Statistical Mechanics of Time Independent Non-Dissipative Nonequilibrium States

We examine the question of whether the formal expressions of equilibrium statistical mechanics can be applied to time independent non-dissipative systems that are not in true thermodynamic equilibrium and are nonergodic. By assuming the phase space may be divided into time independent, locally ergodic domains, we argue that within such domains the relative probabilities of microstates are given by the standard Boltzmann weights. In contrast to previous energy landscape treatments, that have been developed specifically for the glass transition, we do not impose an a priori knowledge of the inter-domain population distribution. Assuming that these domains are robust with respect to small changes in thermodynamic state variables we derive a variety of fluctuation formulae for these systems. We verify our theoretical results using molecular dynamics simulations on a model glass forming system. Non-equilibrium Transient Fluctuation Relations are derived for the fluctuations resulting from a sudden finite change to the system's temperature or pressure and these are shown to be consistent with the simulation results. The necessary and sufficient conditions for these relations to be valid are that the domains are internally populated by Boltzmann statistics and that the domains are robust. The Transient Fluctuation Relations thus provide an independent quantitative justification for the assumptions used in our statistical mechanical treatment of these systems.Comment: 17 pages, 4 figures, minor amendment

The rheology of solid glass

As the glass transition is approached from the high temperature side, viewed as a liquid, the properties of the ever more viscous supercooled liquid are continuous functions of temperature and pressure. The point at which we decide to classify the fluid as a solid is therefore subjective. This subjective decision does, however, have discontinuous consequences for how we determine the rheological properties of the glass. We apply the recently discovered relaxation theorem to the time independent, nondissipative, nonergodic glassy state to derive an expression for the phase space distribution of an ensemble of glass samples. This distribution is then used to construct a time dependent linear response theory for aged glassysolids. The theory is verified using molecular dynamics simulations of oscillatory shear for a realistic model glass former with excellent agreement being obtained between the response theory calculations and direct nonequilibrium molecular dynamics calculations. Our numerical results confirm that unlike all the fluid states, including supercooled liquids, a solidglass (in common with crystalline states) has a nonzero value for the zero frequency shear modulus. Of all the states of matter, a supercooled fluid approaching the glass transition has the highest value for the limiting zero frequency shear viscosity. Finally, solidglasses like dilute gases and crystals have a positive temperature coefficient for the shear viscosity whereas supercooled and normal liquids have a negative temperature coefficient.We thank the National Computational Infrastructure NCI for computational facilities and the Australian Research Council ARC for funding

Verification of time-reversibility requirementfor systems satisfying the Evans-Searles fluctuation theorem

The Evans-Searles fluctuation theorem (ESFT) has been shown to be applicable in the near- and far-from-equilibrium regimes for systems with both constant and time-dependent external fields. The derivations of the ESFT have assumed that the external field has a definite parity under a time-reversal mapping. In the present paper, we confirm that the time-reversibility of the system dynamics is a necessary condition for the ESFT to hold. The manner in which the ESFT fails for systems that are not time-reversible is presented, and results are shown which demonstrate that systems which fail to satisfy the ESFT may still satisfy the Crooks relation (CR)

On the relaxation to nonequilibrium steady states

The issue of relaxation has been addressed in terms of ergodic theory in the past. However, the application of that theory to models of physical interest is problematic, especially when dealing with relaxation to nonequilibrium steady states. Here, we consider the relaxation of classical, thermostatted particle systems to equilibrium as well as to nonequilibrium steady states, using dynamical notions including decay of correlations. We show that the condition known as {\Omega}T-mixing is necessary and sufficient to prove relaxation of ensemble averages to steady state values. We then observe that the condition known as weak T-mixing applied to smooth observables is sufficient for relaxation to be independent of the initial ensemble. Lastly, weak T-mixing for integrable functions makes relaxation independent of the ensemble member, apart from a negligible set of members enabling the result to be applied to observations from a single physical experiment. The results also allow us to give a microscopic derivation of Prigogine's principle of minimum entropy production in the linear response regime. The key to deriving these results lies in shifting the discussion from characteristics of dynamical systems, such as those related to metric transitivity, to physical measurements and to the behaviour of observables. This naturally leads to the notion of physical ergodicity.Comment: 44 pages, 1 figur

On Optimal Binary One-Error-Correcting Codes of Lengths 2m−42^m-4 and 2m−32^m-3

Best and Brouwer [Discrete Math. 17 (1977), 235-245] proved that triply-shortened and doubly-shortened binary Hamming codes (which have length $2^m-4$ and $2^m-3$, respectively) are optimal. Properties of such codes are here studied, determining among other things parameters of certain subcodes. A utilization of these properties makes a computer-aided classification of the optimal binary one-error-correcting codes of lengths 12 and 13 possible; there are 237610 and 117823 such codes, respectively (with 27375 and 17513 inequivalent extensions). This completes the classification of optimal binary one-error-correcting codes for all lengths up to 15. Some properties of the classified codes are further investigated. Finally, it is proved that for any $m \geq 4$, there are optimal binary one-error-correcting codes of length $2^m-4$ and $2^m-3$ that cannot be lengthened to perfect codes of length $2^m-1$.Comment: Accepted for publication in IEEE Transactions on Information Theory. Data available at http://www.iki.fi/opottone/code

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

Fatigue delamination behaviour of unidirectional carbon fibre/epoxy laminates reinforced by Z-Fiber® pinnin

-Pin reinforced carbon-fibre epoxy laminates were tested under Mode I and Mode II conditions, both quasi-statically and in fatigue. Test procedures were adapted from existing standard or pre-standard tests. Samples containing 2% and 4% areal densities of carbon-fibre Z-pins (0.28mm diameter) were compared with unpinned laminates. Quasi-static tests under displacement control yielded a dramatic increase of the apparent delamination resistance. Specimens with 2% pin density failed in Mode I at loads 170N, equivalent to an apparent GIC of 2kJ/m2. Fatigue testing under load control showed that the presence of the through- thickness reinforcement slowed down fatigue delamination propagation

A Way to Dynamically Overcome the Cosmological Constant Problem

The Cosmological Constant problem can be solved once we require that the full standard Einstein Hilbert lagrangian, gravity plus matter, is multiplied by a total derivative. We analyze such a picture writing the total derivative as the covariant gradient of a new vector field (b_mu). The dynamics of this b_mu field can play a key role in the explanation of the present cosmological acceleration of the Universe.Comment: 5 page