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# Relationship between dynamical entropy and energy dissipation far from thermodynamic equilibrium

Edited

^{†}by R. Stephen Berry, The University of Chicago, Chicago, IL, and approved August 29, 2013 (received for review June 26, 2013)

## Significance

The dissipation of energy occurs naturally in systems as diverse as those in biology and as common as those responsible for the weather. Systems that dissipate energy while self-assembling also show promise as a synthetic route to responsive nanoscale materials. Optimizing the energy efficiency of these syntheses requires reducing the theoretical description to only those variables that are essential to an accurate prediction of the energy lost as heat. Here, with computer simulations and a model of assembly, we establish a methodology for identifying those essential variables. With this minimal description of the system we also find a linear relationship between the dynamical entropy of the self-assembling system and the energy dissipated as heat during assembly.

## Abstract

Connections between microscopic dynamical observables and macroscopic nonequilibrium (NE) properties have been pursued in statistical physics since Boltzmann, Gibbs, and Maxwell. The simulations we describe here establish a relationship between the Kolmogorov–Sinai entropy and the energy dissipated as heat from a NE system to its environment. First, we show that the Kolmogorov–Sinai or dynamical entropy can be separated into system and bath components and that the entropy of the system characterizes the dynamics of energy dissipation. Second, we find that the average change in the system dynamical entropy is linearly related to the average change in the energy dissipated to the bath. The constant energy and time scales of the bath fix the dynamical relationship between these two quantities. These results provide a link between microscopic dynamical variables and the macroscopic energetics of NE processes.

Uncovering the laws governing nonequilibrium (NE) energy-dissipating systems is a fundamental challenge, consequential for the design of new dynamic materials (1, 2) and, ultimately, to the understanding of life (3⇓–5). From a theoretical perspective, this challenge is complicated by the inability to use the methods and simplifications familiar from equilibrium thermodynamics and equilibrium statistical mechanics. Nevertheless, there have been numerous important contributions to the understanding of NE phenomena, the earliest of which applied thermodynamics to irreversible processes (6⇓–8) and processes executed in finite time (9, 10) in the near-equilibrium regime, often with assumptions of local equilibrium. More recently, advances have been made in the far-from-equilibrium regime, including what are now collectively called fluctuation theorems (11⇓⇓–14).

Most relevant to the present study is the statistical mechanics of NE phenomena based on the elements of dynamical systems theory (15, 16). Missing from this theoretical framework, and others, is a means of simplifying the description of irreversible processes, when such a simplification is possible. Most generally it is not possible to describe the dynamics of NE systems separately from the dynamics of their surroundings (e.g., a heat bath) (17)––although progress has been made in reducing the number of degrees of freedom of NE systems that obey Markovian dynamics (18, 19). This difficulty is a more general signature of systems in extreme states of disequilibrium: There is not necessarily a clear physical system–bath boundary because all of the microscopic variables of the bath may be important at one time or another in the dynamics of mass, energy, or momentum transport through the system. As a consequence, statistical-mechanical properties of NE systems cannot generally be parameterized by the static properties of the surroundings.

Central to the dynamical systems approach to NE statistical mechanics are the Kolmogorov–Sinai (KS) entropy per unit time (20, 21) and Lyapunov exponents (22). These quantities, for example, can be used to formulate the deterministic fluctuation theorems that characterize the statistical dynamics of systems far from equilibrium (12, 13, 23). In the present study, we build on these two concepts to study the process of NE self-assembly (3, 5, 24). The specific system we analyze is inspired by experiments in which nanoparticles covered with photoactive ligands are irradiated with light, thereby changing their surface properties and modifying interparticle interactions, resulting in particle assembly (25⇓–27). In our simulations, interparticle interactions are similarly modified, displacing the system from equilibrium and causing it to evolve under NE conditions until a final equilibrium self-assembled state is reached. In both cases, the assembly process is accompanied by dissipation of energy during the transient relaxation to an equilibrium state. We analyze this evolution from our simulations both in terms of the dynamical (KS) entropy and the energy of (or heat released by) the system.

## Model System

Our model is a classical, 3D, periodic 12−6 Lennard-Jones (LJ) fluid of *N* particles with a set of 3*N* positions and 3*N* momenta. The Hamiltonian is used with the interaction potential of particles *i* and *j* a distance apart being , where determines the strength of the interaction at time ** t** and

*σ*is the distance at which the attractive and repulsive terms are equal (i.e., a measure of the particle size). The minority of the particles (typically, a set

*S*of 7) are switchable and constitute the system to be assembled into a bound cluster, whereas the majority (typically, a set

*B*of 143) are the bath particles (note: simulations with other particle numbers yield conclusions identical to those described in the following; see Fig. S1 for details).

The method used to simulate NE self-assembly trajectories, shown schematically in Fig. 1, produces a sample of classical trajectories. From a given sample of trajectories, average properties of microscopic dynamical observables can be examined as they evolve in time. Initial phase points for the trajectories of interest are selected from the isothermal–isobaric ensemble. For this model, all interparticle interactions are initially identical. At a time assigned to be *t =* 0, the entire collection of particles is isolated, and the well-depth parameter for the system particles is instantaneously changed from a value corresponding to weak interaction to a new value corresponding to strong attraction. The effect of this change is to generate an initial NE state from which system particles self-assemble and concurrently dissipate their excess kinetic energy to the explicit bath particles, while conserving the total energy, volume, and number of particles. The dynamics of all particles is studied until a new equilibrium state is reached. After the new equilibrium state is reached, the system is returned to its initial equilibrium ensemble by again making all interparticle interactions weakly interacting, and repeating the entire process.

The above procedure can be performed cyclically to generate any desired number of trajectories. Equilibration before each NE trajectory removes the energy dissipated during the previous assembly trajectory. Data shown here for the assembly trajectories at constant total energy, volume, and number of particles will begin at *t =* 0 and end when assembly is complete. For intermediate times, the energy dissipated by the system particles is entirely accounted for in the energy of the bath. Because the initial phase points are consistent with the isothermal–isobaric ensemble, the total energy of each trajectory varies from trajectory to trajectory. In what follows, all distances and times will be expressed in LJ reduced units, defined in *Methods*, and the energy parameter will be expressed in multiples of a constant (Fig. 1).

One of the methodological advances of this work is the calculation of separate KS entropies per unit time (henceforth, simply KS entropies) for bath and system components of a Hamiltonian system. These KS entropies are calculated from each time step (28) of the set of trajectories characterizing particle assembly. To compute KS entropies , orthogonal (Gram–Schmidt) Lyapunov vectors are simulated for the entire collection of particles; the *i*th vector has the form of first variations in position and momentum with evolving under the linearized Hamiltonian dynamics (29). This basis set for tangent space is propagated along with the phase point by the linearized velocity Verlet algorithm, a time step size of , and orthonormalization at every step (for details, see refs. 30⇓–32).

At each time step, Lyapunov exponent spectra are estimated separately for the system and bath particles from the total comoving set of simulated orthogonal Lyapunov vectors. These spectra are computed from subvectors identified in the *i*th Lyapunov vector with and . The metric allows system and bath Lyapunov spectra to be computed from the associated set of Lyapunov subvectors. We use *C* to represent either the set of system *S*, bath *B*, or total particles. Each set of 2*n* finite-time Lyapunov exponents is defined for the total, system, and bath as (29)For closed dynamical systems, Pesin’s theorem defines KS entropies as the sum of positive Lyapunov exponents (33). Assuming the theorem holds for the system and bath separately, their KS entropies are calculated from the system and the bath Lyapunov spectra , respectively,The (finite-time) KS entropies thus defined can be interpreted as an inverse predictability time scale or uncertainty gain resulting from the evolution of a trajectory. Put differently, KS entropies measure the degree of temporal order in dynamics. More disordered dynamics have a larger *h* and the evolution of nearby trajectories is predictable over a shorter time scale. It is emphasized that the KS entropy is not the entropy of microstates familiar from equilibrium thermodynamics: the KS entropy can be small for a structurally disordered system and large for an ordered system.

## Results and Discussion

With these considerations in mind, and with the aim to bridge the micro- and macroscale descriptions of our system, we examined the statistics of KS entropies sampled from ensembles of classical trajectories.

The distributions of KS entropies *f*(*h*), collected from single time steps in two time segments of the trajectory ensemble, are shown in Fig. 2 for the assembling system (*h*_{sys}), nonassembling system (7 randomly chosen particles of the bath, *h*_{nas}), the bath (*h*_{bath}), and the “total” (i.e., system and bath particles, *h*_{tot}). Here, a time segment is defined as an interval of 1,000 numerical time steps from a single trajectory. For each time segment, we analyze the statistics of KS entropies over the entire ensemble of trajectories. Comparison of the distributions in the first (*τ* = 1) and last (*τ* = 75) time segments reveals that all KS entropies distributions shift to larger values during the evolution from the disassembled to assembled states. Interestingly, *f(h*_{sys}*)* is skewed at *τ* = 1 but approximately Gaussian at equilibrium, *τ* = 75. The distributions *f(h*_{nas}*)* for a nonassembling system of 7 randomly chosen particles of the bath are nearly Gaussian at both times and their shift is much smaller than that for *f(h*_{sys}*)*. The deviation of *f(h*_{sys}*)* from Gaussian at *τ* = 1 is attributed to the fact that self-assembly events of the system particles are being sampled in the time segment. In contrast with the system distributions, the bath and total KS entropy distributions in Fig. 2 are nearly identical to each other and of nearly Gaussian form at all times. We also note that the bath distributions are narrower than those of the system. We have found this behavior of the distributions to be true regardless of the magnitude of the work done by changing . To summarize, the features of the KS entropy distributions reflect the variety of dynamical processes the system and the bath undergo after triggering self-assembly.

During the self-assembly process, the KS entropy distribution of the assembling system shifts more than the other distributions considered. To better quantify the differences between the system and the bath, we consider the percentage increase in sample average KS entropies , as shown in Fig. 3. These percentages, computed from the formula , allow a direct comparison of all average KS entropies simultaneously. All of the mean KS entropies were smaller at short times (when the interaction potential was switched on and the system was driven out of equilibrium) than at long times (when the particles were equilibrated in the self-assembled structure). Strictly increasing average KS entropies indicate an increase in the dynamical instability as the system particles are “funneled” predictably toward the assembled state. Conversely, they show that the equilibrium dynamics were less predictable (larger KS entropy) than the out-of-equilibrium dynamics. The difference in this sensitivity to initial conditions at and away from equilibrium is clearly most dramatic for the set of assembling particles. This indicates that the KS entropies we have defined measure the dynamical involvement of phase space variables in the NE process.

Next, to study the evolution of the system displaced to different distances from equilibrium, we performed simulations in which the potential parameter of the system particles was changed from 1*ε* to 5*ε*, 10*ε*, 15*ε*, and 20*ε*. In each case, the bath and the system–bath interaction parameters were maintained at 1*ε*. Fig. 3 *A* and *B* show results for instantaneous switching, at *t* = 0, of the potential well depth from 1*ε* to 5*ε*, and from 1*ε* to 15*ε*. In *B*, the mean KS entropy of the assembling system increases by nearly 60% during the approach to equilibrium. By comparison, the mean KS entropies of the bath , nonassembling system , and total set of particles are not as responsive to the perturbation from equilibrium––increasing by only 25% for this switch of the interaction potential. Importantly, for all perturbations, we observe the monotonic increase of these KS entropies over time, indicating that the behavior of these quantities is consistent across a wide range of perturbation strengths. The more modest increase in these KS entropies, in contrast with those associated with the assembling system, reflects the change in the dynamics associated with the extra kinetic energy produced by self-assembly in the bath, and possibly the rearrangement of the bath particles necessary for the self-assembly processes. The separation of the system KS entropy from the other KS entropies considered suggests it characterizes the entire self-assembly process and, as we show next, the energy dissipation process.

We are now in position to consider the main result of this work: namely, the relationship between KS entropy and energy dissipation. To measure the energy dissipated by the system, we calculated the difference in the (sample mean) total energy of the system over the trajectory ensemble at two consecutive trajectory segments. We scaled the energy change between times *τ* and *τ+*1, , by the natural energy scale of the bath, 2× the thermal energy = 1/*k*_{B}*T*. At the same time, we scaled the average KS entropy difference of the system by the characteristic time scale of the bath (, which for 1*ε*, the bath interaction parameter, in LJ time units is ). This inverse time scale *γ* is determined by the potential of interaction and eliminates the dimensions both of and . Fig. 4 summarizes the results by plotting the scaled energy and KS entropy difference as a function of time. For the two data sets shown, corresponding to two switches of , the time dependencies of the two-scaled quantities coincide with one another. Similar agreement holds for other values of studied (see Fig. S2 for details), leading to a general relationship of the formIn writing this relationship, we assumed that for single time step KS entropies and that the sample mean becomes the ensemble average over all trajectories (in the limit of an infinite number of trajectories). Both sides of this expression are dimensionless and decay to zero at equilibrium, when no entropy is being produced and no energy is being irreversibly dissipated. The dynamics and energetics of the bath, determining the mechanism of energy dissipation, enter only through *β* and *γ*.

Further support for Eq. **3** can be found by minor rearrangement and division by the number of system degrees of freedom 3*n*. These operations lead to a quantity . This *α*, derived from our data, is remarkably similar to a quantity built into the equations of motion often used to study many-particle systems in NE steady states (34, 35). There, *α* is the sum over the entire spectrum of Lyapunov exponents (not just those that are positive). It is interpreted as the (irreversible) entropy production rate, (viscous) heat production rate, and the phase space contraction rate, all per degree of freedom. The first two interpretations are similarly fitting here, given the relation of *α* to and . It is also reasonable to interpret our *α* as the phase space contraction rate of the system compensated for by the phase space expansion rate of the bath, according to Liouville’s theorem. The definition of *α* also provides a justification for the number 2 in Eq. **3**. The change in average total energy of the system is divided by the mean total energy for particles from the equipartition theorem because of our choice of initial phase points selected from the isothermal–isobaric ensemble.

It is emphasized that each trajectory in the ensemble sample evolved under Hamiltonian dynamics, requiring no a priori assumption of thermostatting constraints or NE fluxes, as is commonly done in atomistic simulations of NE processes (34, 35). Despite conserving total energy, energy in the system is irreversibly dissipated to the explicit bath as heat because of the choice of initial NE, disassembled states. The system particles diffuse randomly through the bath, until individual assembly events cause energy fluctuations and a dissipative response in the bath. This picture of self-assembly raises an interesting connection. In Langevin’s theory of Brownian motion, the parameters *β* and *γ* characterize the fluctuation, dissipation, and relaxation to equilibrium. In our work, their product *βγ* fixes the relationship between the dynamical instability of the self-assembling particles and the energy they dissipate. Given previous measurements of KS entropies for Brownian particles (36), experiments should be able to confirm the validity of Eq. **3** in NE self-assembly processes.

## Conclusions

Our results provide direct evidence for the relationship between the system dynamical entropy and the energy dissipated from the NE system to a bath. In analogy with equilibrium statistical mechanics, the role of the bath in the NE process is to act as an energy sink. As at equilibrium, the temperature determines the energy scale, whereas the intrinsic relaxation of the bath, measured by , characterizes the relevant time scale. Because the relationship in Eq. **3** consists of an entropy and an energy, there may be an underlying variational principle for a dynamical “free energy.” In this context, one difficulty with our simulations results, and many other simulations of NE processes, is that to obtain the necessary KS entropy statistics we need to calculate the KS entropy over finite simulation time steps. It is tempting to assume that Eq. **3** is valid in the limit of an infinitesimal time step and thus it would be valid for the time derivatives. If that were the case, then the variational free energy would be a constant of the motion for NE self-assembly processes.

## Methods

We performed simulations of a 3D, periodic 6–12 LJ fluid with shifted and truncated forces (37). Conventional LJ reduced units were used, and . The mass of all particles *m* was set to 1.0. An ensemble of constant energy trajectories was generated by sampling transitions between two Hamiltonians [e.g., and ]. Their initial phase points were sampled from a trajectory consistent with the isothermal–isobaric ensemble with the temperature *T** = 1.0 and pressure *P** = 1.0. From each initial phase point, the potential parameter was instantaneously switched to that of the second Hamiltonian. Constant energy trajectories were then propagated forward in time with the velocity Verlet algorithm in steps of size *t** = 10^{−3}.

A Verlet list with a cutoff of *r** = 2.5 and a “skin” of *r** = 0.1 was updated every 10 time steps. Translational center-of-mass motion was removed every 10 time steps during isothermal–isobaric sampling. Combination rules for interaction parameters were given by the minimum [i.e., ]. In practice, the switching procedure was performed by cycling between Hamiltonian and non-Hamiltonian dynamics. Non-Hamiltonian equilibration trajectory segments were 25,000 time steps long. The Berendsen thermostat and barostat maintained constant temperature and pressure; both used a time constant of . Each Hamiltonian, constant-energy NE trajectory segment was 75,000 time steps long. Phase space volumes during these segments were then preserved such that the dissipation was explicitly retained in the degrees of freedom of the total system. During each constant energy trajectory, tangent space dynamics were also simulated, which limited the system size that could be studied because of their high computational cost.

For each constant energy trajectory, the KS entropies were calculated for each time step from the associated set of Lyapunov exponents. System, bath, and total system exponents were computed from the same set of orthogonal Lyapunov basis vectors (Gram–Schmidt vectors) (29, 38). This tangent space basis is propagated and comoves with the phase point. Propagation orients the first vector parallel to the maximally changing tangent space direction. Propagation with periodic orthogonalization orients the remaining vectors in the maximally changing direction of a substance of tangent space and prevents their collapse onto the maximally changing direction.

The set of Gram–Schmidt vectors was propagated forward in time with the linearized velocity Verlet algorithm, a time step size of 10^{−3}, and orthonormalization at every step. This algorithm is described in detail elsewhere (30, 31). Construction of the linearized velocity Verlet form of the tangent space propagator required second derivative (Hessian) matrices at consecutive steps, which used forward differences of the analytical gradients with a displacement of *r** = 10^{−8}. Initial basis sets were arbitrarily defined as randomly filled, orthonormal matrices at the initial phase point. All calculations were performed in double precision.

## Acknowledgments

This material is based upon work supported as part of the Non-Equilibrium Energy Research Center, an Energy Frontier Research Center funded by the US Department of Energy, Office of Science, Office of Basic Energy Sciences under Award DE-SC0000989.

## Footnotes

- ↵
^{1}To whom correspondence should be addressed. E-mail: igalsz{at}northwestern.edu.

Author contributions: J.R.G., A.B.C., B.A.G., and I.S. designed research; J.R.G. and A.B.C. performed research; J.R.G., A.B.C., and I.S. analyzed data; and J.R.G., A.B.C., B.A.G., and I.S. wrote the paper.

The authors declare no conflict of interest.

↵

^{†}This Direct Submission article had a prearranged editor.This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1312165110/-/DCSupplemental.

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