James Reid

Communication: Beyond Boltzmann's H-theorem: Demonstration of the relaxation theorem for a non-monotonic approach to equilibrium

Relaxation of a system to equilibrium is as ubiquitous, essential, and as poorly quantified as any phenomena in physics. For over a century, the most precise description of relaxation has been Boltzmanns H-theorem, predicting that a uniform ideal gas will relax monotonically. Recently, the relaxation theorem has shown that the approach to equilibrium can be quantified in terms of the dissipation function first defined in the proof of the Evans-Searles fluctuation theorem. Here, we provide the first demonstration of the relaxation theorem through simulation of a simple fluid system that generates a non-monotonic relaxation to equilibrium.

Demonstration of the steady-state fluctuation theorem from a single trajectory

The fluctuation theorem (FT) quantifies the probability of Second Law of Thermodynamics violations in small systems over short timescales. While this theorem has been experimentally demonstrated for systems that are perturbed from an initial equilibrium state, there are a number of studies suggesting that th et heorem applies asymptotically in the long time limit to systems in a nonequilibrium steady state. The asymptotic application of the FT to such nonequilibrium steady-states has been referred to in the literature as the steady-state fluctuation theorem (or SSFT). In 2005 Wang et aldemonstrated experimentally an integrated form of the SSFT using a colloidal bead that was weakly held in ac ircularly translating optical trap. Moreover, they showed that the integrated form of the FT may, for certain systems, hold under non-equilibrium steady states for all time, and not just in the long time limit, as suggested by the SSFT. While demonstration of the integrated forms of these theorems is compact and illustrative, a proper demonstration shows the theorem directly ,r ather than in its integrated form. In this paper, we present experimental results that demonstrate the SSFT directly ,a nd show thatthe FT can hold for all time under nonequilibrium steady states.

The fluctuation theorem and dissipation theorem for Poiseuille flow

The fluctuation theorem and the dissipation theorem provide relationships to describe nonequilibrium systems arbitrarily far from, or close to equilibrium. They both rely on definition of a central property, the dissipation function. In this manuscript we apply these theorems to examine a boundary thermostatted system undergoing Poiseuille flow. The relationships are verified computationally and show that the dissipation theorem is potentially useful for study of boundary thermostatted systems consisting of complex molecules undergoing flow in the nonlinear regime.

The dissipation function: its relationship to entropy production, theorems for nonequilibrium systems and observations on its extrema

In this chapter we introduce the dissipation function, and discuss the behaviour of its extrema. The dissipation function allows the reversibility of a nonequilibrium process to be quantified for systems arbitrarily close to or far from equilibrium. For a system out of equilibrium, the average dissipation over a period, t, will be positive. For field driven flow in the thermodynamic and small field limits, the dissipation function becomes proportional to the rate of entropy production from linear irreversible thermodynamics. It can therefore be considered as an entropy-like quantity that remains useful far from equilibrium and for relaxation processes. The dissipation function also appears in three important theorems in nonequilibrium statistical mechanics: the fluctuation theorem, the dissipation theorem and the relaxation theorem. In this chapter we introduce the dissipation function and the theorems, and show how they quantify the emergence of irreversible behaviour in perturbed, steady state, and relaxing nonequilibrium systems. We also examine the behaviour of the dissipation function in terms of the extrema of the function using numerical and analytical approaches.

Different approaches for evaluating exponentially weighted nonequilibrium relations

The Kawasaki identity (KI) and the Jarzynski equality (JE) are important nonequilibrium relations. Both of these relations take the form of an ensemble average of an exponential function and can exhibit convergence problems when the average of the exponent differs greatly from the log of the average of the exponential function. In this work, we re-express these relations so that only selected regions need to be evaluated in an attempt to avoid these convergence issues. In the context of measuring free energies, we compare our method to the JE and the literature standard approach, the maximum likelihood estimator (MLE), and show that in a system with asymmetric work distributions it can perform as well as the MLE. For the KI, we derive an analog to the MLE to compare with our relation and show that these two new relations improve on the KI and are complimentary to each other.

The free energy of expansion and contraction: Treatment of arbitrary systems using the Jarzynski equality

Thermodynamic integration, free energy perturbation, and slow change techniques have long been utilised in the calculation of free energy differences between two states of a system that has undergone some transformation. With the introduction of the Jarzynski equality and the Crooks relation, new approaches are possible. This paper investigates an important phenomenon - systems undergoing a change in volume/density - and derives both the Jarzynski equality and Crooks relation of such systems using a statistical mechanical approach. These results apply to systems with arbitrary particle interactions and densities. The application of this approach to the expansion/compression of particles confined within a vessel with a piston and within a periodic system is considered.

Influence of Constraints within a Cyclic Polymer on Solution Properties

Cyclic polymers with internal constraints provide new insight into polymer properties in solution and bulk and can serve as a model system to explain the stability and mobility of cyclic biomacromolecules. The model system used in this work consisted of cyclic polystyrene structures, all with a nearly identical molecular weight, designed with 0-3 constraints located at strategic sites within the cyclic polymer, with either 4 or 6 branch points. The total number of branch points (or arms) within the cyclic ranged from 0 to 18. Molecular dynamic (MD) simulations showed that as the number of arms increased within the cyclic structure, the radius of gyration and the hydrodynamic radius generally decreased, suggesting the greater number of constraints resulted in a more compact polymer chain. The simulations further showed that the excluded volume was much greater for the cyclics compared to a linear polymer at the same molecular weight. The spirocyclic, a structure consisting of three rings joined in series, showed significant excluded volume effects in agreement with experimental data; the reason for which is unclear at this stage. Interestingly, under a size exclusion chromatography flow, the radius of hydration for all the cyclic structures increased compared with the DLS data, and could be explained from the greater swelling of the rings perpendicular to the flow found from previous simulations on rings. This data suggests that the greater compactness, greater excluded volume and structural rearrangements under flow of constrained cyclic polymers could be used to provide a physical basis for understanding greater stability and activity of cyclic biological macromolecules.

Applicability of optimal protocols and the Jarzynski equality

The Jarzynski equality is a well-known and widely used identity, relating the free energy difference between two states of a system to the work done over some arbitrary, nonequilibrium transformation between the two states. Despite being valid for both stochastic and deterministic systems, we show that the optimal transformation protocol for the deterministic case seems to differ from that predicated from an analysis of the stochastic dynamics. In addition, it is shown that for certain situations, more dissipative processes can sometimes lead to better numerical results for the free energy differences.

Applying Bi-directional Jarzynski Methods to Quasi-equilibrium States

Measuring free energy differences between states is of fundamental importance to understanding and predicting the behaviour of thermodynamic systems. The Jarzynski equality provides a method for measuring free energy differences using non-equilibrium work paths and represents a major advance of modern thermodynamics. Recent work has extended the theory by using work paths in both directions between the states to improve the accuracy of the free energy measurement. It has also been shown that the Jarzynski equality can be adapted to measure the free energy of quasi-equilibrium systems such as glasses. Here we combine these advances to accurately measure the free energy difference between a glassy state and equilibrium using bi-directional methods. For this system however, the result is not as accurate as that achieved using the work evaluated in a single direction.

Free Energy Calculations with Reduced Potential Cutoff Radii

The Jarzynski Equality, the Crooks Fluctuation Theorem, and the Maximum Likelihood Estimator use a nonequilibrium approach for the determination of free energy differences due to a change in the state of a system. Here, this approach is used in combination with a novel transformation algorithm to increase computational efficiency in simulations with interacting particles, without losing accuracy. The algorithm is shown to work well for a Lennard-Jones fluid undergoing a change in density over three very different density ranges, and for the systems considered the algorithm demonstrates computational savings of up to approximately 90%. The results obtained directly from the Jarzynski Equality and from the Maximum Likelihood Estimator are also compared.