The Macroscopic Fluctuating Theory is presented from a practical and self consistent point of view. We take as starting point the assumption that a system at a mesoscopic scale is described by a field $ϕ(x,t)$ that evolves by a Langevin equation that locally either conserves or not the field. Its dynamic behavior may also depend on the action of external agents on the bulk or/and at the system's boundaries. We derive the corresponding Fokker-Planck equation and the probability of a path and we use them to study general properties of the system's stationary state. In particular we focus on the study of the quasi-potential that defines the stationary distribution at the small noise limit. We argue that the system is at equilibrium when it is macroscopic reversible, that is when the most probable path to create a fluctuation from the stationary state is equal to the time reversed path that relaxes it. When this doesn't occur the system is in a nonequilibrium stationary state whose quasi-potential may present some lack of differentiability and/or long range action. We also derive closed equations for the two-body correlations at the stationary state and we apply them to some typical cases. Finally we obtain generalized Green-Kubo class of formulas by using the Large Deviation Principle.
Starting from the microscopic description of a normal fluid in terms of any kind of local interacting many-particle theory we present a well defined step by step procedure to derive the hydrodynamic equations for the macroscopic phenomena. We specify the densities of the conserved quantities as the relevant hydrodynamic variables and apply the methods of non-equilibrium statistical mechanics with projection operator techniques. As a result we obtain time-evolution equations for the hydrodynamic variables with three kinds of terms on the right-hand sides: reversible, dissipative and fluctuating…
In a description of physical systems with Langevin equations, interacting degrees of freedom are usually coupled through symmetric parameter matrices. This coupling symmetry is a consequence of time-reversal symmetry of the involved conservative forces. If coupling parameters fluctuate randomly, the resulting noise is called multiplicative. For example, mechanical oscillators can be coupled through a fluctuating, symmetric matrix of spring "constants". Such systems exhibit well-studied instabilities. In this note, we study the complementary case of antisymmetric, time-reversal symmetry breakin…
We extend recent results on the exact hydrodynamics of a system of diffusive active particles displaying a motility-induced phase separation to account for typical fluctuations of the dynamical fields. By calculating correlation functions exactly in the homogeneous phase, we find that two macroscopic length scales develop in the system. The first is related to the diffusive length of the particles and the other to the collective behavior of the particles. The latter diverges as the critical point is approached. Our results show that the critical behavior of the model in one dimension belongs t…
A bottom-up approach to fluctuating hydrodynamics: Coarse-graining of stochastic lattice gases and the Dean-Kawasaki equation
Tata Institute of Fundamental Research（塔塔基础研究所）；Institut de Physique Théorique, CEA Saclay and CNRS (UMR 3681)（CEA萨克雷和CNRS（UMR 3681）理论物理研究所）；Laboratoire Charles Coulomb (L2C), Université de Montpellier and CNRS (UMR 5221)（蒙彼利埃大学和CNRS（UMR 5221）夏尔·库仑实验室）
We investigate macroscopic behaviors of fluctuating domain walls in nonequilibrium open systems with the help of the effective field theory based on symmetry. Since the domain wall in open systems breaks the translational symmetry, there appears a gapless excitation identified as the Nambu-Goldstone (NG) mode, which shows the non-propagating diffusive behavior in contrast to those in closed systems. After demonstrating the presence of the diffusive NG mode in the $(2+1)$-dimensional dissipative Josephson junction, we provide a symmetry-based general analysis for open systems breaking the one-d…
We present a numerical investigation of the control of few-particle fluctuating clusters with a macroscopic field. Our goal is to reach a given target cluster shape is minimum time. This question is formulated as a first passage problem in the space of cluster configurations. We find the optimal policy to set the macroscopic field as a function of the observed shape using dynamic programming. Our results show that the optimal policy is non-unique, and its degeneracy is mainly related to symmetries shared by the initial shape, the force and the target shape. The total fraction of shapes for whi…
Despite major advances in the understanding of the formation and dynamics of nano-clusters in the past decades, theoretical bases for the control of their shape are still lacking. We investigate strategies for driving fluctuating few-particle clusters to an arbitrary target shape in minimum time with or without an external field. This question is recast into a first passage problem, solved numerically, and discussed within a high temperature expansion. Without field, large-enough low-energy target shapes exhibit an optimal temperature at which they are reached in minimum time. We then compute …
A quantum spin impurity coupled to a critical free field (the Bose-Kondo model) can be represented as a 0+1D field theory with long-range-in-time interactions that decay as $|t-t'|^{-(2-δ)}$. This theory is a simpler analogue of nonlinear sigma models with topological Wess-Zumino-Witten terms in higher dimensions. In this note we show that the RG flows for the impurity problem exhibit an annihilation between two nontrivial RG fixed points at a critical value $δ_c$ of the interaction exponent. The calculation is controlled at large spin $S$. This clarifies the phase diagram of the Bose-Kondo mo…
At the molecular level fluid motions are, by first principles, described by time reversible laws. On the other hand, the coarse grained macroscopic evolution is suitably described by the Navier-Stokes equations, which are inherently irreversible, due to the dissipation term. Here, a reversible version of three-dimensional Navier-Stokes is studied, by introducing a fluctuating viscosity constructed in such a way that enstrophy is conserved, along the lines of the paradigm of microcanonical versus canonical treatment in equilibrium statistical mechanics. Through systematic simulations we attack …
