Paolo De Gregorio

Exact solution of a jamming transition: Closed equations for a bootstrap percolation problem

Jamming, or dynamical arrest, is a transition at which many particles stop moving in a collective manner. In nature it is brought about by, for example, increasing the packing density, changing the interactions between particles, or otherwise restricting the local motion of the elements of the system. The onset of collectivity occurs because, when one particle is blocked, it may lead to the blocking of a neighbor. That particle may then block one of its neighbors, these effects propagating across some typical domain of size named the dynamical correlation length. When this length diverges, the system becomes immobile. Even where it is finite but large the dynamics is dramatically slowed. Such phenomena lead to glasses, gels, and other very long-lived nonequilibrium solids. The bootstrap percolation models are the simplest examples describing these spatio-temporal correlations. We have been able to solve one such model in two dimensions exactly, exhibiting the precise evolution of the jamming correlations on approach to arrest. We believe that the nature of these correlations and the method we devise to solve the problem are quite general. Both should be of considerable help in further developing this field.

Universality behaviour in ‘ideal’ dynamical arrest transitions of a lattice glass model

Using dynamically available volume (DAV) as an order parameter, we study the ideal dynamical arrest for some simple lattice glass models. For these models the dynamically available volume is expressed as holes, or vacant sites into which particles can move. We find that on approach to the arrest the holes, which are the only mediators of transport, become increasingly rare. Near the arrest, dynamical quantities can be expanded in a series of hole density, in which the leading term is found to quadratic, as opposed to unfrustrated systems which have a linear dependence. Dynamical quantities for the models we have studied show universal behaviour when expressed in terms of the hole density. The dynamically available volume is shown to be a useful characterisation of the slow aging in lattice glasses.

Effects of breaking vibrational energy equipartition on measurements of temperature in macroscopic oscillators subject to heat flux

When the energy content of a resonant mode of a crystalline solid in thermodynamic equilibrium is directly measured, assuming that quantum effects can be neglected it coincides with temperature except for a proportionality factor. This is due to the principle of energy equipartition and the equilibrium hypothesis. However, most natural systems found in nature are not in thermodynamic equilibrium and thus the principle cannot be taken for granted. We measured the extent to which the low-frequency modes of vibration of a solid defy energy equipartition, in the presence of a steady state heat flux, even close to equilibrium. We found, experimentally and numerically, that the energy separately associated with low-frequency normal modes depends strongly on the heat flux, and decouples noticeably from temperature. A relative temperature difference of 4% across the object around room temperature suffices to excite two modes of a macroscopic oscillator, as if they were at equilibrium, separately, at temperatures about 20% and a factor of 3.5 higher. We interpret the result in terms of new flux-mediated correlations between modes in the nonequilibrium state which are absent at equilibrium.

Geometry of dynamically available empty space is the key to near-arrest dynamics

We study several examples of kinetically constrained lattice models using dynamically accessible volume as an order parameter. Thereby we identify two distinct regimes exhibiting dynamical slowing, with a sharp threshold between them. These regimes are identified both by a new response function in dynamically available volume, as well as directly in the dynamics. Results for the self-diffusion constant in terms of the connected hole density are presented, and some evidence is given for scaling in the limit of dynamical arrest.

Steady state fluctuation relations and time reversibility for non-smooth chaotic maps

Steady state fluctuation relations for dynamical systems ar e commonly derived under the assumption of some form of time-reversibility and of chaos. There are, however, cases in which they are observed to hold even if the usual notion of time reversal invariance is violated, e.g. for local fluctuations of Navier-Stokes sy stems. Here we construct and study analytically a simple non-smooth map in which the standard steady state fluctuation relation is valid, although the model violates the Anosov property of chaotic dynamical systems. Particularly, the time reversal operation is performed by a discontinuous involution, and the invariant measure is also discontinuous along the unstable manifolds. This further indicates that the validity of fluctuation relations for dynamical sys tems does not rely on particularly elaborate conditions, usually violated by systems of inter est in physics. Indeed, even an irreversible map is proved to verify the steady state fluctua tion relation.

The geometry of empty space is the key to arresting dynamics

We present the concept of dynamically available volume as a suitable order parameter for dynamical arrest. We show that dynamical arrest can be understood as a de-percolation transition of a vacancy network or available space. Beyond the arrest transition we find that droplets of available space are disconnected and the dynamics is frozen. This connection of the dynamics to the underlying geometrical structure of empty space provides us with a rich framework for studying the arrest transition.

Long tail correlations between hydrophobic solutes in a model solvent

Recently, a one-dimensional model with nearest-neighbour square-well interactions has been studied, in which parameters were chosen to mimic some of the thermodynamic properties of solutions of non-polar solutes in water. Here we investigate the properties of the pair distribution functions, comparing correlations between solvent molecules with those between solute molecules in the limit of infinite dilution. Under some appropriate thermodynamic conditions, both contain the same underlying–exponentially monotonic–decay, with a decay length that is characteristically long-tailed. But while this length is barely of interest for the pure solvent by virtue of the minuteness of the correlations at such long distances, and is almost undetectable for all practical purposes, it appears dramatically in the case of solute–solute correlations, becoming visible when one can access large length separations. The implications of this unexpected feature are discussed, also in relation to the mean force between solute molecules and the osmotic second virial coefficient for the solute. The case of square-well interactions is discussed at some length, and references are made to analogous results for the case of a triangular-well interaction potential.

Cellular automata with rare events; Resolution of an outstanding problem in the bootstrap percolation model

We study the modified bootstrap model as an example of a cellular automaton that is dominated by the physics of rare events. For this reason, the characteristic laws are quite different from the simple power laws often found in well known critical phenomena. Those laws are quite delicate, and are driven by co-operative events at diverging length and time scales. To deal with such problems we introduce a new importance-sampling procedure in simulation, based on rare events around “holes”. This enables us to access bootstrap lengths beyond those previously studied. By studying the paths or processes that lead to emptying of the lattice we are able to develop systematic corrections to the theory, and compare them to simulations. Thereby, for the first time in the literature, it is possible to obtain credible comparisons between theory and simulation in the accessible density range.

Quantum Correlations under Time Reversal and Incomplete Parity Transformations in the Presence of a Constant Magnetic Field

We derive the quantum analogues of some recently discovered symmetry relations for time correlation functions in systems subject to a constant magnetic field. The symmetry relations deal with the effect of time reversal and do not require—as in the formulations of Casimir and Kubo—that the magnetic field be reversed. It has been anticipated that the same symmetry relations hold for quantum systems. Thus, here we explicitly construct the required symmetry transformations, acting upon the relevant quantum operators, which conserve the Hamiltonian of a system of many interacting spinless particles, under time reversal. Differently from the classical case, parity transformations always reverse the sign of both the coordinates and of the momenta, while time reversal only of the latter. By implementing time reversal in conjunction with ad hoc “incomplete” parity transformations (i.e., transformations that act upon only some of the spatial directions), it is nevertheless possible to achieve the construction of the quantum analogues of the classical maps. The proof that the mentioned symmetry relations apply straightforwardly to quantal time correlation functions is outlined.