On Newton's equation of motion with friction and stochastic noise, the Ostrogradsky-instability and the hierarchy of environments, An application of the Onsager-Machlup theory II
Onsager and Machlup proposed a second order variational-principle in order to include inertial effects into the Langevin-equation, giving a Lagrangian with second order derivatives in time. This but violates Ostrogradysky's theorem, which proves that Lagrangians with higher than first order derivatives are meaningless. As a consequence, inertial effects cannot be included in a standard way. By using the canonical formalism, we suggest a solution to this fundamental problem. Furthermore, we provide elementary arguments about the hierarchy of immersions and actions between an ideal system and several environments and show, that the structure of the Lagrangian sensitively depends on this hierarchy.
In this work a generalization of Onsager-Machlup's theory of time-dependent thermal fluctuations of equilibrium systems is proposed, to the case in which the system relaxes irreversibly along a non-equilibrium trajectory that can be approximated as a sequence of stationary states. This generalization is summarized by a canonical description of the dependence of the two-time correlation function C(t+τ,t), and of the equal-time correlation function σ(t)= C(t,t) (the covariance of the fluctuations), on the non-equilibrium relaxation time t.
We straight-forwardly derive the Onsager-Machlup Lagrangian from the Fokker-Planck equation and show that friction and dissipation are a natural property of the equation of motion. We develop a method to calculate the local variance $σ_{2}\,b(q)^{2}$ and identify this function as a Helmholtz-factor. In both meanings the function $b(q)$ describes properties of the environment. For application, we examine the free fall through a barometric medium and model a blow of wind by a solitonic pulse running through the medium. We treat harmonic oscillators immersed in a thermal bath, finding intuitive a…
We study a micro and macroscopic model for chemical reactions with feedback between reactions and temperature of the solute. The first result concerns the quasipotential as the large-deviation rate of the microscopic invariant measure. The second result is an application of modern Onsager-Machlup theory to the pathwise large deviations, in case the system is in detailed balance. The third result is an application of macroscopic fluctuation theory to the reaction flux large deviations, in case the system is in complex balance.
A generalization of the Onsager-Machlup theory from equilibrium to nonequilibrium steady states and its connection with recent fluctuation theorems are discussed for a dragged particle restricted by a harmonic potential in a heat reservoir. Using a functional integral approach, the probability functional for a path is expressed in terms of a Lagrangian function from which an entropy production rate and dissipation functions are introduced, and nonequilibrium thermodynamic relations like the energy conservation law and the second law of thermodynamics are derived. Using this Lagrangian function…
The definition and manipulation of Langevin equations with multiplicative white noise require special care (one has to specify the time discretisation and a stochastic chain rule has to be used to perform changes of variables). While discretisation-scheme transformations and non-linear changes of variable can be safely performed on the Langevin equation, these same transformations lead to inconsistencies in its path-integral representation. We identify their origin and we show how to extend the well-known Itō prescription ($dB^2=dt$) in a way that defines a modified stochastic calculus to be u…
We propose a novel path sampling method based on the Onsager-Machlup (OM) action by generalizing the multiscale enhanced sampling (MSES) technique suggested by Moritsugu and coworkers (J. Chem. Phys. 133, 224105 (2010)). The basic idea of this method is that the system we want to study (for example, some molecular system described by molecular mechanics) is coupled to a coarse-grained (CG) system, which can move more quickly and computed more efficiently than the original system. We simulate this combined system (original + CG system) using (underdamped) Langevin dynamics where different heat …
The chemical birth-death process, whose chemical master equation (CME) is exactly solvable, is a paradigmatic toy problem often used to get intuition for how stochasticity affects chemical kinetics. In a certain limit, it can be approximated by an Ornstein-Uhlenbeck-like process which is also exactly solvable. In this paper, we use this system to showcase eight qualitatively different ways to exactly solve continuous stochastic systems: (i) integrating the stochastic differential equation; (ii) computing the characteristic function; (iii) eigenfunction expansion; (iv) using ladder operators; (…
Neural optimization of the most probable paths of 3D active Brownian particles
Wenzhou Institute, University of Chinese Academy of Sciences（中国科学院大学温州研究院）；Zhejiang University（浙江大学）；Xiangshan Hospital of Wenzhou Medical University（温州医科大学象山县医院）；Xiamen University（厦门大学）；Guangdong Technion – Israel Institute of Technology（广东以色列理工学院）；Technion - Israel Institute of Technology（以色列理工学院）；Research Institute for Mathematical Sciences, Kyoto University（京都大学数理科学研究科研究所）；Department of Mathematics, Kyoto University（京都大学数学系）
We derive the distribution function of work performed by a harmonic force acting on a uniformly dragged Brownian particle subjected to a rotational torque. Following the Onsager and Machlup's functional integral approach, we obtain the transition probability of finding the Brownian particle at a particular position at time $t$ given that it started the journey from a specific location at an earlier time. The difference between the forward and the time-reversed form of the generalized Onsager-Machlup's Lagrangian is identified as the rate of medium entropy production which further helps us deve…
By using recent developments for the Langevin dynamics of spatially asymmetric systems, we routinely generalize the Onsager-Machlup fluctuation theory of the second order in time. In this form, it becomes applicable to fluctuating variables, including hydrodynamic currents, in equilibrium as well as nonequilibrium steady states. From the solution of the obtained stochastic equations we derive an analytical expression for the time autocorrelation function of a general fluctuating quantity. This theoretical result is then tested in a study of a shear flow by molecular dynamics simulations. The p…
In this paper, we generalize the theory of Brownian motion and the Onsager-Machlup theory of fluctuations for spatially symmetric systems to equilibrium and nonequilibrium steady-state systems with a preferred spatial direction, due to an external force. To do this, we extend the Langevin equation to include a bias, which is introduced by the external force and alters the Gaussian structure of the system's fluctuations. By solving this extended equation, we demonstrate that the statistical properties of the fluctuations in these systems can be predicted from physical observables, such as the t…
Stochastic mechanics---the study of classical stochastic systems governed by things like master equations and Fokker-Planck equations---exhibits striking mathematical parallels to quantum mechanics. In this article, we make those parallels more transparent by presenting a quantum mechanics-like formalism for deriving a path integral description of systems described by stochastic differential equations. Our formalism expediently recovers the usual path integrals (the Martin-Siggia-Rose-Janssen-De Dominicis and Onsager-Machlup forms) and is flexible enough to account for different variable domai…
We formulate a dynamical fluctuation theory for stationary non equilibrium states (SNS) which is tested explicitly in stochastic models of interacting particles. In our theory a crucial role is played by the time reversed dynamics. Within this theory we derive the following results: the modification of the Onsager-Machlup theory in the SNS; a general Hamilton-Jacobi equation for the macroscopic entropy; a non equilibrium, non linear fluctuation dissipation relation valid for a wide class of systems; an H theorem for the entropy. We discuss in detail two models of stochastic boundary driven lat…
We present an alternative to the perturbative diagrammatic approach for studying stochastic dynamics. Our approach is based on an auxiliary field loop expansion for the path integral representation for the generating functional of the noise induced correlation functions. We derive two different effective actions, one based on the Onsager-Machlup (OM) approach, and the other on the Martin-Siggia-Rose (MSR) response function approach. In particular we determine the leading order approximation for the effective action and effective potential for arbitrary spatial dimensions for several simple sys…
Using a path integral formulation for correlation functions of stochastic partial differential equations based on the Onsager-Machlup approach, we show how, by introducing a composite auxiliary field one can generate an auxiliary field loop expansion for the correlation functions which is similar to the one used in the $1/N$ expansion for an $O(N)$ scalar quantum field theory. We apply this formalism to the Kardar Parisi Zhang (KPZ) equation, and introduce the composite field $σ= \fracλ{2} \nabla ϕ\cdot \nabla ϕ$ by inserting a representation of the unit operator into the path integral which e…
Neural dynamics is often investigated with tools from bifurcation theory. However, many neuron models are stochastic, mimicking fluctuations in the input from unknown parts of the brain or the spiking nature of signals. Noise changes the dynamics with respect to the deterministic model; in particular bifurcation theory cannot be applied. We formulate stochastic neuronal dynamics in the Martin-Siggia-Rose de Dominicis-Janssen (MSRDJ) formalism and present the fluctuation expansion of the effective action and the functional renormalization group (fRG) as two systematic ways to incorporate correc…
