✎ Continuity Equation [The Physics Travel Guide]

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<WRAP lag> $\color{blue}{\frac{\partial \rho}{\partial t}}  = \color{red}{\sigma} - \color{magenta}{\rho \vec \nabla  \vec v} $</WRAP>

====== Continuity Equation ======

The continuity equation states that the total $\color{blue}{\text{change of some quantity}}$ is equal to the $\color{red}{\text{amount that gets produced}}$ minus the amount that $\color{magenta}{\text{flows out of the volume}}$.

[{{ :equations:venturi.gif?nolink |Image by Thierry Dugnolle}}]

If we are dealing with a conserved quantity, like energy or electric charge, the total amount that is produced or destroyed is exactly zero.

Whenever a system possesses some symmetry we know from [[theorems:noethers_theorems|Noether's theorem]] that some corresponding quantity is conserved. Using Noether's theorem, we can then also derive the corresponding continuity equation that describes how the conserved quantity flows through the system.

----

**Recommended further reading**

* [[https://www.wired.com/2016/08/perfection-continuity-equation-key-foundations-reality/|The Perfection of the Continuity Equation, Key to the Foundations of Reality]] by WIRED magazine
  
<tabbox Concrete> 
  * The continuity equation in __hydrodynamics__ describes the flow of mass. Here $ρ$ is fluid density and $ \vec v$ the fluid flow velocity. In this context it is also known as one of the Euler equations of fluid dynamics.
  * The continuity equation in __electrodynamics__ describes the flow of electric charge. Here $ρ$ is the charge density and $ \vec v$ the electric flow velocity, such that $ρ \vec v = j$ is the electric current.
  * The continuity equation in __quantum mechanics__ describes the flow of probability. Here $ρ = \Psi^\dagger \Psi$ is the probability density and $ \vec v$ the probability flow velocity, such that $ρ \vec v = j =  \frac { \hbar } { 2m i } [ \Psi ^ { * } ( \nabla \Psi ) - \Psi ( \nabla \Psi ^ { * } )]$ is the probability current .

In general, continuity equations can be derived by using [[theorems:noethers_theorems|Noether's theorem]].

-->Derivation of the continuity equation in electrodynamics#

We start with Ampere's law, which is one of the [[equations:maxwell_equations|Maxwell equations]]
$$ \nabla \times  H  =  J  + \frac { \partial  D  } { \partial t }.$$

Next we take the divergence of this equation, which yields
$$\nabla \cdot ( \nabla \times  H  ) = \nabla \cdot  J  + \frac { \partial ( \nabla \cdot  D  ) } { \partial t }
 $$
The divergence of a curl is zero, and therefore we get
$$ \nabla \cdot  J  + \frac { \partial ( \nabla \cdot  D  ) } { \partial t } = 0.
$$
Finally, we use another [[equations:maxwell_equations|Maxwell equation]], namely [[formulas:gauss_law|Gauss law]], 
$$\nabla \cdot  D  = \rho
 $$
and substitute it into the previous equation
$$ \nabla \cdot  J  + \frac { \partial \rho } { \partial t } = 0 .$$
This is exactly the continuity equation.

<--

-->Derivation of the continuity equation in quantum mechanics#

The probability density is quantum mechanics is $\rho = \Psi^* \Psi$. The partial derivative of $\rho$ with respect to time is therefore
$$ \frac { \partial \rho } { \partial t } = \frac { \partial | \Psi | ^ { 2} } { \partial t } = \frac { \partial } { \partial t } ( \Psi ^ { * } \Psi ) = \Psi ^ { * } \frac { \partial \Psi } { \partial t } + \Psi \frac { \partial \Psi ^ { * } } { \partial t } . $$

Next, we consider the [[equations:schroedinger_equation|Schrödinger equation]]

$$ - \frac { \hbar ^ { 2} } { 2m } \nabla ^ { 2} \Psi ^ { * } + U \Psi ^ { * } = - i \hbar \frac { \partial \Psi ^ { * } } { \partial t }.$$

Taking the complex conjugate of it yields

$$- \frac { \hbar ^ { 2} } { 2m } \nabla ^ { 2} \Psi ^ { * } + U \Psi ^ { * } = - i \hbar \frac { \partial \Psi ^ { * } } { \partial t }. $$

Multiplying the Schrödinger equation with $\Psi^\star$ and the complex conjugated Schrödinger equation with $\Psi$ yields the two equations

$$ \Psi \cdot \frac { \partial \Psi } { \partial t } = \frac { 1} { i \hbar } [ - \frac { \hbar ^ { 2} \Psi ^ { * } } { 2m } \nabla ^ { 2} \Psi + U \Psi ^ { * } \Psi ]$$
$$ \Psi \frac { \partial \Psi ^ { * } } { \partial t } = - \frac { 1} { i \hbar } [ - \frac { \hbar ^ { 2} \Psi } { 2m } \nabla ^ { 2} \Psi ^ { * } + U \Psi \Psi ^ { * } ] .$$

Putting these two equations into our equation for $\frac { \partial \rho } { \partial t }$ from above yields
$$ \frac { \partial \rho } { \partial t } = \frac { 1} { i \hbar } [ - \frac { \hbar ^ { 2} \Psi ^ { * } } { 2m } \nabla ^ { 2} \Psi + U \Psi ^ { * } \Psi ] - \frac { 1} { i \hbar } [ - \frac { \hbar ^ { 2} \Psi } { 2m } \nabla ^ { 2} \Psi ^ { * } + U \Psi \Psi ^ { * } ]$$
$$ = \frac { \hbar } { 2i m } [ \Psi \nabla ^ { 2} \Psi ^ { * } - \Psi ^ { * } \nabla ^ { 2} \Psi ] .$$

The second puzzle piece that appears in the continuity equation is the current $j$ which in quantum mechanics is given by
$$j =  \frac { \hbar } { 2m i } [ \Psi ^ { * } ( \nabla \Psi ) - \Psi ( \nabla \Psi ^ { * } )]$. $$
Taking the divergence of it (since $\nabla j$ is what appears in the continuity equation)  yields
$$ \nabla \cdot j = \nabla \cdot [ \frac { \hbar } { 2m i } ( \Phi ^ { * } ( \nabla \Phi ) - \Psi ( \nabla \Psi ^ { * } ) ) ]
 $$
$$ = - \frac { \hbar } { 2m i } [ \Psi ( \nabla ^ { 2} \Psi ^ { * } ) - \Psi ^ { * } ( \nabla ^ { 2} \Psi ) ]. $$

This is exactly, except for the minus sign what we derived above for $\frac { \partial \rho } { \partial t }$ and therefore we can conclude

$$ \frac { \partial \rho } { \partial t } = - \nabla \cdot j $$
$$ \therefore \frac { \partial \rho } { \partial t } +  \nabla \cdot j =0. $$

This is exactly the continuity equation that we wanted to derive.
<--

----

**Integral Form of the continuity equation**

By integrating the continuity equation over some volume $V$, we get

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

For the second term, we can then use [[basic_tools:vector_calculus:gauss_theorem|Gauss divergence theorem]] that tells us that we can replace the volume integral over some divergence by a surface integral

$$ \int_V \frac { \partial \rho } { \partial t } +  \int_S  j =0 , $$

where $S$ denotes the surface of our volume $V$. This is the integral form of the continuity equation.

* The first term simply describes the total amount of the quantity, e.g. electric charge or mass, inside our volume. 
  * The second term describes the amount that flows into the surface minus the amount that flows out of the surface.
<tabbox Abstract>

In relativistic theories, the charge density and the current live in one object called the current four-vector (or four-current) $j_\mu = (c \rho, \vec j),$ where $c$ denotes the speed of light and is inserted such that all components have the same dimensions. Using this definition, the continuity equation reads

$$ \partial_\mu j^\mu = 0. $$

The continuity equation can also be written using the [[advanced_tools:differential_forms|3-form]] of charge density

$$ d \gamma = 0$$

where
$$ \gamma  = - \frac { 1} { c } ( 1,d x ^ { 2} \wedge d x ^ { 3} + i _ { 2} d x ^ { 3} \wedge d x ^ { 1} + j _ { 3} d x ^ { 1} \wedge d x ^ { 2} ) \wedge d x ^ { 0} + \rho d x ^ { \prime } \wedge d x ^ { 2} \wedge d x ^ { 3}.$$
<tabbox Why is it interesting?> 
  
A continuity equation is important whenever we are dealing with a system where some quantity is conserved. The continuity helps us to keep track how the conserved quantity moves through the system.

Important examples are

* the continuity equation in electrodynamics that encodes the conservation of electric charge,
  * the continuity equation in hydrodynamics that encodes the conservation of mass.
  * the continuity equation in quantum mechanics that encodes the conservation of probability.

</tabbox>

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