Thermodynamic formalism for the Lorentz gas with open boundaries in $d$ dimensions

A Lorentz gas may be defined as a system of fixed dispersing scatterers, with a single light particle moving among these and making specular collisions on encounters with the scatterers. For a dilute Lorentz gas with open boundaries in $d$ dimensions we relate the thermodynamic formalism to a random flight problem. Using this representation we analytically calculate the central quantity within this formalism, the topological pressure, as a function of system size and a temperaturelike parameter $\beta$. The topological pressure is given as the sum of the topological pressure for the closed system and a diffusion term with a $\beta$-dependent diffusion coefficient. From the topological pressure we obtain the Kolmogorov-Sinai entropy on the repeller, the topological entropy, and the partial information dimension.

Figure 1

Sketch of a trajectory in two dimensions of the light particle starting form the initial point and following the first two collisions.

Figure 2

(a) Topological pressure divided by the collision frequency as a function of $\beta$ for different dimensions $d$, with the parameter choice $a=1$, $v=1$, $n=0.001$, and $k_0=0.001$. (b) A closeup of this for $\beta$ values around 1.

Figure 3

(a) Dynamical entropy divided by the collision frequency, as function of $\beta$ for different dimensions $d$ and with the parameter choice $a=1$, $v=1$, $n=0.001$, and $k_0=0.001$. (b) A closeup of this for $\beta$ values around 1.

Figure 4

Correction terms proportional to $k_0^2$ of (a) the topological pressure [see Eq. (9)] and (b) the absolute value of the dynamical entropy [see Eq. (17)] as a function of $\beta$ for different dimensions $d$. The inset in (b) shows a closeup of $[D\left(\beta \right)-\beta D^{\prime }\left(\beta \right)]$ around $\beta =0.4$, where this function crosses zero. All results are for $a=1$, $v=1$, and $n=0.001$.

Figure 5

Relative corrections to the topological and to the KS entropy. (a) shows the natural logarithm of $(h_{\mathit{top}}-h_{\mathit{top}}^0){\nu }_d∕\left(\gamma h_{\mathit{top}}^0\right)$ and (b) shows $(h_{KS}-h_{KS}^0){\nu }_d∕\left(\gamma h_{KS}^0\right)$, both as functions of $d$, for parameter values $a=1$, $v=1$, and $n=0.001$.