EoM

this page is about the Equations of Motion for viscous hydrodynamics in the SPH approach.
=== Equations of Motion ===
:
T^{\mu\nu}{}_{;\mu} = 0
:
\partial_\mu T^{\mu\nu} + \Gamma^{\mu}_{\mu\alpha} T^{\alpha\nu} + \Gamma^{\nu}_{\mu\alpha} T^{\mu\alpha} = 0
:
\frac{1}{\sqrt{-g}}\partial_\mu ( \sqrt{-g} T^{\mu\nu} ) + \Gamma^{\nu}_{\mu\alpha} T^{\mu\alpha} = 0
=== Hyperbolic coordinates ===
:
g_{\mu\nu} = diag( 1, -1, -1, -\tau^2 )
:
\sqrt{-g} = \tau
\\
\Gamma^{\tau}_{\eta\eta} = \tau
\\
\Gamma^{\eta}_{\tau\eta} = \Gamma^{\eta}_{\eta\tau} = \frac{1}{\tau}
:
\frac{1}{\sqrt{-g}}\partial_\mu ( \sqrt{-g} T^{\mu\nu} ) + \Gamma^{\nu}_{\eta\eta} T^{\eta\eta} + \Gamma^{\nu}_{\tau\eta} T^{\tau\eta} + \Gamma^{\nu}_{\eta\tau} T^{\eta\tau} = 0
:
\frac{1}{\sqrt{-g}}\partial_\mu ( \sqrt{-g} T^{\mu\nu} ) + \Gamma^{\nu}_{\eta\eta} T^{\eta\eta} + 2\Gamma^{\nu}_{\tau\eta} T^{\tau\eta} = 0
:
\frac{1}{\tau}\partial_\mu ( \tau T^{\mu\nu} ) + \delta^{\nu\tau} \tau T^{\eta\eta} + \frac{2 \delta^{\nu\eta} }{\tau} T^{\tau\eta} = 0
=== Ideal hydrodynamics ===
The energy-momentum tensor of ideal hydrodynamics is expressed as
:
T^{\mu\nu} = \omega u^\mu u^\nu - Pg^{\mu\nu}
==== First derivation ====
Now we apply the covariant derivative and open the metric derivative terms
:
\frac{1}{\tau}\partial_\mu ( \tau \omega u^\mu u^\nu ) - \frac{1}{\tau} \partial_\mu (\tau P g^{\mu\nu}) + \delta^{\nu\tau} \tau (\omega (u^\eta) ^2 + \frac{P}{\tau^2}) + \frac{2 \delta^{\nu\eta} }{\tau} ( \omega u^\tau u^\eta) = 0
it is convenient to show the expressions for each component
:
\frac{1}{\tau}\partial_\mu ( \tau \omega u^\mu u^\tau ) = \frac{1}{\tau} \partial_\tau (\tau P) - \tau (\omega (u^\eta) ^2 + \frac{P}{\tau^2})
\\
\frac{1}{\tau}\partial_\mu ( \tau \omega u^\mu u^i ) = - \partial_i P
\\
\frac{1}{\tau}\partial_\mu ( \tau \omega u^\mu u^\eta ) = - \frac{1}{\tau^2} \partial_\eta P - \frac{2}{\tau}( \omega u^\tau u^\eta)
By introducing a conserved quantity \sigma one gets
:
(\sigma u^\mu)_{;\mu} = \frac{1}{\sqrt{-g}} \partial_\mu (\sqrt{-g} \sigma u^\mu) = 0
\\
\frac{1}{\tau} \partial_\mu (\tau \sigma u^\mu) = 0
:
\sigma u^\mu \partial_\mu ( \frac{\omega u^\tau }{\sigma} ) = \frac{1}{\tau} \partial_\tau (\tau P) - \tau \omega (u^\eta) ^2 - \frac{P}{\tau} = \partial_\tau P - \tau \omega (u^\eta) ^2
:
\sigma u^\mu \partial_\mu ( \frac{\omega u^i }{\sigma} ) = - \partial_i P
:
\sigma u^\mu \partial_\mu ( \frac{\omega u^\eta }{\sigma} ) = - \frac{1}{\tau^2} \partial_\eta P - \frac{2}{\tau}( \omega u^\tau u^\eta)
==== Second derivation ====
From the energy-momentum conservation
:
T^{\mu\nu}{}_{;\mu} = 0
we can write
:
(\omega u^\mu u^\nu){}_{;\mu} - g^{\mu\nu} P{}_{;\mu} = 0
using that g^{\mu\nu}{}_{;\mu} = 0.
Now we introduce a conserved quantity \sigma
:
\sigma u^\mu (\frac{\omega u^\nu}{\sigma})_{;\mu} - g^{\mu\nu} \partial_\mu P = 0
:
\sigma u^\mu (\partial_\mu (\frac{\omega u^\nu}{\sigma}) + \Gamma^{\nu}_{\mu\alpha} \frac{\omega u^\alpha}{\sigma} ) - g^{\mu\nu} \partial_\mu P = 0
:
\sigma u^\mu \partial_\mu (\frac{\omega u^\nu}{\sigma}) = - \Gamma^{\nu}_{\mu\alpha} \omega u^\mu u^\alpha + g^{\mu\nu} \partial_\mu P
the components are
:
\sigma u^\mu \partial_\mu (\frac{\omega u^\tau}{\sigma}) = - \Gamma^{\tau}_{\eta\eta} \omega (u^\eta)^2 + \partial_\tau P = - \tau \omega (u^\eta)^2 + \partial_\tau P
\\
\sigma u^\mu \partial_\mu (\frac{\omega u^i}{\sigma}) = - \partial_i P
\\
\sigma u^\mu \partial_\mu (\frac{\omega u^\eta}{\sigma}) = - 2\Gamma^{\eta}_{\eta\tau} \omega u^\eta u^\tau - \frac{1}{\tau^2} \partial_\eta P = - \frac{2}{\tau}\omega u^\eta u^\tau - \frac{1}{\tau^2} \partial_\eta P
one can absorb the first term in the rhs of the \eta-component as such
:
\sigma u^\mu \partial_\mu (\frac{\omega \tau^2 u^\eta}{\sigma}) = - \partial_\eta P
=== Hyperbolic coordinates ===
:
\partial_\mu (\tau \sigma u^\mu) = 0
\\
v^\mu \partial_\mu ( \tau \sigma u^\tau ) + \tau \sigma u^\tau \partial_\mu v^\mu = 0
we introduce the quantity \bar\sigma = \tau \sigma u^\tau 
:
v^\mu \partial_\mu \bar\sigma + \bar\sigma \partial_\mu v^\mu = 0
:
\frac{ \sigma u^\mu }{\tau} \partial_\mu (\frac{\tau\omega u^\nu}{\sigma}) - g^{\mu\nu} \partial_\mu P = 0
\\
v^\mu \partial_\mu (\frac{\tau\omega u^\nu}{\sigma}) = \frac{ \tau }{ u^\tau \sigma } g^{\mu\nu} \partial_\mu P
we introduce the quantity q^\mu = \frac{\tau\omega u^\nu}{\sigma} 
:
\frac{ \sigma u^\mu }{\tau} \partial_\mu (\frac{\tau\omega u^\tau}{\sigma} ) - \partial_\tau P = 0
\\
\frac{ \sigma u^\mu }{\tau} \partial_\mu ( \frac{\tau\omega u^i }{\sigma} ) + \partial_i P = 0
\\
\frac{ \sigma u^\mu }{\tau} \partial_\mu ( \frac{\tau\omega u^\eta }{\sigma} ) + \frac{1}{\tau^2}\partial_\eta P = 0 
:
v^\mu \partial_\mu (\frac{\tau\omega u^\tau}{\sigma} ) = \frac{ \tau }{ u^\tau \sigma}\partial_\tau P
\\
v^\mu \partial_\mu ( \frac{\tau\omega u^i }{\sigma} ) = - \frac{ \tau }{u^\tau \sigma} \partial_i P 
\\
v^\mu \partial_\mu ( \frac{\tau\omega u^\eta }{\sigma} ) = -\frac{ 1}{\tau u^\tau \sigma}\partial_\eta P
so the evolution equations are
:
v^\mu \partial_\mu ( \frac{\tau\omega u^i }{\sigma} ) = - \frac{ \tau }{u^\tau \sigma} \partial_i P 
\\
v^\mu \partial_\mu ( \frac{\tau\omega u^\eta }{\sigma} ) = -\frac{ 1}{\tau u^\tau \sigma}\partial_\eta P
\\
v^\mu \partial_\mu \bar\sigma + \bar\sigma \partial_\mu v^\mu = 0
\\
\dot x^i = v^i
\\
\dot \eta = v^\eta
=== SPH ===
==== Conserved density ====
:
j^\mu(x) = \sum_a \dot x_a^\mu W_a(x)
:
j^\tau = \sum_a W_a(x)
since \dot \tau_a = 1
:
j^\mu(x) = \tau \sigma u^\mu = j^\tau v^\mu
:
v^\mu = \frac{j^\mu}{j^\tau} = \frac{\sum_a \dot x_a^\mu W_a(x)}{\sum_a W_a(x)}
:
\partial_\mu j^\mu(x) = 0
\\
\partial_\tau j^\tau + \partial_i j^i + \partial_\eta j^\eta = 0
:
\sum_a \dot \tau_a \partial_\tau W_a(x) + \sum_a \dot x_a^i \partial_i W_a(x) + \sum_a \dot \eta_a \partial_\eta W_a(x) = 0
:
\partial_\tau W_a(x) = - \dot x_a^i \partial_i W_a(x) - \dot \eta_a \partial_\eta W_a(x)
==== Lagrangean ====
:
I = \int \tau dxdy d\eta d\tau ( \varepsilon(\sigma) + A \partial_\mu (\tau \sigma u^\mu) + B(g_{\mu\nu} u^\mu u^\nu - 1) )
===== Case 1 =====
:
\varepsilon = \sum_a \frac{\varepsilon(\sigma_a)}{\rho_a} W_a(x)
:
I = \int d\tau \sum_a \frac{\tau \varepsilon(\sigma_a)}{\rho_a} + \int \tau dxdy d\eta (-\tau \sigma u^\mu \partial_\mu A) + B(g_{\mu\nu} u^\mu u^\nu - 1)
by using
:
j^\mu(x) = \sum_a \dot x_a^\mu W(x-x_a)
the conservation is satistied and by setting v^\mu = \dot x^\mu the normalization is also automatically satisfied.
====== \delta x_m ======
:
\sum_a \partial_m ( \frac{\tau_a \varepsilon_a}{\rho_a} ) 
\\
= \sum_a \frac{\tau_a}{\rho^2_a} ( \rho_a \partial_m \varepsilon_a - \varepsilon_a \partial_m \rho_a )
\\
= \sum_a \frac{\tau_a}{\rho^2_a} ( \rho_a \frac{\omega_a}{\sigma_a} \partial_m \sigma_a - \varepsilon_a \partial_m \rho_a )
\\
= \sum_a \frac{\tau_a P_a }{\rho^2_a} \partial_m \rho_a
\\
= \sum_a \frac{\tau_a P_a }{\rho^2_a} \sum_b \partial_m W_{ba}
\\
= \sum_a \frac{\tau_a P_a }{\rho^2_a} \sum_b W'_{ba} r_{ba} (\delta_{bm} - \delta_{am})
\\
= \sum_a \frac{\tau_a P_a }{\rho^2_a} \sum_b W'_{ba} r_{ba} \delta_{bm} - \sum_a \frac{\tau_a P_a }{\rho^2_a} \sum_b W'_{ba} r_{ba} \delta_{am}
\\
= \sum_a \frac{\tau_a P_a }{\rho^2_a} W'_{ma} r_{ma} - \frac{\tau_m P_m }{\rho^2_m} \sum_a W'_{am} r_{am}
\\
= - ( \frac{\tau_m P_m }{\rho^2_m} \sum_a W'_{am} r_{am} + \sum_a \frac{\tau_a P_a }{\rho^2_a} W'_{am} r_{am} )
\\
= - \sum_a (\frac{\tau_m P_m }{\rho^2_m} + \frac{\tau_a P_a }{\rho^2_a}) W'_{am} r_{am}
:
- \sum_a \partial_m ( \frac{\tau_a \varepsilon_a}{\rho_a} ) 
= - (\frac{\tau_m P_m }{\rho^2_m} (\nabla\rho)_m + ( \nabla \frac{\tau P}{\rho})_m)
====== \delta \dot x_m ======
:
\sum_a \partial_{\dot x_m} ( \frac{\tau_a \varepsilon_a}{\rho_a} )
\\
= \sum_a \frac{ \tau_a }{ \rho_a } \partial_{\dot x_m}\varepsilon_a 
\\
= \sum_a \frac{ \tau_a }{ \rho_a } \frac{\omega_a}{\sigma_a} \partial_{\dot x_m}\sigma_a
\\
= \sum_a \frac{ \tau_a }{ \rho_a } \frac{\omega_a}{\sigma_a} \frac{\rho_a}{\tau_a} \partial_{\dot x_m} \gamma^{-1}_a
\\
= - \frac{\omega_m}{\sigma_m} \gamma_a g_{ij} \dot x_a^j \delta_{am}
\\
= - \frac{\omega_m}{\sigma_m} \gamma_m g_{ij} \dot x_m^j
where we used that
:
\partial_{\dot x_m} \gamma^{-1}_a
\\
= \partial_{\dot x_m} ( 1 + g_{ij} \dot x_a^i \dot x_a^j)^{1/2}
\\
= \frac{1}{2}( 1 + g_{ij} \dot x_a^i \dot x_a^j)^{-1/2} g_{ij} \partial_{\dot x_m}( \dot x_a^i \dot x_a^j)
\\
= \gamma_a g_{ij} \dot x_a^j \delta_{am}
:
\sum_a \partial_{\dot x_m} ( \frac{\tau_a \varepsilon_a}{\rho_a} )
= - \frac{\omega_m}{\sigma_m} \gamma_m g_{ij} \dot x_m^j = q_j{}_m = - \frac{\tau_m \omega_m}{\rho_m} g_{ij} \gamma_m^2 \dot x_m^j = - g_{ij} \frac{\tau_m T^{\tau j}_m}{\rho_m}
In order to extract the velocity out of the momentum one first computes
:
q_j = - \frac{\tau \gamma \omega}{\rho} g_{ij} u^j
:
\sqrt{ -q^2 } = \sqrt{ -q^i q_i } = \sqrt{ -g^{ij}q_i q_j } = \frac{\tau \gamma \omega}{\rho} \sqrt{ -g_{ij} u^i u^j } = \frac{\tau \gamma \omega}{\rho} \sqrt{ -u^2 } \gt 0
where we used that \gamma^2 = 1 - g_{ij} \gamma^2 \dot x^i \dot x^j.
Therefore, 
:
\frac{q_i}{ \sqrt{-q^2} } = - \frac{1}{ \sqrt{ \gamma^2 - 1 } } g_{ij}\gamma \dot x^j
or
:
u^i = \frac{ \sqrt{ -u^2 } }{ \sqrt{-q^2} } ( - g^{ij} q_j )
====== equations of motion ======
:
\frac{d}{d\tau} (-\frac{\omega_m}{\sigma_m} \gamma_m g_{ij} \dot x_m^j) = -\sum_a (\frac{\tau_m P_m }{\rho^2_m} + \frac{\tau_a P_a }{\rho^2_a}) W'_{am} r_{am}
:
\frac{d}{d\tau} q_m^j = - (\frac{\tau_m P_m }{\rho^2_m} (\nabla \rho)_m + (\nabla \frac{\tau P}{\rho})_m)
where
:
\rho_m = \sum_a W_{am}
\\
(\nabla \rho)_m = \sum_a W'_{am} r_{am}
Furthermore
:
E = q^\tau = \sum_a v^i_a q_a{}_i - L
\\
= \sum_a v^i_a q_a{}_i + \frac{\tau \varepsilon}{\rho}
:
v^\mu \partial_\mu \frac{ \omega g_{ij} u^j }{ \sigma }
\\
= \frac{\tau P}{\rho^2} \partial_i \rho + \partial_i \frac{\tau P}{\rho}
\\
= \frac{1}{\rho} ( \frac{\tau P}{\rho} \partial_i \rho + \rho \partial_i \frac{\tau P}{\rho} ) 
\\
= \frac{1}{\rho} \partial_i( \tau P )
:
\sigma u^\mu \partial_\mu \frac{ \omega g_{ij} u^j }{ \sigma }
= \partial_i P
===== Case 2 =====
:
j^\tau \varepsilon = \sum_a \varepsilon_a W_a(x)
\\
\varepsilon = \frac{\sum_a \varepsilon_a W_a(x)}{\sum_a W_a(x)}
:
\varepsilon = j^\tau E = \sum_a E_a W_a(x)
=== SPH properties ===
We start we the identity (in n dimensions)
:
\phi(x) = \int d^d x \phi(x') \delta^d(x-x')
Using the distribution W we rewrite this as
:
\phi(x) = \lim_{h\to 0} \int d^d x \phi(x') W^d(x-x')
:
\phi(x) = \lim_{h\to 0}\lim_{N\to \infty} \sum_a^N V_a \phi(x_a) W^d(x-x_a)
Given a conserved density
:
\rho(x) = \sum_a \nu_a W(x - r_a)
we can calculate its integral to be
:
\int d^3x \rho(x) = \sum_a \nu_a \int d^3x W(x - r_a) = \sum_a \nu_a
We can relate another quantity with the conserved quantity by
:
A(x) = \sum_a \frac{\nu_a A_a}{\rho_a} W(x-r_a)
in this way we get
:
\int d^3x A(x) = \sum_a \frac{\nu_a A_a}{\rho_a}
we first assume 
:
\rho_a = \sum_b \nu_b W(r_a - r_b) = \nu_a W(0) + \sum_{b\neq a} \nu_b W(r_a - r_b)
and therefore,
:
A_a = \sum_b \frac{\nu_b A_b}{\rho_b} W(r_a - r_b)
now if we use this is the previous definition of A(x) for self-consistency we get
:
\int d^3x A(x) = \sum_a \frac{\nu_a A_a}{\rho_a} = \sum_a \frac{\nu_a}{\rho_a} \sum_b \frac{\nu_b A_b}{\rho_b} W(r_a-r_b)
\\
= \sum_b \frac{\nu_b A_b}{\rho_b} \sum_a \frac{\nu_a}{\rho_a} W(r_a-r_b)
so that, 
:
\sum_a \frac{\nu_a}{\rho_a} W(r_a-r_b) = 1
:
\sum_a \frac{\nu_a}{\rho_a} \int d^3x W(r_a - x) = \int d^3x 1
:
\sum_a \frac{\nu_a}{\rho_a} = V = \sum_a V_a