From hard spheres and cubes to nonequilibrium maps with thirty-some years of thermostatted molecular dynamics

Maximum speed of dissipation

We derive statistical-mechanical speed limits on dissipation from the classical, chaotic dynamics of many-particle systems. In one, the rate of irreversible entropy production in the environment is the maximum speed of a deterministic system out of equilibrium, S[over ¯]_{e}/k_{B}≥1/2Δt, and its inverse is the minimum time to execute the process, Δt≥k_{B}/2S[over ¯]_{e}. Starting with deterministic fluctuation theorems, we show there is a corresponding class of speed limits for physical observables measuring dissipation rates. For example, in many-particle systems interacting with a deterministic thermostat, there is a trade-off between the time to evolve between states and the heat flux, Q[over ¯]Δt≥k_{B}T/2. These bounds constrain the relationship between dissipation and time during nonstationary processes, including transient excursions from steady states.

Density matrix formulation of dynamical systems

Physical systems that are dissipating, mixing, and developing turbulence also irreversibly transport statistical density. However, predicting the evolution of density from atomic and molecular scale dynamics is challenging for nonsteady, open, and driven nonequilibrium processes. Here, we establish a theory to address this challenge for classical dynamical systems that is analogous to the density matrix formulation of quantum mechanics. We show that a classical density matrix is similar to the phase-space metric and evolves in time according to generalizations of Liouville's theorem and Liouville's equation for non-Hamiltonian systems. The traditional Liouvillian forms are recovered in the absence of dissipation or driving by imposing trace preservation or by considering Hamiltonian dynamics. Local measures of dynamical instability and chaos are embedded in classical commutators and anticommutators and directly related to Poisson brackets when the dynamics are Hamiltonian. Because the classical density matrix is built from the Lyapunov vectors that underlie classical chaos, it offers an alternative computationally tractable basis for the statistical mechanics of nonequilibrium processes that applies to systems that are driven, transient, dissipative, regular, and chaotic.

Conservative dynamics in a novel class of 3D generalized thermostatted systems

This paper reports a method to derive a novel class of 3D generalized thermostatted oscillators from a simple damped harmonic oscillator. Its detailed procedure is obtained through a mathematical derivation. Then, we propose an example system to show the effectiveness of the method. Furthermore, the numerical analysis is performed to investigate its rich conservative dynamics, including chaotic sea, hierarchical invariant tori, and the coexistence of chaotic seas and invariant tori with islands-around-islands hierarchy, even if there are three dissipative terms in the example system. To verify the existence of conservative chaos at the physical level, an analog circuit is presented to observe the existing conservative chaotic flows in National Instruments multisim. We finally provide two systems, which can produce conservative chaotic flows with more complicated topologies, and draw our conclusions.

Nonequilibrium Time Reversibility with Maps and Walks

Time-reversible dynamical simulations of nonequilibrium systems exemplify both Loschmidt's and Zermélo's paradoxes. That is, computational time-reversible simulations invariably produce solutions consistent with the irreversible Second Law of Thermodynamics (Loschmidt's) as well as periodic in the time (Zermélo's, illustrating Poincaré recurrence). Understanding these paradoxical aspects of time-reversible systems is enhanced here by studying the simplest pair of such model systems. The first is time-reversible, but nevertheless dissipative and periodic, the piecewise-linear compressible Baker Map. The fractal properties of that two-dimensional map are mirrored by an even simpler example, the one-dimensional random walk, confined to the unit interval. As a further puzzle the two models yield ambiguities in determining the fractals' information dimensions. These puzzles, including the classical paradoxes, are reviewed and explored here.

Fcc vs. hcp competition in colloidal hard-sphere nucleation: on their relative stability, interfacial free energy and nucleation rate

Hard-sphere crystallization has been widely investigated over the last six decades by means of colloidal suspensions and numerical methods. However, some aspects of its nucleation behaviour are still under debate. Here, we provide a detailed computational characterisation of the polymorphic nucleation competition between the face-centered cubic (fcc) and the hexagonal-close packed (hcp) hard-sphere crystal phases. By means of several state-of-the-art simulation techniques, we evaluate the melting pressure, chemical potential difference, interfacial free energy and nucleation rate of these two polymorphs, as well as of a random stacking mixture of both crystals. Our results highlight that, despite the fact that both polymorphs have very similar stability, the interfacial free energy of the hcp phase could be marginally higher than that of the fcc solid, which in consequence, mildly decreases its propensity to nucleate from the liquid compared to the fcc phase. Moreover, we analyse the abundance of each polymorph in grown crystals from different types of inserted nuclei: fcc, hcp and stacking disordered fcc/hcp seeds, as well as from those spontaneously emerged from brute force simulations. We find that post-critical crystals fundamentally grow maintaining the polymorphic structure of the critical nucleus, at least until moderately large sizes, since the only crystallographic orientation that allows stacking close-packed disorder is the fcc (111) plane, or equivalently the hcp (0001) one. Taken together, our results contribute with one more piece to the intricate puzzle of colloidal hard-sphere crystallization.

Structural and thermodynamic properties of hard-sphere fluids

This Perspective article provides an overview of some of our analytical approaches to the computation of the structural and thermodynamic properties of single-component and multicomponent hard-sphere fluids. For the structural properties, they yield a thermodynamically consistent formulation, thus improving and extending the known analytical results of the Percus-Yevick theory. Approximate expressions linking the equation of state of the single-component fluid to the one of the multicomponent mixtures are also discussed.