While classical models assume random, uncorrelated jumps, the finite difference approach acknowledges that the location of a second collision is correlated with the first (occurring most likely within a specific "green" zone relative to the first collision). The duration of motion between collisions (non-stochastic) often significantly exceeds the collision time (stochastic), introducing memory effects that differential equations cannot capture.

Significance and Claims
The paper concludes that reducing the diffusion model to a differential equation effectively "kills" nano-scale and multi-scale physics by removing the mean free path as an independent variable. The author claims that:

The finite difference diffusion equation is the mathematically correct representation of Einstein's evolution equation, preserving the physical scale of the fluid structure.
This model resolves fundamental defects in classical theory, such as causality problems and the inability to model adsorption or density fluctuations at the molecular scale.
The step size in the derived finite difference equation is not a computational parameter but a physical characteristic (the mean free path) determined by collision cross-sections and fluid structure.
Real fluid diffusion involves correlations between consecutive collisions, making the process Non-Markovian and necessitating a model that does not rely on the limit of infinitesimal steps ( $\lambda \to 0$ ).

The work suggests that this corrected framework is essential for applications in nanotechnology and turbulent flows where fluid structure and multi-scale variations are critical.

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