Strong shock waves and nonequilibrium response in a one-dimensional gas: A Boltzmann equation approach

We investigate the nonequilibrium behavior of a one-dimensional binary fluid on the basis of Boltzmann equation, using an infinitely strong shock wave as probe. Density, velocity, and temperature profiles are obtained as a function of the mixture mass ratio μ. We show that temperature overshoots near the shock layer, and that heavy particles are denser, slower, and cooler than light particles in the strong nonequilibrium region around the shock. The shock width ω(μ), which characterizes the size of this region, decreases as ω(μ)∼μ^1∕3 for μ→0. In this limit, two very different length scales control the fluid structure, with heavy particles equilibrating much faster than light ones. Hydrodynamic fields relax exponentially toward equilibrium: ϕ(x)∼exp[−x∕λ]. The scale separation is also apparent here, with two typical scales, λ₁ and λ₂, such that λ₁∼μ^1∕2 as μ→0, while λ₂, which is the slow scale controlling the fluid’s asymptotic relaxation, increases to a constant value in this limit. These results are discussed in light of recent numerical studies on the nonequilibrium behavior of similar one-dimensional binary fluids.