On the Zero Mass Limit of Tagged Particle Diffusion in the 1-d Rayleigh-Gas

Abstract

We consider the M → 0 limit for tagged particle diffusion in a 1-dimensional Rayleigh-gas, studied originaly by Sinai and Soloveichik [14], respectively by Szász and Tóth [19]. In this limit we derive a new type of model for tagged paricle diffusion, with Calogero-Moser-Sutherland (i.e. inverse quadratic) interaction potential between the two central particles. Computer simulations on this new model reproduce exactly the numerical value of the limiting variance obtained by Boldrighini, Frigio and Tognetti in [4].

Laws Relating Runs, Long Runs, and Steps in Gambler’s Ruin, with Persistence in Two Strata

“…Thus α = 0 in (1.13). Therefore by (1.13), q * (a)[1] = lim α→0 2 − α {(2a + α) − (2a − α) } = a −1 .Thus, by the second formula of (1.13), we obtain w * (a)[1] by x a [1] = a 2 , so 3 is proved. Now apply (1.54), also at (r, y, z) = 1.…”

“…Since any downward lattice path from n to 0 must first reach the level m = 1, we have an initial factor g n,1 in a product formula for g n,0 . Any section of a lattice path for g n,1 , which must end in DD, is followed by a sequence of steps of the form (U D) U U , or by a terminal sequence (U D) D. To handle transitions that do not start U U or DD we introduce: [1] is as well. In our discussion of g n,0 , if f ≥ 3, then k a = k(a, a) accounts for a generating factor for an upward preamble (U D) from level m = 1, succeeding DD and preceding U U ; in this case…”

“…Physical models of persistence often consider the velocity of a particle either staying the same or being changed according to a random collision process [1,17,21]; in our model the velocity only takes values ±1. Our introduction of strata corresponds to a change in medium over which the persistence parameter, or likelihood of the velocity staying the same, would deterministically change.…”

“…The process starts at some fixed level m ∈ (−N, N ) and terminates at an epoch j when |X j | = N . For initial probabilities, take π + = P (ε j = 1) = π − = P (ε j = −1) = 1 2 . We call the two ranges of values |k| ≤ f − 1 and f ≤ |k| < N as strata for the two persistence parameter values a and b, respectively.…”

“…q * (a)[1] = a −1 , w * (a)[1] = a −1 ( − ( − 1)a) ; ∀ ≥ 1.Proof. At (r, y, z) = 1 we have β a = 2a and x a = a 2 .…”