Heat conduction in a chain of colliding particles with a stiff repulsive potential

One-dimensional billiards, i.e., a chain of colliding particles with equal masses, is a well-known example of a completely integrable system. Billiards with different particle masses is generically not integrable, but it still exhibits divergence of a heat conduction coefficient (HCC) in the thermodynamic limit. Traditional billiards models imply instantaneous (zero-time) collisions between the particles. We relax this condition of instantaneous impact and consider heat transport in a chain of stiff colliding particles with the power-law potential of the nearest-neighbor interaction. The instantaneous collisions correspond to the limit of infinite power in the interaction potential; for finite powers, the interactions take nonzero time. This modification of the model leads to a profound physical consequence—the probability of multiple (in particular triple) -particle collisions becomes nonzero. Contrary to the integrable billiards of equal particles, the modified model exhibits saturation of the heat conduction coefficient for a large system size. Moreover, the identification of scattering events with triple-particle collisions leads to a simple definition of the characteristic mean free path and a kinetic description of heat transport. This approach allows us to predict both the temperature and density dependencies for the HCC limit values. The latter dependence is quite counterintuitive—the HCC is inversely proportional to the particle density in the chain. Both predictions are confirmed by direct numerical simulations.

Figure 1

Exponential decay of the autocorrelation function $C\left(t\right)$ for the chain of length $N=10000$ for temperature $T=0.033$ and density $p=0.25$. Curves 1, 2, 3, and 4 correspond to $\alpha =2$, 2.5, 4, and 6, respectively.

Figure 2

Dependence of the heat conduction coefficient $\kappa$ on the distance between thermostats $L$ for temperatures $T=0.001$ (curve 1), 0.01 (curve 2), 0.1 (curve 3), and fixed density. The system includes particles with diameter $d=1$ and density $p=0.5$ (the average distance between the particles $a=1/p=2$), where the power of the potential function is $\alpha =5/2$. Straight dashed lines correspond to the results obtained from the Green-Kubo formula.

Figure 3

Temperature dependence of the HCC for density (a) $p=0.375$ and (b) $p=0.25$. Curves 1, 2, 3, and 4 correspond to $\alpha =2$, 2.5, 4, and 6, respectively. Straight dashed lines correspond to the power functions $\kappa =b{T}^{\beta }$.

Figure 4

Dependence of the HCC on inverse density $a=1/p$ for fixed temperature and parameter (a) $\alpha =4$ and (b) $\alpha =6$. Curves 1, 2, 3, and 4 correspond to $T=0.01$, 0.033, 0.1, and 0.33, respectively.

Figure 5

Dependence of the scaling function $f\left(\alpha \right)$ on power $\alpha$ for all simulated values of temperatures and particle densities $p=0.125$, 0.1875, 0.25, 0.375, and 0.5 (curves 1, 2, 3, 4, and 5, respectively, to guide the eye). The fine details of the data collapse are presented in the inset. The best linear fit (dashed line 6) corresponds to $f\left(\alpha \right)=1.12{\alpha }^{3.6}$.