Continuity equation
A continuity equation is a partial differential equation determined that belongs to (see below) and the temporal change of the size corresponding to this density by the spatial change in its power density in the following manner linked to a conserved:
For the mathematical definition of see divergence ( mathematics).
Associated with a conserved quantity
The contained in a volume V " charge " (the volume integral of the density) may change due to the continuity equation only in that unbalanced currents from the surface of the volume flow out. Accordingly, the total charge does not change for time and is conserved if there is no ( net ) currents flow through the surface of the observed volume.
For the change in the charge in a time-invariant volume
Is because of the continuity equation for the integral theorem of Gauss
Equal to the surface integral over the peripheral surface of the volume of the proportion of the current density flowing in the direction of the surface normal to the outside. The charge in the volume changes only, if unbalanced currents flow in the manner specified by the boundary surface.
The continuity equation occurs in all field theories of physics. The obtained values can, as in the following examples, the mass, the electric charge, the energy of the pulse, the probability of particles and some ( leptons, baryon ) be.
Hydrodynamics
Changes in the hydrodynamic mass density, as the fluid flows with the speed along the path curves, as is
The corresponding current density, and the continuity equation is
For the change in the density at a particle that passes through the web, this implies:
Along a trajectory that is, the density changes with the divergence of the flow, the flow is incompressible, the density along a trajectory remains constant. It follows that the divergence of the flow is zero.
Electrodynamics
In electrodynamics, the continuity equation for the electric charge density and the electric current density results using the identity from the Maxwell equations
In semiconductors describes the violation of the continuity equation
The change of the space charge density due to the recombination rate per unit volume, and the generation rate.
From the Maxwell equations of electrodynamics follows (in CGS units ) for the energy density
And the energy flux density (also Poynting vector )
Almost a continuity equation:
The continuity equation for the energy in the electromagnetic field is met, where the electric current density vanishes, for example in vacuo. There, energy density can only be changed by energy flows. Where the electric current density does not vanish, making the electric field work and exchanges energy with the charge carriers from.
The continuity equation for the electromagnetic field energy is the set of Poynting.
In the relativistic formulation of electrodynamics with Minkowski vectors cρ we group and j to a four-vector together. As above, it follows from the Maxwell equations that the four divergence vanishes This formulation is independent of the chosen Minkowski signature, equivalent to the continuity equation and can be generalized to relativistic field theories.
Quantum mechanics
In quantum mechanics, the state of a particle, such as a single electron, is described by a wave function.
The magnitude-squared
Gives the probability density to ensure a particle to be found currently in place. With the associated probability current density
Applies as a result of the Schrödinger equation, the continuity equation
Other applications: General conserved quantities
It is seen from the analogy to the " electric " case that continuity equations must always apply when a charge -like size and a current-type size related as indicated above. As another concrete example, one could specify about the important in thermodynamics heat flow. The " charge density " must when integrated over the whole area a conserved quantity arising, for example, the total electric charge, or - in the case of quantum mechanics - the overall probability 1, or in the third case, the total heat supplied, for systems with heat content can be considered " received " as (eg thermal diffusion).