62 research outputs found
Entropy production and the geometry of dissipative evolution equations
Purely dissipative evolution equations are often cast as gradient flow
structures, , where the variable
of interest evolves towards the maximum of a functional
according to a metric defined by an operator . While the functional often
follows immediately from physical considerations (e.g., the thermodynamic
entropy), the operator and the associated geometry does not necessarily so
(e.g., Wasserstein geometry for diffusion). In this paper, we present a
variational statement in the sense of maximum entropy production that directly
delivers a relationship between the operator and the constraints of the
system. In particular, the Wasserstein metric naturally arises here from the
conservation of mass or energy, and depends on the Onsager resistivity tensor,
which, itself, may be understood as another metric, as in the Steepest Entropy
Ascent formalism. This new variational principle is exemplified here for the
simultaneous evolution of conserved and non-conserved quantities in open
systems. It thus extends the classical Onsager flux-force relationships and the
associated variational statement to variables that do not have a flux
associated to them. We further show that the metric structure is intimately
linked to the celebrated Freidlin-Wentzell theory of stochastically perturbed
gradient flows, and that the proposed variational principle encloses an
infinite-dimensional fluctuation-dissipation statement
Formation of Nanotwin Networks during High-Temperature Crystallization of Amorphous Germanium
Germanium is an extremely important material used for numerous functional
applications in many fields of nanotechnology. In this paper, we study the
crystallization of amorphous Ge using atomistic simulations of critical
nano-metric nuclei at high temperatures. We find that crystallization occurs by
the recurrent transfer of atoms via a diffusive process from the amorphous
phase into suitably-oriented crystalline layers. We accompany our simulations
with a comprehensive thermodynamic and kinetic analysis of the growth process,
which explains the energy balance and the interfacial growth velocities
governing grain growth. For the crystallographic
orientation, we find a degenerate atomic rearrangement process, with two
zero-energy modes corresponding to a perfect crystalline structure and the
formation of a twin boundary. Continued growth in this direction
results in the development a twin network, in contrast with all other growth
orientations, where the crystal grows defect-free. This particular mechanism of
crystallization from amorphous phases is also observed during solid-phase
epitaxial growth of semiconductor crystals, where growth is
restrained to one dimension. We calculate the equivalent X-ray diffraction
pattern of the obtained nanotwin networks, providing grounds for experimental
validation
A statistical mechanics framework for constructing non-equilibrium thermodynamic models
Far-from-equilibrium phenomena are critical to all natural and engineered
systems, and essential to biological processes responsible for life. For over a
century and a half, since Carnot, Clausius, Maxwell, Boltzmann, and Gibbs,
among many others, laid the foundation for our understanding of equilibrium
processes, scientists and engineers have dreamed of an analogous treatment of
non-equilibrium systems. But despite tremendous efforts, a universal theory of
non-equilibrium behavior akin to equilibrium statistical mechanics and
thermodynamics has evaded description. Several methodologies have proved their
ability to accurately describe complex non-equilibrium systems at the
macroscopic scale, but their accuracy and predictive capacity is predicated on
either phenomenological kinetic equations fit to microscopic data, or on
running concurrent simulations at the particle level. Instead, we provide a
framework for deriving stand-alone macroscopic thermodynamics models directly
from microscopic physics without fitting in overdamped Langevin systems. The
only necessary ingredient is a functional form for a parameterized, approximate
density of states, in analogy to the assumption of a uniform density of states
in the equilibrium microcanonical ensemble. We highlight this framework's
effectiveness by deriving analytical approximations for evolving mechanical and
thermodynamic quantities in a model of coiled-coil proteins and double stranded
DNA, thus producing, to the authors' knowledge, the first derivation of the
governing equations for a phase propagating system under general loading
conditions without appeal to phenomenology. The generality of our treatment
allows for application to any system described by Langevin dynamics with
arbitrary interaction energies and external driving, including colloidal
macromolecules, hydrogels, and biopolymers
Second-order fast-slow dynamics of non-ergodic Hamiltonian systems:Thermodynamic interpretation and simulation
A class of fast-slow Hamiltonian systems with potential
describing the interaction of non-ergodic fast and slow degrees of freedom is
studied. The parameter indicates the typical timescale ratio of
the fast and slow degrees of freedom. It is known that the Hamiltonian system
converges for to a homogenised Hamiltonian system. We study
the situation where is small but positive. First, we rigorously
derive the second-order corrections to the homogenised (slow) degrees of
freedom. They can be decomposed into explicitly given terms that oscillate
rapidly around zero and terms that trace the average motion of the corrections,
which are given as the solution to an inhomogeneous linear system of
differential equations. Then, we analyse the energy of the fast degrees of
freedom expanded to second-order from a thermodynamic point of view. In
particular, we define and expand to second-order a temperature, an entropy and
external forces and show that they satisfy to leading-order, as well as on
average to second-order, thermodynamic energy relations akin to the first and
second law of thermodynamics. Finally, we analyse for a specific fast-slow
Hamiltonian system the second-order asymptotic expansion of the slow degrees of
freedom from a numerical point of view. Their approximation quality for short
and long time frames and their total computation time are compared with those
of the solution to the original fast-slow Hamiltonian system of similar
accuracy
Harnessing fluctuations to discover dissipative evolution equations
This dataset contains the source codes for for the paper "Harnessing fluctuations to discover dissipative evolution equations". The code computes the macroscopic evolution operator associated with many-particle systems (hydrodynamic limit) from particle simulations. The method is based on fluctuation-dissipation theory and is described in the paper. A test case considered is the zero-range process.The zip file contains the code and a README fil
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