A função relativística de distribuição de velocidades de Maxwell-Jüttner

Damião, Gilberto J.; Rodrigues, Clóves G.

doi:10.1590/1806-9126-RBEF-2021-0271

Figura 1: Função de distribuição de velocidades de Maxwell-Boltzmann para dois valores de temperatura. O valor adotado para a massa m foi a massa de repouso do próton.

Thumbnail

Figura 2: Distribuição de velocidades de Maxwell-Jüttner para prótons (

m_{0} = 1, 6726 \times 10^{- 27} kg

) à temperatura da superfície do Sol (

T = 5.800 K

).

Thumbnail

Figura 3: Distribuições de Maxwell-Jütner (curva em azul) e de Maxwell-Boltzmann (curva em vermelho) para elétrons à temperatura do critério de Lawson (

T = 1, 5 \times 10^{8} K

);

ζ = 39, 533

.

Thumbnail

Figura 4: Distribuições de Maxwell-Boltzamann (em vermelho) e de Maxwell-Jüttner (em azul) para elétrons com

T = 1, 5 \times 10^{9} K

;

ζ = 3, 9533

.

Thumbnail

Figura 5: Área perdida (quadriculada) pelo uso da distribuição de Maxwell-Boltzmann com

ζ = 3, 9533

, para elétrons (

m_{0} = 9, 1094 \times 10^{- 31} kg

) com

T = 1, 5 \times 10^{9} K

.

Thumbnail

Figura 6: Comparativo entre as distribuições de Maxwell-Boltzmann (em vermelho) e de Maxwell-Jüttner (em azul) para prótons (

m_{0} = 1, 6726 \times 10^{- 27} kg

) com

T = 10^{13} K

;

ζ = 1, 0888

.

Thumbnail

Figura 7: Distribuição de Maxwell-Boltzmann para prótons (

m_{0} = 1, 6726 \times 10^{- 27} kg

) com

T = 10^{13} K

;

ζ = 1, 0888

.

Thumbnail

Figura 8: Distribuição de Maxwell-Jüttner para prótons à temperatura do quasar 3C 273.

Thumbnail

Tabela 1: Comparativo entre as distribuições de Maxwell-Boltzmann e Maxwell-Jüttner.

Thumbnail

Tabela 2: Valores esperados de velocidade (em m/s) de prótons para

T = 5.800 K

;

ζ ≃ 1, 877 \times 10^{9}

.

Thumbnail

Tabela 3: Parâmetro

ζ

(reação D-D).

Thumbnail

Tabela 4: Valores esperados de

β

para elétrons (

m_{0} = 9, 1094 \times 10^{- 31} kg

) com

T = 1, 5 \times 10^{8} K

,

ζ = 39, 533

.

Thumbnail

Tabela 5: Valores esperados de

β

para elétrons (

m_{0} = 9, 1094 \times 10^{- 31} kg

) com

T = 1, 5 \times 10^{9} K

;

ζ = 3, 9533

.

Thumbnail

Tabela 6: Valores esperados de

β

para prótons com

T = 10^{13} K

;

ζ = 1, 0888

.

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

(12)

(13)

(14)

(15)

(16)

(17)

(18)

(19)

(20)

(21)

(22)

(23)

(24)

(25)

(26)

(27)

(28)

(29)

(30)

(31)

(32)

(33)

(34)

(35)

(36)

(37)

(38)

(39)

(40)

(41)

(A.1)

(A.2)

(A.3)

(A.4)

(A.5)

(A.6)

(A.7)

(A.8)

(A.9)

(A.10)

(A.11)

(A.12)

(A.13)

(A.14)

(A.15)

(A.16)

(A.17)

(A.18)

(H.1)

	Maxwell-Boltzmann	Maxwell-Jüttner
Distribuição	$\sqrt{2 / π} ζ^{3 / 2} β^{2} e^{- ζ β^{2} / 2}$	$\frac{ζ β^{2} γ^{5} e^{- ζ γ}}{K_{2} (ζ)}$
$β_{p}$	$\sqrt{\frac{2}{ζ}}$	$\sqrt{σ_{1} + σ_{2} + σ_{3} + σ_{4}}$
$⟨ β ⟩$	$\sqrt{\frac{8}{π ζ}}$	$\frac{ζ}{K_{2} (ζ)} (\frac{Γ (3, ζ)}{ζ^{3}} - \frac{e^{- ζ}}{ζ})$
$β_{r m s}$	$\sqrt{\frac{3}{ζ}}$	$\sqrt{1 - \frac{ζ K_{1} (ζ)}{K_{2} (ζ)} + \frac{ζ π}{2 K_{2} (ζ)} [1 - ζ K_{0} (ζ) L_{- 1} (ζ) - ζ K_{1} (ζ) L_{0} (ζ)]}$

Valores esperados	Maxwell-Boltzmann	Maxwell-Jüttner	Diferença (%)
$v_{p}$	9.786,554	9.785,136	0,01
$⟨ v ⟩$	11.042,943	11.042,943	0
$v_{r m s}$	11.986,032	11.986,032	0

	Massa (kg)	Temperatura (K)	$ζ$
Dêuteron	$3, 3436 \times 10^{- 27}$	$1, 5 \times 10^{8}$	$1, 45 \times 10^{5}$
Elétron	$9, 1094 \times 10^{- 31}$	$1, 5 \times 10^{8}$	$39, 533$

Valores esperados	Maxwell-Boltzmann	Maxwell-Jüttner	Diferença (%)
$β_{p}$	0,2249	0,2305	2,4
$⟨ β ⟩$	0,2538	0,2483	2,2
$β_{r m s}$	0,2755	0,2672	3,0

Valores esperados	Maxwell-esperados	Maxwell-Jüttner	Diferença (%)
$β_{p}$	0,7113	0,7786	8,6
$⟨ β ⟩$	0,8027	0,6604	17,7
$β_{r m s}$	0,8712	0,6828	21,6

Valores esperados	Maxwell-Boltzmann	Maxwell-Jüttner	Diferença (%)
$β_{p}$	1,3553	0,9769	27,9
$⟨ β ⟩$	1,5293	0,8953	41,5
$β_{r m s}$	1,6599	0,9024	45,6

f_{M J} (p) d p = Φ (p_{x}, p_{y,} p_{z}) d p_{x} d p_{y} d p_{z},

f_{M J} (p) d p = Φ (p) p^{2} d Ω d p,

f_{M J} (p) d p = 4 π Φ (p) p^{2} d p .

Z = 4 π \int_{0}^{+ \infty} p^{2} \exp {- \frac{m_{0} c^{2}}{k T} \sqrt{1 + \frac{p^{2}}{m_{0}^{2} c^{2}}}} d p .

f_{M J} (p) d p = \frac{p^{2} \exp {- \frac{m_{0} c^{2}}{k T} \sqrt{1 + \frac{p^{2}}{m_{0}^{2} c^{2}}}}}{\int_{0}^{+ \infty} p^{2} \exp {- \frac{m_{0} c^{2}}{k T} \sqrt{1 + \frac{p^{2}}{m_{0}^{2} c^{2}}}} d p} d p .

f_{M J} (v) d v = \frac{v^{2} γ^{5} e^{(- \frac{m_{0} c^{2}}{k T} γ)}}{\int_{0}^{c} v^{2} γ^{5} e^{(- \frac{m_{0} c^{2}}{k T} γ)} d v} d v .

\frac{\partial}{\partial β} f_{M J} (β) = 0,

\frac{\partial}{\partial β} (\frac{ζ β^{2} γ^{5} e^{- ζ γ}}{K_{2} (ζ)}) = 0,

\frac{ζ}{K_{2} (ζ)} \frac{\partial}{\partial β} [\frac{β^{2} e^{- ζ / \sqrt{1 - β^{2}}}}{{(1 - β^{2})}^{5 / 2}}] = 0,

\frac{ζ}{K_{2} (ζ)} (\frac{e^{- ζ / \sqrt{1 - β^{2}}} [β^{2} (1 - ζ \sqrt{1 - β^{2}}) - 3 β^{4} + 2] β}{{(1 - β^{2})}^{9 / 2}}) = 0,

e^{- ζ / \sqrt{1 - β^{2}}} [β^{2} (1 - ζ \sqrt{1 - β^{2}}) - 3 β^{4} + 2] β = 0 .

⟨ β ⟩ = \int_{0}^{1} \frac{ζ β^{3} γ^{5} e^{- ζ γ}}{K_{2} (ζ)} d β,

⟨ β ⟩ = \frac{ζ}{K_{2} (ζ)} \int_{0}^{1} β^{3} γ^{5} e^{- ζ γ} d β,

⟨ β ⟩ = \frac{ζ}{K_{2} (ζ)} \int_{0}^{1} \frac{β^{3} e^{- ζ / \sqrt{1 - β^{2}}}}{{(1 - β^{2})}^{5 / 2}} d β,

⟨ β ⟩ = {\frac{ζ}{K_{2} (ζ)} (\frac{e^{- ζ / \sqrt{1 - β^{2}}}}{ζ} - \frac{Γ (3, ζ / \sqrt{1 - β^{2}})}{ζ^{3}}) |}_{0}^{1},

⟨ β^{2} ⟩ = 1 - \frac{ζ K_{1} (ζ)}{K_{2} (ζ)} + \frac{ζ π}{2 K_{2} (ζ)}

\times [1 - ζ K_{0} (ζ) L_{- 1} (ζ) - ζ K_{1} (ζ) L_{0} (ζ)],

β_{r m s} = \sqrt{\begin{matrix} 1 - \frac{ζ K_{1} (ζ)}{K_{2} (ζ)} + \frac{ζ π}{2 K_{2} (ζ)} \\ [1 - ζ K_{0} (ζ) L_{- 1} (ζ) - ζ K_{1} (ζ) L_{0} (ζ)] \end{matrix}} .

β_{p (M B)} = \sqrt{\frac{2 k T}{m_{0} c^{2}}} = \sqrt{\frac{2}{ζ}},

{⟨ β ⟩}_{M B} = \sqrt{\frac{8 k T}{π m_{0} c^{2}}} = \sqrt{\frac{8}{π ζ}},

\cosh (2 x) \cosh (x) = \cosh^{3} (x) + \sinh^{2} (x) \cosh (x),

\cosh (2 x) \cosh (x) - \sinh^{2} (x) \cosh (x) = \cosh^{3} (x) .

\cosh (2 x) \cosh (x) - 2 s i n h^{2} (x) \cosh (x) = \cosh (x),

2 s i n h^{2} (x) \cosh (x) = \cosh (2 x) \cosh (x) - \cosh (x) .

2 (\frac{\sinh (2 x) \sinh (x)}{2}) = \cosh (2 x) \cosh (x) - \cosh (x),

\cosh (2 x) \cosh (x) - \sinh (2 x) \sinh (x) - \cosh (x) = 0 .

(- 3 β^{4} + 2 + β^{2} - β^{2} ζ \sqrt{1 - β^{2}})

(\frac{- 3 β^{4} + 2 + β^{2} + β^{2} ζ \sqrt{1 - β^{2}}}{- 3 β^{4} + 2 + β^{2} + β^{2} ζ \sqrt{1 - β^{2}}}) = 0,

9 β^{8} + (ζ^{2} - 6) β^{6} - (ζ^{2} + 11) β^{4} + 4 β^{2} + 4 = 0 .

\sinh (θ) = β γ,

\cosh^{2} (θ) - {(β γ)}^{2} = 1,

\cosh (θ) = \sqrt{1 + {(β γ)}^{2}},

\cosh (θ) = γ = \frac{1}{\sqrt{1 - β^{2}}},

\tanh (θ) = β,

d β = \frac{d θ}{\cosh^{2} (θ)} .

⟨ β^{2} ⟩ = \frac{ζ}{K_{2} (ζ)} \int_{0}^{+ \infty} e^{- ζ \cosh (θ)} \sinh^{3} (θ) \tanh (θ) d θ,

⟨ β^{2} ⟩ = \frac{ζ}{K_{2} (ζ)} \int_{0}^{+ \infty} e^{- ζ \cosh (θ)} \sinh^{2} (θ) \tanh^{2} (θ) \times \cosh (θ) d θ .

⟨ β^{2} ⟩ = \frac{ζ}{K_{2} (ζ)} \int_{0}^{+ \infty} e^{- ζ \cosh (θ)} \cosh (θ) \times [\sinh^{2} (θ) - \tanh^{2} (θ)] d θ,

⟨ β^{2} ⟩ = \frac{ζ}{K_{2} (ζ)} \int_{0}^{+ \infty} [e^{- ζ \cosh (θ)} \sinh^{2} (θ) \cosh (θ) - e^{- ζ \cosh (θ)} \tanh^{2} (θ) \cosh (θ)] d θ .

⟨ β^{2} ⟩ = \frac{ζ}{K_{2} (ζ)} [\frac{K_{2} (ζ)}{ζ} - \int_{0}^{+ \infty} e^{- ζ \cosh (θ)} \cosh (θ) d θ + \int_{0}^{+ \infty} e^{- ζ \cosh (θ)} sech (θ) d θ],

f (α) = \int_{0}^{+ \infty} e^{- α ζ \cosh (θ)} sech (θ) d θ,

\frac{d (f (α))}{d α} = f^{'} (α) = - \int_{0}^{+ \infty} ζ e^{- α ζ \cosh (θ)} \cosh (θ) sech (θ) d θ,

f^{'} (α) = - ζ \int_{0}^{+ \infty} e^{- α ζ \cosh (θ)} d θ,

f^{'} (α) = - ζ K_{0} (α ζ),

f (α) = - ζ \int K_{0} (α ζ) d α .

f (α) = - \frac{π ζ α}{2} [L_{- 1} (ζ α) K_{0} (ζ α) + L_{0} (ζ α) K_{1} (ζ α)] + C,

f (α) = \int_{0}^{+ \infty} e^{- α ζ \cosh (θ)} sech (θ) d θ

= - \frac{π ζ α}{2} [L_{- 1} (ζ α) K_{0} (ζ α) + L_{0} (ζ α) K_{1} (ζ α)] + C .

f (0) = \int_{0}^{+ \infty} sech (θ) d θ,

f (0) = {2 a r c t a n [\tanh (\frac{θ}{2})] |}_{0}^{+ \infty} = \frac{π}{2} .

f (α) = - \frac{π ζ α}{2} [L_{- 1} (ζ α) K_{0} (ζ α) + L_{0} (ζ α) K_{1} (ζ α)] + C

lim_{α \to 0} K_{0} (ζ α) α = 0,

lim_{α \to 0} L_{- 1} (ζ α) K_{0} (ζ α) α = [lim_{α \to 0} K_{0} (ζ α) α] L_{- 1} (0) = 0

[L_{- 1} (0)] = 0,

lim_{α \to 0} L_{0} (ζ α) K_{1} (ζ α) = \frac{2}{π},

lim_{α \to 0} L_{0} (ζ α) K_{1} (ζ α) α = [lim_{α \to 0} L_{0} (ζ α) K_{1} (ζ α)] 0 = 0 .

f (1) = \int_{0}^{+ \infty} e^{- ζ \cosh (θ)} sech (θ) d θ

= - \frac{π ζ}{2} [L_{- 1} (ζ) K_{0} (ζ) + L_{0} (ζ) K_{1} (ζ)] + \frac{π}{2},

⟨ β^{2} ⟩ = \frac{ζ}{K_{2} (ζ)} {\frac{K_{2} (ζ)}{ζ} - K_{1} (ζ)

+ \frac{π}{2} [1 - ζ K_{0} (ζ) L_{- 1} (ζ) - ζ K_{1} (ζ) L_{0} (ζ)]},

⟨ β^{2} ⟩ = 1 - \frac{ζ K_{1} (ζ)}{K_{2} (ζ)} + \frac{ζ π}{2 K_{2} (ζ)}

\times [1 - ζ K_{0} (ζ) L_{- 1} (ζ) - ζ K_{1} (ζ) L_{0} (ζ)] .

z^{2} \frac{d^{2} w}{d z^{2}} + z \frac{d w}{d z} + (z^{2} - ν^{2}) w = \frac{4 {(z / 2)}^{ν + 1}}{Γ (ν + 1 / 2) \sqrt{π}} .

z^{2} \frac{d^{2} w}{d z^{2}} + z \frac{d w}{d z} - (z^{2} + ν^{2}) w = \frac{4 {(z / 2)}^{ν + 1}}{Γ (ν + 1 / 2) \sqrt{π}} .

f_{M B} (β) d β = 4 π {(\sqrt{\frac{m_{0}}{2 π k T}})}^{3} β^{2} c^{2} e^{- \frac{m_{0} β^{2} c^{2}}{2 k T}} c d β,

f_{M B} (β) d β = 4 π {(\sqrt{\frac{m_{0}}{2 π k T}})}^{3} β^{2} c^{3} e^{- \frac{m_{0} c^{2} β^{2}}{2 k T}} d β,

f_{M B} (β) d β = 4 π {(\sqrt{\frac{m_{0} c^{2}}{2 π k T}})}^{3} β^{2} e^{- \frac{m_{0} c^{2} β^{2}}{2 k T}} d β,

f_{M B} (β) d β = \frac{4 π}{{(\sqrt{2 π})}^{3}} {(\sqrt{\frac{m_{0} c^{2}}{k T}})}^{3} β^{2} e^{- \frac{m_{0} c^{2} β^{2}}{2 k T}} d β .

{⟨ β ⟩}_{MB} = \sqrt{\frac{8 k T}{π m_{0} c^{2}}} = \sqrt{\frac{8}{π ζ}},

β_{p (M B)} = \sqrt{\frac{2 k T}{m_{0} c^{2}}} = \sqrt{\frac{2}{ζ}},

A função relativística de distribuição de velocidades de Maxwell-Jüttner

The relativistic Maxwell-Jüttner velocity distribution function

Resumo