Global random attractor of stochastic hydrodynamical equation for the Heisenberg paramagnet with multiplicative noise on ℝd

The stochastic hydrodynamical equation for the Heisenberg paramagnet with multiplicative noise defined on the entire [Formula: see text] is mainly investigated. The global random attractor for the random dynamical system associated with the equation is obtained. The method is to transform the stochastic equation into the corresponding partial differential equations with random coefficients by Ornstein–Uhlenbeck process. The uniform priori estimates for far-field values of solutions have been studied via a truncation function, and then the asymptotic compactness of the random dynamical system is established.

A NOTE ON THE ANALOGY BETWEEN SUPERFLUIDS AND COSMOLOGY

A new analogy between superfluid systems and cosmology is here presented, which relies strongly on the following ingredient: the back-reaction of the vacuum to the quanta of sound waves. We show how the presence of thermal phonons, the excitations above the quantum vacuum for T > 0, enable us to deduce an hydrodynamical equation formally similar to the one obtained for a perfect fluid in a Universe obeying the Friedmann–Robertson–Walker metric.

WAVE EQUATION OF THE SCALAR FIELD AND SUPERFLUIDS

The new formal analogy between superfluid systems and cosmology, which emerges by taking into account the back-reaction of the vacuum to the quanta of sound waves,1 enables us to put forward some common features between these two different areas of physics. We find the condition that allows us to justify a General Relativity (GR) derivation of the hydrodynamical equation for the superfluid in a four-dimensional space whose metric is the Unruh one.2 Furthermore, we show how, in the particular case taken into account, our hydrodynamical equation can be deduced within a four-dimensional space from the wave equation of a massless scalar field.