Conservation laws and continuity equations

Question

I'm a bit messed up with conservation laws and continuity equations.

This is my understanding:

• A conservation law describes that a physical quantity $G$ is conserved with time. It does not prevent "quantity teleportation", as long as at a given time, the created quantity and the disappeared quantity are equal. In Wikipedia's wordings: "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 continuity equation is stronger: It implies that there is no "quantity teleportation".

• A global conservation law describes that globally, a physical quantity $G$ is conserved with time: $$\dfrac{\mathrm{d}G}{\mathrm{d}t}=0$$ Using mathematics, this can be written as the sum of an integral over a volume and an integral over a surface, or as a single integral over a volume using Stokes theorem (and introducing a divergence)

• A local conservation law is the result of writing that the integrand of the global law are equal.

Questions:

1. Is "quantity transportation" possible in a local conservation law? If not is there any difference between the two?
2. In the equation (i.e. mathematically), where do you see the differences between continuity equations and conservation laws?

Answer 1

Continuity equations are an embodiment of local conservation laws, and they both reflect the fact that there is no 'quantity teleportation'. That said, the local transport of a quantity is perfectly possible within local conservation laws and it is precisely this that the continuity equation models.

Your distinction between global and local conservation laws could use some refinement, though. Consider a quantity $G$ whose local density is $g(\mathbf r)$, and whose flow density (i.e. flux) is $\mathbf j(\mathbf r)$. With this notation, a global conservation law establishes only that the total amount of $G$, i.e. $$ G=\int g(\mathbf r)\text d\mathbf r, $$ where the integral is over all space, is constant over time.

Saying that $G$ additionally obeys a local conservation law is a stronger statement, and it is exactly the statement that $g$ and $\mathbf j$ obey the continuity equation. This one comes in two flavours:

• differential, $$\frac{\partial g}{\partial t}+\nabla\cdot\mathbf j=0,$$

• and integral, $$\frac{d}{dt}\int_Vg(\mathbf r)\text d\mathbf r +\bigcirc \!\!\!\!\!\!\!\!\!\iint_{\partial V}\mathbf j(\mathbf r)\cdot \text d\mathbf a.$$

It is important to note that both of these forms are completely equivalent (modulo technical assumptions on point, line and surface charges). The differential form holds at every point $\mathbf r$, and the integral form holds for every volume $V$, and one can use appropriate calculations to translate between both forms and therefore between both freedoms.

The reason that we say continuity equations embody local conservation laws is that they make precise the intuition that all the $g$ that "comes out" of some region can be "seen crossing the boundary", which is measured by the surface integral / the divergence term.

This is as opposed to, for example, a quantity with a density of the form $$ g(\mathbf r,t)=g_0 \cos^2(\omega t)e^{-(\mathbf r-\mathbf r_1)^2/\sigma^2} + g_0 \sin^2(\omega t)e^{-(\mathbf r-\mathbf r_2)^2/\sigma^2} $$ where $\mathbf r_1$ and $\mathbf r_2$ are in principle far apart. Here $G$ stays constant, but between $t=0$ and $\pi/2\omega$, all of the $G$ near $\mathbf r_1$ has disappeared without there being any flux through the plane between it and $\mathbf r_2$. Here $G$ obeys a global conservation law, but not a local one.

For clarity, I should note that your statement that "A local conservation law is the result of writing that the integrand of the global law are equal" is incorrect, and depending on exactly what you mean by it, there may be exceptionally few systems that obey that.

Answer 2

In the equation (i.e. mathematically), where do you see the differences between continuity equations and conservation laws?

The continuity equation is not sufficient to derive conservation of something. For example, continuity equation for fluid flow in non-relativistic theory is

$$ \partial_t \rho + \nabla \cdot (\rho \mathbf v) = 0 $$

wherer $\rho$ is density of fluid and $\mathbf v$ is its velocity. Integrating this equation over some region $V$ with boundary surface $\Sigma$, switching the order of integration and differentiation and using the Gauss-Ostrogradskii theorem we obtain $$ \frac{d}{dt}\int_V \rho \,dV = -\oint_\Sigma \rho \mathbf v \cdot d\boldsymbol{\Sigma}. $$ As the region $V$ is expanded to contain all space in a limit, the left-hand side gives rate of range of mass in the whole space. This is not necessarily zero, for the right hand side may be generally have non-zero limit.

To get conservation, we have to impose additional condition that there is a surface $\Sigma$ such that value of the surface integral is zero or at least goes to zero as the surface is expanded to infinity. This may not be always possible.

For example, the Poynting theorem for current-free region

$$ \partial_t \left(\frac{1}{2}\epsilon_0E^2 + \frac{1}{2\mu_0}B^2 \right) + \nabla \cdot (\mathbf E\times \mathbf B/\mu_0) = 0 $$ has the same form as the above equation of continuity, but the integral version $$ \frac{d}{dt}\int_V \left(\frac{1}{2}\epsilon_0E^2 + \frac{1}{2\mu_0}B^2 \right)\,dV = -\oint_\Sigma \mathbf E\times \mathbf B/\mu_0 \cdot d\boldsymbol \Sigma $$ may have nonvanishing surface integral on the right-hand side, if there are EM waves present at infinity. From this it follows that the rate on the left-hand side is not zero and there is no conservation of value of the integral there (Poynting energy).

So continuity does not automatically mean conservation.

Similarly, conservation does not imply continuity - mass or energy can jump from place to place suddenly while the net value remains the same.