If I have a differential equation that is of the form,

\begin{align} \frac{\partial f(x,p,t)}{\partial t}=a_{10}\frac{\partial f}{\partial p}+a_{01}\frac{\partial f}{\partial x}+a_{11}\frac{\partial f}{\partial x\partial p}+a_{20}\frac{\partial^2 f}{\partial p^2}+a_{02}\frac{\partial^2 f}{\partial x^2}+\cdots\,, \end{align}

such that the full equation can be represented as (for finite $m$, $n$) \begin{align} \frac{\partial f(x,p,t)}{\partial t}=\sum_{m,n}a_{m,n}\frac{\partial^m}{\partial p^m}\frac{\partial^n}{\partial x^n}f(x,p,t)\,, \end{align}

how can I know the physical meaning of an arbitrary derivative of $f(x,p,t)$?

For example, if I look at Fick's laws of diffusion, can I then say that my $\partial/\partial x$ and $\partial/\partial p$ terms represent flux in position and momentum respectively, even though there are more terms than single derivatives acting on $f(x,p,t)$?

If this holds true, then the second derivatives would represent diffusion. How would I then interpret mixed partial derivatives or higher order derivatives?



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