# Hydrodynamics, non-equilibrium thermodynamics and equations of states

Why is it that in non-equilibrium hydrodynamic modelling, e.g. when the solution is time dependent, people use equations of state that are derived from equilibrium considerations (i.e. the ideal gas law, or an ideal fermi gas EoS etc.)?

• In some cases, it'd have to do with timescales (e.g., the timescale for non-equilibrium processes to being energetically dominant being smaller/longer than hydrodynamics timescales). Not sure that that works for all cases. – Kyle Kanos Nov 9 '15 at 20:11
• Hi. Edited my question so that it reflects that Im dealing with time dependent solutions. – mathdummy Nov 9 '15 at 20:40
• My statement was assuming time dependence. – Kyle Kanos Nov 9 '15 at 20:56

I think that the key requirement is that the material be in local thermodynamic equilibrium. Even if it is a dynamic situation with mass flow or shock waves running through the material under consideration, if the material is in local thermodynamic equilibrium at every instant in time, then equilibrium thermodynamic concepts such as temperature, pressure, and equations of state are valid and can be applied to describe any local element of the material.

So your question basically reduces to the question of whether there is enough time for the material to reach local thermodynamic equilibrium in any given hydrodynamic situation. I'll give an example based on my own research of compressing and metallizing fluid hydrogen with a strong shock wave (P=0.9 to 1.8 Mbar). The vibrational period of a hydrogen molecule is about $10^{-14}$ second. So within the 5 nsec measurement resolution of our equipment, each hydrogen molecule undergoes about 500,000 collisions and vibrations with neighboring hydrogen molecules. That's plenty of collisions to thoroughly distribute the vibrational and thermal energies among the hydrogen molecules and ensure local thermal equilibrium. As a result, it was valid for us to use hydrodynamic modeling and a hydrogen equation of state to simulate our experiment.

For the particular case of your hydrodynamic modeling setup, you will have to determine if the various internal degrees of freedom in your material (e.g., vibrational, electronic) have time to equilibrate with each other within the characteristic time scale for motion for the material in your setup.

• How should one be sure that the system has reached a state of local equilibrium? are there any quantitative criteria related to this notion? – dedekindCuttage Oct 1 '16 at 12:26

You seem to only have a blurry idea of the hydrodynamic approach, so I will add a tad more about the whole idea, mainly to give you a better intuition. Hopefully this will be a useful addition to Samuel Weir's wonderful answer.

A hydrodynamic state is described by the variables: mass density field, energy density field and momentum density field. These are obtained from the microscopic description by averaging over suitable ranges. More importantly this means that such hydrodynamic variables will still satisfy continuity equations. (similar to microscopic quantities)

In order to obtain a weakly coarse grained density $\rho_h$, the microscopic density is averaged over a small volume $v_0:$ $$\rho_h(\mathbf{r},t)=\frac{1}{v_0}\int_{v_0}\rho(\mathbf{r'}-\mathbf{r},t)d\mathbf{r'}$$

$v_0$ is chosen to be large enough such that fluctuations of particle numbers are negligible within this domain. Now what is large enough, simply depends on the details of the system considered.

To study the dominant fluctuations, we can e.g. look at the density-density correlation $N^{-1} \langle \hat{\rho_h}(\mathbf{k}), \hat{\rho_h}(-\mathbf{k})\rangle$ of different fourier modes given by ($\mathbf{k}$ is taken here as conjugate to $\mathbf{r}$): $$\hat{\rho}_h(t)=\int \rho_h e^{-i\mathbf{r} \cdot \mathbf{k}}d\mathbf{r}$$

The averaging in the density-density correlations is no longer to be taken as an ensemble average, being at the macroscopic scale, $\langle,\rangle$ is an average over initial conditions where it is assumed that all subvolumes are in equilibrium. In other words, although the system evolves, it only does so on long time and large length (small $\mathbf{k}$) scales, such that all the subvolumes $v_0$ stay in thermal equilibrium. Thus to conclude, all hydrodynamic approaches are taken to work on time scales that are longer than the time needed for local thermalization to take place.