The Glass Transition and the Jarzynski Equality

A simple model featuring a double well potential is used to represent a liquid that is quenched from an ergodic state into a history dependent glassy state. Issues surrounding the application of the Jarzynski Equality to glass formation are investigated. We demonstrate that the Jarzynski Equality gives the free energy difference between the initial state and the state we would obtain if the glass relaxed to true thermodynamic equilibrium. We derive new variations of the Jarzynski Equality which are relevant to the history dependent glassy state rather than the underlying equilibrium state. It is shown how to compute the free energy differences for the nonequilibrium history dependent glassy state such that it remains consistent with the standard expression for the entropy and with the second law inequality.Comment: 16 pages, 5 figure

Note on the Kaplan-Yorke dimension and linear transport coefficients

A number of relations between the Kaplan-Yorke dimension, phase space contraction, transport coefficients and the maximal Lyapunov exponents are given for dissipative thermostatted systems, subject to a small external field in a nonequilibrium stationary state. A condition for the extensivity of phase space dimension reduction is given. A new expression for the transport coefficients in terms of the Kaplan-Yorke dimension is derived. Alternatively, the Kaplan-Yorke dimension for a dissipative macroscopic system can be expressed in terms of the transport coefficients of the system. The agreement with computer simulations for an atomic fluid at small shear rates is very good.Comment: 12 pages, 5 figures, submitted to J. Stat. Phy

The Steady State Fluctuation Relation for the Dissipation Function

We give a proof of transient fluctuation relations for the entropy production (dissipation function) in nonequilibrium systems, which is valid for most time reversible dynamics. We then consider the conditions under which a transient fluctuation relation yields a steady state fluctuation relation for driven nonequilibrium systems whose transients relax, producing a unique nonequilibrium steady state. Although the necessary and sufficient conditions for the production of a unique nonequilibrium steady state are unknown, if such a steady state exists, the generation of the steady state fluctuation relation from the transient relation is shown to be very general. It is essentially a consequence of time reversibility and of a form of decay of correlations in the dissipation, which is needed also for, e.g., the existence of transport coefficients. Because of this generality the resulting steady state fluctuation relation has the same degree of robustness as do equilibrium thermodynamic equalities. The steady state fluctuation relation for the dissipation stands in contrast with the one for the phase space compression factor, whose convergence is problematic, for systems close to equilibrium. We examine some model dynamics that have been considered previously, and show how they are described in the context of this work.Comment: 30 pages, 1 figur

Wall mediated transport in confined spaces: Exact theory for low density

We present a theory for the transport of molecules adsorbed in slit and cylindrical nanopores at low density, considering the axial momentum gain of molecules oscillating between diffuse wall reflections. Good agreement with molecular dynamics simulations is obtained over a wide range of pore sizes, including the regime of single-file diffusion where fluid-fluid interactions are shown to have a negligible effect on the collective transport coefficient. We show that dispersive fluid-wall interactions considerably attenuate transport compared to classical hard sphere theory

Stationary and Transient Work-Fluctuation Theorems for a Dragged Brownian Particle

Recently Wang et al. carried out a laboratory experiment, where a Brownian particle was dragged through a fluid by a harmonic force with constant velocity of its center. This experiment confirmed a theoretically predicted work related integrated (I) Transient Fluctuation Theorem (ITFT), which gives an expression for the ratio for the probability to find positive or negative values for the fluctuations of the total work done on the system in a given time in a transient state. The corresponding integrated stationary state fluctuation theorem (ISSFT) was not observed. Using an overdamped Langevin equation and an arbitrary motion for the center of the harmonic force, all quantities of interest for these theorems and the corresponding non-integrated ones (TFT and SSFT, resp.) are theoretically explicitly obtained in this paper. While the (I)TFT is satisfied for all times, the (I)SSFT only holds asymptotically in time. Suggestions for further experiments with arbitrary velocity of the harmonic force and in which also the ISSFT could be observed, are given. In addition, a non-trivial long-time relation between the ITFT and the ISSFT was discovered, which could be observed experimentally, especially in the case of a resonant circular motion of the center of the harmonic force.Comment: 20 pages, 3 figure

On the Fluctuation Relation for Nose-Hoover Boundary Thermostated Systems

We discuss the transient and steady state fluctuation relation for a mechanical system in contact with two deterministic thermostats at different temperatures. The system is a modified Lorentz gas in which the fixed scatterers exchange energy with the gas of particles, and the thermostats are modelled by two Nos\'e-Hoover thermostats applied at the boundaries of the system. The transient fluctuation relation, which holds only for a precise choice of the initial ensemble, is verified at all times, as expected. Times longer than the mesoscopic scale, needed for local equilibrium to be settled, are required if a different initial ensemble is considered. This shows how the transient fluctuation relation asymptotically leads to the steady state relation when, as explicitly checked in our systems, the condition found in [D.J. Searles, {\em et al.}, J. Stat. Phys. 128, 1337 (2007)], for the validity of the steady state fluctuation relation, is verified. For the steady state fluctuations of the phase space contraction rate \zL and of the dissipation function \zW, a similar relaxation regime at shorter averaging times is found. The quantity \zW satisfies with good accuracy the fluctuation relation for times larger than the mesoscopic time scale; the quantity \zL appears to begin a monotonic convergence after such times. This is consistent with the fact that \zW and \zL differ by a total time derivative, and that the tails of the probability distribution function of \zL are Gaussian.Comment: Major revision. Fig.10 was added. Version to appear in Journal of Statistical Physic

Comparison of work fluctuation relations

We compare two predictions regarding the microscopic fluctuations of a system that is driven away from equilibrium: one due to Crooks [J. Stat. Phys. 90, 1481 (1998)] which has gained recent attention in the context of nonequilibrium work and fluctuation theorems, and an earlier, analogous result obtained by Bochkov and Kuzovlev [Zh. Eksp. Teor. Fiz. 72(1), 238247 (1977)]. Both results quantify irreversible behavior by comparing probabilities of observing particular microscopic trajectories during thermodynamic processes related by time-reversal, and both are expressed in terms of the work performed when driving the system away from equilibrium. By deriving these two predictions within a single, Hamiltonian framework, we clarify the precise relationship between them, and discuss how the different definitions of work used by the two sets of authors gives rise to different physical interpretations. We then obtain a extended fluctuation relation that contains both the Crooks and the Bochkov-Kuzovlev results as special cases.Comment: 14 pages with 1 figure, accepted for publication in the Journal of Statistical Mechanic

Moderated peer assessment of individual contribution to group work

UCL Engineering trains students to use engineering knowledge within extended group practical activities to prepare them for their careers after graduation. However, despite the substantial educational benefits of getting students to work in teams, providing individual assessment can be challenging. Students frequently express dissatisfaction if all members of a team are given the same mark regardless of the individual effort. Here, we aim to promote student engagement and improve student experience during group work by giving each student an individual mark. The individual mark results from multiplying the overall “group mark” by a personal contribution factor. This personal contribution is assessed directly by peers, who are aware of each team member’s contribution, encouraging self-reflection, and moderated by tutors when necessary. This practice has been well received by students in other universities. We are working with a student committee to identify and evaluate various methods and e-learning systems that would aid us to run this practice efficiently even for large numbers of students. This includes rules to flag cases requiring moderation. This project, partially funded by ELDG 2015, fits with our aim of increasing students’ satisfaction and engagement with assessment. We have combined it with our ‘360 degrees peer assessment method’, which we presented at last year’s conference, to provide a reliable and individual peer assessment of group work. We provide a novel approach to group assessment which encourages self-reflection and is intended to improve the learning experience and student satisfaction during group work, in line with UCL 2034

Experimental demonstration of violations of the second law of thermodynamics for small systems and short time scales

We experimentally demonstrate the fluctuation theorem, which predicts appreciable and measurable violations of the second law of thermodynamics for small systems over short time scales, by following the trajectory of a colloidal particle captured in an optical trap that is translated relative to surrounding water molecules. From each particle trajectory, we calculate the entropy production/consumption over the duration of the trajectory and determine the fraction of second law–defying trajectories. Our results show entropy consumption can occur over colloidal length and time scales

Thermodynamic entropy production fluctuation in a two dimensional shear flow model

We investigate fluctuations in the momentum flux across a surface perpendicular to the velocity gradient in a stationary shear flow maintained by either thermostated deterministic or by stochastic boundary conditions. In the deterministic system the Gallavotti-Cohen (GC)relation for the probability of large deviations, which holds for the phase space volume contraction giving the Gibbs ensemble entropy production, never seems to hold for the flux which gives the hydrodynamic entropy production. In the stochastic case the GC relation is found to hold for the total flux, as predicted by extensions of the GC theorem but not for the flux across part of the surface. The latter appear to satisfy a modified GC relation. Similar results are obtained for the heat flux in a steady state produced by stochastic boundaries at different temperatures.Comment: 9 postscript figure