Recently, a variational approach has been introduced for the paradigmatic Kardar--Parisi--Zhang (KPZ) equation. Here we review that approach, together with the functional Taylor expansion that the KPZ nonequilibrium potential (NEP) admits. Such expan
sion becomes naturally truncated at third order, giving rise to a nonlinear stochastic partial differential equation to be regarded as a gradient-flow counterpart to the KPZ equation. A dynamic renormalization group analysis at one-loop order of this new mesoscopic model yields the KPZ scaling relation alpha+z=2, as a consequence of the exact cancelation of the different contributions to vertex renormalization. This result is quite remarkable, considering the lower degree of symmetry of this equation, which is in particular not Galilean invariant. In addition, this scheme is exploited to inquire about the dynamical behavior of the KPZ equation through a path-integral approach. Each of these aspects offers novel points of view and sheds light on particular aspects of the dynamics of the KPZ equation.
The local, uncorrelated multiplicative noises driving a second-order, purely noise-induced, ordering phase transition (NIPT) were assumed to be Gaussian and white in the model of [Phys. Rev. Lett. textbf{73}, 3395 (1994)]. The potential scientific an
d technological interest of this phenomenon calls for a study of the effects of the noises statistics and spectrum. This task is facilitated if these noises are dynamically generated by means of stochastic differential equations (SDE) driven by white noises. One such case is that of Ornstein--Uhlenbeck noises which are stationary, with Gaussian pdf and a variance reduced by the self-correlation time (tau), and whose effect on the NIPT phase diagram has been studied some time ago. Another such case is when the stationary pdf is a (colored) Tsallis (q)--emph{Gaussian} which, being a emph{fat-tail} distribution for (q>1) and a emph{compact-support} one for (q<1), allows for a controlled exploration of the effects of the departure from Gaussian statistics. As done before with stochastic resonance and other phenomena, we now exploit this tool to study--within a simple mean-field approximation and with an emphasis on the emph{order parameter} and the ``emph{susceptibility}--the combined effect on NIPT of the noises statistics and spectrum. Even for relatively small (tau), it is shown that whereas fat-tail noise distributions ((q>1)) counteract the effect of self-correlation, compact-support ones ((q<1)) enhance it. Also, an interesting effect on the susceptibility is seen in the last case.
We analyze several aspects of the phenomenon of stochastic resonance in reaction-diffusion systems, exploiting the nonequilibrium potentials framework. The generalization of this formalism (sketched in the appendix) to extended systems is first carri
ed out in the context of a simplified scalar model, for which stationary patterns can be found analytically. We first show how system-size stochastic resonance arises naturally in this framework, and then how the phenomenon of array-enhanced stochastic resonance can be further enhanced by letting the diffusion coefficient depend on the field. A yet less trivial generalization is exemplified by a stylized version of the FitzHugh-Nagumo system, a paradigm of the activator-inhibitor class. After discussing for this system the second aspect enumerated above, we derive from it -through an adiabatic-like elimination of the inhibitor field- an effective scalar model that includes a nonlocal contribution. Studying the role played by the range of the nonlocal kernel and its effect on stochastic resonance, we find an optimal range that maximizes the systems response.
