Related Experiment Video
Updated: Jul 12, 2026
06:55
Synthesis of Cyclic Polymers and Characterization of Their Diffusive Motion in the Melt State at the Single Molecule Level
Published on: September 26, 2016
Particles, trajectories, and diffusion: Random walks in cooling granular gases.
Santos Bravo Yuste1, Rubén Gómez González2, Vicente Garzó1
1Universidad de Extremadura, Departamento de Física and Instituto de Computación Científica Avanzada (ICCAEx), E-06006 Badajoz, Spain.
Physical Review. E
|February 20, 2026
Summary
We derived a new analytical expression for the mean-square displacement (MSD) of a tracer particle in a granular gas. This formula accurately predicts diffusion behavior, outperforming simpler approximations.
Area of Science:
- Physics
- Statistical Mechanics
- Granular Materials
Background:
- Understanding particle diffusion in granular gases is crucial for various applications.
- Tracer particles often have different mechanical properties than the surrounding granular gas.
- Previous models struggled to accurately capture diffusion dynamics under cooling conditions.
Purpose of the Study:
- To develop an accurate analytical expression for the mean-square displacement (MSD) of a tracer particle in a 3D granular gas.
- To validate the derived expression against numerical simulations.
- To compare the new analytical results with existing approximations.
Main Methods:
- Series expansion of the MSD based on successive displacements.
- Approximation of the series as a geometric series.
- Derivation of an analytical expression for the geometric series ratio (Ω).
- Validation using the direct simulation Monte Carlo (DSMC) method.
Main Results:
- The MSD series approximates a geometric series with ratio Ω.
- An explicit analytical expression for Ω in 3D granular gases was derived.
- The derived MSD formula accurately predicts diffusion, validated by DSMC.
- The new analytical results show improved accuracy over the first-Sonine approximation.
Conclusions:
- The derived analytical expression for MSD provides a simple yet accurate method for predicting tracer diffusion in granular gases.
- The results offer a valuable tool for analyzing granular gas dynamics.
- The findings suggest that simpler analytical models can achieve high accuracy in complex systems.
Related Concept Videos
02:43Basic Postulates of Kinetic Molecular Theory: Particle Size, Energy, and Collision
The ideal-gas equation, which is empirical, describes the behavior of gases by establishing relationships between their macroscopic properties. For example, Charles’ law states that volume and temperature are directly related. Gases, therefore, expand when heated at constant pressure. Although gas laws explain how the macroscopic properties change relative to one another, it does not explain the rationale behind it.
03:48Behavior of Gas Molecules: Molecular Diffusion, Mean Free Path, and Effusion
Although gaseous molecules travel at tremendous speeds (hundreds of meters per second), they collide with other gaseous molecules and travel in many different directions before reaching the desired target. At room temperature, a gaseous molecule will experience billions of collisions per second. The mean free path is the average distance a molecule travels between collisions. The mean free path increases with decreasing pressure; in general, the mean free path for a gaseous molecule will be...
01:27Distribution of Molecular Speeds
The motion of molecules in a gas is random in magnitude and direction for individual molecules, but a gas of many molecules has a predictable distribution of molecular speeds. This predictable distribution of molecular speeds is known as the Maxwell-Boltzmann distribution. The distribution of molecular speeds in liquids is comparable to that of gases but not identical and can help to understand the phenomenon of the boiling and vapor pressure of a liquid. Consider that a molecule requires a...
01:20Maxwell-Boltzmann Distribution: Problem Solving
Individual molecules in a gas move in random directions, but a gas containing numerous molecules has a predictable distribution of molecular speeds, which is known as the Maxwell-Boltzmann distribution, f(v).
This distribution function f(v) is defined by saying that the expected number N (v1,v2) of particles with speeds between v1 and v2 is given by
This distribution function f(v) is defined by saying that the expected number N (v1,v2) of particles with speeds between v1 and v2 is given by
01:22Mean free path and Mean free time
Consider the gas molecules in a cylinder. They move in a random motion as they collide with each other and change speed and direction. The average of all the path lengths between collisions is known as the "mean free path."
01:24The Kinetic Model of Gases
The kinetic model of gases explains the properties of a perfect gas using three main assumptions: molecules move in ceaseless random motion, their size is negligible compared to the distances between them, and they do not interact except during perfectly elastic collisions. The total energy of a gas is the sum of the kinetic energies of all its constituent molecules. The pressure exerted by the gas arises from the continual bombardment of the container walls by billions of colliding molecules.