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This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

In this review, we discuss the Dynamical approach to Non-Equilibrium Molecular Dynamics (D-NEMD), which extends stationary NEMD to time-dependent situations, be they responses or relaxations. Based on the original Onsager regression hypothesis, implemented in the nineteen-seventies by Ciccotti, Jacucci and MacDonald, the approach permits one to separate the problem of dynamical evolution from the problem of sampling the initial condition. D-NEMD provides the theoretical framework to compute time-dependent macroscopic dynamical behaviors by averaging on a large sample of non-equilibrium trajectories starting from an ensemble of initial conditions generated from a suitable (equilibrium or non-equilibrium) distribution at time zero. We also discuss how to generate a large class of initial distributions. The same approach applies also to the calculation of the rate constants of activated processes. The range of problems treatable by this method is illustrated by discussing applications to a few key hydrodynamic processes (the “classical” flow under shear, the formation of convective cells and the relaxation of an interface between two immiscible liquids).

The most widespread use of Molecular Dynamics (MD) [

At variance with Monte Carlo, the dynamical approach of Molecular Dynamics can be directly extended to sample distributions corresponding to stationary non-equilibrium conditions, where there exists a stationary distribution but, at variance with equilibrium, its expression is not explicitly known. However, the statistical problem of sampling a time-dependent ensemble cannot be solved by generating states along a single dynamical non-equilibrium trajectory, as long as time cannot be taken as homogeneous and averages over time make no sense.

Generally, to compute macroscopic dynamical behaviors, as, e.g., in hydrodynamics, the assumption of time-scale separation is made and rigorous ensemble averages are substituted with short-time averages equivalent to local smoothing. This may not be the case, sometimes. Moreover, the statistical error implied by this procedure cannot be made as small as desirable and possible. These difficulties can be faced and solved.

In the nineteen-thirties, Lars Onsager [

In the case of Kubo’s procedure one does not need to make reference to an initial _{0} = 0 of the system. This result has an important consequence for Molecular Dynamics simulations, since it allows one to separate the problem of dynamical evolution from the problem of sampling the initial condition.

Starting from the mid-nineteen-seventies, the direct numerical simulation of the response was used in conjunction with a sample of initial conditions extracted from an equilibrium trajectory [

Some time later on, it was realized that the same approach could be used to calculate dynamical properties for rare events (e.g., transmission coefficients) by averaging the dynamical response over time-dependent trajectories started from initial conditions sampled from a constrained/conditional equilibrium ensemble [

Quite recently, finally, the idea of creating a large sample of non-equilibrium trajectories starting from a given initial distribution has been extended to cover whatever distribution that can be sampled starting from an equilibrium or a non-equilibrium, but stationary, dynamics. In particular stationary non-equilibrium ensembles can be generated by suitably restraining standard MD simulations.

In particular, we will illustrate the approach by reporting the results of a study of the time evolution of classical fields, including the onset of convective cells and the relaxation of hydrodynamic interfaces in simple liquids. In this context, we will also briefly address a conceptual difficulty of the approach, due to the possible existence of more than one macroscopic state associated with specific perturbations. In particular cases the problem can be circumvented.

The structure of the paper is as follows. In Section 2 we derive the general framework and specify the possible forms for the initial ensemble. In Section 3 we present a few successful applications of the method. Finally, in Section 4 we try to assess the situation and sketch an outlook.

We start considering, in a very general way, a (classical) dynamical system with _{1}, _{1}, _{2}, _{2},...,_{n}_{n}^{2n}Γ) in phase space [_{0} to time

Time evolution in phase space can be alternatively expressed in term of the Jacobian _{0})) of the time transformation from Γ⃗(_{0}) to Γ⃗(^{2n}Γ(_{0}) at time _{0} transforms into the volume element ^{2n}Γ(_{0})) ^{2n}Γ(_{0}) at time _{0})) = 1 and the dynamics preserves volume in phase space (Liouville Theorem). More generally, when ^{2n}Γ is no longer a dynamical invariant and one needs to introduce a metric factor to define the invariant measure of the phase space under the dynamical evolution. Starting from the general expression for the Jacobian determinant, one gets
^{2n}Γ, by using
^{2n}Γ.

The solution of ^{†}(_{0}) of the previously defined time evolution operator _{0}) acting on the phase space variables Γ⃗ and the phase density _{0} = _{0}) at the initial time _{0}.

The average over the (non-)equilibrium ensemble of a physical observable _{t}_{t}_{j}Ô_{j}_{t}

We can make the time evolution explicit by means of the adjoint time evolution operator ^{†}(_{0})_{0}) and then, by taking advantage of the fact that ^{†} is the adjoint of the dynamics, we can transfer the effect of time evolution to the physical observables
_{0})_{0}) at time _{0}. We have introduced the shorthand notation, 〈· · · 〉_{ρ0}, for the averages over the ensemble described by the space density _{0} at the initial time _{0}.

Despite the apparent complexity of the time evolution operator _{0}) in

In the following, we will deal with fluid systems where the relevant macroscopic fields are [

_{0} = _{0}).

If the ensemble at the initial time _{0} can be simulated by a dynamical system in stationary conditions, then such a probability density function can be sampled by MD, generating a set of (possibly independent) phase space points distributed according to _{0}. From each of these points, one can then start an independent dynamical trajectory along which the observables

In order to use MD to sample the appropriate initial ensemble at time _{0}, one needs to define, for any specific problem, the dynamical evolution,

We will now list a number of cases, which will later be illustrated with the corresponding application. Transport properties, like viscosity, thermal conductivity, _{0} with the statistical mechanics equilibrium ensemble, while the dynamical trajectories are carried out under the influence of an external (time-dependent) force field.

More generally, we can generate (and sample) initial ensembles by less trivial procedures, e.g., in the case of the formation of convective cells, gravity is considered as the external perturbation to be applied on a system initially in a steady state under the effect of a thermal gradient. The ensemble at time _{0} no longer corresponds to the equilibrium one, but it is set up by introducing a stationary boundary perturbation which, in the specific case, is just an

Another possible case we will consider is the relaxation to equilibrium of an interface between two immiscible liquids, starting from an imposed, non-equilibrium, condition in which the curvature of the interface is maintained by a macroscopic restraint fixing the shape of the initial interface. The ensemble at time _{0} is described by a conditional probability density in which an

Linear Response Theory is a nice result of the nineteen-fifties in the theory of irreversible processes [_{0}.

For a system of particles in three dimensions described by the usual set of Cartesian coordinates and momenta, {_{j}_{j}_{j}_{j}_{j}_{0}) (_{0}) = 1, _{0}) = 0) or a Dirac delta impulse _{0}), at _{0}, after which the system is left free to relax. In the linear regime, the general response can be computed as the superposition of these impulsive responses. One then derives the equations of motion using the standard Hamiltonian route, where we start by separating in the Hamiltonian _{0}(Γ⃗) + _{p}_{0} is the equilibrium Hamiltonian to which one can possibly add the coupling to a thermostat or a barostat, something that can be done in a variety of ways that we do not need to specify here. Indicating generically the possible presence of such couplings to different baths with ellipses, the equations of motion for particle

The structure of the equations of motion can be broken into the two terms of the Liouville operator defined in _{0}(Γ⃗) + _{p}_{0} defining the dynamical evolution in phase space for the sampling of the ensemble at time _{0}. Accordingly, the corresponding evolution operator for the stationary dynamics will be called _{0}(_{0}-(time dependent) evolution operator _{0}), obeying the (usual) Dyson equation

A more general scheme has also been used for bulk perturbations, where the new equations of motion, which cannot be derived from a time-dependent Hamiltonian in a way that remains consistent with applied (periodic) boundary conditions, are obtained from _{p}_{j}_{j}_{j}_{j}_{j}_{j}

In the typical setup for a planar Couette flow, one establishes a gradient of the _{xy}_{ij}_{i}_{j}_{0} = 0, one can measure the viscous time-dependent response _{xy}_{ρ0}/

For the purpose of illustrating the method in the original applications, when the ensemble at the initial time _{0} is an equilibrium ensemble, we will restrict ourselves to the simple case of shear (Couette) flow. We would like to mention, however, that also elongational flows [

In Panel (A) of _{B}T_{p}_{p}_{0} = 0 impulsive perturbation with a _{0}) term was used to investigate the range of validity of the Linear Response Theory for very small shear rates by comparison with the running time integral of the corresponding stress autocorrelation at equilibrium [

The D-NEMD approach can be used also to follow the transient evolution of a system, which, starting from an out-of-equilibrium state under the effect of a stationary thermodynamic field, reaches a final (different) non-equilibrium state in response to an additional external perturbation. Below, we illustrate the approach with a case worked out in [

The 2D system is composed of _{m}^{1/6}_{m}_{f}_{wall}_{w}_{m}_{n}^{n}^{+1} (^{−mv⃗2/(2kBTi)}/(2_{B}T_{i}_{i}_{T}_{m}_{1} = 1.5 and _{2} = 9.9, corresponding to a thermal gradient ∇_{0} was sampled. Time-dependent trajectories have been generated and then suitable properties averaged at times 0 ⩽ _{i}_{i}_{i}^{2} for particles inside the cell labeled by (_{j}_{k}_{m}_{jk}_{P}_{0} = 0; its value grows with time and, after a small overshooting, reaches its plateau, stationary, value at

We have seen how D-NEMD can be used to illustrate the build up of a convective roll when a gravity field is instantaneously switched on in a system where a stationary (non-equilibrium) thermal gradient was already present. This is not the only case in which a convective roll can be observed. Indeed, keeping the same geometry for the system,

Complications arise if the system is setup with a different geometry, e.g., in the case of Rayleigh–Bénard convection when the direction of the thermal gradient is parallel to the direction of the gravity field. The system has a higher symmetry and rolls rotating both in the counterclockwise and clockwise directions are possible. This implies that the D-NEMD averages cannot be carried out directly, as in the case we have described so far. In fact, now, in the ensemble of individual trajectories, one samples with equal probability the initial conditions leading to clockwise or to counterclockwise rolls. Performing ensemble averages without paying attention to the direction of rotation would give a wrong result. One needs either to enforce a mechanism that breaks this symmetry or to weight trajectories differently, according to the direction of rotation of the convective roll. The latter was the choice applied in [

Probably the most interesting application of the D-NEMD procedure is when the non-equilibrium dynamical trajectories start from states corresponding to a very unlikely fluctuation and we want to follow dynamically the way in which the system relaxes back to equilibrium. An efficient sampling of the points in phase space representing the initial condition cannot be achieved just by waiting long enough for the desired event to occur during a standard MD trajectory, but more advanced methods are required to enhance the sampling. Whenever the conditions can be described using an “order parameter”,

As a more advanced illustration of D-NEMD when sampling from a conditional probability density, we describe the case of the hydrodynamic relaxation to equilibrium of the interface between two immiscible liquids [^{A}^{B}_{0} corresponds to the stationary conditions of the system subject to a macroscopic restraint that forces a non-equilibrium geometry for the interface. This requires the implementation of a method, like the Blue Moon one, that allows one to sample the conditional probability density associated with the constraint. However, using the Blue Moon approach for vector or field-like constraints can become considerably cumbersome and rather inconvenient in practice, especially for molecular systems where constraints are already used in the force field to impose molecular geometries. A much more practical alternative is to use restrained MD, where one substitutes the constraint with an equivalent restraining potential in terms of an additional coupling parameter, asymptotically reproducing unbiased constrained conditions. Let us summarize the restraint MD approach for the case in which the constraint is imposed on a field-like observable, as for the density difference Δ_{S}_{S}

According to Irving and Kirkwood [_{0}) = Δ_{0}. However, in the numerical approach, one cannot deal directly with a continuous (vector) variable _{α}_{α}_{α}_{α}_{α}_{α}_{α}

Consider, now, a system described by the Hamiltonian
_{0}
^{2n}Γ^{−βℋ(Γ⃗)} are the canonical partition functions and

The idea of using a biasing potential to sample unlikely points in configuration space was pioneered by Torrie and Valleau for MC simulation [

The choice of a restraining potential, which depends only on the coordinates of the particles, as in this case, does not influence the probability density in the momentum space, which remains the Maxwellian (equilibrium) distribution and, at variance with the Blue Moon approach, independent points along the stationary restrained MD trajectory can be directly taken as initial configurations representative of the probability density at time _{0}. Moreover, if needed, the restrained MD approach can be further generalized to enforce a more general macroscopic constraint affecting also the momenta of the particles, for example, coupling it with the

The definition of the microscopic field in _{j}_{α}_{j}^{1} ≡ ^{2} ≡ ^{3} ≡

The two fluids, A and B, are modeled using identical Lennard–Jones particles with mass _{AA}_{BB}_{AB}_{AA}_{BB}_{AB}^{A}^{B}^{12}. The simulation was performed at a fixed temperature _{B}T_{0}, the initial configuration for the interface between Fluid A and Fluid B is defined by selecting the _{α}_{α}_{0}. Periodic boundary conditions are applied in all directions, so that a second flat interface (the condition that minimizes the surface tension) is created at the same time at the sides of the box along the ^{−4} in LJ units. Such a rather small value for

A long, 10^{6} MD restrained trajectory was then carried out, with that same time step, taking out, at regular intervals of 25,000 steps, the configuration of the system in phase space. A set of 40 independent initial conditions was collected in this way, and from each of them was started, now with a regular time step ^{−3}, a 25,000-steps unrestrained MD trajectory at constant energy,

In the right panel of

One can see that the interface curvature diminishes progressively towards the flat, equilibrium condition, while maintaining (approximatively) both the initial uniformity along the direction of the _{max}_{0} of the interface. If one takes the LJ parameters of argon (for which ^{−10} m and the unit of time corresponds to ^{−12} s), this value translates to an experimentally convincing velocity of ≈ 80 m · s^{−1}.

The second interface on the sides of the MD box remains flat, with even smaller deviations, all along the 25,000 time steps. There seem to be no significant effects on it as a result of the relaxation process, which takes place in the middle of the box. The reason for this will become evident after looking at the time-dependent behavior of the velocity field. We have shown, in fact, how the D-NEMD approach provides very detailed information on the hydrodynamic behavior of the system and unravels the underlying physical mechanisms.

Starting from _{0} probability density along the restrained MD trajectory. The results of the calculation are shown in the left panels of

Snapshots at three successive times are shown. In the snapshot taken 500 time steps after _{0}, one can notice the build up of some coherence in the velocity field, which becomes structured in a region that is, along the _{0}, when the curvature of the interface is significantly reduced (Panel C).

One can notice that the size and the intensity of the field has decreased, but the profile remains highly symmetrical along the mirror symmetry plane. All along, one can notice also that the field in the extreme sides of the simulation box remains essentially unperturbed, which explains why the second flat interface at the boundary is not affected in any significant way and remains stable during the whole relaxation process of the curved interface. It is very interesting to compare these insights on the hydrodynamic processes underlying the interface relaxation, as given by the time-dependent behavior of the dynamical response calculated using the D-NEMD procedure, with the standard approach used on single trajectory simulations of hydrodynamic processes. Starting from the local equilibrium hypothesis of hydrodynamic theory, based on the assumption of a time scale separation between the fast microscopic motion of the particles and the slower hydrodynamic processes, the macroscopic fields are computed as local time averages on the short time scale

By applying this approach to the time evolution from a single initial condition, one obtains the results shown in the right panels of _{0}, contrary to the more convincing evidence, given from the D-NEMD results, of a relaxation mechanism satisfying, on average, such symmetry at all times along the dynamical trajectory.

In this paper, we have presented a dynamical approach to non-equilibrium MD, which makes it possible to compute, numerically, but, otherwise, rigorously, time-dependent non-equilibrium responses,

We illustrated a few applications of the method starting from the early, historical, approach to the calculation of transport properties in the lines of Linear Response Theory and beyond, to a couple of recent atomistic simulations of hydrodynamic processes: the establishing of a convective cell, when gravity is switched on in the presence of a stationary thermal gradient, and the relaxation of an initially curved interface between two immiscible liquids. We have shown that the method generates rigorous time-dependent non-equilibrium averages, providing valuable insights on the mechanisms of hydrodynamic processes that can be missed using a method like the local time average, which cannot have a rigorous justification, presents a statistical error that cannot be reduced at will and, finally, as we have seen above, can bias the statistical response. A word of caution is needed, though. The time-dependent ensemble averages are meaningful only if the thermodynamical response is unique [

In summary, with the outlined exceptions, D-NEMD is a method ready for challenging applications, by which it is possible to study complex time-dependent phenomena using only the fundamental laws of Statistical Mechanics,

We acknowledge the Science Foundation Ireland SFI Grant No. 08-IN.1-I1869 and the Istituto Italiano di Tecnologia under the SEED Project grant No. 259 SIMBEDD-Advanced Computational Methods for Biophysics, Drug Design and Energy Research for financial support.

The authors declare no conflict of interest.

^{6}V·cm

^{−1}or thermal gradients up to the order of 10

^{8}K·cm

^{−1}and, finally, for viscous phenomena, the shear rate applicable to simple fluids up to the order of 10

^{12}s

^{−1}, as compared with an intercollisional frequency of the order of 10

^{13}s

^{−1}.

Phase space representation of the ensemble of dynamical side-trajectories providing the non-equilibrium statistical averages: in blue, the Molecular Dynamics (MD) trajectory sampling the ensemble at time _{0}; in black, the individual non-equilibrium trajectories sampling the Non-Equilibrium Molecular Dynamics (D-NEMD) ensemble, over which one can average the time behavior of the observable

We distinguish three different classes for the sampling of the initial distribution: equilibrium, direct stationary non-equilibrium simulations and advanced conditional sampling. They are shown to be associated with the corresponding sampling techniques and test-case applications.

The Lees–Edwards periodic boundary conditions (Panel _{0}. Periodic boundary conditions can be effectively imposed using the equivalent non-orthogonal reference cell, highlighted in red (the actual inclination increases uniformly with time).

Panel (_{0}) perturbation with ^{−4}, averaged over 4000 trajectories versus the running-time integral (dashed line) of the stress autocorrelation function shows the agreement of D-NEMD results with the Green-Kubo linear reponse theory [

The simulation setup (Panel

The build up of the convective flow is shown by visualizing the local velocity field averaged over 1,000 independent initial configurations as a function of time: (

(

The behavior of the velocity field in the two liquid regions as a function of time: comparison of the results obtained using the D-NEMD approach averaging over 40 initial conditions (Panels