18 research outputs found
Deep learning as closure for irreversible processes: A data-driven generalized Langevin equation
The ultimate goal of physics is finding a unique equation capable of
describing the evolution of any observable quantity in a self-consistent way.
Within the field of statistical physics, such an equation is known as the
generalized Langevin equation (GLE). Nevertheless, the formal and exact GLE is
not particularly useful, since it depends on the complete history of the
observable at hand, and on hidden degrees of freedom typically inaccessible
from a theoretical point of view. In this work, we propose the use of deep
neural networks as a new avenue for learning the intricacies of the unknowns
mentioned above. By using machine learning to eliminate the unknowns from GLEs,
our methodology outperforms previous approaches (in terms of efficiency and
robustness) where general fitting functions were postulated. Finally, our work
is tested against several prototypical examples, from a colloidal systems and
particle chains immersed in a thermal bath, to climatology and financial
models. In all cases, our methodology exhibits an excellent agreement with the
actual dynamics of the observables under consideration
A Finite-Volume Method for Fluctuating Dynamical Density Functional Theory
We introduce a finite-volume numerical scheme for solving stochastic
gradient-flow equations. Such equations are of crucial importance within the
framework of fluctuating hydrodynamics and dynamic density functional theory.
Our proposed scheme deals with general free-energy functionals, including, for
instance, external fields or interaction potentials. This allows us to simulate
a range of physical phenomena where thermal fluctuations play a crucial role,
such as nucleation and other energy-barrier crossing transitions. A
positivity-preserving algorithm for the density is derived based on a hybrid
space discretization of the deterministic and the stochastic terms and
different implicit and explicit time integrators. We show through numerous
applications that not only our scheme is able to accurately reproduce the
statistical properties (structure factor and correlations) of the physical
system, but, because of the multiplicative noise, it allows us to simulate
energy barrier crossing dynamics, which cannot be captured by mean-field
approaches
A Brownian Model for Crystal Nucleation
In this work a phenomenological stochastic differential equation is proposed
to model the time evolution of the radius of a pre-critical molecular cluster
during nucleation (the classical order parameter). Such a stochastic
differential equation constitutes the basis for the calculation of the
(nucleation) induction time under Kramers' theory of thermally activated escape
processes. Considering the nucleation stage as a Poisson rare-event, analytical
expressions for the induction time statistics are deduced for both steady and
unsteady conditions, the latter assuming the semiadiabatic limit. These
expressions can be used to identify the underlying mechanism of molecular
cluster formation (distinguishing between homogeneous or heterogeneous
nucleation from the nucleation statistics is possible) as well as to predict
induction times and induction time distributions. The predictions of this model
are in good agreement with experimentally measured induction times at constant
temperature, unlike the values obtained from the classical equation, but
agreement is not so good for induction time statistics. Stochastic simulations
truncated to the maximum waiting time of the experiments confirm that this fact
is due to the time constraints imposed by experiments. Correcting for this
effect, the experimental and predicted curves fit remarkably well. Thus, the
proposed model seems to be a versatile tool to predict cluster size
distributions, nucleation rates, (nucleation) induction time and induction time
statistics for a wide range of conditions (e.g. time-dependent temperature,
supersaturation, pH, etc.) where classical nucleation theory is of limited
applicability.Comment: 20 pages, 3 figure
General framework for fluctuating dynamic density functional theory
We introduce a versatile bottom-up derivation of a formal theoretical framework to describe (passive) soft-matter systems out of equilibrium subject to fluctuations. We provide a unique connection between the constituent-particle dynamics of real systems and the time evolution equation of their measurable (coarse-grained) quantities, such as local density and velocity. The starting point is the full Hamiltonian description of a system of colloidal particles immersed in a fluid of identical bath particles. Then, we average out the bath via Zwanzig's projection-operator techniques and obtain the stochastic Langevin equations governing the colloidal-particle dynamics. Introducing the appropriate definition of the local number and momentum density fields yields a generalisation of the Dean-Kawasaki (DK) model, which resembles the stochastic Navier-Stokes (NS) description of a fluid. Nevertheless, the DK equation still contains all the microscopic information and, for that reason, does not represent the dynamical law of observable quantities. We address this controversial feature of the DK description by carrying out a nonequilibrium ensemble average. Adopting a natural decomposition into local-equilibrium and nonequilibrium contribution, where the former is related to a generalised version of the canonical distribution, we finally obtain the fluctuating-hydrodynamic equation governing the time-evolution of the mesoscopic density and momentum fields. Along the way, we outline the connection between the ad-hoc energy functional introduced in previous DK derivations and the free-energy functional from classical density-functional theory (DFT). The resultant equation has the structure of a dynamical DFT (DDFT) with an additional fluctuating force coming from the random interactions with the bath. We show that our fluctuating DDFT formalism corresponds to a particular version of the fluctuating NS equations, originally derived by Landau and Lifshitz. Our framework thus provides the formal apparatus for ab-initio derivations of fluctuating DDFT equations capable of describing the dynamics of soft-matter systems in and out of equilibrium. We believe that the derivation offered here represents the current state of the art in the field
Dynamical density functional theory for orientable colloids including inertia and hydrodynamic interactions
Over the last few decades, classical density-functional theory (DFT) and its
dynamic extensions (DDFTs) have become powerful tools in the study of colloidal
fluids. Recently, previous DDFTs for spherically-symmetric particles have been
generalised to take into account both inertia and hydrodynamic interactions,
two effects which strongly influence non-equilibrium properties. The present
work further generalises this framework to systems of anisotropic particles.
Starting from the Liouville equation and utilising Zwanzig's
projection-operator techniques, we derive the kinetic equation for the Brownian
particle distribution function, and by averaging over all but one particle, a
DDFT equation is obtained. Whilst this equation has some similarities with
DDFTs for spherically-symmetric colloids, it involves a
translational-rotational coupling which affects the diffusivity of the
(asymmetric) particles. We further show that, in the overdamped (high friction)
limit, the DDFT is considerably simplified and is in agreement with a previous
DDFT for colloids with arbitrary shape particles.Comment: dynamical density functional theory ; colloidal fluids ;
arbitrary-shape particles ; orientable colloid
Understanding Soaring Coronavirus Cases and the Effect of Contagion Policies in the UK
The number of new daily SARS-CoV-2 infections experienced an abrupt increase during the last quarter of 2020 in almost every European country. The phenomenological explanation offered was a new mutation of the virus, first identified in the UK. We use publicly available data in combination with a time-delayed controlled SIR model, which captures the effects of preventive measures on the spreading of the virus. We are able to reproduce the waves of infection occurred in the UK with a unique transmission rate, suggesting that the new SARS-CoV-2 variant is as transmissible as previous strains. Our findings indicate that the sudden surge in cases was, in fact, related to the relaxation of preventive measures and social awareness. We also simulate the combined effects of restrictions and vaccination campaigns in 2021, demonstrating that lockdown policies are not fully effective to flatten the curve. For effective mitigation, it is critical that the public keeps on high alert until vaccination reaches a critical threshold