Computing Nonequilibrium Transport from Short-Time Transients: From Lorentz Gas to Heat Conduction in One Dimensional Chains

Abstract

We test the Transient Time Correlation Function (TTCF) method to compute nonequilibrium transport coefficients, highlighting its conceptual and practical difference from the standard time-average approach. While time averages extract transport properties from long stationary trajectories and discard transient dynamics, TTCF adopts the complementary strategy: it exploits the information contained in short-time transients following the onset of an external perturbation, while discarding the long-time evolution once stationarity is reached. We revisit the theoretical framework of TTCF and assess its numerical performance through representative case studies, the Lorentz gas and a many-body system, namely a chain of oscillators with anharmonic pinning potential. By direct comparison with time averages, we show that for the Lorentz gas TTCF yields consistent transport coefficients in both linear and nonlinear regimes at a reduced computational cost. Moreover, the TTCF displays superior precision in the linear-response regime, and remains reliable in non-ergodic situations, revealing the presence of regions of phase space corresponding to different behaviors, as well as the possibility of phase transitions. For the anharmonic chain, we show that TTCF is a scalable and efficient alternative for the numerical study of nonequilibrium transport.

Figures (8)

• Figure 1: Transient time correlation function (TTCF). $\langle v_x(t)\,v_x(0)\rangle_0$ (left panel) and its time integral $\int_0^t \langle v_x(s)\,v_x(0)\rangle_0\,\mathrm{d}s$ (right panel) for increasing values of the external field $E$ and for an ensemble of $10^6$ particles. Curves are ordered from bright to dark colors according to increasing field strength. The cumulative integral approaches a plateau as expected from response theory. The periodic oscillations at higher fields indicates a residual correlation with initial conditions, due to finite size effect.
• Figure 2: Left: Comparison between time averages and TTCF predictions in the linear-response regime (small external fields). In this regime, time averages display large fluctuations as the signal is small compared to the intrinsic fluctuations of the system. Center: Zoom on the TTCF results in the linear regime, showing that, despite the small magnitude of the flux, the response remains linear in the applied field. The value of the fitted response coefficient is $0.17$ (intercept $7\cdot10^{-11}$) and $R^2=1$ for the fit. Right: Comparison between time averages (TA) and TTCF predictions in the nonlinear-response regime (strong external fields). A very good agreement between TA and TTCF is observed over most of the explored field range. Noticeable discrepancies arise in the region around $E_x \simeq 2.5$, where a splitting of phase space occurs, leading to the emergence of periodic orbits with zero flux.
• Figure 3: Nonequilibrium Lorentz gas. Left: Examples of typical trajectory (sky blue) and zero-flux island trajectory (orange). Right: Comparison of the non-equilibrium momentum flux $\mathbb{E}_{neq}[v_x]$ for a driving field $E_x = 2.528$ obtained from different methods: transient time correlation function (TTCF, green), time average (TA) over positive-flux trajectories (sky blue), TA over zero-flux trajectories (orange), and TA computed from $10^5$ trajectories (yellow). Error bars indicate the standard deviation for TA datasets and semidispersion for TTCF. Percentages in the legend denote the fraction of trajectories contributing to each TA category.
• Figure 4: Pinned anharmonic chain. Left: Transient time correlation function (TTCF) $\langle J(t)\,\Omega^{(0)}\rangle_0$ for several chain lengths at a temperature gradient $\Delta T/N = 10^{-3}$. Right: Comparison between TA and TTCF predictions in the linear-response regime (small temperature gradient). In this regime, the system exhibits normal heat transport.
• Figure 5: Pinned anharmonic chain. Left: Transient time correlation function (TTCF) $\langle J(t)\,\Omega^{(0)}\rangle_0$ for several chain lengths at a temperature gradient $\Delta T/N = 1$. Right: Comparison between TA and TTCF predictions in the nonlinear-response regime. In this regime, deviations from linear response appear and the system displays anomalous heat transport.