Measurement of nonequilibrium entropy from space-time thermodynamic integration

The entropy of a system transiently driven out of equilibrium by a time-inhomogeneous stochastic dynamics is first expressed as a transient response function generalizing the nonlinear Kawasaki-Crooks response. This function is then reformulated into three statistical averages defined over ensembles of nonequilibrium trajectories. The first average corresponds to a space-time thermodynamic perturbation relation, while the two following ones correspond to space-time thermodynamic integration relations. Provided that trajectories are initiated starting from a distribution of states that is analytically known, the ensemble averages are computationally amenable to Markov chain Monte Carlo methods. The relevance of importance sampling in path ensembles is confirmed in practice by computing the nonequilibrium entropy of a driven toy system. We finally study a situation where the dynamics produces entropy. In this case, we observe that space-time thermodynamic integration still yields converged estimates, while space-time thermodynamic perturbation turns out to converge very slowly.     

L-fuzzy ⋎ and ⋏ hyperoperations and the associated l-fuzzy hyperalgebras

In this paper we study two fuzzy hyperoperations, denoted by ⋎ (which can be seen as a generalization of ∨) and ⋏ (which can be seen as a generalization of ∧). ⋎ is obtained from a family of crisp ∨; _p hyperoperations and ⋏ is obtained from a family of crisp ∧ _p hyperoperations. The hyperstructure (X, ⋎, ∧) resembles ahyperlattice and the hyperstructure (X, ∨, ⋏) resembles adual hyperlattice     

Low-Dimensional Architectures in Isomeric cis-PtCl2{Ph2PCH2N(Ar)CH2PPh2} Complexes Using Regioselective-N(Aryl)-Group Manipulation

The solid-state behaviour of two series of isomeric, phenol-substituted, aminomethylphosphines, as the free ligands and bound to Pt^II, have been extensively studied using single crystal X-ray crystallography. In the first library, isomeric diphosphines of the type Ph₂PCH₂N(Ar)CH₂PPh₂ [1a–e; Ar = C₆H₃(Me)(OH)] and, in the second library, amide-functionalised, isomeric ligands Ph₂PCH₂N{CH₂C(O)NH(Ar)}CH₂PPh₂ [2a–e; Ar = C₆H₃(Me)(OH)], were synthesised by reaction of Ph₂PCH₂OH and the appropriate amine in CH₃OH, and isolated as colourless solids or oils in good yield. The non-methyl, substituted diphosphines Ph₂PCH₂N{CH₂C(O)NH(Ar)}CH₂PPh₂ [2f, Ar = 3-C₆H₄(OH); 2g, Ar = 4-C₆H₄(OH)] and Ph₂PCH₂N(Ar)CH₂PPh₂ [3, Ar = 3-C₆H₄(OH)] were also prepared for comparative purposes. Reactions of 1a–e, 2a–g, or 3 with PtCl₂(η⁴-cod) afforded the corresponding square-planar complexes 4a–e, 5a–g, and 6 in good to high isolated yields. All new compounds were characterised using a range of spectroscopic (¹H, ³¹P{¹H}, FT–IR) and analytical techniques. Single crystal X-ray structures have been determined for 1a, 1b∙CH₃OH, 2f∙CH₃OH, 2g, 3, 4b∙(CH₃)₂SO, 4c∙CHCl₃, 4d∙½Et₂O, 4e∙½CHCl₃∙½CH₃OH, 5a∙½Et₂O, 5b, 5c∙¼H₂O, 5d∙Et₂O, and 6∙(CH₃)₂SO. The free phenolic group in 1b∙CH₃OH, 2f∙CH₃OH_, 2g, 4b∙(CH₃)₂SO, 5a∙½Et₂O, 5c∙¼H₂O, and 6∙(CH₃)₂SO exhibits various intra- or intermolecular O–H∙∙∙X (X = O, N, P, Cl) hydrogen contacts leading to different packing arrangements.     

Accessibility and Reliability for Connected Bipartite Graphs

In the recently published atlas of graphs [9] the general listing of graphs with diagrams went up to V=7 vertices but the special listing for connected bipartite graphs carried further up to V=8. In this paper we wish to study the random accessibility of these connected bipartite graphs by means of random walks on the graphs using the degree of the gratis starting point as a weighting factor. The accessibility is then related to the concept of reliability of the graphs with only edge failures. Exact expectation results for accessibility are given for any complete connected bipartite graph N₁ cbp N₂ (where cbp denotes connected bipartite) for several values of J (the number of new vertices searched for). The main conjecture in this paper is that for any complete connected bipartite graph N₁ cbp N₂: if |N₁–N₂| 1, then the graph is both uniformly optimal in reliability and optimal in random accessibility within its family. Numerical results are provided to support the conjecture.     

Simulating structural transitions by direct transition current sampling: the example of LJ38

Reaction paths and probabilities are inferred, in a usual Monte Carlo or molecular dynamic simulation, directly from the evolution of the positions of the particles. The process becomes time-consuming in many interesting cases in which the transition probabilities are small. A radically different approach consists of setting up a computation scheme where the object whose time evolution is simulated is the transition current itself. The relevant timescale for such a computation is the one needed for the transition probability rate to reach a stationary level, and this is usually substantially shorter than the passage time of an individual system. As an example, we show, in the context of the "benchmark" case of 38 particles interacting via the Lennard-Jones potential ("LJ(38)" cluster), how this method may be used to explore the reactions that take place between different phases, recovering efficiently known results, and uncovering new ones with small computational effort.     

Web ensemble averages for retrieving relevant information from rejected Monte Carlo moves

We study the relevance of including information about rejected Monte-Carlo moves in path-sampling computations of free energies. For this purpose, we define webs as sets of paths linked by the path-sampling scheme and introduce an associated statistical ensemble. Within this web ensemble, we derive and test several statistical averages enabling to include information about configurational and path quantities belonging to the unselected trial moves. We numerically observe that retrieving this information does not always result in variance reduction, as theoretically predicted by Delmas and Jourdain. To explain the possible detrimental effect of information-retrieving from web sampling, an action for the webs is introduced. The behaviour of the statistical variance is observed to correlate to an overlapping area of a web action histogram. This area represents the probability that a generated web is such that the difference of its action between the targeted and reference ensembles is lower than the corresponding difference of free energy. Variance reductions are numerically observed for increased areas, as it is the case for the residence weight method proposed previously. More generally, web ensembles yield a rigorous framework for rationalizing existing methods and also for deriving potentially new methods enabling to retrieve relevant information from rejected trial moves.     

Gibbs free-energy estimates from direct path-sampling computations

We have implemented a path-sampling scheme enabling a direct estimation of Gibbs free energy. This scheme consists of a Monte Carlo sampling of constant-pressure Langevin paths, followed by an ensemble averaging carried out over the Markov chain of paths. In practice, we sample an umbrella path ensemble, which requires to rigorously define a statistical weight for the paths, equivalent of the Boltzmann weight. This statistical weight is a function of an effective work related to the path. The umbrella ensemble is chosen so that its work histogram overlaps with the histograms corresponding to the ensembles of forward and backward paths. We have finally investigated the relations between numerical efficiency and overlapping properties of the various work histograms. This analysis yields a built-in criterion for diagnosing the convergence during a single-run simulation.     

Waste-recycling Monte Carlo with optimal estimates: application to free energy calculations in alloys

The estimator proposed recently by Delmas and Jourdain for waste-recycling Monte Carlo achieves variance reduction optimally with respect to a control variate that is evaluated directly using the simulation data. Here, the performance of this estimator is assessed numerically for free energy calculations in generic binary alloys and is compared to those of other estimators taken from the literature. A systematic investigation with varying simulation parameters of a simplified system, the anti-ferromagnetic Ising model, is first carried out in the transmutation ensemble using path-sampling. We observe numerically that (i) the variance of the Delmas-Jourdain estimator is indeed reduced compared to that of other estimators; and that (ii) the resulting reduction is close to the maximal possible one, despite the inaccuracy in the estimated control variate. More extensive path-sampling simulations involving an FeCr alloy system described by a many-body potential additionally show that (iii) gradual transmutations accommodate the atomic frustrations; thus, alleviating the numerical ergodicity issue present in numerous alloy systems and eventually enabling the determination of phase coexistence conditions.     

A note on the congruences of the nakano superlattice and some properties of the associated quotients

In this paper we explore theNakano superlattice (H, ⊔, ⊓), where ⊔, ⊓ are the Nakanohyperoperations x⊔y={z:x∨z=y∨z=x∨y},x⊓y={z:x∧z=y∧z=x∧y}. In particular, we study the properties of congruences on the Nakano superlattice and the associated quotients. New hyperoperations are introduced on the quotient and their properties studied.