Stochastic thermodynamics as reviewed here systematically provides a framework for extending the notions of classical thermodynamics such as work, heat and entropy production to the level of individual trajectories of well-defined non-equilibrium ensembles. It applies whenever a non-equilibrium process is still coupled to one (or several) heat bath(s) of constant temperature. Paradigmatic systems are single colloidal particles in time-dependent laser traps, polymers in external flow, enzymes and molecular motors in single molecule assays, small biochemical networks and thermoelectric devices involving single electron transport. For such systems, a first-law like energy balance can be identified along fluctuating trajectories. For a basic Markovian dynamics implemented either on the continuum level with Langevin equations or on a discrete set of states as a master equation, thermodynamic consistency imposes a local-detailed balance constraint on noise and rates, respectively. Various integral and detailed fluctuation theorems, which are derived here in a unifying approach from one master theorem, constrain the probability distributions for work, heat and entropy production depending on the nature of the system and the choice of non-equilibrium conditions. For non-equilibrium steady states, particularly strong results hold like a generalized fluctuation–dissipation theorem involving entropy production. Ramifications and applications of these concepts include optimal driving between specified states in finite time, the role of measurement-based feedback processes and the relation between dissipation and irreversibility. Efficiency and, in particular, efficiency at maximum power can be discussed systematically beyond the linear response regime for two classes of molecular machines, isothermal ones such as molecular motors, and heat engines such as thermoelectric devices, using a common framework based on a cycle decomposition of entropy production. (Some figures may appear in colour only in the online journal) This article was invited by Erwin Frey.

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Stochastic thermodynamics, fluctuation theorems and molecular machines

In 1984, H. Risken authored a book (H. Risken, The Fokker-Planck Equation: Methods of Solution, Applications, Springer-Verlag, Berlin, New York) discussing the Fokker-Planck equation for one variable, several variables, methods of solution and its applications, especially dealing with laser statistics. There has been a considerable progress on the topic as well as the topic has received greater clarity. For these reasons, it seems worthwhile again to summarize previous as well as recent developments, spread in literature, on the topic. The Fokker-Planck equation describes the evolution of conditional probability density for given initial states for a Markov process, which satisfies the Ito stochastic differential equation. The structure of the Fokker-Planck equation for the vector case is

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The Fokker-Planck Equation

Matrix Differential Calculus with Applications in Statistics and Econometrics

Jets, ie collimated streams of matter, occur from the microscale up to the large-scale structure of the universe Our focus will be mostly on surface tension effects, which result from the cohesive properties of liquids Paradoxically, cohesive forces promote the breakup of jets, widely encountered in nature, technology and basic science, for example in nuclear fission, DNA sampling, medical diagnostics, sprays, agricultural irrigation and jet engine technology Liquid jets thus serve as a paradigm for free-surface motion, hydrodynamic instability and singularity formation leading to drop breakup In addition to their practical usefulness, jets are an ideal probe for liquid properties, such as surface tension, viscosity or non-Newtonian rheology They also arise from the last but one topology change of liquid masses bursting into sprays Jet dynamics are sensitive to the turbulent or thermal excitation of the fluid, as well as to the surrounding gas or fluid medium The aim of this review is to provide a unified description of the fundamental and the technological aspects of these subjects

Physics of liquid jets

In systems possessing spatial or dynamical symmetry breaking, Brownian motion combined with unbiased external input signals, deterministic and random alike, can assist directed motion of particles at submicron scales. In such cases, one speaks of ``Brownian motors.'' In this review the constructive role of Brownian motion is exemplified for various physical and technological setups, which are inspired by the cellular molecular machinery: the working principles and characteristics of stylized devices are discussed to show how fluctuations, either thermal or extrinsic, can be used to control diffusive particle transport. Recent experimental demonstrations of this concept are surveyed with particular attention to transport in artificial, i.e., nonbiological, nanopores, lithographic tracks, and optical traps, where single-particle currents were first measured. Much emphasis is given to two- and three-dimensional devices containing many interacting particles of one or more species; for this class of artificial motors, noise rectification results also from the interplay of particle Brownian motion and geometric constraints. Recently, selective control and optimization of the transport of interacting colloidal particles and magnetic vortices have been successfully achieved, thus leading to the new generation of microfluidic and superconducting devices presented here. The field has recently been enriched with impressive experimental achievements in building artificial Brownian motor devices that even operate within the quantum domain by harvesting quantum Brownian motion. Sundry akin topics include activities aimed at noise-assisted shuttling other degrees of freedom such as charge, spin, or even heat and the assembly of chemical synthetic molecular motors. This review ends with a perspective for future pathways and potential new applications.

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Artificial Brownian motors: Controlling transport on the nanoscale

In classical mechanics, the two-body problem has been well studied. The governing equations form a system of two-coupled second-order nonlinear differential equations for the radial and angular coordinates. The perturbation induced by the astronomical disturbance like ‘dust’ is normally not considered in the orbit dynamics. Distributed dust produces an additional random force on the orbiting particle, which can be modelled as a random force having ‘Gaussian statistics’. The estimation of accurate positioning of the orbiting particle is not possible without accounting for the stochastic perturbation felt by the orbiting particle. The objective of this paper is to use the stochastic differential equation (SDE) formalism to study the effect of such disturbances on the orbiting body. Specifically, in this paper, we linearize SDEs about the mean of the state vector. The linearization operation performed above, transforms the system of SDEs into another system of SDEs that resembles a bilinear system, as described in signal processing and control literature. However, the mean trajectory of the resulting bilinear stochastic differential model does not preserve the perturbation effect felt by the orbiting particle; only the variance trajectory includes the perturbation effect. For this reason, the effectiveness of the dust-perturbed model is examined on the basis of the bilinear and second-order approximations of the system nonlinearity. The bilinear and second-order approximations of the system nonlinearity allow substantial simplifications for the numerical implementation and preserve some of the properties of the original stochastically perturbed model. Most notably, this paper reveals that the Brownian motion process is accurate to model and study the effect of dust perturbation on the orbiting particle. In addition, analytical findings are supported with finite difference method-based numerical simulations.

Dynamics of a stochastically perturbed two-body problem

The stochastic Duffing-van der Pol (SDvdP) system is an appealing case in stochastic system theory, since it involves linear and non-linear vector fields, i.e. system non-linearities, state-independent and state-dependent stochastic accelerations. The equation describing the Duffing-van der Pol system is a second-order non-linear autonomous differential equation. Stochastic estimation procedures, without accounting for possible small stochastic accelerations, may cause an inaccurate estimation of positioning of dynamical systems. In this paper, the estimation-theoretic scenarios of the stochastic version of the Duffing-van der Pol system, which accounts for a state-independent perturbation as well as a state-dependent stochastic perturbation of the order n, where n ≥ 1, is the subject of investigation. This paper discusses the notion of a qualitative analysis for the non-linear stochastic differential system using the Ito differential rule, and subsequently the method is applied to the stochastic system of this paper. This paper develops and explores the efficacy of three different approximate estimation procedures for analyzing the non-linear problem of concern here as well. The theory of the estimation procedures of this paper is developed using the Kolmogorov-Fokker-Planck equation. Numerical simulations involving two different sets of initial conditions are given, since approximate estimation procedures are hard to evaluate theoretically. This paper suggests that the higher-order approximate estimation procedure will lead to the more accurate state estimates.

A Kolmogorov-Fokker-Planck approach for a stochastic Duffing-van der Pol system

Technical communique: A Kushner approach for small random perturbations of the Duffing-van der Pol system

The analytical and numerical solutions of the non-linear exact Kushner filter are not possible, since the mean and variance evolutions are infinite dimensional and require the knowledge of the higher-order moment evolutions. The approximate filters seem to preserve some of the qualitative characteristics of the exact filter. In this paper, evolutions of conditional mean and conditional covariance of the third-order approximate filter for estimating the states of a non-linear dynamical system, especially accounting state-dependent and state-independent noise perturbations, are derived. In this analysis, we make a comparison of this filter with second-order Gaussian filter discussed in standard textbooks on non-linear filtering. This paper discusses Duffing filter, by taking up two different non-linear observation equations to demonstrate the effectiveness of the higher-order filters, i.e. third order, and second-order filters. Most notably, this paper is about examining the ability of the higher-order filt...

Shambhu N. Sharma

Papers

The Fokker-Planck Equation

Dynamics of a stochastically perturbed two-body problem

A Kolmogorov-Fokker-Planck approach for a stochastic Duffing-van der Pol system

Technical communique: A Kushner approach for small random perturbations of the Duffing-van der Pol system

Third-order approximate Kushner filter for a non-linear dynamical system