Dynamical symmetries of Markov processes with multiplicative white noise

Abstract

We analyse various properties of stochastic Markov processes with multiplicative white noise. We take a single-variable problem as a simple example, and we later extend the analysis to the LandauLifshitz-Gilbert equation for the stochastic dynamics of a magnetic moment. In particular, we focus on the non-equilibrium transfer of angular momentum to the magnetization from a spin-polarised current of electrons, a technique which is widely used in the context of spintronics to manipulate magnetic moments. We unveil two hidden dynamical symmetries of the generating functionals of these Markovian multiplicative white-noise processes. One symmetry only holds in equilibrium and we use it to prove generic relations such as the fluctuation-dissipation theorems. Out of equilibrium, we take profit of the symmetry-breaking terms to prove fluctuation theorems. The other symmetry yields strong dynamical relations between correlation and response functions which can notably simplify the numerical analysis of these problems. Our construction allows us to clarify some misconceptions on multiplicative white-noise stochastic processes that can be found in the literature. In particular, we show that a first-order differential equation with multiplicative white noise can be transformed into an additive-noise equation, but that the latter keeps a non-trivial memory of the discretisation prescription used to define the former.

Keldysh field theory for driven open quantum systems

“…Thus, the symmetry of the Keldysh action under this transformation is a direct proof of the presence of thermodynamic equilibrium conditions. The existence of a symmetry that is related to thermal equilibrium has first been realized in the context of classical stochastic models [197,198,[206][207][208], and these considerations have been extended to the realm of quantum systems in Refs. [189,209].…”

“…(205), the couplingsū 2,3 can be obtained by taking further derivatives with respect toρ c . However, instead of projecting the flow equation (208) in this way onto equations forū 2,3 , it is more convenient to introduce rescaled quantities as outlined above.…”

(Non) equilibrium dynamics: a (broken) symmetry of the Keldysh generating functional

“…In the framework of the Martin-Siggia-Rose-Janssen-deDominicis (MSRJD) path-integral formulation of classical stochastic dynamics [3,4,5], the idea of an equilibrium symmetry of the action was first introduced by Janssen in the late 70s [6,7,8]. Recently, the authors showed that the (zero-source) MSRJD generating functional of equilibrium Langevin processes is invariant under a model-independent discrete field transformation combining time reversal with a thermal translation in the complex plane [9,10,11]. The above-mentioned consequences of this equilibrium symmetry (and its out-of-equilibrium breaking) were derived within this path-integral formalism, and they were generalized with little cost to virtually any type of classical equilibrium dynamics: from simple deterministic Newtonian dynamics, or basic Markov processes, to complex stochastic processes with colored and multiplicative noise.…”

Rules of calculus in the path integral representation of white noise Langevin equations: the Onsager–Machlup approach

“…The important role of the time discretisation in the Langevin equation is now clearly elucidated [5] and many results have been obtained for the construction of an associated path-integral formalism, whose functional action and Jacobian correctly take into account the choice of discretisation [6,7,8,9,10,11,12,13].…”

“…Although such manipulations are well understood at the Langevin equation level, the situation is less clear for the transformation of fields performed inside the action functional corresponding to the Langevin equation. It is known, for instance, that the use of the stochastic chain rule in the action can yield unsolved inconsistencies, both in statistical field theory [14,7] and in quantum field theory [15,16,17,14,18,19].…”