equations:continuity_equation
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equations:continuity_equation [2018/04/19 09:40] jakobadmin [Concrete] |
equations:continuity_equation [2020/03/03 10:38] (current) 128.179.254.165 |
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| - | <WRAP lag> $\color{blue}{\frac{\partial \rho}{\partial t}} + \color{magenta}{\rho \vec \nabla \vec v} = \color{red}{\sigma} $</WRAP> | + | <WRAP lag> $\color{blue}{\frac{\partial \rho}{\partial t}} = \color{red}{\sigma} - \color{magenta}{\rho \vec{\nabla} \cdot \vec{v}} $</WRAP> |
| ====== Continuity Equation ====== | ====== Continuity Equation ====== | ||
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| <tabbox Intuitive> | <tabbox Intuitive> | ||
| - | The continuity equation states that the $\color{red}{\text{total amount of a quantity (like water) that is produced (or destroyed) inside some volume}}$ is proportional to the $\color{blue}{\text{change of the quantity}}$ plus the $\color{magenta}{\text{total amount that flows in minus the amount that flows out of the volume}}$. | + | 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 $\color{magenta}{\text{flows out of the volume}}$. |
| - | + | ||
| - | Or formulated differently, the total $\color{blue}{\text{change of some quantity}}$ is equal to the $\color{red}{\text{amount that gets produced}}$ plus the amount that $\color{magenta}{\text{flows in minus the amount that flows out of the volume}}$. | + | |
| [{{ :equations:venturi.gif?nolink |Image by Thierry Dugnolle}}] | [{{ :equations:venturi.gif?nolink |Image by Thierry Dugnolle}}] | ||
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| 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. | 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> | <tabbox Concrete> | ||
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| $$ \nabla \cdot J + \frac { \partial ( \nabla \cdot D ) } { \partial t } = 0. | $$ \nabla \cdot J + \frac { \partial ( \nabla \cdot D ) } { \partial t } = 0. | ||
| $$ | $$ | ||
| - | Finally, we use another [[equations:maxwell_equations|Maxwell equation]], namely [[equations:yang_mills_equations:gauss_law|Gauss law]], | + | Finally, we use another [[equations:maxwell_equations|Maxwell equation]], namely [[formulas:gauss_law|Gauss law]], |
| $$\nabla \cdot D = \rho | $$\nabla \cdot D = \rho | ||
| $$ | $$ | ||
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| This is exactly the continuity equation that we wanted to derive. | 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> | <tabbox Abstract> | ||
| - | <note tip> | + | 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 |
| - | The motto in this section is: //the higher the level of abstraction, the better//. | + | |
| - | </note> | + | $$ \partial_\mu j^\mu = 0. $$ |
| + | |||
| + | 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?> | <tabbox Why is it interesting?> | ||
| | | ||
equations/continuity_equation.1524123631.txt.gz · Last modified: 2018/04/19 07:40 (external edit)
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