Local equilibrium of slow time varying thermal system

Question

I'm trying to differentiate a thermal system in local equilibrium (and slow time varying) v/s a non-equilibrium system.

For a thermal system which is slowly time varying, how does one define local thermalization/local equilibrium/local temperature?

-> One thought is take a snapshot of the (energy) distribution of the particles (bosons or fermions) in a small volume of the system. Average it over a time $\Delta t$, where $\Delta t$ > particle mean free path. Try to fit a bose einstein distribution (or fermi dirac) which minimizes the rms error with the actual distribution. This would then give the local temperature.

-> Local thermalization: One can say that the small volume has reached local thermalization if the rms error is < some epsilon.

-> But the problem is that the value of epsilon is subjective (not a well defined number). So, the concept of local thermalization/temperature seems not an extremely well-defined concept.

Is there a more formal and quantitative definition of a thermal system being in "local equilibrium" instead of a "non-equilibrium" system?

Answer 1

There is an effective description of systems in local thermal equilibrium, called "fluid dynamics". This means that we can test local equilibration by checking whether fluid dynamics describe the evolution.

There are many dimensionless parameters in fluid dynamics that govern the validity of the theory (these are described in standard text books). The general idea is this: Fluid dynamics describes the evolution of conserved quantities, and the densities of these objects are expanded in gradients of local thermodynamic variables. We have to check the convergence of this expansion.

Take the stress of a fluid (this is current associated with the momentum density of the fluid). At leading order $$ \Pi^{(0)}_{ij}=P\delta_{ij}+\rho u_i u_j $$ where $P$ is the (local) pressure, $\rho$ is the density, and $u_i$ is the local velocity. All of these quantities can be extracted from local measurements (mean energy, mean density, and mean momentum). The next correction is $$ \Pi^{(1)}_{ij} = \eta(\nabla_iu_j+\nabla_ju_u-\frac{2}{3}(\nabla\cdot u)) +\zeta\delta_{ij}(\nabla\cdot u) $$ where $\eta,\zeta$ are shear and bulk viscosity. Fluid dynamics (and local equilibration) is valid if $$ |\Pi_{ij}^{(0)}|\gg |\Pi_{ij}^{(1)} \gg \ldots $$
The ratio of the first two terms (under some mild assumptions) is called the inverse Reynolds number.

Postscript (kinetic picture): What does that mean from a more microscopic point of view?

Local equilibrium (in a system described by kinetic theory) is established by collisions, and local equlibration requires the mean free path to be short compared to the distance over which the local temperature, density, and fluid velocity varies $$ l_{\it mfp} \ll L . $$ That this criterion is the same as the one given above can be seen from standard kinetic estimates for the transport coefficient $$ \eta \simeq n l_{\it mfp}\bar{p} $$ where $n$ is the density and $\bar{p}$ is the mean momentum. (So $\eta$ is really a measure of the mean free path).