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Phase transitions in non-equilibrium stationary states driven by multiplicative stochastic processes

Daniel G. Barci 1 ; Miguel V. Moreno 1 ; Zochil González Arenas 2
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Stochastic processes with multiplicative noise often leads to stationary out-of-equilibrium states.
They are characterized by the presence of probability currents and, in general, time-reversal is a broken symmetry and usual equilibrium properties, such as detailed balance, are not satisfied.
In this type of stationary states, symmetry-breaking phase transitions could take place, induced by noise\footnote{C. Van den Broeck, J. M. R. Parrondo, R. Toral, R. Kawai, Phys. Rev. Lett. {\bf 73}, 3395 (1994); Phys. Rev. {\bf E55}, 4084 (1997).
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That is, for weak noise the stationary state is usually disordered. However, an ordered state sets in when the noise intensity is increased. There are two necessary ingredients to produce this class of phase transitions: multiplicative noise and out-of-equilibrium stationary states.

In this work, we present a study on out-of-equilibrium phase transitions induced by multiplicative noise.
Recently, we have presented a functional formalism\footnote{Zochil Gonz?lez Arenas and Daniel G. Barci, Phys. Rev. {\bf E81}, 051113 (2010); Phys. Rev. {\bf E 85}, 041122 (2012); J. Stat. Mech. P12005.(2012). }~\footnote{Miguel V. Moreno, Zochil Gonz?lez Arenas and Daniel G. Barci, Phys. Rev. {\bf E91}, 042103 (2015).}
to compute correlations functions in these systems. Based on that, we built up a ``dynamical potential'' written in terms of an order parameter capable to describe non-equilibrium phase transitions.

As an example, we applied our formalism to a particularly simple model which captures the physics of non-equilibrium phase transition. The model is defined by a set of stochastic variables arranged in a hyper-cubic lattice satisfying a system of interacting Langevin equations with multiplicative noise, where we consider first neighbors interactions. We computed a ``dynamical potential'' for the stationary state in the saddle-point plus Gaussian fluctuations approximation. From this, we have built up a phase-diagram in terms of the lattice interaction and the noise. We discovered a phase transition with reentrant behavior for sufficiently strong lattice coupling. We computed the phase diagram for different dimensions and for different values of the stochastic prescription that defines the multiplicative stochastic process. At the level of this approximation we found that the phase transition is continuous and we computed critical exponents. Even thought, the concept of universality is not completely developed in out-of-equilibrium transitions, the computed exponents are in the universality class of the dynamical Ising model.

This work was partially supported by CNPq, CAPES and FAPERJ.