Função distribuição

Nota: Este artigo é sobre conceito de função de distribuição como tratado em Física, em especial na Teoria cinética. Para função distribuição em Teoria da Probabilidade, veja Função distribuição acumulada.

Em teoria cinética molecular em física, a função distribuição de uma partícula é a função de sete variáveis, $f(x,y,z,t;v_{x},v_{y},v_{z})$ , a qual dá o número de partículas por unidade de volume num espaço de fase. É o número de partículas tendo aproximadamente a velocidade $(v_{x},v_{y},v_{z})$ próxima ao local $(x,y,z)$ e o tempo $(t)$ . A normalização usual desta função é

n(x,y,z,t)=\int f\,dv_{x}\,dv_{y}\,dv_{z}

N(t)=\int n\,dx\,dy\,dz

Aqui, N é o número total de partículas e n é o número densidade de partículas - o número de partículas por unidade de volume, ou a densidade dividida pela massa de partículas individuais.

As funções distribuição de partículas são frequentemente usadas em física de plasma para descrever interações onda-partícula e instabilidades velocidade-espaço. Funções distribuição são também usadas em mecânica dos fluidos e mecânica estatística.

A função distribuição básica usa a constante de Boltzmann $k$ e temperatura $T$ com o número densidade para modificas a distribuição normal:

f={\frac {n}{\sqrt {(2\pi RT)^{3}}}}\exp \left({-{\frac {m(v_{x}^{2}+v_{y}^{2}+v_{z}^{2})}{2kT}}}\right)

Funções distribuição relacionadas devem permitir um fluxo fluido maior, nos casos em que a velocidade original é fixada, então que o numerador do expoente é $m((v_{x}-u_{x})^{2}+(v_{y}-u_{y})^{2}+(v_{z}-u_{z})^{2})$ ; $(u_{x},u_{y},u_{z})$ é a maior velocidade do fluido. Funções distribuição podem também representar temperaturas não isotrópicas, nas quais cada termo no expoente é dividido por uma diferente temperatura.

Teorias sobre plasma tais como a magnetoidrodinâmica podem considerar as partículas como estando em equilíbrio termodinâmico. Neste caso, a função distribuição é Maxwelliana. Esta função distribuição trata o fluxo fluido e diferentes temperaturas em direções paralelas a, e perpendiculares a, o campo magnético local. Funções fistribuição mais complexas podem também ser usadas dado que plasmas raramente estão em equilíbrio térmico.

O análogo matemático da distribuição é uma medida; a evolução no tempo de uma medida num estado de fase é o tópico estudado em sistemas dinâmicos.

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