The One-dimensional KPZ Equation and the Airy Process

Our previous work on the one-dimensional KPZ equation with sharp wedge initial data is extended to the case of the joint height statistics at n spatial points for some common fixed time. Assuming a particular factorization, we compute an n-point generating function and write it in terms of a Fredholm determinant. For long times the generating function converges to a limit, which is established to be equivalent to the standard expression of the n-point distribution of the Airy process.Comment: 15 page

A Stochastic Process Approach of the Drake Equation Parameters

The number N of detectable (i.e. communicating) extraterrestrial civilizations in the Milky Way galaxy is usually done by using the Drake equation. This equation was established in 1961 by Frank Drake and was the first step to quantifying the SETI field. Practically, this equation is rather a simple algebraic expression and its simplistic nature leaves it open to frequent re-expression An additional problem of the Drake equation is the time-independence of its terms, which for example excludes the effects of the physico-chemical history of the galaxy. Recently, it has been demonstrated that the main shortcoming of the Drake equation is its lack of temporal structure, i.e., it fails to take into account various evolutionary processes. In particular, the Drake equation doesn't provides any error estimation about the measured quantity. Here, we propose a first treatment of these evolutionary aspects by constructing a simple stochastic process which will be able to provide both a temporal structure to the Drake equation (i.e. introduce time in the Drake formula in order to obtain something like N(t)) and a first standard error measure.Comment: 22 pages, 0 figures, 1 table, accepted for publication in the International Journal of Astrobiolog

Smoluchowski's equation: rate of convergence of the Marcus-Lushnikov process

We derive a satisfying rate of convergence of the Marcus-Lushnikov process toward the solution to Smoluchowski's coagulation equation. Our result applies to a class of homogeneous-like coagulation kernels with homogeneity degree ranging in $(-\infty,1]$. It relies on the use of a Wasserstein-type distance, which has shown to be particularly well-adapted to coalescence phenomena.Comment: 34 Page

Analysis of the Relaxation Process using Non-Relativistic Kinetic Equation

We study the linearized kinetic equation of relaxation model which was proposed by Bhatnagar, Gross and Krook (also called BGK model) and solve the dispersion relation. Using the solution of the dispersion relation, we analyze the relaxation of the macroscopic mode and kinetic mode. Since BGK model is not based on the expansion in the mean free path in contrast to the Chapman-Enskog expansion, the solution can describe accurate relaxation of initial disturbance with any wavelength. This non-relativistic analysis gives suggestions for our next work of relativistic analysis of relaxation.Comment: 18 pages, 14 figures, accepted for publication in Prog. Theor. Phys

Markov vs. nonMarkovian processes A comment on the paper Stochastic feedback, nonlinear families of Markov processes, and nonlinear Fokker-Planck equations by T.D. Frank

The purpose of this comment is to correct mistaken assumptions and claims made in the paper Stochastic feedback, nonlinear families of Markov processes, and nonlinear Fokker-Planck equations by T. D. Frank. Our comment centers on the claims of a nonlinear Markov process and a nonlinear Fokker-Planck equation. First, memory in transition densities is misidentified as a Markov process. Second, Frank assumes that one can derive a Fokker-Planck equation from a Chapman-Kolmogorov equation, but no proof was given that a Chapman-Kolmogorov equation exists for memory-dependent processes. A nonlinear Markov process is claimed on the basis of a nonlinear diffusion pde for a 1-point probability density. We show that, regardless of which initial value problem one may solve for the 1-point density, the resulting stochastic process, defined necessarily by the transition probabilities, is either an ordinary linearly generated Markovian one, or else is a linearly generated nonMarkovian process with memory. We provide explicit examples of diffusion coefficients that reflect both the Markovian and the memory-dependent cases. So there is neither a nonlinear Markov process nor nonlinear Fokker-Planck equation for a transition density. The confusion rampant in the literature arises in part from labeling a nonlinear diffusion equation for a 1-point probability density as nonlinear Fokker-Planck, whereas neither a 1-point density nor an equation of motion for a 1-point density defines a stochastic process, and Borland misidentified a translation invariant 1-point density derived from a nonlinear diffusion equation as a conditional probability density. In the Appendix we derive Fokker-Planck pdes and Chapman-Kolmogorov eqns. for stochastic processes with finite memory