Supersymmetric formulation of multiplicative white-noise stochastic processes

Supersymmetries in nonequilibrium Langevin dynamics

Stochastic phenomena are often described by Langevin equations, which serve as a mesoscopic model for microscopic dynamics. It has been known since the work of Parisi and Sourlas that reversible (or equilibrium) dynamics present supersymmetries (SUSYs). These are revealed when the path-integral action is written as a function not only of the physical fields, but also of Grassmann fields representing a Jacobian arising from the noise distribution. SUSYs leave the action invariant upon a transformation of the fields that mixes the physical and the Grassmann ones. We show that contrary to common belief, it is possible to extend the known reversible construction to the case of arbitrary irreversible dynamics, for overdamped Langevin equations with additive white noise-provided their steady state is known. The construction is based on the fact that the Grassmann representation of the functional determinant is not unique, and can be chosen so as to present a generalization of the Parisi-Sourlas SUSY. We show how such SUSYs are related to time-reversal symmetries and allow one to derive modified fluctuation-dissipation relations valid in nonequilibrium. We give as a concrete example the results for the Kardar-Parisi-Zhang equation.

State-dependent diffusion in a bistable potential: Conditional probabilities and escape rates

We consider a simple model of a bistable system under the influence of multiplicative noise. We provide a path integral representation of the overdamped Langevin dynamics and compute conditional probabilities and escape rates in the weak noise approximation. The saddle-point solution of the functional integral is given by a diluted gas of instantons and anti-instantons, similar to the additive noise problem. However, in this case, the integration over fluctuations is more involved. We introduce a local time reparametrization that allows its computation in the form of usual Gaussian integrals. We found corrections to the Kramers escape rate produced by the diffusion function which governs the state-dependent diffusion for arbitrary values of the stochastic prescription parameter. Theoretical results are confirmed through numerical simulations.

Symmetry and its breaking in a path-integral approach to quantum Brownian motion

We study the Caldeira-Leggett model where a quantum Brownian particle interacts with an environment or a bath consisting of a collection of harmonic oscillators in the path-integral formalism. Compared to the contours that the paths take in the conventional Schwinger-Keldysh formalism, the paths in our study are deformed in the complex time plane as suggested by the recent study by C. Aron, G. Biroli, and L. F. Cugliandolo [SciPost Phys. 4, 008 (2018)10.21468/SciPostPhys.4.1.008]. This is done to investigate the connection between the symmetry properties in the Schwinger-Keldysh action and the equilibrium or nonequilibrium nature of the dynamics in an open quantum system. We derive the influence functional explicitly in this setting, which captures the effect of the coupling to the bath. We show that in equilibrium the action and the influence functional are invariant under a set of transformations of path-integral variables. The fluctuation-dissipation relation is obtained as a consequence of this symmetry. When the system is driven by an external time-dependent protocol, the symmetry is broken. From the terms that break the symmetry, we derive a quantum Jarzynski-like equality for a quantum mechanical worklike quantity given as a function of fluctuating quantum trajectory. In the classical limit, the transformations becomes those used in the functional integral formalism of the classical stochastic thermodynamics to derive the classical fluctuation theorem.

Conditional probabilities in multiplicative noise processes

We address the calculation of transition probabilities in multiplicative noise stochastic differential equations using a path integral approach. We show the equivalence between the conditional probability and the propagator of a quantum particle with variable mass. Introducing a time reparametrization, we are able to transform the problem of multiplicative noise fluctuations into an equivalent additive one. We illustrate the method by showing the explicit analytic computation of the conditional probability of a harmonic oscillator in a nonlinear multiplicative environment.

Langevin dynamics for vector variables driven by multiplicative white noise: A functional formalism

We discuss general multidimensional stochastic processes driven by a system of Langevin equations with multiplicative white noise. In particular, we address the problem of how time reversal diffusion processes are affected by the variety of conventions available to deal with stochastic integrals. We present a functional formalism to build up the generating functional of correlation functions without any type of discretization of the Langevin equations at any intermediate step. The generating functional is characterized by a functional integration over two sets of commuting variables, as well as Grassmann variables. In this representation, time reversal transformation became a linear transformation in the extended variables, simplifying in this way the complexity introduced by the mixture of prescriptions and the associated calculus rules. The stochastic calculus is codified in our formalism in the structure of the Grassmann algebra. We study some examples such as higher order derivative Langevin equations and the functional representation of the micromagnetic stochastic Landau-Lifshitz-Gilbert equation.

Nonlinear inhomogeneous Fokker-Planck equation within a generalized Stratonovich prescription

We deduce a nonlinear and inhomogeneous Fokker-Planck equation within a generalized Stratonovich, or stochastic α, prescription (α=0, 1/2, and 1, respectively, correspond to the Itô, Stratonovich and anti-Itô prescriptions). We obtain its stationary state p(st)(x) for a class of constitutive relations between drift and diffusion and show that it has a q-exponential form, p(st)(x)=N(q)[1-(1-q)βV(x)](1/(1-q)), with an index q which does not depend on α in the presence of any nonvanishing nonlinearity. This is in contrast with the linear case, for which the index q is α dependent.

Nonequilibrium work relation beyond the Boltzmann-Gibbs distribution

The presence of multiplicative noise can alter measurements of forces acting on nanoscopic objects. Taking into account of multiplicative noise, we derive a series of nonequilibrium thermodynamical equalities as generalization of the Jarzynski equality, the detailed fluctuation theorem and the Hatano-Sasa relation. Our result demonstrates that the Jarzynski equality and the detailed fluctuation theorem remains valid only for systems with the Boltzmann-Gibbs distribution at the equilibrium state, but the Hatano-Sasa relation is robust with respect to different stochastic interpretations of multiplicative noise.