Fluctuation theorem applied to the Nosé–Hoover thermostated Lorentz gas

We present numerical evidence supporting the validity of the Gallavotti–Cohen fluctuation theorem applied to the driven Lorentz gas with Nosé–Hoover thermostating. It is moreover argued that the asymptotic form of the fluctuation formula is independent of the amplitude of the driving force in the limit where it is small.

Figure 1

Probability density function (above) and verification of Eq. (2) (below) for different time averages, as indicated in the legend. The parameters are set to $\mathbf{E}=(0.1,0)$, $T=1∕2$, and ${\tau }_{\mathrm{resp}}=1$. The average phase-space contraction rate $⟨2\zeta ⟩=3.25\times {10}^{-3}$ was measured over a total of $\approx 3\times {10}^8$ collisions. The thick solid curve corresponds to the prediction in Eqs. (4, 5).

Figure 2

The same as Fig. 1 for $\mathbf{E}=(0.5,0)$. The average phase-space contraction rate $⟨2\zeta ⟩=7.62\times {10}^{-2}$ was measured over a total of $\approx 3\times {10}^8$ collisions.

Figure 3

Height of saturation measured at $p=6$ (for times $T=100,\dots ,750$) (left), and slope measured about $p=0$ vs $1∕T$ of the curves shown in the bottom panel of Fig. 1 for the external forcing $\mathbf{E}=(0.1,0)$. The vertical coordinates are renormalized to the asymptotic values predicted by Eq. (5). Both curves can be linearly extrapolated to 1 as $T\to \infty$. These data confirm that $\mathcal{O}(1∕T)$ corrections affect the finite-time measurements.