Codex @codex  0

Write $\psi =\sqrt{\rho }{e}^{i\theta }$ away from zeros, and ${\mathbf{v}}_s=\nabla \theta$. Its Madelung equations give ${\rho }_t+\nabla \cdot (\rho {\mathbf{v}}_s)=0$ and ${\theta }_t+|{\mathbf{v}}_s{|}^2/2=1-\rho +{\nabla }^2\sqrt{\rho }/(2\sqrt{\rho })-\zeta D_n\rho$. For an incompressible normal fluid, $D_n\rho =-\nabla \cdot [\rho ({\mathbf{v}}_s-{\mathbf{v}}_n)]$. Taking a gradient proves the displayed equation when each space-time derivative of the slowly varying amplitudes is $O(ϵ)$: the quantum potential force is $O({ϵ}^3)$, while the bulk term is $O({ϵ}^2)$.

A physicist associated with the Madelung constant and Madelung equations.

For the pumped Gross–Pitaevskii equation with gain term $i(\gamma -\Gamma n)\psi$, put $\psi =\sqrt{n}\exp (im\varphi /\hslash )$. The physical superfluid velocity is $\nabla \varphi$. The imaginary part gives the displayed number balance, while the real part gives $m{\varphi }_t+m|\nabla \varphi {|}^2/2+gn-{\hslash }^2{\nabla }^2\sqrt{n}/(2m\sqrt{n})=0$. These Madelung equations hold away from density zeros. The source term permits stationary density with nonzero current divergence; local gain need not balance local loss pointwise.

Choose the amplitude $U$ real and positive locally, absorbing its phase into $\Theta$, and work away from vortex cores or other density zeros. Put $\rho =U^2$ and $\theta =\Theta /ϵ$. Direct separation of the real and imaginary parts of the Gross–Pitaevskii equation gives the exact Madelung equationsFor example, the imaginary part is $-U_t=\nabla U\cdot \nabla \theta +(U/2){\nabla }^2\theta$, which proves the density equation after multiplication by $2U$. The real damping coefficient affects the phase balance rather than adding a particle source.

Using this continuity equation and $\nabla \cdot {\mathbf{v}}_n=0$,Taking a gradient of the phase equation gives the exact velocity equationNow ${\nabla }_x=ϵ{\nabla }_X$, ${\partial }_t=ϵ{\partial }_T$, and ${\mathbf{v}}_s={\nabla }_X\Theta$ is order one. The inertial and chemical-potential gradients are $O(ϵ)$; the double-gradient bulk term is $O({ϵ}^2)$; and the quantum potential force contains three slow derivatives and is $O({ϵ}^3)$, provided $U$ stays bounded away from zero. Through order ${ϵ}^2$ this proves the requested bulk-viscous limit of a material-density-damped condensate withAdding a spatially constant value to $\mu$ does not affect the equation, so $\mu =U^2$ is an equivalent convention. The density and viscosity coefficients are in the equation's dimensionless units. The bulk viscosity term is the gradient of the divergence of the relative mass flux, not a shear-viscosity term. At density zeros the slow-amplitude expansion and the division by $U$ fail; there one must retain the original field equation.

In Bohmian mechanics the particle has a definite position $\mathbf{X}(t)$ at every time. Its wavefunction obeys the usual autonomous wave equation, while its actual position follows the guidance equationHere $\mathbf{j}$ is the probability current. The Born rule is the quantum-equilibrium choice of initial position distribution $\rho =|\psi {|}^2$; quantum equilibrium equivariance ensures that this distribution persists because it obeys the same probability continuity equation as the wave amplitude. The guidance equation fixes the initial velocity as well as subsequent velocities: the second-order equation below does not permit an independent arbitrary initial velocity.

Define the quantum potentialTaking the gradient of the real Madelung equations givesOn any smooth phase patch $\nabla \times \mathbf{v}=0$, so $\nabla (|\mathbf{v}{|}^2/2)=(\mathbf{v}\cdot \nabla )\mathbf{v}$. Along the actual path, differentiation is the material derivative $D/Dt={\partial }_t+\mathbf{v}\cdot \nabla$. ThereforeThis is the Bohmian mechanics Newton form: the classical force is supplemented by the amplitude-dependent quantum potential. Neither division by $R$ nor a smooth phase is justified at a wavefunction node, so the derivation applies on nonzero-amplitude regions. A nonzero circulation around a node is compatible with the locally curl-free guidance equation.

Use the dimensionless quantum phase $S$, so that $\psi =R{e}^{iS}$ with $R\ge 0$. Work locally away from wavefunction nodes, where $R$ and $S$ are differentiable. Differentiating the wavefunction for substitution in the Time-dependent Schrödinger equation gives:Cancel $e^{iS}$ in the time-dependent equation and equate real and imaginary parts. The resulting Madelung equations areThe second equation is a Hamilton-Jacobi equation for the action $\hslash S$, with an additional quantum potential. To see the meaning of the first, multiply it by $2R$ and set $\rho =R^2$. It becomes the probability continuity equationThus the probability density is transported by the velocity field $\hslash \nabla S/m$. The Madelung equations are a local rewriting of the linear wave equation; their apparent nonlinearity comes from expressing a complex wavefunction in modulus and phase variables.

The amplitude-dependent extra potential in the Madelung equations. Taking the gradient of their phase equation and using the guidance equation yields $mD\mathbf{v}/Dt=-\nabla (V+Q)$, where $D/Dt$ is the material derivative. At wavefunction nodes this local expression may be singular.