We develop a microscopic theory for reaction-difusion (R-D) processes based on a generalization of Einstein's master equation with a reactive term and we show how the mean field formulation leads to a generalized R-D equation with non-classical solutions. For the n-th order annihilation reaction A + A + A + ... + A → 0, we obtain a nonlinear reaction-diffusion equation for which we discuss scaling and non-scaling formulations. We find steady states with either solutions exhibiting long range power law behavior showing the relative dominance of sub-diffusion over reaction effects in constrained systems, or conversely solutions with finite support of the concentration distribution describing situations where diffusion is slow and extinction is fast. Theoretical results are compared with experimental data for morphogen gradient formation.
Fix α, θ > 0, and consider the sequence (αn θ mod 1) n≥1 . Since the seminal work of Rudnick-Sarnak (1998), and due to the Berry-Tabor conjecture in quantum chaos, the fine-scale properties of these dilated mononomial sequences have been intensively studied. In this paper we show that for θ ≤ 1/3, and α > 0, the pair correlation function is Poissonian. While (for a given θ = 1) this strong pseudo-randomness property has been proven for almost all values of α, there are next-to-no instances where this has been proven for explicit α. Our result holds for all α > 0 and relies solely on classical Fourier analytic techniques. This addresses (in the sharpest possible way) a problem posed by Aistleitner-El-Baz-Munsch (2021).
We prove the invariance principle for a random Lorentz-gas particle in 3 dimensions under the Boltzmann-Grad limit and simultaneous diffusive scaling. That is, for the trajectory of a point-like particle moving among infinite-mass, hard-core, spherical scatterers of radius r, placed according to a Poisson point process of density $$\varrho $$ ϱ , in the limit $$\varrho \rightarrow \infty $$ ϱ → ∞ , $$r\rightarrow 0$$ r → 0 , $$\varrho r^{2}\rightarrow 1$$ ϱ r 2 → 1 up to time scales of order $$T=o(r^{-2}\left| {\log r}\right| ^{-2})$$ T = o ( r - 2 log r - 2 ) . To our knowledge this represents the first significant progress towards solving rigorously this problem in classical nonequilibrium statistical physics, since the groundbreaking work of Gallavotti (Phys Rev 185:308–322, 1969, Nota Interna Univ di Roma 358, 1970, Statistical mechanics. A short treatise. Theoretical and mathematical physics series, Springer, Berlin, 1999), Spohn (Commun Math Phys 60:277–290, 1978, Rev Mod Phys 52:569–611, 1980) and Boldrighini–Bunimovich–Sinai (J Stat Phys 32:477–501, 1983). The novelty is that the diffusive scaling of particle trajectory and the kinetic (Boltzmann-Grad) limit are taken simultaneously. The main ingredients are a coupling of the mechanical trajectory with the Markovian random flight process, and probabilistic and geometric controls on the efficiency of this coupling. Similar results have been earlier obtained for the weak coupling limit of classical and quantum random Lorentz gas, by Komorowski–Ryzhik (Commun Math Phys 263:277–323, 2006), respectively, Erdős–Salmhofer–Yau (Acta Math 200:211–277, 2008, Commun Math Phys 271:1–53, 2007). However, the following are substantial differences between our work and these ones: (1) The physical setting is different: low density rather than weak coupling. (2) The method of approach is different: probabilistic coupling rather than analytic/perturbative. (3) Due to (2), the time scale of validity of our diffusive approximation—expressed in terms of the kinetic time scale—is much longer and fully explicit.
Abstract:A microscopic theory for reaction-difusion (R-D) processes is developed from Einstein's master equation including a reactive term. The mean field formulation leads to a generalized R-D equation for the -th order annihilation reaction A + A + A + + A → 0, and the steady state solutions exhibit long range power law behavior showing the relative dominance of sub-diffusion over reaction effects in constrained systems, or conversely short range concentration distribution with finite support describing situations where diffusion is slow and extinction is fast. We apply the theory to analyze experimental data for morphogen gradient formation in the wing disc of the Drosophila embryo.PACS ( MotivationWhen diffusion and reaction are coupled, these processes are described phenomenologically by reaction-diffusion (R-D) equations. For instance, the evanescence process (A → 0) of suspended particles diffusing in a non-reactive medium the concentration of species A, ( ; ), is described by the classical R-D equation Nevertheless while many systems observed in nature seem to be logically described in terms of a reaction-diffusion formulation, quite often non-classical distributions are found: the steady state spatial distributions are nonexponential. This happens when the particles encounter obstacles or are retarded in their diffusive motion, or because the reactive process is hindered or enhanced by concentration effects.
In this paper we consider the fractional parts of a general sequence, for example the sequence α √ n or αn 2 . We give a general method, which allows one to show that long-range correlations (correlations where the support of the test function grows as we consider more points) are Poissonian. We show that these statements about convergence can be reduced to bounds on associated Weyl sums. In particular we apply this methodology to the aforementioned examples. In so doing, we recover a recent result of Technau-Walker (2020) for the triple correlation of αn 2 and generalize the result to higher moments. For both of the aforementioned sequences this is one of the only results which indicates the pseudo-random nature of the higher level (m ≥ 3) correlations.
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