Maxwellian

Maxwellian distribution

Description

The Maxwellian (or Maxwell-Boltzmann) distribution object defines a Maxwell-Boltzmann distribution function with the provided mass and temperature parameters. The probability density function (PDF) of the Maxwellian distribution is given by the Eq. (1).

(1)var element = document.getElementById("moose-equation-beafc5c6-08b5-464a-93a4-ab39e8bc373d");katex.render("f(v \\; | \\; m, T) = \\sqrt{ \\frac{ m }{ 2 \\pi k_B T }} \\; \\exp\\left( - \\frac{m v^2}{ 2 k_B T } \\right)", element, {displayMode:true,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});

where $var element = document.getElementById("moose-equation-224b8c12-2667-4084-90f5-7fbea6b82b56");katex.render("v", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ is a single component of a velocity vector, $var element = document.getElementById("moose-equation-a91028c6-3fa1-4c95-a623-9eae37f0b828");katex.render("m", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ is the mass of the species in kg, and $var element = document.getElementById("moose-equation-86f204db-19d6-4fc4-a96f-b7d9bdf5ca1a");katex.render("T", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ is the temperature the component of velocity is at in K

Since the single component Maxwellian is a special case of a normal distribution, where the mean $var element = document.getElementById("moose-equation-a5ced83d-9c00-4314-8120-7d8e6f5c8c6e");katex.render("\\mu = 0", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ , and the standard deviation is given by Eq. (2),

(2)var element = document.getElementById("moose-equation-4f014e60-aab9-479c-83c1-cbe305bf2c47");katex.render("\\sigma = \\sqrt{ \\frac{ k_B T }{ m } }", element, {displayMode:true,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});

this class is a specialization of Normal.

Example Input Syntax

The following input file defines a maxwellian distribution that is equivalent to a normal distribution with a mean of 0 and a standard deviation of 1.

[Distributions<<<{"href": "../../syntax/Distributions/index.html"}>>>]
  [distribution]
    type = Maxwellian<<<{"description": "Maxwellian distribution", "href": "Maxwellian.html"}>>>
    mass<<<{"description": "The mass of the species that are the given temperature in kg."}>>> = 2.0
    # this combination of mass and temperature
    # result in a distribution equivalent to a
    # normal distribution with a mean of 0 and
    # a standard deviation of 1
    temperature<<<{"description": "The temperature of the distribution in K."}>>> = '${fparse 2.0 / k_B}'
  []
[]

Input Parameters

massThe mass of the species that are the given temperature in kg.
C++ Type:double
Unit:(no unit assumed)
Range:mass > 0
Controllable:No
Description:The mass of the species that are the given temperature in kg.
temperatureThe temperature of the distribution in K.
C++ Type:double
Unit:(no unit assumed)
Range:temperature > 0
Controllable:No
Description:The temperature of the distribution in K.

Required Parameters

control_tagsAdds user-defined labels for accessing object parameters via control logic.
C++ Type:std::vector<std::string>
Controllable:No
Description:Adds user-defined labels for accessing object parameters via control logic.
enableTrueSet the enabled status of the MooseObject.
Default:True
C++ Type:bool
Controllable:No
Description:Set the enabled status of the MooseObject.

Advanced Parameters

Input Files

(modules/stochastic_tools/test/tests/distributions/maxwellian.i)

(modules/stochastic_tools/test/tests/distributions/maxwellian.i)

# Boltzmann constant in J/K
k_B = 1.380649e-23

[StochasticTools]
[]

[Distributions]
  [distribution]
    type = Maxwellian
    mass = 2.0
    # this combination of mass and temperature
    # result in a distribution equivalent to a
    # normal distribution with a mean of 0 and
    # a standard deviation of 1
    temperature = ${fparse 2.0 / k_B}
  []
[]

[Postprocessors]
  [cdf]
    type = TestDistributionPostprocessor
    distribution = distribution
    value = 0
    method = cdf
    execute_on = initial
  []
  [pdf]
    type = TestDistributionPostprocessor
    distribution = distribution
    value = 0
    method = pdf
    execute_on = initial
  []
  [quantile]
    type = TestDistributionPostprocessor
    distribution = distribution
    value = 0.5
    method = quantile
    execute_on = initial
  []
[]

[Outputs]
  execute_on = 'INITIAL'
  csv = true
[]

(modules/stochastic_tools/test/tests/distributions/maxwellian.i)

# Boltzmann constant in J/K
k_B = 1.380649e-23

[StochasticTools]
[]

[Distributions]
  [distribution]
    type = Maxwellian
    mass = 2.0
    # this combination of mass and temperature
    # result in a distribution equivalent to a
    # normal distribution with a mean of 0 and
    # a standard deviation of 1
    temperature = ${fparse 2.0 / k_B}
  []
[]

[Postprocessors]
  [cdf]
    type = TestDistributionPostprocessor
    distribution = distribution
    value = 0
    method = cdf
    execute_on = initial
  []
  [pdf]
    type = TestDistributionPostprocessor
    distribution = distribution
    value = 0
    method = pdf
    execute_on = initial
  []
  [quantile]
    type = TestDistributionPostprocessor
    distribution = distribution
    value = 0.5
    method = quantile
    execute_on = initial
  []
[]

[Outputs]
  execute_on = 'INITIAL'
  csv = true
[]

Description
Example Input Syntax
Input Parameters
Input Files