Standard Maxwell-Boltzmann distribution: Definition and Properties (original) (raw)
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ForsChem Research Reports, 2019
In this report, the probability distribution of molecular kinetic energy and molecular temperature for systems with known velocity probability distributions is presented. Also, conditional probabilities are determined when one or more velocity components of the molecular are known. Particularly, the mathematical expressions obtained for Maxwell-Boltzmann systems (zero-mean normal distribution of molecular velocity components) are included. When all velocity components are random, the thermal kinetic energy distribution corresponds to a Chi-squared distribution with 3 degrees of freedom.
Standard Maxwell-Boltzmann Distribution: Additional Nonlinear and Multivariate Properties
ForsChem Research Reports, 2017
The standard Maxwell-Boltzmann distribution corresponds to the norm of a three-dimensional vector of independent standard normal random variables. The standard Maxwell-Boltzmann distribution is useful for describing the speed of molecules in a system, as well as their related properties: Momentum, kinetic energy, temperature and other thermodynamic properties of the system, collision rates, molecular fluxes, and diffusion and other transport properties. Some of these applications involve non-linear and multivariate functions of independent standard Maxwell-Boltzmann random variables. The purpose of this paper is to summarize the most important properties of non-linear and multivariate functions of the standard Maxwell-Boltzmann distribution.
The Maxwell-Boltzmann distribution has been a very useful statistical distribution for understanding the molecular motion of ideal gases. In this work, it will be shown that it is also possible to obtain a generalized Maxwell-Boltzmann distribution valid for any macroscopic system of any composition and having any arbitrary state of aggregation. This distribution is the result of the large number of collisions and molecular interactions taking place in such macroscopic system. On the other hand, in order to better understand the concept of thermal equilibrium, a mathematical interpretation of the zeroth law of Thermodynamics is included.
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The Boltzmann-Gibbs energy distribution is one of the cornerstones of modern statistical mechanics and thermodynamics. According to this distribution, the energy of a molecular system follows an exponential probability model. However, in this report the validity of the Boltzmann-Gibbs energy distribution is questioned since different mathematical inconsistencies are observed. As a result, the Boltzmann distribution of molecular kinetic energy should be replaced by a Chi-squared distribution with 3 degrees of freedom (for monoatomic ideal gases). Similarly, the Gibbs distribution of energy for a system of size N should be replaced by a Chi-squared distribution with 3N degrees of freedom, for monoatomic ideal gases. The proposed distributions are shown to be mathematically consistent, where the Boltzmann-Gibbs distribution fails.
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A new derivation to the so-called Flux Corrected Maxwell-Boltzmann distribution [4, 5] is discussed and presented. We used the discussed expression as a daily bread in the collision physics, combustion, nanoscale heat transfer & other theoretical Mathematical & Physical community [1, 2]. However, the origin of this expression is not clearly given nor derived in full length anywhere in the any of the existing documents the author is aware of. The author uses it for sampling velocity distribution to be assigned in the direction normal to the colliding surface in a usual MD collision simulation.
Determination of the Maxwell-Boltzmann Distribution Probability for Different Gas Mixtures
Eng. &Tech. Journal, 2014
From the Maxwell speed distribution can be calculated several characteristic molecular speeds, namely, the most probable speed νp , the mean speed <ν>, and the root-mean-square speed νrms , furthermore satisfy the condition of speed for percentage different gaseous mixtures at 3000K. The obtained results have been drawn as functions for its variables and appeared in good agreement with the literature.