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When does the macroscopic continuum description of a medium like a fluid break down? Say I'm interested in a scattering process of some particles with momentum $p$ and energy $E$ off a fluid of temperature $T$, volume $V$, and pressure $p$: when should I consider the single fluid particles rather than the collective modes?

For a solid with a lattice, there is a natural cutoff, but is there for a fluid?

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4 Answers 4

When you consider scales of time and distance beyond the Hydrodynamic regime given by the hydrodynamic time $\tau_H$ and the hydrodynamic length $l_H$. For instance in an ordinary gas under ordinary conditions the mean free path for the particles is much shorter than $l_H$ and the hydrodynamic description characterizes the behaviour of the gas. But for an ordinary gas under a shock wave or for a plasma at very high temperatures the mean free path becomes very long, of the order of meters and you need to go beyond a hydrodynamic description.

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how are these hydrodynamic time and length defined? beside, the mean free path I guess it refers to a particle of the fluid, rather than to other type of particles scattering off the fluid. –  argopulos Nov 3 '12 at 14:40
    
Yes, the mean free path refers to fluid particles. You need to consider the fluid molecular parameters to know if a hydrodynamic description of the fluid is enough or not. The hydrodynamic length is defined as $l_H \approx \max(|\nabla f| / f)$ in kinetic theory. The corresponding time scale satisfies $l_H = V \tau_H$ where $V$ is a typical particle velocity, say the thermal velocity. –  juanrga Nov 4 '12 at 12:30

From the particle physics point of view...

As a general rule, when the length scale associated with the interaction drops much below the inter-molecular distance in the liquid you can treat the interaction as a point like interaction between two particles.

Possible there are special cases when you could generate coherent effects even at those energies, but these will exception not the rule.

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I think the relevant parameter is the scattering vector $q = 4 \pi/\lambda \sin (\theta)$, given by the particle energy (or, equivalently, its wavelength $\lambda$) and the scattering angle $2\theta$.

  • At small $q$, when the distance $2\pi/q$ is much larger than a typical intermolecular length scale, the scattering process is due to hydrodynamic (in the wide acceptance of the term) modes.
  • When the two distances are comparable, one probes the structure factor of the liquids (given by the interaction between molecules).
  • At very large $q$, one probes the form factor of the molecules, i.e. their atomic structure.

As expected, the continuum model is no longer sufficient when trying to describe the system over length scales comparable to the distance between its constituents.

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I am not sure the parameters you mentioned are enough. For example, in some viscous liquids, one can observe the Mössbauer effect, where gamma-ray absorption differs dramatically for the liquid and a single particle; therefore, even details of the nuclear spectra can be important.

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