Continuity equation

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Continuity equations are a stronger, local form of conservation laws. For example, it is true that "the total energy in the universe is conserved". But this statement does not immediately rule out the possibility that energy could disappear from Earth while simultaneously appearing in another galaxy. A stronger statement is that energy is locally conserved: Energy can neither be created nor destroyed, nor can it "teleport" from one place to another—it can only move by a continuous flow. A continuity equation is the mathematical way to express this kind of statement.




General equation[edit]

Definition of flux[edit]

Main article: Flux

Before we can write down the continuity equation (below), we must first define flux, a quantity specifying flow or motion.

The continuity equation is applicable when there is some quantity q which can flow or move, such as mass, energy, electric charge, momentum, number of molecules, etc. Let ρ be the volume density of this property, i.e., the amount of q per unit volume.


  • The dimension of flux is "amount of q per unit time, per unit area". For example, in the mass continuity equation for flowing water, if 1 gram per second of water is flowing through a pipe with cross-sectional area 1 cm2, then the average mass flux j inside the pipe is (1 gram / second) / cm2, pointing along the pipe in the direction that the water is flowing. Outside the pipe, where there is no water, the flux is zero.
\mathbf{j} = \rho \mathbf{u}
For example, in the mass continuity equation for flowing water, u would be the water's velocity at each point, ρ would be the water's density at each point, and then j would be the mass flux.

 (\text{Rate that }q\text{ is flowing through the imaginary surface }S) = \int\!\!\!\!\int_S \mathbf{j} \cdot d\mathbf{S}

in which \int\!\!\!\!\int_S d\mathbf{S} is a surface integral.

Integral form[edit]

The integral form of the continuity equation states that:


Mathematically, the integral form of the continuity equation is:

\frac{d q}{d t} + \oiint\scriptstyle S\mathbf{j} \cdot d\mathbf{S} = \Sigma


In the integral form of the continuity equation, S is any imaginary closed surface that fully encloses a volume V, like any of the surfaces on the left. S can not be a surface with boundaries, like those on the right. (Surfaces are blue, boundaries are red.)
  • \oiint\scriptstyle Sd\mathbf{S} denotes a surface integral over that closed surface,
  • j is the flux of q,
  • t is time,
  • \Sigma is the net rate that q is being generated inside the volume V. (When q is being generated, it is called a "source" of q, and it makes Σ more positive. When q is being destroyed, it is called a "sink" of q, and it makes Σ more negative.)

In a simple example, V could be a building, and q could be the number of people in the building. The surface S would consist of the walls, doors, roof, and foundation of the building. Then the continuity equation states that the number of people in the building increases when people enter the building (an inward flux through the surface), decreases when people exit the building (an outward flux through the surface), increases when someone in the building gives birth (a "source", Σ > 0), and decreases when someone in the building dies (a "sink", Σ < 0).

Differential form[edit]


\frac{\partial \rho}{\partial t} + \nabla \cdot \mathbf{j} = \sigma\,


  • ∇⋅ is divergence,
  • j is the flux of q,
  • t is time,
  • σ is the generation of q per unit volume per unit time. Terms that generate (σ > 0) or remove (σ < 0) q are referred to as a "sources" and "sinks" respectively.





Main article: Charge conservation

In electromagnetic theory, the continuity equation is an empirical law expressing (local) charge conservation. Mathematically it is an automatic consequence of Maxwell's equations, although charge conservation is more fundamental than Maxwell's equations. It states that the divergence of the current density J (in amperes per square meter) is equal to the negative rate of change of the charge density ρ (in coulombs per cubic metre),

 \nabla \cdot \mathbf{J} = - {\partial \rho \over \partial t}



Fluid dynamics[edit]

In fluid dynamics, the continuity equation states that, in any steady state process, the rate at which mass enters a system is equal to the rate at which mass leaves the system.[1][2]

The differential form of the continuity equation is:[1]




If ρ is a constant, as in the case of incompressible flow, the mass continuity equation simplifies to a volume continuity equation:[1]

\nabla \cdot \mathbf{u} = 0,

which means that the divergence of velocity field is zero everywhere. Physically, this is equivalent to saying that the local volume dilation rate is zero.

Energy and heat[edit]

Conservation of energy says that energy cannot be created or destroyed. (See below for the nuances associated general relativity.) Therefore there is a continuity equation for energy flow:

 \frac{ \partial u}{\partial t} + \nabla \cdot \mathbf{q} = 0


  • u = local energy density (energy per unit volume),
  • q = energy flux (transfer of energy per unit cross-sectional area per unit time) as a vector,


Probability distributions[edit]


Quantum mechanics[edit]


 \rho(\mathbf{r},t) = \Psi^{*}(\mathbf{r},t) \Psi(\mathbf{r},t) = |\Psi(\mathbf{r},t)|^2 \,\!
  • The probability of finding the particle within V at t is denoted and defined by:
P=P_{\mathbf{r} \in V}(t) = \int_V \Psi^{*} \Psi d V = \int_V |\Psi|^2 d V \,
 \mathbf{j}(\mathbf{r},t) = \frac{\hbar}{2mi} \left [ \Psi^{*} \left ( \nabla \Psi \right ) - \Psi \left ( \nabla \Psi^{*} \right ) \right ].

With these definitions the continuity equation reads:

 \nabla \cdot \mathbf{j} + \frac{\partial \rho}{\partial t} = 0 \rightleftharpoons \nabla \cdot \mathbf{j} + \frac{\partial |\Psi|^2}{\partial t} = 0.


Relativistic version[edit]

Special relativity[edit]

See also: 4-vector



J = \left(c \rho, j_x, j_y, j_z \right)


\partial_\mu J^\mu = \frac{\partial \rho}{\partial t} + \nabla \cdot \mathbf{j}


\partial_\mu J^\mu = 0

in the usual case where there are no sources or sinks, i.e. for perfectly conserved quantities like energy or charge. This continuity equation is manifestly ("obviously") Lorentz invariant.

Examples of continuity equations often written in this form include electric charge conservation \partial_\mu J^\mu = 0 where J is the electric 4-current; and energy-momentum conservation \partial_\nu T^{\mu\nu} = 0 where T is the stress-energy tensor.

General relativity[edit]



 T^{\mu}_{\;\;  \nu ; \mu} = 0.

This is an important constraint on the form the Einstein field equations take in general relativity.[4]

However, the ordinary divergence of the stress-energy tensor does not necessarily vanish:[5]

  - \Gamma^{\mu}_{\mu \lambda} T^{\lambda \nu} - \Gamma^{\nu}_{\mu \lambda} T^{\mu \lambda},



Particle physics[edit]


There are many other quantities in particle physics which are often or always conserved: baryon number (proportional to the number of quarks minus the number of anti-quarks), electron number, mu number, tau number, isospin, and others.[7] Each of these has a corresponding continuity equation, possibly including source / sink terms.

Conserved currents from Noether's theorem[edit]

For more detailed explanations and derivations, see Noether's theorem.

If a physical field described by a function of space and time, ϕ(r, t), is varied by a small amount;

\phi \rightarrow \phi + \delta\phi

then Noether's theorem states the Lagrangian L, or rather the Lagrangian density ℒ for fields, is invariant (does not change):

\mathcal{L} \rightarrow \mathcal{L} + \delta\mathcal{L}, \ \delta\mathcal{L}=0

under a continuous symmetry, in which the field is a continuous variable. The field ϕ(r, t) can be a scalar field, often a scalar potential like the classical gravitational potential ζ(r, t) or electric potential ϕ(r, t), or a vector field like the Newtonian gravitational field g(r, t) or vector potentials like the magnetic potential A(r, t), or a tensor field of any order such as the electromagnetic tensor F(r, t), or even a spinor field which arise in describing particles as quantum fields. The Lagrangian density is a function of the field and its space and time derivatives: ℒ[ϕ(r, t), ϕ(r, t)/t, ∇ϕ(r, t)].

The above considerations leads to conserved current densities in a completely general form:[8]

J^\mu (\mathbf{r},t) = \frac{\partial \mathcal{L}}{\partial [\partial_\mu \phi(\mathbf{r},t)]}\delta\phi(\mathbf{r},t)

because they satisfy the continuity equation

\partial_\mu J^\mu = 0

Integrating the density of the conserved quantity J0/c, over a spacelike region of volume V, gives the total amount of conserved quantity within that volume:

q(t) = \int_V \frac{1}{c} J^0(\mathbf{r},t) \, dV

Equivalently, integrating the current density of the conserved quantity j = (J1, J2, J3), through a closed spacelike surface S (see above descriptions) over some time duration, also gives the conserved quantity flowing through that surface in that time duration:

q(t_2 - t_1) = \int_{t_1}^{t_2} \iint_S \mathbf{j}(\mathbf{r},t) \cdot d\mathbf{S} \,dt\,

See also[edit]


  1. ^ a b c Pedlosky, Joseph (1987). Geophysical fluid dynamics. Springer. pp. 10–13. ISBN 978-0-387-96387-7. 
  4. ^ D. McMahon (2006). Relativity DeMystified. Mc Graw Hill (USA). ISBN 0-07-145545-0. 
  8. ^ D. McMahon, Mc Graw Hill (USA) (2008). Quantum Field Theory. ISBN 978-0-07-154382-8. 

Further reading[edit]
