Contributions to Non‐Equilibrium Thermodynamics. I. Theory of Hydrodynamical Fluctuations
TL;DR: In this article, the Langevin equation was used to derive the Navier-Stokes equations for the Brownian motion of a particle of arbitrary shape, and these terms and their correlation properties are presented, and then used to obtain the Lagrangian Lagrangians for linearized hydrodynamical equations, which were first proposed by Landau and Lifshitz.
Abstract: The velocity of a particle in Brownian motion as described by the Langevin equation is a stationary Gaussian–Markov process. Similarly, in the formulation of the laws of non‐equilibrium thermodynamics by Onsager and Machlup, the macroscopic variables describing the state of a system lead to an n‐component stationary Gaussian–Markov process, which, in addition, these authors assumed to be even in time. By dropping this assumption, the most general stationary Gaussian–Markov process is discussed. As a consequence, the theory becomes applicable to the linearized hydrodynamical equations and suggests that the Navier–Stokes equations require additional fluctuation terms which were first proposed by Landau and Lifshitz. Such terms and their correlation properties are presented, and these equations are then used to derive the Langevin equation for the Brownian motion of a particle of arbitrary shape.
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01 Jan 1993TL;DR: In this paper, the authors propose a theory which goes beyond the classical formulation of thermodynamics by enlarging the space of basic independent variables, through the introduction of non-equilibrium variables, such as the dissipative fluxes appearing in the balance equations.
Abstract: Our aim is to propose a theory which goes beyond the classical formulation of thermodynamics. This is achieved by enlarging the space of basic independent variables, through the introduction of non-equilibrium variables, such as the dissipative fluxes appearing in the balance equations. The next step is to find evolution equations for the dissipative fluxes. Whereas the evolution equations for the classical variables are given by the usual balance laws, no general criteria exist concerning the evolution equations of the dissipative fluxes, with the exception of the restrictions imposed on them by the second law of thermodynamics.
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TL;DR: In this article, the authors review some applications of fractional calculus developed by the author (partly in collaboration with others) to treat some basic problems in continuum and statistical mechanics.
Abstract: We review some applications of fractional calculus developed by the author (partly in collaboration with others) to treat some basic problems in continuum and statistical mechanics. The problems in continuum mechanics concern mathematical modelling of viscoelastic bodies (Sect. 1), and unsteady motion of a particle in a viscous fluid, i.e. the Basset problem (Sect. 2). In the former analysis fractional calculus leads us to introduce intermediate models of viscoelasticity which generalize the classical spring-dashpot models. The latter analysis induces us to introduce a hydrodynamic model suitable to revisit in Sect. 3 the classical theory of the Brownian motion, which is a relevant topic in statistical mechanics. By the tools of fractional calculus we explain the long tails in the velocity correlation and in the displacement variance. In Sect. 4 we consider the fractional diffusion-wave equation, which is obtained from the classical diffusion equation by replacing the first-order time derivative by a fractional derivative of order $0< \beta <2$. Led by our analysis we express the fundamental solutions (the Green functions) in terms of two interrelated auxiliary functions in the similarity variable, which turn out to be of Wright type (see Appendix), and to distinguish slow-diffusion processes ($0 < \beta < 1$) from intermediate processes ($1 < \beta < 2$).
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TL;DR: In this article, the authors define stochastic differential equations (SDEs) and their occurrence in physics, and present an alternative treatment, applicable only in a special case, but not confined to small ατ c.
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TL;DR: Atomistic molecular dynamics simulations reveal the formation of nanojets with velocities up to 400 meters per second, created by pressurized injection of fluid propane through nanoscale convergent gold nozzles with heating or coating of the nozzle exterior surface to prevent formation of thick blocking films.
Abstract: Atomistic molecular dynamics simulations reveal the formation of nanojets with velocities up to 400 meters per second, created by pressurized injection of fluid propane through nanoscale convergent gold nozzles with heating or coating of the nozzle exterior surface to prevent formation of thick blocking films. The atomistic description is related to continuum hydrodynamic modeling through the derivation of a stochastic lubrication equation that includes thermally triggered fluctuations whose influence on the dynamical evolution increases as the jet dimensions become smaller. Emergence of double-cone neck shapes is predicted when the jet approaches nanoscale molecular dimensions, deviating from the long-thread universal similarity solution obtained in the absence of such fluctuations.
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