Lorentz gas

Figure 2. The geometry of the Lorentz gas channel model. The two heat reservoirs at temperatures Tl and Tr are indicated.

When applied to spatially extended dynamical systems, the PoUicott-Ruelle resonances give the dispersion relations of the hydrodynamic and kinetic modes of relaxation toward the equilibrium state. This can be illustrated in models of deterministic diffusion such as the multibaker map, the hard-disk Lorentz gas, or the Yukawa-potential Lorentz gas [1, 23]. These systems are spatially periodic. Their time evolution Frobenius-Perron operator... 

The periodic hard-disk Lorentz gas is a two-dimensional billiard in which a point particle undergoes elastic collisions on hard disks which are fixed in the plane in the form of a spatially periodic lattice. Bunimovich and Sinai have proved that... 

Figure 9. Two trajectories of the periodic hard-disk Lorentz gas. They start from the same position but have velocities that differ by one part in a million, (a) Both trajectories depicted on large spatial scales, (b) Initial segments of both trajectories showing the sensitivity to initial conditions.

Figure 10. The diffusive inodes of the periodic hard-disk Lorentz gas represented by their cumulative function depicted in the complex plane (ReF, ImF ) versus the wavenumber A .

This other Lorentz gas is similar to the previous one except that the hard disks are replaced by Yukawa potentials centered here at the vertices of a square lattice. The Hamiltonian of this system is given by... 

For diffusion in the open two-dimensional periodic Lorentz gas with parallel absorbing walls separated by the distance L, Eq. (95) shows that the diffusion coefficient is given by [38]... 

In the Lorentz gas approximation, this term is proportional to the number densities of atoms of type A and B, nA and nB, because the probability of finding an atom of the light species with a speed between vA and vA = dvA is given by the Maxwellian distribution function,... 

Exercise. Apply the same method to the Lorentz gas (although in that case it is a priori clear that the fluctuations are Poissonian according to VII.6). 

Das and Bhattacharjee236 derive the frequency and shear dependent viscosity of a simple fluid at the critical point and find good agreement with recent experimental measurements of Berg et al.237 Ernst238 calculates universal power law tails for single and multi-particle time correlation functions and finds that the collisional transfer component of the stress autocorrelation function in a classical dense fluid has the same long-time behaviour as the velocity autocorrelation function for the Lorentz gas, i.e. 

In the final part of this section, we present another simple toy model that describes the nonequilibrium nature of transport [40], The Lorentz gas model, as shown in Fig. 3, is a two-dimensional billiard where a point particle... 

The periodic Lorentz gas model with finite-horizon configuration is of particular interest since it resembles the situation of diffusive motions in multi-valley potential surfaces of molecular systems. The local dynamics inside the array of disks is strongly hyperbolic as mentioned above, which reminds us of the intrabasin mixing within a potential well. If the distance between disks is sufficiently small, the channel between arrays might play the role of the bottleneck on the configuration space. The interbasin diffusion process may be modeled by the large-scale diffusion represented by the Lorentz gas model. The similarity will be discussed more closely in the final section. 

However, there are essential differences between the Lorentz gas model and IS structures in many-dimensional molecular systems. The most obvious difference is the size distribution of basins. In the Lorentz gas model, the size of unit cells is identical but the basin size, as well as the depth of potentials, of molecular systems is believed to range quite broadly, and possibly distributed in a self-similar way, reflecting that local potential minima increase exponentially as a function of the system size. As a result, one expects that the diffusion among multibasin structures bears different characters. The situation of the latter is... 

Matyas, L., Gaspard, P Entropy production in diffusion-reaction systems the reactive random Lorentz gas. Phys. Rev. E 71(3), 036147 (2005). http //link.aps. org/abstract/ PRE/v71/e036147... 

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