Thermodynamic Entropy as a Noether Invariant from Contact Geometry.

We use a formulation of Noether's theorem for contact Hamiltonian systems to derive a relation between the thermodynamic entropy and the Noether invariant associated with time-translational symmetry. In the particular case of thermostatted systems at equilibrium, we show that the total entropy of the system plus the reservoir are conserved as a consequence thereof. Our results contribute to understanding thermodynamic entropy from a geometric point of view.

Monte Carlo sampling in diffusive dynamical systems.

We introduce a Monte Carlo algorithm to efficiently compute transport properties of chaotic dynamical systems. Our method exploits the importance sampling technique that favors trajectories in the tail of the distribution of displacements, where deviations from a diffusive process are most prominent. We search for initial conditions using a proposal that correlates states in the Markov chain constructed via a Metropolis-Hastings algorithm. We show that our method outperforms the direct sampling method and also Metropolis-Hastings methods with alternative proposals. We test our general method through numerical simulations in 1D (box-map) and 2D (Lorentz gas) systems.

Geometric integrator for simulations in the canonical ensemble.

We introduce a geometric integrator for molecular dynamics simulations of physical systems in the canonical ensemble that preserves the invariant distribution in equations arising from the density dynamics algorithm, with any possible type of thermostat. Our integrator thus constitutes a unified framework that allows the study and comparison of different thermostats and of their influence on the equilibrium and non-equilibrium (thermo-)dynamic properties of a system. To show the validity and the generality of the integrator, we implement it with a second-order, time-reversible method and apply it to the simulation of a Lennard-Jones system with three different thermostats, obtaining good conservation of the geometrical properties and recovering the expected thermodynamic results. Moreover, to show the advantage of our geometric integrator over a non-geometric one, we compare the results with those obtained by using the non-geometric Gear integrator, which is frequently used to perform simulations in the canonical ensemble. The non-geometric integrator induces a drift in the invariant quantity, while our integrator has no such drift, thus ensuring that the system is effectively sampling the correct ensemble.

Localization properties of the sparse Barrat-Mézard trap model.

Inspired by works on the Anderson model on sparse graphs, we devise a method to analyze the localization properties of sparse systems that may be solved using cavity theory. We apply this method to study the properties of the eigenvectors of the master operator of the sparse Barrat-Mézard trap model, with an emphasis on the extended phase. As probes for localization, we consider the inverse participation ratio and the correlation volume, both dependent on the distribution of the diagonal elements of the resolvent. Our results reveal a rich and nontrivial behavior of the estimators across the spectrum of relaxation rates and an interplay between entropic and activation mechanisms of relaxation that give rise to localized modes embedded in the bulk of extended states. We characterize this route to localization and find it to be distinct from the paradigmatic Anderson model or standard random matrix systems.

Thermodynamic cost for classical counterdiabatic driving.

Motivated by the recent growing interest about the thermodynamic cost of shortcuts to adiabaticity, we consider the cost of driving a classical system by the so-called counterdiabatic driving (CD). To do so, we proceed in three steps: first we review a general definition recently put forward in the literature for the thermodynamic cost of driving a Hamiltonian system; then we provide a new complementary definition of cost, which is of particular relevance for cases where the average excess work vanishes; finally, we apply our general framework to the case of CD. Interestingly, we find that in such a case our results are the exact classical counterparts of those reported by Funo et al. [Phys. Rev. Lett. 118, 100602 (2017)PRLTAO0031-900710.1103/PhysRevLett.118.100602]. In particular we show that a universal trade-off between speed and cost for CD also exists in the classical case. To illustrate our points we consider the example of a time-dependent harmonic oscillator subject to different strategies of adiabatic control.