How do I calculate variance for volume, 𝑉 (i.e, ⟨Δ𝑉2⟩=⟨𝑉2⟩−⟨𝑉⟩2)?

[bumblebee77]

Homework Statement

I have to calculate Δ𝑉2 (where the "2" means "squared") to calculate something else. But I don't know if I should treat V as a scalar or vector.

Relevant Equations

⟨Δ𝑉2⟩=⟨𝑉2⟩−⟨𝑉⟩2
"2" is "squared."
⟨⟩ means the average of a column of values (i.e., collected over time).

This is not actually a homework problem. I'm old but having trouble with something that's probably at student level because it's so long since I learned this stuff. I would be grateful if someone would please take pity on me and help me out!

I am trying to calculate something that includes this term: ⟨Δ𝑉²⟩. It means "variance of volume." I'm getting lost though because I don't understand if I should treat volume as a scalar or vector here.

What I mean is, for any other parameter made up of three component directions (x, y, z), I would calculate variance by breaking the parameter into its x, y, z components and then using a process that involves dotting them together (I can write out the details if anyone is interested). However, I'm not sure if this is how I should treat V.

If I use Python's np.var(V) function on V (not its components), I get an answer that seems reasonable to the calculation that uses the result of ⟨Δ𝑉²⟩. If I use the component method, I don't. Does anyone know what is going on with this? Thank you.

 

[mjc123]

Volume is a scalar, not a vector. It doesn't have components. Are you thinking of a rectangular container with length, width and height? These are not "components" of the volume, in a vectorial sense. They don't behave the same way. If you have e.g. a velocity vector, then
v = √(v_x²+v_y²+v_z²)
But the volume is given by V = xyz. You can't analyse it into components in the same way.

 

[bumblebee77]

@mjc123, thank you very much!

I don't know why I'm having so much trouble with this. I guess I'm fixated on components because I'm doing simulations at fixed pressure where the volume of my system varies and I'm getting volume output in terms of x, y, z components.

I have done similar calculations for properties that are vectors--where I have to dot the components together as part of the variance calculation.

Yes, I was thinking of the coordinate directions as vector components.

Thanks a lot. I understand your words but still feel like I'm missing something fundamental. At least it sounds like I'm on the right track because my result is right when I treat volume as a scalar! Really appreciate your help. If there's a way to credit you in addition to liking your reply, please feel free to let me know!