TY - JOUR

T1 - Refined central limit theorem and infinite density tail of the Lorentz gas from Levy walk

AU - Fouxon, Itzhak

AU - Ditlevsen, Peter

PY - 2020/10/16

Y1 - 2020/10/16

N2 - We consider point particle that collides with a periodic array of hard-core elastic scatterers where the length of the free flights is unbounded due to infinitely long corridors between the scatterers (the infinite-horizon Lorentz gas, LG). The Bleher central limit theorem(CLT) states that the distribution of the particle displacement divided by root t ln t is Gaussian in the limit of infinite time t. However this result, describing the bulk of the distribution, is not as powerful as in normal diffusion cases. The Gaussian peak fails to describe the displacement's moments of order higher than two and gives only half of the true value of the dispersion. These demand the tail of the distribution which is formed by rare long flights along the corridors where the particle propagates much further than in typical diffusive displacements due to collisions with close scatterers. We derive the tail using a L ' evy walk (LW) model of the LG, demonstrating extreme sensitivity to single flight event. The tail depends on whether between the steps the particlemoves ballistically or jumps instantaneously and it obeys the single-bigjump principle i.e. large deviations are formed fully in only one step of the walk. The validity of the LW model in the LG's description was proved recently in studying another deficiency of the CLT- the slow (logarithmic) convergence to the Gaussian peak hindering its observability. Using the LW model, it was proposed that rescaling the LG's displacement by a Lambert function factor instead of root t ln t provides a fast convergent observable CLT, which was confirmed by the LG simulations. Here we demonstrate that this result can be simplified giving 'mixed CLT' where the scaling factor combines normal and anomalous diffusions. This fits the previous observation that if infinite corridors are narrow then for long time the diffusion is normal. We conjecture that for observables determined by single flight events the LW results can be transferred to the LG.

AB - We consider point particle that collides with a periodic array of hard-core elastic scatterers where the length of the free flights is unbounded due to infinitely long corridors between the scatterers (the infinite-horizon Lorentz gas, LG). The Bleher central limit theorem(CLT) states that the distribution of the particle displacement divided by root t ln t is Gaussian in the limit of infinite time t. However this result, describing the bulk of the distribution, is not as powerful as in normal diffusion cases. The Gaussian peak fails to describe the displacement's moments of order higher than two and gives only half of the true value of the dispersion. These demand the tail of the distribution which is formed by rare long flights along the corridors where the particle propagates much further than in typical diffusive displacements due to collisions with close scatterers. We derive the tail using a L ' evy walk (LW) model of the LG, demonstrating extreme sensitivity to single flight event. The tail depends on whether between the steps the particlemoves ballistically or jumps instantaneously and it obeys the single-bigjump principle i.e. large deviations are formed fully in only one step of the walk. The validity of the LW model in the LG's description was proved recently in studying another deficiency of the CLT- the slow (logarithmic) convergence to the Gaussian peak hindering its observability. Using the LW model, it was proposed that rescaling the LG's displacement by a Lambert function factor instead of root t ln t provides a fast convergent observable CLT, which was confirmed by the LG simulations. Here we demonstrate that this result can be simplified giving 'mixed CLT' where the scaling factor combines normal and anomalous diffusions. This fits the previous observation that if infinite corridors are narrow then for long time the diffusion is normal. We conjecture that for observables determined by single flight events the LW results can be transferred to the LG.

KW - rare events

KW - Lorentz gas

KW - Levy walk

KW - tail

KW - anomalous diffusion

KW - FREE-PATH LENGTHS

KW - STATISTICAL PROPERTIES

KW - ANOMALOUS DIFFUSION

KW - SINAI BILLIARD

KW - DECAY

KW - SYSTEMS

U2 - 10.1088/1751-8121/abadb6

DO - 10.1088/1751-8121/abadb6

M3 - Journal article

VL - 53

JO - Journal of Physics A: Mathematical and Theoretical

JF - Journal of Physics A: Mathematical and Theoretical

SN - 1751-8113

IS - 41

M1 - 415004

ER -