Fundamental Relationship Between Time and Space Derivatives

[ElectricKitchen]

Many physical laws involve relationships between time derivatives to space derivatives of one or more quantities. For example, thermal conduction relates the thermal energy time rate of change [dQ/dt] to temperature space rate of change [dT/dx]. In fluid flow, the Navier-Stokes Theorem relates the time rate of change of velocity [du/dt] to the first and second space derivatives of velocity [u' =du/dx and du'/dx]. In quantum mechanics, Schroedinger equation relates the time rate of change of the wave function [d(psi)/dt] to the second space derivative the of the wave function [d^2(psi)/dx^2].
In general, what does it say about a phenomenon that time and space derivatives are proportional?

 

[Dale]

Hi ElectricKitchen, welcome to PF.

I am not certain exactly what sort of answer you are looking for. Are you asking about the possible forms of solutions to first and second order differential equations of that sort? Or are you asking something more along the lines of a discussion about why they are differential equations at all?

 

[ElectricKitchen]

Hi ElectricKitchen, welcome to PF.

I am not certain exactly what sort of answer you are looking for. Are you asking about the possible forms of solutions to first and second order differential equations of that sort? Or are you asking something more along the lines of a discussion about why they are differential equations at all?

I'm interested in the latter question. I do understand the mathematics (calculus) for finding solutions. Many physical phenomena have the property that time variation is proportional to spatial variations such as its gradient, divergence, curl, or Laplacian. I'm wondering whether there is any deeper qualitative insight to a phenomenon that exhibit this property. Maybe there's no answer or it's so trivial I'm missing it.

 

[Dale]

To me the fact that the laws of physics can be written as partial differential equations means that they are local. Changes here are only determined by fields here, not fields there.

I am not sure if that is a standard approach, but that is how I think of it.

 

[ElectricKitchen]

I am asking something slightly different. To illustrate, I'll use the one-dimensional wave equation, say, for a vibrating string.

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where y is the displacement (a function of both x and t), t is time, and x is the independent space variable.

This equation says is that the acceleration of the string at a fixed point over time is directly proportional to the second derivative of the displacement at a fixed time through space. In other words, a plot of the string's acceleration at a fixed point versus time (t), looks just like a plot of its "spatial acceleration" at fixed time versus location (x), except for a proportionality constant (c^2).
Other phenomena have a FIRST time derivative (dy/dt) that is proportional to SECOND spatial derivative. For example, the fluid flow velocity from the Navier-Stokes equation and the wave function in the Schroedinger equation. These equation indicate, simplistically, that the "plot" (kind of hard to do with the wave function) of the time rate of change looks like the plot of spatial rates of change except for proportionality constant (i.e., amplitude and possibly phase differences).
Other phenomena have have a FIRST time derivative (dy/dt) that is proportional to FIRST spatial derivative (dy/dx). Again the plots look the same except for some scale factor.
These seem to be rather remarkable properties that are not universal.
My question is whether there is some fundamental insight about phenomena in which the time derivatives are proportional to spatial derivatives.

 

[Dale]

I posted the only such insight that I have. Sorry if it wasn't satisfactory to you for some reason.

 

[ElectricKitchen]

Your in sight is something I had not thought of and I'll give it some more thought. I appreciate your time and effect in engaging me on this. I feel that, when studying some phenomenon, I sometimes get lost in the math solving problems and miss some greater truth. Thanks again.