The spread of individuals of species is often described by so-called dispersal kernels. The main parameter of spread is then often the variance defined as the average squared movement distance of a specimen per time step (assuming all specimen start from $x=0$ at $t=0$; $\mu=0$):

$$\sigma = \frac{1}{n} \sum_{i=1}^n (x_i - \mu)^2$$

which is a central part of the gaussian/normal distribution that is often used describing random walk models: $$N \; = \; f(x \; | \; \mu, \sigma) = \frac{1}{\sigma\sqrt{2\pi} } \; e^{ -\frac{(x-\mu)^2}{2\sigma^2} }$$

The spread over multiple time steps can be calculated by the convolution of the corresponding (e.g. gaussian) dispersal kernel.

However, how exactly is the diffusion coefficient $D$ of a 1-d diffusion equation defined and how does it relate to the variance of a distribution/dispersal kernel? For example, the heat kernel/equation, a special form of diffusion, which seems related to the normal distribution [1].

How to calculate a diffusion coefficient $D$ from a probability density function? For example what is the the diffusion coefficient $D$ of a double normal distribution (1-d Gaussian-mixture model) describing a dispersal kernel of the form:

$$N_1(x \; | \; \mu=0, \sigma_1)\times p \;+\; N_2(x \; | \; \mu=0, \sigma_2)\times (1-p) $$ where $0\leq\;p\;\leq\;1$ and $\sigma_{1,2}$ are the average spreads of individual from a stationary and mobile component for one time step $t_0\rightarrow t_{1}$. I'd like to calculate $D\;[m^2/s]$ for this double-normal dispersal kernel with real values for $\sigma_1=35$, $\sigma_2=400$ and $p=0.67$.

[1] https://en.wikipedia.org/wiki/Heat_equation#Some_Green.27s_function_solutions_in_1D


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