Deriving perfect fluid energy tensor from point particles

In summary, a perfect fluid energy tensor is a symmetric tensor used in fluid dynamics to describe the energy, momentum, and stress of a fluid. It can be derived from the energy-momentum tensor of point particles and is significant because it allows for the understanding of fluid behavior at a microscopic level. However, it only applies to ideal fluids and involves some assumptions. Real fluids cannot be fully described by a perfect fluid energy tensor due to their dissipative effects.
  • #1
mjordan2nd
177
1

Homework Statement


[
For a system of discrete point particles the energy momentum takes the form

[tex]T_{\mu \nu} = \sum_a \frac{p_\mu^{(a)}p_\nu^{(a)}}{p^{0(a)}} \delta^{(3)}(\vec{x}-\vec{x}^{(a)}),[/tex]

where the index a labels the different particles. Show that, for a dense collection of particles with isotropically distributed velocities, we can smooth over the individual particle worldlines to obtain the perfect-fluid energy-momentum tensor.

Homework Equations


[/B]
Energy-momentum tensor of a perfect fluid:

[tex]T^{\mu \nu} = (\rho + p)U^\mu U^\nu + p \eta^{\mu \nu}[/tex].

Here [itex]\rho[/itex] is the rest-frame energy density, p the isotropic rest-frame pressure, and [itex]U[/itex] the four-velocity.

The Attempt at a Solution



I'm not really sure how to approach this problem. I would assume "smooth over" means average, so the only thing I can think of trying is

[tex]\Delta s = \int \sqrt{\eta_{\mu \nu} \frac{dx^\mu}{d \lambda} \frac{dx^\nu}{d \lambda}} d \! \lambda,[/tex]

[tex]T_{\mu \nu} = \frac{1}{\Delta s} \int \sum_a \frac{p_\mu^{(a)}p_\nu^{(a)}}{p^{0(a)}} \delta^{(3)}(\vec{x}-\vec{x}^{(a)}) \sqrt{\eta_{\mu \nu} \frac{dx^\mu}{d \lambda} \frac{dx^\nu}{d \lambda}} d \! \lambda, [/tex]

but I'm not sure how to proceed with this integral, or if this is even the right approach. Can someone help me figure out how to approach this problem?
 
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  • #3
Go ahead and post your solution and we can go from there.
 
  • #4

1. What is a perfect fluid energy tensor?

A perfect fluid energy tensor is a mathematical tool used in fluid dynamics to describe the energy, momentum, and stress of a fluid. It is a symmetric tensor that describes the energy density, pressure, and velocity of the fluid at a specific point in space and time.

2. How is a perfect fluid energy tensor derived from point particles?

A perfect fluid energy tensor can be derived from point particles using the energy-momentum tensor of the particles. This involves summing up the individual energy-momentum tensors of the particles and taking the continuum limit, which results in a perfect fluid energy tensor.

3. What is the significance of deriving a perfect fluid energy tensor from point particles?

The derivation of a perfect fluid energy tensor from point particles is significant because it allows us to understand and describe the behavior of a fluid at a microscopic level. It also allows for the application of fluid dynamics equations to systems with a large number of particles, such as gases and liquids.

4. Are there any assumptions involved in deriving a perfect fluid energy tensor from point particles?

Yes, there are a few assumptions involved in this derivation, such as assuming that the particles are so small that their individual sizes can be neglected, and that the interactions between the particles are negligible except for collisions.

5. Can a perfect fluid energy tensor be used to describe all types of fluids?

No, a perfect fluid energy tensor is only applicable to ideal fluids, which are those that have no viscosity, heat conduction, or other dissipative effects. Real fluids, such as gases and liquids, have some level of viscosity and other dissipative effects, and thus a perfect fluid energy tensor cannot fully describe their behavior.

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