Continuity equation

A continuity equation is a certain partial differential equation that belongs to a conserved quantity (see below). It links the temporal change in the density associated with this conservation quantity with the spatial change in its current density : ${\displaystyle {\frac {\partial }{\partial t}}}$  ${\displaystyle \rho }$ ${\displaystyle {\vec {j}}}$

${\displaystyle {\frac {\partial \rho }{\partial t}}+{\vec {\nabla }}\cdot {\vec {j}}=0}$

For the mathematical definition of see divergence of a vector field . ${\displaystyle {\vec {\nabla }}\cdot }$

The continuity equation occurs in all field theories of physics . The sizes obtained can be:

• the crowd ,
• the electric charge ,
• the energy ,
• the probability and
• some particle numbers ( lepton number , baryon number ).

The generalization of the continuity equation to physical quantities that are not conserved quantities is the balance equation . An additional source term appears in it on the right-hand side of the equation .

Connection with a conservation quantity

The “charge” contained in a volume V (the volume integral over the density) can only change due to the continuity equation in that unbalanced currents flow out of the surface of the volume. Accordingly, the total charge does not change over time and is a conservation quantity if no (net) currents flow through the surface of the volume under consideration. ${\displaystyle V\to \infty }$

Because the change in charge over time , given by ${\displaystyle Q_{V}}$

${\displaystyle Q_{V}=\iiint _{V}\mathrm {d} ^{3}x\,\rho (t,{\vec {x}})}$

in a volume that does not change over time , is because of the continuity equation according to Gauss's integral theorem ${\displaystyle V}$

${\displaystyle {\frac {\mathrm {d} Q_{V}}{\mathrm {d} t}}=\iiint _{V}\mathrm {d} ^{3}x\,{\frac {\partial \rho }{\partial t}}=-\iiint _{V}\!\mathrm {d} ^{3}x\,{\vec {\nabla }}\cdot {\vec {j}}=-\oint _{\partial V}\,{\vec {j}}\;\cdot {\vec {n}}\,\mathrm {d} S\,,}$

equal to the area integral over the edge area of the volume over the proportion of the current density that flows outward in the direction of the surface normal . The charge in the volume only changes if unbalanced currents flow through the edge surface in the specified manner. ${\displaystyle \partial V}$ ${\displaystyle {\vec {j}}}$ ${\displaystyle {\vec {n}}}$

Special continuity equations

Hydrodynamics

If the mass density changes in hydrodynamics because the liquid flows with the speed along the trajectories , then the corresponding current density is ${\displaystyle \rho (t,{\vec {x}})}$ ${\displaystyle {\vec {u}}={\tfrac {\mathrm {d} {\vec {x}}}{\mathrm {d} t}}}$ ${\displaystyle {\vec {x}}(t)}$

${\displaystyle {\vec {j}}=\rho \,{\vec {u}}}$

and the equation of continuity is

${\displaystyle {\begin{alignedat}{2}&{\frac {\partial \rho }{\partial t}}+{\vec {\nabla }}\cdot (\rho {\vec {u}})&&=0\\\Leftrightarrow &{\frac {\partial \rho }{\partial t}}+{\vec {\nabla }}\rho \cdot {\vec {u}}+\rho {\vec {\nabla }}\cdot {\vec {u}}&&=0\end{alignedat}}}$(Reason: product rule )

For the change in density over time for a particle passing through the orbit , this says: ${\displaystyle {\vec {x}}(t)}$

${\displaystyle {\begin{alignedat}{2}&{\frac {\partial \rho }{\partial t}}+{\vec {\nabla }}\rho \cdot {\frac {\mathrm {d} {\vec {x}}}{\mathrm {d} t}}&&=-\rho \cdot {\vec {\nabla }}\cdot {\vec {u}}\\\Leftrightarrow &{\frac {\mathrm {d} }{\mathrm {d} t}}\rho (t,{\vec {x}}(t))&&=-\rho \cdot {\vec {\nabla }}\cdot \,{\vec {u}}\end{alignedat}}}$(Reason: total differential ).

Along a trajectory, the density changes with the divergence of the flow ${\displaystyle {\vec {u}}.}$

The flow is incompressible if the density remains constant along a trajectory:

${\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}\rho (t,{\vec {x}}(t))=0}$

It follows that in this case the divergence of the flow is zero:

${\displaystyle \Rightarrow {\vec {\nabla }}\cdot {\vec {u}}={\frac {\partial u}{\partial x}}+{\frac {\partial v}{\partial y}}+{\frac {\partial w}{\partial z}}=0}$

Electrodynamics

In electrodynamics , the continuity equation for the electrical charge density and the electrical current density results from the identity and the two inhomogeneous Maxwell equations ${\displaystyle \rho }$ ${\displaystyle {\vec {j}}}$ ${\displaystyle {\vec {\nabla }}\cdot {\vec {\nabla }}\times \dots =0}$

${\displaystyle 0\ \ {\stackrel {\operatorname {div} \ \operatorname {rot} \ =\ 0}{=}}\ \ {\vec {\nabla }}\cdot \left({\vec {\nabla }}\times {\vec {H}}\right)\ \ {\stackrel {\text{Maxwell}}{=}}\ \ {\vec {\nabla }}\cdot \left({\frac {\partial }{\partial t}}{\vec {D}}+{\vec {j}}\right)={\frac {\partial }{\partial t}}{\vec {\nabla }}\cdot {\vec {D}}+{\vec {\nabla }}\cdot {\vec {j}}={\frac {\partial \rho }{\partial t}}+{\vec {\nabla }}\cdot {\vec {j}}\ ,}$

d. H. it follows with the other inhomogeneous Maxwell equation

${\displaystyle {\frac {\partial \rho }{\partial t}}+{\vec {\nabla }}\cdot {\vec {j}}=0\ .}$

In semiconductors describes the violation of the continuity equation

${\displaystyle {\frac {\partial \rho }{\partial t}}+{\vec {\nabla }}\cdot {\vec {j}}=-r+g}$

the change in the space charge density due to the recombination rate per volume,, and the generation rate . ${\displaystyle \rho }$ ${\displaystyle r}$ ${\displaystyle g}$

From the Maxwell equations of electrodynamics it follows (in CGS units) for the energy density

${\displaystyle u={\frac {1}{8\pi }}\left({\vec {E}}^{2}+{\vec {B}}^{2}\right)}$

and the energy flux density (also Poynting vector )

${\displaystyle {\vec {S}}={\frac {c}{4\pi }}\left({\vec {E}}\times {\vec {H}}\right)}$

almost a continuity equation:

${\displaystyle {\frac {\partial u}{\partial t}}+{\vec {\nabla }}\cdot {\vec {S}}=-{\vec {j}}\cdot {\vec {E}}\,.}$

The continuity equation for the energy in the electromagnetic field is fulfilled where the electrical current density disappears, for example in a vacuum. There energy density can only change through energy flows. Where the electrical current density does not disappear, the electrical field does work and exchanges energy with the charge carriers. ${\displaystyle {\vec {j}}}$ ${\displaystyle {\vec {j}}}$ ${\displaystyle {\vec {E}}}$

The continuity equation for electromagnetic field energy is Poynting's theorem .

In the relativistic formulation of electrodynamics with Minkowski vectors , cρ and j are combined into a four-vector . As above, it follows from the Maxwell equations that its four-way divergence vanishes. This formulation is independent of the Minkowski signature chosen, equivalent to the continuity equation and can be generalized to relativistic field theories. ${\displaystyle (j^{\alpha })=(c\rho ,j_{x},j_{y},j_{z})\,,}$ ${\displaystyle \partial _{\alpha }j^{\alpha }={\frac {c\partial \rho }{c\partial t}}+{\frac {\partial j_{x}}{\partial x}}+{\frac {\partial j_{y}}{\partial y}}+{\frac {\partial j_{z}}{\partial z}}=0\,.}$

Quantum mechanics

In non-relativistic quantum mechanics, the state of a particle, such as a single electron , is described by a wave function . ${\displaystyle \Psi ({\vec {x}},t)}$

The square of the amount

${\displaystyle \rho ({\vec {x}},t)=|\Psi ({\vec {x}},t)|^{2}}$

is the probability density to ensure a particle at the time at the location to be found. With the associated probability current density ${\displaystyle t}$ ${\displaystyle {\vec {x}}}$

${\displaystyle {\vec {j}}=-{\frac {i\hbar }{2m}}(\Psi ^{*}{\vec {\nabla }}\Psi -\Psi {\vec {\nabla }}\Psi ^{*})}$

Without an external magnetic field, the continuity equation applies as a consequence of the Schrödinger equation

${\displaystyle {\frac {\partial }{\partial t}}\rho +{\vec {\nabla }}\cdot {\vec {j}}=0\,}$.

If there is an external magnetic field, the Pauli equation must be used and this results

${\displaystyle {\vec {j}}=-{\frac {i\hbar }{2m}}(\Psi ^{\dagger }{\vec {\nabla }}\Psi -({\vec {\nabla }}\Psi ^{\dagger })\Psi )-{\frac {q}{m}}{\vec {A}}\Psi ^{\dagger }\Psi +{\frac {\hbar }{2m}}{\vec {\nabla }}\times (\Psi ^{\dagger }{\vec {\sigma }}\Psi )}$

where stand for the Pauli matrices . The last term disappears when the divergence is formed and cannot be derived directly from the Pauli equation, but results from the non-relativistic limit case of the Dirac equation. ${\displaystyle \sigma }$

In the context of relativistic quantum mechanics, particles obey the Klein-Gordon equation (for scalar bosons ) or the Dirac equation (for fermions ). Since the equations obey the special theory of relativity, the continuity equations for these cases can be manifestly covariant

${\displaystyle \partial _{\mu }j^{\mu }=\partial _{t}\rho +{\vec {\nabla }}\cdot {\vec {j}}=0}$

be written and it arises

${\displaystyle j_{\text{KG}}^{\mu }=\mathrm {i} \left(\phi ^{*}\partial ^{\mu }\phi -\phi \partial ^{\mu }\phi ^{*}\right)}$

${\displaystyle j_{\text{Dirac}}^{\mu }=\psi ^{\dagger }\gamma ^{0}\gamma ^{\mu }\psi }$

where and respectively stand for the scalar bosonic / vector valued fermionic wave function and are the Dirac matrices . ${\displaystyle \phi }$ ${\displaystyle \psi }$ ${\displaystyle \gamma }$

In the context of the Klein-Gordon continuity equation - in contrast to the nonrelativistic or fermion case - the quantity cannot be interpreted as a probability density, since this quantity is not positive semidefinite. ${\displaystyle j^{0}={\frac {1}{c}}\mathrm {i} \left(\phi ^{*}\partial _{t}\phi -\phi \partial _{t}\phi ^{*}\right)}$

Further applications: general conserved quantities

The analogy to the “electrical” case shows that continuity equations must always apply when a charge-like quantity and a current-like quantity are related as stated above. Another concrete example could be the heat flow , which is important in thermodynamics . When integrated over the entire space, the “charge density” must result in a conservation quantity, e.g. B. the total electrical charge, or - in the case of quantum mechanics - the total probability, 1 , or in the third case, the total heat supplied, in systems whose heat content can be viewed as "preserved" (e.g. heat diffusion ).

In fluid mechanics, the continuity law for (incompressible) fluids follows from the continuity equation .

literature

• Batchelor, GK : An introduction to fluid dynamics, Cambridge university press, 2000, ISBN 0-521-66396-2

Web links

• Video: equation of continuity . Institute for Scientific Film (IWF) 2004, made available by the Technical Information Library (TIB), doi : 10.3203 / IWF / C-14818 .

References and footnotes

1. ↑ When deriving u. a. the divergence of the so-called Maxwell's complement is formed and the interchangeability of the partial derivative with the divergence operator is used. ${\displaystyle {\frac {\partial {\vec {D}}}{\partial t}}}$ ${\displaystyle {\frac {\partial }{\partial t}}}$
2. ↑ Torsten Fließbach : Electrodynamics Spectrum Akademischer Verlag, 3rd edition, p. 159.