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Related Concept Videos

Distribution of Molecular Speeds01:27

Distribution 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...
Maxwell-Boltzmann Distribution: Problem Solving01:20

Maxwell-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
Entropy02:39

Entropy

Salt particles that have dissolved in water never spontaneously come back together in solution to reform solid particles. Moreover, a gas that has expanded in a vacuum remains dispersed and never spontaneously reassembles. The unidirectional nature of these phenomena is the result of a thermodynamic state function called entropy (S). Entropy is the measure of the extent to which the energy is dispersed throughout a system, or in other words, it is proportional to the degree of disorder of a...
Bernoulli's Equation00:59

Bernoulli's Equation

In the middle of the nineteenth century, it was observed that two trains passing each other at a high relative speed get pulled towards each other. The same occurs when two cars pass each other at a high relative speed. The reason is that the fluid pressure drops in the region where the fluid speeds up. As the air between the trains or the cars increases in speed, its pressure reduces. The pressure on the outer parts of the vehicles is still the atmospheric pressure, while the resultant...
The Kinetic Model of Gases01:24

The 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.
Energy Conservation and Bernoulli's Equation01:16

Energy Conservation and Bernoulli's Equation

Applying the conservation of energy principle or the work-energy theorem to an incompressible, inviscid fluid in laminar, steady, irrotational flow leads to Bernoulli's equation. It states that the sum of the fluid pressure, potential, and kinetic energy per unit volume is constant along a streamline.
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Updated: Jun 14, 2026

An Analog Macroscopic Technique for Studying Molecular Hydrodynamic Processes in Dense Gases and Liquids
11:03

An Analog Macroscopic Technique for Studying Molecular Hydrodynamic Processes in Dense Gases and Liquids

Published on: December 4, 2017

Boltzmann equation and hydrodynamic fluctuations.

Matteo Colangeli1, Martin Kröger, Hans Christian Ottinger

  • 1Polymer Physics, Department of Materials, ETH Zürich, CH-8093 Zürich, Switzerland.

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|April 7, 2010
PubMed
Summary
This summary is machine-generated.

This study derives generalized hydrodynamics equations from the Boltzmann equation using invariant manifolds, yielding exact transport coefficients. Findings are validated with Maxwell molecules, showing accurate density fluctuation spectra for finite Knudsen numbers.

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Published on: February 22, 2018

Area of Science:

  • Physics
  • Statistical Mechanics
  • Fluid Dynamics

Background:

  • The linearized Boltzmann equation is a fundamental tool for describing dilute gases.
  • Generalized hydrodynamics extends classical fluid dynamics to systems with memory effects.
  • Transport coefficients are crucial for understanding macroscopic material properties.

Purpose of the Study:

  • To derive generalized hydrodynamics equations from the linearized Boltzmann equation.
  • To determine exact transport coefficients using invariant manifold methods.
  • To investigate density fluctuations and hydrodynamic regimes with finite Knudsen numbers and frequencies.

Main Methods:

  • Application of the invariant manifold method.
  • Derivation of generalized hydrodynamics equations.
  • Numerical calculations for Maxwell molecules.
  • Comparison with experimental data and previous theoretical approaches.

Main Results:

  • Exact transport coefficients obeying Green-Kubo formulas were determined.
  • The spectrum of density fluctuations was investigated.
  • The model successfully addresses finite Knudsen numbers and finite frequencies in hydrodynamics.

Conclusions:

  • The invariant manifold method provides a rigorous framework for deriving generalized hydrodynamics.
  • The derived transport coefficients are exact and consistent with Green-Kubo relations.
  • The study offers insights into non-equilibrium statistical mechanics and fluid behavior under specific conditions.