Optimal transport from a point-like source

Abstract

We present a dynamical interpretation of the Monge–Kantorovich theory in a stationary regime. This new principle, akin to the Fermat principle of geometric optics, captures the geodesic character of many distribution networks such as plant roots, river basins and the physiological transportation network of metabolites in living systems. Our general continuum framework allows us to map a previously proposed phenomenological principle into a stationary Monge optimization principle in the Kantorovich relaxed format.

Appendices

Appendix A: stationary continuity equation with production $f^+$ and dissipation $f^-$

For any subset $\varDelta \varOmega \subseteq \varOmega$ involved in the stationary motion $x=x(t,y)$, the balance law reads:

$$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}t}\int _{x\in x({\varDelta \varOmega },t)}\rho (x)\mathrm{d}x=\int _{x\in x({\varDelta \varOmega },t)}(f^+ -f^-)(x)\mathrm{d}x. \end{aligned}$$

(41)

Denoting $J(t,y)=\det \frac{\partial x}{\partial y}(t,y)$,

$$\begin{aligned}&\frac{\mathrm{d}}{\mathrm{d}t}\int _{y\in {\varDelta \varOmega }}\rho (x(t,y))J(t,y)\mathrm{d}y=\int _{y\in {\varDelta \varOmega }} (f^+ -f^-) (x(t,y))J(t,y)\mathrm{d}y \nonumber \\&\quad \int _{y\in {\varDelta \varOmega }}\Big [ \frac{\partial \rho }{\partial x}(x(t,y))\cdot \frac{\partial x}{\partial t}(t,y)J(t,y)+ \rho (x(t,y))\frac{\partial }{\partial t}J(y,t) \Big ]\mathrm{d}y \nonumber \\&\quad =\int _{y\in {\varDelta \varOmega }} (f^+ -f^-) (x(t,y))J(t,y)\mathrm{d}y. \end{aligned}$$

(42)

Recalling that

$$\begin{aligned} \frac{\partial }{\partial t}J(t,y)=J(t,y)\nabla \cdot v(t,x)\big |_{x=x(t,y)},\qquad v(t,x)=\frac{\partial x}{\partial t}(t,y)\big |_{y=y(t,x)}=v(x), \end{aligned}$$

we obtain, for any measurable $\varDelta \varOmega$,

$$\begin{aligned} \int _{y\in {\varDelta \varOmega }}\Big [\nabla \cdot (\rho (x) v(x))-(f^+ -f^-)(x)\Big ]\big |_{x=x(y,t)}J(t,y)\mathrm{d}y=0, \end{aligned}$$

(43)

and therefore, under standard smoothness requirements, Eq. (13) does hold.

Appendix B: Fermat-like geometry for the networks

We consider a stationary scenario, in which the distribution of metabolites is independent of time.

Given the rate of local consumption, $f^-(x)\ge 0,\ x\in {\varOmega }$, the ‘living system’ ${\varOmega }$ is characterized by a network of blood and/or lymphatic vasculature that carries the food metabolite, of density $\rho (x)> 0$, from the source point $x_0 \in \varOmega$ to every other point x precisely along channels (pipes), a transportation network, that, we will see as a theorem, is structured as geodesics of a suitable Riemannian metric.

$f^-(x)$ generates, just as in geometric optics, a sort of refraction indexn(x) inducing a Riemann metric g:

$$\begin{aligned} n(x)=\breve{n}[f^-(\cdot )](x),\qquad g_{ij}(x)=n^2(x)\delta _{ij}. \end{aligned}$$

(44)

The ‘length’ function, from the source point $x_0$ to x and generalizing $L_X$ of [1], is⁸

$$\begin{aligned} S(x_0,x):=\inf _{\begin{matrix} h(\cdot )\in H^1_0 ([0,1];{\varOmega })\\ \gamma (t):=x_0+t(x-x_0)+h(t) \end{matrix}} \int _0^1\sqrt{ g_{ij}(\gamma (t))\dot{\gamma }^i (t) \dot{\gamma }^j(t) }\mathrm{d}t. \end{aligned}$$

(45)

When the infimum in (45) is realized by the solutions of the Euler–Lagrange equation of the Lagrangian function

$$\begin{aligned} L(x,\dot{x})=\sqrt{ g_{ij}(x)\dot{x}^i \dot{x}^j }, \end{aligned}$$

we compute the differential of S with respect to x, which is the final point of $\gamma$:

$$\begin{aligned} \frac{\partial S}{\partial x^l}= & {} \frac{\partial }{\partial x^l} \int _0^1\sqrt{ g_{ij}(\gamma (t))\dot{\gamma }^i (t) \dot{\gamma }^j(t) }\mathrm{d}t\Big |_{d\int _0^1 L=0}\\= & {} \int _0^1 \left( \frac{\partial L}{\partial x^l}t+\frac{\partial L}{\partial \dot{x}^l}\right) \mathrm{d}t\Big |_{d\int _0^1 L=0}\\= & {} \int _0^1 \frac{\mathrm{d}}{\mathrm{d}t}\left( \frac{\partial L}{\partial \dot{x}^l}t\right) \mathrm{d}t \Big |_{d\int _0^1 L=0} \\= & {} \frac{\partial L}{\partial \dot{x}^l}\Big |_{t=1}=:p_l=\frac{g_{lm}(x)\dot{\gamma }^m}{\sqrt{ g_{ij}(x)\dot{\gamma }^i \dot{\gamma }^j }}\qquad \qquad (x=\gamma (1),\ \ \ x_0=\gamma (0)). \end{aligned}$$

This implies that $g^{-1}(x)(p,p)=1$, or

$$\begin{aligned} g^{-1}(x)(\nabla S(x),\nabla S(x))= 1. \end{aligned}$$

This is equivalent to

$$\begin{aligned} \parallel \nabla S(x)\parallel ^2 =n^2(x) \end{aligned}$$

(46)

that is, the function

$$\begin{aligned} x\longmapsto S(x)=S(x_0,x) \end{aligned}$$

solves the Hamilton–Jacobi equation (46), which is an eikonal-like equation and $|\nabla S(x)|$ is the Euclidean norm of $\nabla S(x)$.