Macroscopic vs. microscopic continuity equation

[Shadowz]

Homework Statement

Derive a microscopic version of the continuity equation given
$$\rho(\vec{r},t) = \sum_{i=1}^N \delta(\vec{r}-\vec{q}_i(t))$$

and $$\rho$$ is dynamic variables

Homework Equations

I wonder if someone can point out the difference (in general) between the macroscopic and microscopic continuity equation. In addition, the given definition of $$\rho$$ here is a delta function but only depends on q, but not p. Why is that?

The Attempt at a Solution

I know that
$$\frac{dM}{dt} = \int_V -\nabla (\rho \vec{v}) = \int_V \frac{\partial \rho}{\partial t}dr$$

and get $$\frac{\partial \rho}{\partial t} = -\nabla (\rho\vec{v})$$
I am not sure if that's what the answer should be? Should it be more specific?

Thanks,

 

[mark726]

I can provide some clarification and additional information regarding the continuity equation and its macroscopic and microscopic versions.

Firstly, the continuity equation is a fundamental equation in fluid dynamics that describes the conservation of mass in a fluid system. It states that the rate of change of mass in a given volume is equal to the net flow of mass into or out of that volume. In mathematical form, it can be written as:
\frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \vec{v}) = 0
where \rho is the density of the fluid and \vec{v} is the velocity field.

The macroscopic version of the continuity equation is the one commonly used in fluid dynamics, where the density \rho is treated as a continuous variable. This version is applicable to systems where the fluid is well-mixed and the individual particles are not distinguishable. In this case, the density is a function of space and time, and the equation describes the conservation of mass at a macroscopic level.

On the other hand, the microscopic version of the continuity equation is derived for systems where the fluid is composed of individual particles that can be distinguished from one another. In this case, the density \rho is a function of the positions and velocities of these particles, and the equation describes the conservation of mass at a microscopic level.

To derive the microscopic version of the continuity equation, we can start with the equation given in the forum post:
\rho(\vec{r},t) = \sum_{i=1}^N \delta(\vec{r}-\vec{q}_i(t))
where \vec{q}_i(t) is the position of the ith particle at time t. This equation states that the density at a given point is equal to the sum of delta functions centered at the positions of all the particles.

To incorporate the velocity field \vec{v} into this equation, we can use the concept of material derivative, which takes into account the change in density due to the motion of the particles. This can be written as:
\frac{D\rho}{Dt} = \frac{\partial \rho}{\partial t} + \vec{v} \cdot \nabla \rho
where \frac{D}{Dt} represents the material derivative.

Substituting the expression for \rho from the forum post into this equation, we get:
\