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The one-dimensional Galerkin-truncated Burgers equation, with both dissipation and noise terms included, is studied using spectral methods. When the truncation-scale Reynolds number [Formula: see text] is varied, from very small values to order 1 values, the scale-dependent correlation time [Formula: see text] is shown to follow the expected crossover from the short-distance [Formula: see text] Edwards-Wilkinson scaling to the universal long-distance Kardar-Parisi-Zhang scaling [Formula: see text]. In the inviscid limit, [Formula: see text], we show that the system displays another crossover to the Galerkin-truncated inviscid-Burgers regime that admits thermalized solutions with [Formula: see text]. The scaling forms of the time-correlation functions are shown to follow the known analytical laws and the skewness and excess kurtosis of the interface increments distributions are characterized. This article is part of the theme issue 'Scaling the turbulence edifice (part 2)'.

We show, for the first time, the explicit form of the nonlinear partial differential equations (PDEs) subject to the correct boundary conditions that have to be satisfied by transport coefficients having a vanishing skew-symmetric piece. We also report, for the first time, the nonlinear PDEs (with the appropriate boundary conditions) for transport coefficients when the thermodynamic system is subject to two thermodynamic forces. Since the proposed PDEs have been derived without neglecting any term present in the dynamical equations (i.e., the energy, mass, and momentum balance equations), we propose them as a good candidate for describing transport in thermodynamic systems also far from equilibrium (e.g., in the turbulent regime). The preliminary test was carried out by analyzing a concrete example where Onsager's relationships manifestly disagree with experience: magnetically confined Tokamak-plasmas. More specifically, we focus our calculations to compute mass and energy transports in Frascati Tokamak Upgrade-plasmas subject to two thermodynamic forces. We show a good agreement between the theoretical predictions and the experimental data. The aim of this study is to apply our approach to the Divertor Tokamak Test Facility, to be built in Italy, and to the International Thermonuclear Experimental Reactor.

We provide numerical solutions based on the path integral representation of stochastic processes for non-gradient drift Langevin forces in the presence of noise, to follow the temporal evolution of the probability density function and to compute exit times even for arbitrary noise. We compare the results with theoretical calculations, obtaining excellent agreement in the weak noise limit.This article is part of the theme issue 'Dissipative structures in matter out of equilibrium: from chemistry, photonics and biology (part 2)'., (© 2018 The Author(s).)

The main objective of this work [previously appeared in literature, the thermodynamical field theory (TFT)] is to determine the nonlinear closure equations (i.e., the flux-force relations) valid for thermodynamic systems out of Onsager's region. The TFT rests upon the concept of equivalence between thermodynamic systems. More precisely, the equivalent character of two alternative descriptions of a thermodynamic system is ensured if, and only if, the two sets of thermodynamic forces are linked with each other by the so-called thermodynamic coordinate transformations (TCT). In this work, we describe the Lie group and the group representations associated to the TCT. The TCT guarantee the validity of the so-called thermodynamic covariance principle (TCP): The nonlinear closure equations, i.e., the flux-force relations, everywhere and in particular outside the Onsager region, must be covariant under TCT. In other terms, the fundamental laws of thermodynamics should be manifestly covariant under transformations between the admissible thermodynamic forces, i.e., under TCT. The TCP ensures the validity of the fundamental theorems for systems far from equilibrium. The symmetry properties of a physical system are intimately related to the conservation laws characterizing that system. Noether's theorem gives a precise description of this relation. We derive the conserved (thermodynamic) currents and, as an example of calculation, a system out of equilibrium (tokamak plasmas) where the validity of TCP imposed at the level of the kinetic equations is also analyzed.

Thin fluid films can have surprising behavior depending on the boundary conditions enforced, the energy input and the specific Reynolds number of the fluid motion. Here we study the equations of motion for a thin fluid film with a free boundary and its other interface in contact with a solid wall. Although shear dissipation increases for thinner layers and the motion can generally be described in the limit as viscous, inertial modes can always be excited for a sufficiently high input of energy. We derive the minimal set of equations containing inertial effects in this strongly dissipative regime.

Sound emission produced by the interaction of several vortices in a two-dimensional homogeneous system obeying the nonlinear Schrödinger (NLS) equation is considered. The radiation effect is explicitly computed in terms of assumed vortex motion. The results are applied to a simple test case of two corotating vortices. The prediction is compared to the result of numerical simulations of the NLS equation. The numerical data give support to the estimate of radiation.

We study the robust dynamical behaviors of reaction-diffusion systems where the transport gives rise to non-Fickian diffusion. A prototype model describing the deposition of molecules in a surface is used to show the generic appearance of Turing structures which can coexist with homogeneous states giving rise to localized structures through the pinning mechanism. The characteristic lengths of these structures are in the nanometer region in agreement with recent experimental observations.

Multistable systems can exhibit stochastic resonance which is characterized by the amplification of small periodic signals by additive noise. Here we consider a nonmultistable linear system with a multiplicative noise forced by an external periodic signal. The noise is the sum of a colored noise of mean value zero and a noise with a definite sign. We show that the system exhibits stochastic resonance through the numerical study of an exact analytical expression for the mean value obtained by functional integral techniques. This is proof of the effect for a very general kind of noise which can even have a definite sign.

The effect of noise in a motionless front between a periodic spatial state and an homogeneous one is studied. Numerical simulations show that noise induces front propagation. From the subcritical Swift-Hohenberg equation with noise, we deduce an adequate equation for the envelope and the core of the front. The equation of the core of the front is characterized by an asymmetrical periodic potential plus additive noise. The conversion of random fluctuations into direct motion of the core of the front is responsible of the propagation. We obtain an analytical expression for the velocity of the front, which is in good agreement with numerical simulations.

The effect of additive noise on a static front that connects a stable homogeneous state with an also stable but spatially periodic state is studied. Numerical simulations show that noise induces front propagation. The conversion of random fluctuations into direct motion of the front's core is responsible of the propagation; noise prefers to create or remove a bump, because the necessary perturbations to nucleate or destroy a bump are different. From a prototype model with noise, we deduce an adequate equation for the front's core. An analytical expression for the front velocity is deduced, which is in good agreement with numerical simulations.

The dynamics of one-dimensional Cahn-Hilliard model is studied. The stationary and particle-type solutions, the bubbles, are perused as a function of initial conditions, boundary conditions, and system size. We characterize the bubble solutions which are involved in the coarsening dynamics and establish the bifurcation scenarios of the system. A set of ordinary differential equation permits us to describe the coarsening dynamics in very good agreement with numerical simulations. We also compare these dynamics with the bubble dynamics deduced from the classical kink interaction computation where our model seems to be more appropriated. In the case of two bubbles, we deduce analytical expressions for the bubble's position and the bubble's width. Besides, a simple description of the ulterior dynamics is presented.

We study stationary, localized solutions in the complex subcritical Ginzburg-Landau equation in the region where there exists coexistence of homogeneous attractors. Using a matching approach, we report on the fact that the appearance of pulses are related to a saddle-node bifurcation. Numerical simulations are in good agreement with our theoretical predictions.

We recall our approach through discretizations for path integrals and its general results for representations of probability densities. It is shown that the result of Arnold [P. Arnold, Phys. Rev. E 61, 6099 (2000)] is a particular case of our work.

We study the dynamics of bubbles in the one dimensional Cahn-Hilliard equation. For a gas of diluted bubbles we find ordinary differential equations describing their interaction which permits us to describe the ulterior dynamics of the system in very good agreement with numerical simulations.