Founding a mathematical diffusion model in linguistics: the case study of German syntactic features in the North-Eastern Italian dialects

Abstract

The initial motivation for this work was the linguistic case of the spread of Germanic syntactic features into Romance dialects of the North-Eastern Italy, which occurred after the immigration of German people in the Tyrol during the High Middle Age. To obtain a representation of the data over the territory suitable for a mathematical formulation, an interactive map is produced as a first step, using tools of what is called Geographic Data Science. A smooth two-dimensional surface $\mathcal {G}$ is introduced, expressing locally which fraction of territory uses a given German language feature: it is obtained by a piece-wise cubic curvature-minimizing interpolant of the discrete function that says if at any surveyed locality that feature is used or not. This surface $\mathcal {G}$ is thought of as the value at the present time of a function describing a diffusion–convection phenomenon in two dimensions (here said tidal mode), which is subjected in a very natural way to the same equation used in physics, introducing a contextual diffusivity concept: it is shown that with two different assumptions about diffusivity, solutions of this equation, evaluated at the present time, fit well with the data interpolated by $\mathcal {G}$, thus providing two convincing different pictures of diffusion–convection in the case under study, albeit simplifications and approximations. Very importantly, it is shown that the linguistic diffusion model known to linguists as Schmidt ‘waves’ can be counted among the solutions of the diffusion equation: to look also at more general then the present study case, superimposing Schmidt ‘waves’ generated at different due times and localities and with a ‘tidal linguistic flooding’ just around the main region of linguistic diffusion can reproduce complexities of real events, thus probing diffusivity assumptions based on historical, local cultural, social and geographical grounds. The present work is motivating a long term research plan, seeking answers to fundamental questions of linguistics as a science, which are recalled in the article.

Appendices

Appendix

A Some maths

2.1 A.1 The one-dimensional approximation of the diffusion equation

By its very definition the gradient of a function is in fact the directional derivative along a gradient line. Denoting by $\vec {\gamma }:\mathbb {R}\rightarrow \mathcal {M}$ a gradient line over the (geographic) map $\mathcal {M}$, by $\vec {u}(s)$ the tangent versor to $\vec {\gamma }$ in s and with $D_{\vec {u}(s)}$ the directional derivative along $\vec {u}(s)$, one has

$$\begin{aligned} \vec {\nabla } \mathfrak {G}(\gamma (s)) \,=\, \vec {u}(s)\, D_{\vec {u}(s)} \mathfrak {G} (\vec {\gamma }(s)) = \vec {u}(s)\, (\frac{d}{ds} \mathfrak {G} \circ \vec {\gamma })(s). \end{aligned}$$

(37)

The parameter $s \in \mathbb {R}$ will be distance (in Km) along $\vec {\gamma }$ from a start point on the contour line of level 0.9. From Eq. (4) one obtains

$$\begin{aligned} - \,(\eta \circ \vec {\gamma })(s)\; \frac{d}{ds} (\mathfrak {G} \circ \gamma )(s) \,=\, \vec {u}\,\cdot \, \Phi (\vec {\gamma }(s))\,=\, (\left| \vec {\Phi }\right| \circ \vec {\gamma })(s). \end{aligned}$$

(38)

Introducing

$$\begin{aligned} \mathcal {G}_\gamma (s) :=(\mathfrak {G} \circ \vec {\gamma })(s), \quad \varphi _\gamma (s) :=(\left| \vec {\Phi } \right| \circ \vec {\gamma })(s), \quad \eta _\gamma (s) :=\left( \eta \circ \vec {\gamma }\right) (s), \end{aligned}$$

(39)

Eq. (38) is written as

$$\begin{aligned} - \eta _\gamma \,\frac{d\mathcal {G}_\gamma }{ds}(s)\, =\, \varphi _\gamma (s), \end{aligned}$$

(40)

and Eq. (4) as

$$\begin{aligned} \frac{\partial \mathfrak {G}_\gamma }{\partial t}(s) = \vec {\nabla } \cdot \left( \eta _\gamma (s)\, \vec {u}(s)\, \frac{\hbox {d}\mathfrak {G}_\gamma }{\hbox {d}s}(s) \right). \end{aligned}$$

(41)

Assuming $\vec {\gamma }$ almost a straight line, that is $\vec {u}$ almost a constant, one has

$$\begin{aligned} \frac{\partial \mathfrak {G}_\gamma }{\partial t}(s)&\,=\, \vec {u}(s)\cdot \vec {\nabla } \left( \eta _\gamma (s)\,\frac{\hbox {d}\mathfrak {G}_\gamma }{\hbox {d}s}(s)\right) \, +\, \eta _\gamma (s)\,\frac{\hbox {d}\mathfrak {G}_\gamma }{\hbox {d}s}(s)\;(\vec {\nabla } \cdot \vec {u}) \end{aligned}$$

(42)

$$\begin{aligned}&\,\approx \, \vec {u}(s)\cdot \vec {\nabla } \left( \eta _\gamma (s)\,\frac{\hbox {d}\mathfrak {G}_\gamma }{\hbox {d}s}(s)\right) \, =\, \frac{d}{ds}\left( \eta _\gamma (s)\,\frac{\hbox {d}\mathfrak {G}_\gamma }{\hbox {d}s}(s)\right) \,. \end{aligned}$$

(43)

2.2 A.2 The error function complement

From Eq. (9), with $\eta$ constant, one has

$$\begin{aligned} -\,2\,\frac{z}{\eta }\,\frac{\hbox {d}\mathcal {G}}{\hbox {d}z} \,=\, \frac{\hbox {d}^{2}\mathcal {G}}{\hbox {d}z^{2}} \end{aligned}$$

and consequently

$$\begin{aligned} -\,2\,\zeta \,\frac{\hbox {d}\mathcal {G}}{\hbox {d}\zeta } \,=\, \frac{\hbox {d}^{2}\mathcal {G}}{\hbox {d}\zeta ^{2}}, \qquad \zeta = \frac{s}{2\,\sqrt{\eta \,t}}, \end{aligned}$$

or

$$\begin{aligned} -\,2\,\zeta \,=\, \frac{\hbox {d}}{\hbox {d}\zeta }\left( \ln \left( \frac{\hbox {d}\mathcal {G}}{\hbox {d}\zeta }\right) \right), \end{aligned}$$

hence

$$\begin{aligned} \frac{\hbox {d}\mathcal {G}}{\hbox {d}\zeta } \,=\, \mathcal {C}\, e^{-\zeta ^2}, \end{aligned}$$

thus

$$\begin{aligned} \mathcal {G}(s,\,t) \,=\, A \,+\, \mathcal {C}\,\int _0^{\;s/(2\,\sqrt{\eta \,t})} \textrm{d} u\; e^{-u^2}. \end{aligned}$$

Imposing

$$\begin{aligned} 1&\,=\, \lim _{s\rightarrow -\infty }\mathcal {G}(s,\,t\ne 0) \,=\, A - \mathcal {C}\,\frac{\sqrt{\pi }}{2},\\ 0&\,=\, \lim _{s\rightarrow +\infty }\mathcal {G}(s,\,t\ne 0) \,=\, A + \mathcal {C}\,\frac{\sqrt{\pi }}{2},\\ \end{aligned}$$

one gets

$$\begin{aligned} \mathcal {G}(s,\,t) \,=\, \frac{1}{2}\, \left( 1 \,-\,\frac{2}{\sqrt{\pi }}\,\int _0^{\;s/(2\,\sqrt{\eta \,t})}\textrm{d} u\;e^{-u^2}\right) \,\equiv \, \frac{1}{2}\,{{\,\textrm{erfc}\,}}{\left( \frac{s}{2\,\sqrt{\eta \,t}}\right) }, \end{aligned}$$

thus proportional to the function known in literature as error function complement (${{\,\textrm{erfc}\,}}$).

It is to be noted that for $s>0$ as $t\rightarrow 0^+$ this function approximate more and more the step Heaviside distribution.

Notice that $\mathcal {G}(0,\,t\ne 0) = 1/2$.

2.3 A.3 Obtaining solutions of the diffusion–convection equation from solutions of the pure diffusion equation

Let

$$\begin{aligned} -\,2\,z\,\frac{\hbox {d}g}{\hbox {d}z} \,=\, \frac{\hbox {d}}{\hbox {d}z}\left( \eta \,\frac{\hbox {d}g}{\hbox {d}z} \right). \end{aligned}$$

(44)

If

$$\begin{aligned} g=g\left( \frac{s-s_0}{2\sqrt{t}}\right) \qquad \text {and} \quad z\equiv \frac{s-s_0}{2\sqrt{t}}, \end{aligned}$$

using the chain rule of the derivatives, one has

$$\begin{aligned} \frac{\partial g}{\partial t} \,=\, \frac{\partial }{\partial t}\left( \frac{s -s_0}{2\sqrt{t}}\right) \; \frac{\hbox {d}g}{\hbox {d}z}\,=\, \frac{1}{4t} \left( -2z\,\frac{\hbox {d}g}{\hbox {d}z}\right) \,=\, \frac{1}{4t}\, \frac{\hbox {d}}{\hbox {d}z}\left( \eta \,\frac{\hbox {d}g}{\hbox {d}z} \right) \,=\, \frac{\partial }{\partial s}\left( \eta \,\frac{\partial g}{\partial s}\right). \end{aligned}$$

If instead

$$\begin{aligned} g = g\left( \frac{s -s_0 - \lambda \,f(t/\tau )}{2\sqrt{t}}\right) \qquad \text {and} \quad z\equiv \, \frac{s -s_0 - \lambda \,f(t/\tau )}{2\sqrt{t}}, \end{aligned}$$

(45)

where $\lambda$ and $\tau$ are, respectively, a length and a time constant parameters, one has

$$\begin{aligned} \frac{\partial g}{\partial t}&\,=\, \frac{\partial }{\partial t}\left( \frac{s -s_0 -\lambda \,f(t/\tau )}{2\sqrt{t}}\right) \; \frac{\hbox {d}g}{\hbox {d}z}\,=\, \frac{\partial }{\partial t}\left( \frac{s -s_0}{2\sqrt{t}}-\frac{\lambda }{2\,\sqrt{t}}\,f(t/\tau )\right) \; \frac{\hbox {d}g}{\hbox {d}z} \\&\,=\,\left( - \frac{s -s_0}{4t\sqrt{t}} \right) \;\frac{\hbox {d}g}{\hbox {d}z} \,+\, \frac{\lambda }{4\,t\,\sqrt{t}}\,f(t/\tau ) \;\frac{\hbox {d}g}{\hbox {d}z}\,-\, \frac{\lambda }{\tau }\,\frac{1}{2\,\sqrt{t}} \,\left. \frac{\hbox {d}f(t')}{\hbox {d}t'}\right| _{t'= t/\tau }\,\;\frac{\hbox {d}g}{\hbox {d}z} \\&\,=\, \left( - \frac{s -s_0 - \lambda \,f(t/\tau )}{4t\sqrt{t}} \right) \;\frac{\hbox {d}g}{\hbox {d}z} \,-\, \frac{\lambda }{\tau }\, \left. \frac{\hbox {d}f(t')}{\hbox {d}t'}\right| _{t'= t/\tau }\,\;\,\frac{1}{2\,\sqrt{t}}\,\frac{\hbox {d}g}{\hbox {d}z} \\&\,=\, \frac{1}{4t}\left( -2z\,\frac{\hbox {d}g}{\hbox {d}z}\right) \,-\, \frac{\lambda }{\tau }\left. \frac{\hbox {d}f(t')}{\hbox {d}t'}\right| _{t'= t/\tau }\,\frac{\partial g}{\partial s}\\&\,=\,\frac{1}{4t}\frac{\hbox {d}}{\hbox {d}z}\left( \eta \,\frac{\hbox {d}g}{\hbox {d}z}\right) \,-\, \frac{\lambda }{\tau }\left. \frac{\hbox {d}f(t')}{\hbox {d}t'}\right| _{t'= t/\tau }\,\frac{\partial g}{\partial s}\\&\,=\, \frac{\partial }{\partial s}\left( \eta \,\frac{\partial g}{\partial s}\right) \,-\, \frac{\lambda }{\tau }\left. \frac{\hbox {d}f(t')}{\hbox {d}t'}\right| _{t'= t/\tau }\,\frac{\partial g}{\partial s}\,. \end{aligned}$$

Thus

$$\begin{aligned} \frac{\partial g}{\partial t} \,=\, \frac{\partial }{\partial s}\left( \eta \,\frac{\partial g}{\partial s}\right) \,-\, \frac{\lambda }{\tau }\,\left. \frac{\hbox {d}f(t')}{\hbox {d}t'}\right| _{t'= t/\tau }\,\frac{\partial g}{\partial s}, \end{aligned}$$

(46)

Taking, for example,

$$\begin{aligned} f\left( \frac{t}{\tau }\right) = \frac{t}{\tau } \end{aligned}$$

(45) becomes

$$\begin{aligned} g = g\left( \frac{s -s_0 - (\lambda /\tau )\,t}{2\,\sqrt{t}}\right), \end{aligned}$$

(47)

with convection term simply

$$\begin{aligned} v\,\frac{\partial g}{\partial s}. \end{aligned}$$

(48)

In this case the gradient front progresses at the constant speed $v= \lambda /\tau$.

As a different, instructive example, one sets

$$\begin{aligned} f\left( t/\tau \right) \,=\,\frac{t}{\tau }\, \exp \left\{ \frac{\tau }{\theta }\,\left( 1\,-\,\frac{t}{\tau }\right) \right\} \end{aligned}$$

(49)

for which the convection term changes to

$$\begin{aligned} \frac{\lambda }{\tau }\, \left( 1\,-\,\frac{t}{\theta }\right) \, \exp \left\{ \frac{\tau }{\theta }\,\left( 1\,-\,\frac{t}{\tau }\right) \right\}. \end{aligned}$$

(50)

In this second case, for $t\ll \tau$, f has approximately has approximately the same growth as in the previous choice, however bending to flatten until $t= \theta \, (\theta > \tau )$, where it begins to decrease: in view of (45) this means that the gradient front initially progresses at a speed approximately constant, then slows down, stops and finally slowly begins to regress.

2.4 A.4 Searching the diffusivity function for data-driven shapes

As shown above, the data suggest some possible functional laws of $\mathcal {G}(z)$. The problem is to check whether there is a diffusivity function that allows Eq. (9) to have such function as a solution.

Taking indeed

$$\begin{aligned} \eta (z ; \mathcal {G}(z)) \,=\, 2\;\left( \frac{\hbox {d}\mathcal {G}(z)}{\hbox {d}z}\right) ^{-1}\, \int _0^z \!\textrm{d} \xi \,\xi \,\frac{\hbox {d}\mathcal {G}(\xi )}{\hbox {d}\xi }, \end{aligned}$$

(51)

in the domain of z where the above is well defined, it follows

$$\begin{aligned}&\frac{\hbox {d}}{\hbox {d}z}\left( \left[ 2\;\left( \frac{\hbox {d}\mathcal {G}(z)}{\hbox {d}z}\right) ^{-1}\,\int _0^z \!\textrm{d} \xi \,\xi \, \frac{\hbox {d}\mathcal {G}(\xi )}{\hbox {d}\xi }\right] \,\left( \frac{\hbox {d}\mathcal {G}(z)}{\hbox {d}z}\right) \right) \,=\,\,2\,\frac{\hbox {d}}{\hbox {d}z}\left( \int _0^z \!\textrm{d} x\,x\,\frac{\textrm{d} \mathcal {G}(z)}{\textrm{d} x} \right) \\&\quad = 2\,z\, \frac{\textrm{d} \mathcal {G}(z)}{\textrm{d} z}, \end{aligned}$$

because in the first term of the chain of equalities $(\text{d}\mathcal {G}/\text{d}z)^{-1}$ and $\text{d}\mathcal {G}/\text{d}z$ elide each other.