Kinetic and hydrodynamic regimes in multi-particle-collision dynamics of a one-dimensional fluid with thermal walls

We study the nonequilibrium steady states of a one-dimensional fluid in a finite-space region of length $L$. Particles interact among themselves by multi-particle collisions and are in contact with two thermal-wall heat reservoirs, located at the boundaries of the region. After an initial ballistic regime, we find a crossover from a normal (kinetic) transport regime to an anomalous (hydrodynamic) one, above a characteristic size $L_{*}$. We argue that $L_{*}$ is proportional to the cube of the collision time among particles. Motivated by the physics of emissive divertors in fusion plasma, we consider the more general case of thermal walls injecting particles with given average (nonthermal) velocity. For fast and relatively cold particles, short systems fail to establish local equilibrium and display non-Maxwellian distributions of velocities.

Figure 1

Simulations of MPC fluid with density $d=5$, $T_0=4$, and $T_L=2$. (a) Scaling of the energy fluxes $Q$ as a function of the size $L$ for increasing collision times $\delta t=0.1,0.5,1.0,2.0,5.0$, and 10 (bottom to top); the upper and the lower dashed lines correspond to the scaling of normal and anomalous transport, respectively. (b) Scaling of the heat conductivity $K=QL/\mathrm{\Delta }T$ with the system size $L$; the dot-dashed blue line is a fit of the data with $\delta t=10$ with the functional form $34.2x/(12.2+x)$; see Eq. (56) in Ref. [3].

Figure 2

Scaling of the anomalous component of the energy currents $Q_A$ for $d=5$, $T_0=4$, and $T_L=2$ and different collision times. The flux $Q_A$ is computed subtracting the normal component $Q_A$ as detailed in the text. The dashed line is a best fit of the data with $\delta t=2$, yielding a decay $Q_A\propto L^{-0.66\left(5\right)}$.

Figure 3

Simulations of MPC fluid with density $d=5$, $T_0=4$, and $T_L=2$. Kinetic regime ($\delta t=10$): temperature (a) and density (b) profiles for $L=160,320,1280$ (bottom to top); dashed lines are fits with the function in Eq. (8) yielding $T_0=3.9$, $T_L=2.0$. Hydrodynamic ($\delta t=0.5$) regime: temperature (c) and density (d) profiles for $L=640,1280,2560$; the inset in (c) shows the singularity at the right edge of the temperature profile, the dashed line corresponding to a power-law behavior $T\left(x\right)\sim {(1-x)}^{3/4}$ (see text for details).

Figure 4

Simulations with thermal wall given by Eq. (9) on the left-hand boundary for different values of the parameter $V$, with $T_0=T_L=3$, $d=5$, and (a) $\delta t=0.1$ or (b) $\delta t=5.$ $V=1,2,4$, and 8 from bottom to top. The dashed lines denote the asymptotic decay corresponding to normal and anomalous transport.

Figure 5

Simulations with thermal wall defined by Eq. (9). (a) and (b) Particle distributions $f(x,v)$ in the steady state, and (c) and (d) their cuts at different positions. The arrows indicate the typical injection velocity $v_0$; $T_0=2$, $T_L=2$, $L=160$, density $d=5$, and $\delta t=10$. (a) and (c) Injection of slow particles ($V=0.5$), with $v_0=1.87$ and $\mathrm{\Delta }v=1.0$. (b) and (d) Injection of fast and cold particles ($V=4$), with $v_0=5.81$ and $\mathrm{\Delta }v=1.24$. Note that even close to the right bath, $x=L$, the velocity distribution significantly deviates from a Maxwellian.

Figure 6

Energy current reversal. (a) Contour plot of the flux $Q$ as a function of the thermal-wall parameters $V$ and $T_0$; simulations with other parameters fixed at $T_L=2$, $L=160$, $d=5$, and $\delta t=10$. The dashed red line is the curve $T_0=2-V^2$, and the energy flux is negative on the left side of it (see text for details). (b) and (c) The velocity distribution functions $f(x,v)$, and (d) and (e) temperature and density profiles (lower and upper curves, respectively). (b) and (d) are for the parameter values corresponding to $T_0=1.6$ and $V=0.2$, and (c) and (e) are for those corresponding to $T_0=1.6$ and $V=1$ [see the two white dots in (a)], showing change in the sign of the gradients.