Understanding the evolution of complex multiscale systems: Dynamic renormalization, non-equilibrium entropy and stochasticity

November 2022 - Finding the dynamical law of observable quantities lies at the core of physics. Within the particular field of statistical mechanics, the generalized Langevin equation (GLE) comprises a general model for the evolution of observables covering a great deal of physical systems with many degrees of freedom and an inherently stochastic nature. Although formally exact, GLE brings its own great challenges. It depends on the complete history of the observables under scrutiny, as well as the microscopic degrees of freedom, all of which are often inaccessible. We show that these drawbacks can be overcome by adopting elements of machine learning from empirical data, in particular coupling a multilayer perceptron (MLP) with the formal structure of GLE and calibrating the MLP with the data. This yields a powerful computational tool capable of describing noisy complex systems beyond the realms of statistical mechanics. It is exemplified with a number of representative examples from different fields: from a single colloidal particle and particle chains in a thermal bath to climatology and finance, showing in all cases excellent agreement with the actual observable dynamics. The new framework offers an alternative perspective for the study of nonequilibrium processes opening also a new route for stochastic modeling. IEEE trans. neural netw. paper [link]
June 2013 - We have developed a novel methodology that enables the study the complex dynamics of dissipative systems characterized from the very beginning by dissipation of energy at any relevant scales. By means of a generic reduced equation which is also computationally efficient we tackle a fundamental problem in science and engineering: Many time-dependent problems are generally too complex to be fully resolved and hence some information (degrees of freedom) needs to be neglected. A central question is then how can one systematically and reliably reduce the complexity of such high-dimensional systems without neglecting essential information. Popular examples of this are models for weather and climate prediction, cell biology processes, non-linear networks, or economics. Phys. Rev. Lett. paper [pdf]

Dynamics of moving contact lines

January 2015 - We revisited the classical matched asymptotic analysis of the moving contact line problem. The main result was to show that the inner and outer regions can match without the need for an intermediate region, as e.g. in the classical Hocking-Cox analysis. This new asymptotic framework was successfully exemplified in a wide spectrum of moving contact line problems. Not only we clarified the classical Hocking-Cox analysis which is cumbersome and difficult to implement in practice, but it also provided an elegant correction to a problem that has been treated incorrectly for several decades. J. Fluid Mech. paper [pdf]
July 2014 - We examine the nanoscale behavior of an equilibrium three-phase contact line in the presence of long-ranged intermolecular forces by employing a statistical mechanics of fluids approach, namely, density functional theory (DFT) together with fundamental measure theory (FMT). This enables us to evaluate the predictive quality of effective Hamiltonian models in the vicinity of the contact line. The work was selected in the Research Highlights from Physics of Fluids. Phys. Fluids paper [link]
February 2013 - For solid-liquid-gas systems the seminal study of Seppecher is often referred to when suggesting that diffuse-interface models resolve the moving contact line singularity. Whilst Seppecher’s work contains some discussion of the asymptotics, the analysis was largely incomplete, with asymptotic regions being probed without careful justification and the crucial behaviour close to the contact line only investigated numerically (a number of constraints were also imposed, e.g. 90° contact angles and fluids of equal viscosity). Moreover, Seppecher’s study has some errors in the solution forms given in an intermediate region (where the classical equations are assumed to hold). We demonstrated analytically that a diffuse-interface model can alleviate the moving contact line singularity with no-slip applied. The model also allows for rolling motion (due to mass transfer through the contact line through diffusion)  and microscopic contact angle variation dependent on flow conditions. E. Phys. J. E. paper [pdf]
February 2010 - We proved rigorously for the first time that the classical "Wenzel law", according to which substrate roughness enhances wetting, is in fact incorrect. Phys. Rev. Lett. paper [pdf]

Statistical mechanics of inhomogeneous classical fluids and dynamic density-functional theory (DDFT)

February 2022 - We develop a novel data-driven approach to the inverse problem of classical statistical mechanics: Given the experimental data on the collective motion of a classical many-body system, how does one characterize the free energy landscape of that system? By combining non-parametric Bayesian inference with physically motivated constraints, we develop an efficient learning algorithm that automates the construction of approximate free-energy functionals. In contrast to optimization-based machine learning approaches, which seek to minimize a cost function, the central idea of the proposed Bayesian inference is to propagate a set of prior assumptions through the model, derived from physical principles. The experimental data are used to probabilistically weigh the possible model predictions. This naturally leads to humanly interpretable algorithms with full uncertainty quantification of predictions. In our case, the output of the learning algorithm is a probability distribution over a family of free energy functionals, consistent with the observed particle data. We find that surprisingly small data samples contain sufficient information for inferring highly accurate analytic expressions of the underlying free-energy functionals, making our algorithm highly data efficient. In particular, we consider classical particle systems with excluded volume interactions, which are ubiquitous in nature, while being highly challenging in terms of free energy modeling. We validate our approach on the paradigmatic case of one-dimensional fluid and develop inference algorithms for the canonical and grand-canonical statistical–mechanical ensembles. Extensions to higher dimensional systems are conceptually straightforward, while standard coarse-graining techniques allow one to easily incorporate attractive interactions. J. Chem. Phys. paper [link]
March 2021 - We present a fully microscopic study of the interfacial thermodynamics of nanodrops and nanobubbles, adsorbed on flat substrates with first-order wetting. We show that both nanodrops and nanobubbles are thermodynamically accessible in regions, demarcated by the spinodals of planar wetting films, with nanobubbles occupying a relatively bigger portion of the phase space. While nanodrops can be described as near-spherical caps of Laplace radius, the radius of nanobubbles is very different from the Laplace value. Additionally, nanobubbles are accompanied by a thin gas film adsorbed on the substrate. By computing the interface binding potential, we relate the sphericity of nanodrops to the thin–thick liquid film coexistence (prewetting transition), whereas nanobubble shapes are determined only by the decay of the fluid–substrate forces. J. Fluid Mech. paper [link]
October 2020 - We introduce a theoretical framework to describe the dynamics of reacting multi-species fluid systems in-and-out of equilibrium. Our starting point is the system of generalised Langevin equations which describes the evolution of the positions and momenta of the constituent particles. One particular difficulty that this system of generalised Langevin equations exhibits is the presence of a history-dependent (i.e. non-Markovian) term, which in turn makes the system's dynamics dependent on its own past history. With the appropriate definitions of the local number density and momentum fields, we are able to derive a non-Markovian Navier–Stokes-like system of equations constituting a generalisation of the Dean–Kawasaki model. These equations, however, still depend on the full set of particles phase-space coordinates. To remove this dependence on the microscopic level without washing out the fluctuation effects characteristic of a mesoscopic description, we need to carefully ensemble-average our generalised Dean–Kawasaki equations. The outcome of such a treatment is a set of non-Markovian fluctuating hydrodynamic equations governing the time evolution of the mesoscopic density and momentum fields. Moreover, with the introduction of an energy functional which recovers the one used in classical density-functional theory and its dynamic extension (DDFT) under the local-equilibrium approximation, we derive a novel non-Markovian fluctuating DDFT (FDDFT) for reacting multi-species fluid systems. With the aim of reducing the fluctuating dynamics to a single equation for the density field, in the spirit of classical DDFT, we make use of a deconvolution operator which makes it possible to obtain the overdamped version of the non-Markovian FDDFT. A finite-volume discretization of the derived non-Markovian FDDFT is then proposed. With this, we validate our theoretical framework in-and-out-of-equilibrium by comparing results against atomistic simulations. Finally, we illustrate the influence of non-Markovian effects on the dynamics of non-linear chemically reacting fluid systems with a detailed study of memory-driven Turing patterns. J. Phys. A: Math. Theor. paper [link]
August 2018 - A great deal of experimental evidence suggests that a wide spectrum of phase transitions occur in a multistage manner via the appearance and subsequent transformation of intermediate metastable states. Such multistage mechanisms cannot be explained within the realm of the classical nucleation framework. Hence, there is a strong need to develop new theoretical tools to explain the occurrence and nature of these ubiquitous intermediate phases. We outline a unified and self-consistent theoretical framework to describe both classical and nonclassical nucleation. Our framework provides a detailed explanation of the whole multistage nucleation pathway showing in particular that the pathway involves a single energy barrier and it passes through a dense phase, starting from a low-density initial phase, before reaching the final stable state. Moreover, we demonstrate that the kinetics of matter inside subcritical clusters favors the formation of nucleation clusters with an intermediate density, i.e. nucleation precursors. Remarkably, these nucleation precursors are not associated with a local minimum of the thermodynamic potential, as commonly assumed in previous phenomenological approaches. On the contrary, we find that they emerge due to the competition between thermodynamics and kinetics of cluster formation. Thus, the mechanism uncovered for the formation of intermediate phases can be used to explain recently reported experimental findings in crystallization: up to now such phases were assumed a consequence of some complex energy landscape with multiple energy minima. Using fundamental concepts from kinetics and thermodynamics, we provide a satisfactory explanation for the so-called nonclassical nucleation pathways observed in experiments. New J. Phys. paper [link]
June 2016 - We generalised the dynamic DFT framework to systems of anisotropic particles to take into account both inertia and hydrodynamic interactions. Starting from the Liouville equation and utilising Zwanzig’s projection-operator techniques, we derived the kinetic equation for the Brownian particle distribution function, eventually obtaining a DDFT equation by utilising the tools of statistical mechanics. Whilst this equation has some similarities with previous DDFTs, it involves a translational-rotational coupling which affects the diffusivity of the (asymmetric) particles. Moreover, in the overdamped (high friction) limit, we recover a prefect agreement with previous DDFTs. J. Stat. Phys. paper [pdf]
June 2016 - Using microscopic classical density functional theory to model systems with realistic Lennard-Jones fluid–fluid and fluid–substrate intermolecular potentials, we unveiled continuous interfacial transitions, analagous to two-dimensional complete wetting, associated with the first-order prewetting line, which can occur on steps, patterned walls, grooves and wedges, and which are sensitive to both the range of the intermolecular forces and interfacial fluctuation effects.  The work was highlighted (link here) by the reviewers as being particularly significant to the community and featured at JPhys+ blog. J. Phys.: Condens. Matter paper [pdf]
February 2013 - In a very recent effort we have discovered and fully characterise a new phase transition in confined fluids. It is driven purely by geometry and we have referred to it as “capillary prewetting”. Phys. Rev. E paper [pdf]
December 2012 - The recent breakthrough here was to unify previous DDFTs and to rigorously formulate a general DDFT that takes into account inertia and hydrodynamic interactions, the combined effect of which was neglected in previous theories, even though they strongly influence non-equilibrium properties of the system. The work was highlighted as a labtalk news article for J. Phys.: Condens. Matter, as well as for inclusion in IOPselect. Phys. Rev. Lett. paper [pdf] J. Phys.: Condens. Matter paper [pdf] SIAM Multiscale Model. Simul. paper [pdf]
December 2011 - In a related project we showed that although DFT is a microscopic approach, it allows for the construction of a macroscopic quantity such as contact angle and hence it could offer a bridge between the micro-scale and macroscale in the moving contact line problem. J. Fluid Mech. paper [pdf]

Upscaling of the Cahn-Hilliard (CH) equation for interfacial dynamics in perforated/strongly heterogeneous domains

October 2012 - An effective macroscopic CH equation for such domains was derived rigorously by homogenisation theory for the first time. Our results were applied to wetting dynamics in porous media and to a single channel with strongly heterogeneous walls, also justifying rigorously phenomenological phase-field models introduced previously on an ad-hoc basis. Classical results such as Taylor-Aris dispersion are simple byproducts of our analysis. Proc. R. Soc. A paper [pdf]

Multiscale analysis and stochastic PDEs

October 2015 - We systematically developed a new framework and novel techniques for extracting coarse-grained models from time series with multiscale structure. Analysing complex phenomena from historical data comprises a tremendous challenge, since available data is often insufficient either because of the rare-event nature of the process or the coexistence of multiple time scales. A theoretical-computational framework capable of providing a systematic and rational estimation of relevant statistical quantitites was developed. The framework was tested with a wide spectrum of examples (from marine biology to paleoclimatic data), showing the accuracy and reliability of the algorithm, which obtained unbiased estimates. Phys. Rev. E. paper [pdf], J. Comp. Phys. paper [pdf], Multiscale Model. Simul paper [pdf]
February 2011 - We analysed the effects of noise on the long-term dynamics of spatially extended systems and showed that depending on the strength of noise, such systems can undergo several non-trivial state transitions including on-off intermittency and stabilised states. Phys. Rev. Lett. paper [pdf]