Druyd:Knowledge

Estimación de Propiedades

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Concentración de partículas

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La concentración de partículas en una posición \vec{x} se puede obtener integrando la función de distribución f(\vec{x},\vec{v},t) sobre todas las velocidades posibles:

$c(\vec{x},t)=\displaystyle\int d\vec{v} f(\vec{x},\vec{v},t)$

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Valor esperado de una magnitud

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Si uno desea estimar un parámetro macroscopico debe promediar su valor microscópico ponderado con la función de distribución f integrando sobre todas las velocidades y dividiendo por el numero de partículas en el volumen

$c(\vec{x},t)=\displaystyle\int d\vec{v} f(\vec{x},\vec{v},t)$

por lo que se expresa como

$\chi_k(\vec{x},t) =\displaystyle\frac{1}{c(\vec{x},t)}\displaystyle\int d\vec{v} f(\vec{x},\vec{v},t) \chi_k(\vec{x},\vec{v},t)$

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Density

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If the parameters are calculated by averaging over the speed using

$\chi_k(\vec{x},t) =\displaystyle\frac{1}{c(\vec{x},t)}\displaystyle\int d\vec{v} f(\vec{x},\vec{v},t) \chi_k(\vec{x},\vec{v},t)$

the mass density estimation is obtained by:

$\rho(\vec{x},t) = m\displaystyle\int f(\vec{x},\vec{v},t)d\vec{v}$

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Speed of the Flow

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If the parameters are calculated by averaging over the speed using

$\chi_k(\vec{x},t) =\displaystyle\frac{1}{c(\vec{x},t)}\displaystyle\int d\vec{v} f(\vec{x},\vec{v},t) \chi_k(\vec{x},\vec{v},t)$

the velocity of the flow is calculated by integrating the velocity distribution function on all velocities by weighing the velocities:

$\vec{u}(\vec{x},t) = \displaystyle\frac{m}{\rho}\int \vec{v}f(\vec{x},\vec{v},t)d\vec{v}$

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Temperature

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If the parameters are calculated by averaging over the speed using

$\chi_k(\vec{x},t) =\displaystyle\frac{1}{c(\vec{x},t)}\displaystyle\int d\vec{v} f(\vec{x},\vec{v},t) \chi_k(\vec{x},\vec{v},t)$

and the equipartition theorem is considered, the temperature can be estimated by integrating the kinetic energy weighted by the velocity distribution divided by the gas constant:

$T(\vec{x},t) = \displaystyle\frac{m}{3R\rho}\displaystyle\int (\vec{v}\cdot\vec{v})f(\vec{x},\vec{v},t)d\vec{v}$

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Tension tensor

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If the parameters are calculated by averaging over the speed using

$\chi_k(\vec{x},t) =\displaystyle\frac{1}{c(\vec{x},t)}\displaystyle\int d\vec{v} f(\vec{x},\vec{v},t) \chi_k(\vec{x},\vec{v},t)$

the flow tensor is calculated by integrating the velocity distribution function on all velocities by weighing the velocity differences:

$\sigma_{ij} = m\displaystyle\int (v_i-u_i)(v_j-u_j)f(\vec{x},\vec{v},t)d\vec{v}$

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