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The 1970 paper, Decay of the Velocity Correlation Function [Phys. Rev. A1, 18 (1970), see also Phys. Rev. Lett. 18, 988 (1967)] by Berni Alder and Tom Wainwright, demonstrated, by means of computer simulations, that the velocity autocorrelation function for a particle in a gas of hard disks decays algebraically in time as $t^{-1},$ and as $t^{-3/2}$ for a gas of hard spheres. These decays appear in non-equilibrium fluids and have no counterpart in fluids in thermodynamic equilibrium. The work of Alder and Wainwright stimulated theorists to find explanations for these long time tails using kinetic theory or a mesoscopic mode-coupling theory. This paper has had a profound influence on our understanding of the non-equilibrium properties of fluid systems. Here we discuss the kinetic origins of the long time tails, the microscopic foundations of mode-coupling theory, and the implications of these results for the physics of fluids. We also mention applications of the long time tails and mode-coupling theory to other, seemingly unrelated, fields of physics. We are honored to dedicate this short review to Berni Alder on the occasion of his 90th birthday!
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In this paper we propose a new formalism to map history-dependent metadynamics in a Markovian process. We apply this formalism to a model Langevin dynamics and determine the equilibrium distribution of a collection of simulations. We demonstrate that the reconstructed free energy is an unbiased estimate of the underlying free energy and analytically derive an expression for the error. The present results can be applied to other history-dependent stochastic processes such as Wang-Landau sampling.
Recent thermophoretic experiments on colloidal suspensions revived an old debate, namely whether the Soret effect is properly described by thermostatics, or necessarily requires non-equilibrium thermodynamics. Based on colloidal transport theory and the entropy production of the related viscous flow, our analysis leads to the conclusion that the equilibrium approach may work for small ions, yet fails for colloidal particles and polymers. Regarding binary molecular mixtures, our results shed some doubt on the validity of thermostatic approaches that derive the Soret coefficient from equilibrium potentials.
We study the structure of stationary non equilibrium states for interacting particle systems from a microscopic viewpoint. In particular we discuss two different discrete geometric constructions. We apply both of them to determine non reversible transition rates corresponding to a fixed invariant measure. The first one uses the equivalence of this problem with the construction of divergence free flows on the transition graph. Since divergence free flows are characterized by cyclic decompositions we can generate families of models from elementary cycles on the configuration space. The second construction is a functional discrete Hodge decomposition for translational covariant discrete vector fields. According to this, for example, the instantaneous current of any interacting particle system on a finite torus can be canonically decomposed in a gradient part, a circulation term and an harmonic component. All the three components are associated to functions on the configuration space. This decomposition is unique and constructive. The stationary condition can be interpreted as an orthogonality condition with respect to an harmonic discrete vector field and we use this decomposition to construct models having a fixed invariant measure.
The Jarzynski identity can be applied to instances when a microscopic system is pulled repeatedly but quickly along some coordinate, allowing the calculation of an equilibrium free energy profile along the pulling coordinate from a set of independent non-equilibrium trajectories. Using the formalism of Wiener stochastic path integrals in which we assign temperature-dependent weights to Langevin trajectories, we derive exact formulae for the temperature derivatives of the free energy profile. This leads naturally to analytical expressions for decomposing a free energy profile into equilibrium entropy and internal energy profiles from non-equilibrium pulling. This decomposition can be done from trajectories evolved at a unique temperature without repeating the measurement as done in finite-difference decompositions. Three distinct analytical expressions for the entropy-energy decomposition are derived: using a time-dependent generalization of the weighted histogram analysis method, a quasi harmonic spring limit, and a Feynman-Kac formula. The three novel formulae of reconstructing the pair of entropy-energy profiles are exemplified by Langevin simulations of a two-dimensional model system prototypical for force-induced biomolecular conformational changes. Connections to single-molecule experimental means to probe the functionals needed in the decomposition are suggested.
These notes are based on lectures given during the Summer School `Active matter and non-equilibrium statistical physics, held in Les Houches in September 2018. In these notes, we have merged our lectures into a single chapter broadly dedicated to `Non-equilibrium active systems. We start with a discussion of generic features of non-equilibrium statistical mechanics, followed by a description of selected examples of the possible consequences of not being at thermal equilibrium. We then introduce the topic of dense glassy materials with a short review of glassy dynamics, rheology and jamming transitions for systems that are not active. We then discuss dense active materials, from simple mean-field theories to numerical models and experimental realizations. Finally, we discuss two examples of materials driven out of equilibrium by an oscillatory driving force.
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