Emergence of fluctuating traveling front solutions in macroscopic theory of noisy invasion fronts

Abstract

The position of an invasion front, propagating into an unstable state, fluctuates because of the shot noise coming from the discreteness of reacting particles and stochastic character of the reactions and diffusion. A recent macroscopic theory [Meerson and Sasorov, Phys. Rev. E 84, 030101(R) (2011)] yields the probability of observing, during a long time, an unusually slow front. The theory is formulated as an effective Hamiltonian mechanics which operates with the density field and the conjugate "momentum" field. Further, the theory assumes that the most probable density field history of an unusually slow front represents, up to small corrections, a traveling front solution of the Hamilton equations. Here we verify this assumption by solving the Hamilton equations numerically for models belonging to the directed percolation universality class.

Extreme current fluctuations in a nonstationary stochastic heat flow

“…These two classes are determined by the sign of σ ′′ (n). For gases of the elliptic class, σ ′′ (n) < 0, the extreme current statistics exhibits a universal decay, ln P(J → ∞, T ) ∼ −J 3 /T [18,23,28]. This decay can be tracked down to the fact that, at sufficiently large currents, one can neglect the diffusion contribution to the current compared with the contribution coming from noise.…”

“…The boundary layer solution must describe the formation of coupled q and p pulses, starting from the flat profile q = 1. Boundary-layer-type effects at small times (although not involving formation of pulses) are also observed for the non-interacting random walkers (where the whole problem is exactly soluble) [18,30], and for the SSEP [23,28]. A defining feature of the boundary layer solution is that all terms of Eqs.…”

“…Furthermore, for the SSEP [actually, for any model with D(n) = 1 and σ(n) = an − bn 2 , where a > 0 and b > 0] the function f (n − , n + ) can be found analytically [23,28]. This progress was possible because, at large J, the diffusion terms in the MFT equations (see below) can be neglected compared with the terms due to fluctuations.…”

“…For σ(n) = an−bn 2 the ensuing mathematical problem is exactly soluble. The complete solution also include regions where the effective fluid density vanishes, and shock discontinuities develop, but these do not contribute to ln P(J → ∞, T ) [23,28].…”

Irreversible reactions and diffusive escape: stationary properties

“…to study front propagation in reaction-diffusion processes [14,15] and shock waves in driven exclusion processes [16], it is also popular in experimental [17,18] and theoretical (see e.g. [19][20][21][22]) studies of cold quantum gases, and in analyses of quantum spin chains ( [23][24][25][26] and references therein).…”