Multidimensional integration through Markovian sampling under steered function morphing: a physical guise from statistical mechanics

We present a computational strategy for the evaluation of multidimensional integrals on hyper-rectangles based on Markovian stochastic exploration of the integration domain while the integrand is being morphed by starting from an initial appropriate profile. Thanks to an abstract reformulation of Jarzynski's equality applied in stochastic thermodynamics to evaluate the free-energy profiles along selected reaction coordinates via non-equilibrium transformations, it is possible to cast the original integral into the exponential average of the distribution of the pseudo-work (that we may term "computational work") involved in doing the function morphing, which is straightforwardly solved. Several tests illustrate the basic implementation of the idea, and show its performance in terms of computational time, accuracy and precision. The formulation for integrand functions with zeros and possible sign changes is also presented. It will be stressed that our usage of Jarzynski's equality shares similarities with a practice already known in statistics as Annealed Importance Sampling (AIS), when applied to computation of the normalizing constants of distributions. In a sense, here we dress the AIS with its "physical" counterpart borrowed from statistical mechanics.Comment: 3 figures Supplementary Material (pdf file named "JEMDI_SI.pdf"

Attracting subspaces in a hyper-spherical representation of autonomous dynamical systems

In this work, we focus on the possibility to recast the ordinary differential equations (ODEs) governing the evolution of deterministic autonomous dynamical systems (conservative or damped and generally non-linear) into a parameter-free universal format. We term such a representation \u201chyper-spherical\u201d since the new variables are a \u201cradial\u201d norm having physical units of inverse-of-time and a normalized \u201cstate vector\u201d with (possibly complex-valued) dimensionless components. Here we prove that while the system evolves in its physical space, the mirrored evolution in the hyper-spherical space is such that the state vector moves monotonically towards fixed \u201cattracting subspaces\u201d (one at a time). Correspondingly, the physical space can be split into \u201cattractiveness regions.\u201d We present the general concepts and provide an example of how such a transformation of ODEs can be achieved for a class of mechanical-like systems where the physical variables are a set of configurational degrees of freedom and the associated velocities in a phase-space representation. A one-dimensional case model (motion in a bi-stable potential) is adopted to illustrate the procedure

A Low-Computational-Cost Strategy to Localize Points in the Slow Manifold Proximity for Isothermal Chemical Kinetics

Dimensionality reduction for the modeling of reacting chemical systems can represent a fundamental achievement both for a clear understanding of the complex mechanisms under study and also for the practical calculation of quantities of interest. To tackle the problem, different approaches have been proposed in the literature. Among them, particular attention has been devoted to the exploitation of the so-called slow manifolds (SMs). These are lower dimensional hypersurfaces where the slow part of the evolution takes place. In this study, we present a low-computational-cost algorithm (based on a previously developed theoretical framework) for the localization of candidate points in the proximity of the SM. A parallel implementation (called DRIMAK) of such an approach has been developed, and the source code is made freely available. We tested the performance of the code on two model schemes for hydrogen combustion, being able to localize points that fall very close to the perceived SM with limited computational effort. The method can provide starting points for other more accurate but computationally demanding strategies; this can be a great help especially when no information about the SM is available a priori, and very many species are involved in the reaction mechanism

Remarks on the chemical Fokker-Planck and Langevin equations: Nonphysical currents at equilibrium

The chemical Langevin equation and the associated chemical Fokker-Planck equation are wellknown continuous approximations of the discrete stochastic evolution of reaction networks. In this work, we show that these approximations suffer from a physical inconsistency, namely, the presence of nonphysical probability currents at the thermal equilibrium even for closed and fully detailedbalanced kinetic schemes. An illustration is given for a model case

"L'emergere del secondo Principio della Termodinamica alla (nano)scala dei sistemi fluttuanti" - Capitolo 4 in "La freccia del tempo - Un ciclo di conferenze sui recenti sviluppi della termodinamica" (Brescia, 10, 12, 17, 19 aprile 2018)

La termodinamica alla nanoscala deve necessariamente tenere conto delle fluttuazioni strutturali alle quali un sistema \ue8 soggetto nel corso di una trasformazione esternamente guidata. A tale scala la nuova termodinamica, denominata `stocastica', sorge dall'unione tra la termodinamica statistica e la teoria dei processi stocastici. Con riferimento al caso di nanosistemi trasformati rimanendo a contatto con un mezzo fluido che funge da termostato (ad esempio singole macromolecole in fase liquida meccanicamente manipolate), mostreremo che il Secondo Principio nella forma della disuguaglianza di Clausius \ue8 valido solo in media, mentre pu\uf2 essere occasionalmente violato in singole realizzazioni. L'aspetto rilevante \ue8 che il Secondo Principio, un'evidenza empirica alla macroscala, emerge come corollario nell'ambito pi\uf9 ampio della termodinamica stocastica

Dissipation, lag, and drift in driven fluctuating systems

This work deals with thermostated fluctuating systems subjected to driven transformations of the internal energetics. The main focus is on generally multidimensional systems with continuous configurational degrees of freedom over which overdamped Markovian fluctuations take place (diffusive regime of the motion). Mutual bounds are established between the average energy dissipation, the deviation between nonequilibrium probability density and underlying equilibrium distribution due to the system\u2019s lag, and the statistical properties of the components of the directed flow induced by the transformation itself. The directed flow is here expressed in terms of time-dependent \u201cdrift velocity\u201d associated with the probability current in a advection-like formulation of the nonstationary Fokker-Planck equation. Consideration of the drift makes that the bounds achieved here extend the inequality derived by Vaikuntanathan and Jarzynski [Europhys. Lett. 87, 60005 (2009)] involving only dissipation and lag. The key relations are then specified for the so-called stochastic pumps, i.e., systems that reach a periodic steady state in response of cyclic transformations and that are prototypes of nonautonomous dissipative converters of input energy into directed motion; a one-dimensional case model is adopted to illustrate the main features. Complementary results concerning bounds between the evolution rates of dissipation and lag, valid for both overdamped and underdamped dynamics, are also presented

Probability inequalities for direct and inverse dynamical outputs in driven fluctuating systems

When a fluctuating system is subjected to a time-dependent drive or nonconservative forces, the direct-inverse symmetry of the dynamics can be broken so inducing an average bias. Here we start from the fluctuation theorem, a cornerstone of stochastic thermodynamics, for inspecting the unbalancing between direct and inverse dynamical outputs, here called "events," in a bidirectional forward-backward setup. The occurrence of an event might correspond to the realization of a quantitative output, or to the realization of a sequence of acts that compose a complex "narrative." The focus is on mutual bounds between the probabilities of occurrence of direct and inverse events in the forward and backward mode. The inspection is made for systems in contact with a thermal bath, and by assuming Markov dynamics on the uncontrolled degrees of freedom. The approach comprises both the case of systems under a time-dependent drive and time-independent external forces. The general formulation is then used to derive (or re-derive) specialized results valid for finite-time processes, and for systems taken into steady conditions (either periodic steady states or steady states) starting from equilibrium. Among the results, we find already known forms of "generalized" thermodynamic uncertainty relations, and derive useful constraints concerning the work distribution function for systems in steady conditions

Stationary Markov jump processes in terms of average transition times: setup and some inequalities of kinetic and thermodynamic kind

The parametrization of continuous-time stationary Markov jump processes is worked out in terms of average times at which the site-to-site transitions take place again (recurrence) or occur starting from a given initial localization of the system (occurrence). The foremost result is the solution of the inverse problem of achieving the rate constants from an essential set of average occurrence/recurrence times. Then we provide the expression of the average entropy production rate at the stationary state in terms of average recurrence times only, elaborate the randomness parameter (squared coefficient of variation) which quantifies the relative precision of the timing of a given transition of interest, and derive some inequalities in which only a partial amount information about the network does enter. In particular, we get lower bounds on the randomness parameter and derive inequalities of both kinetic and thermodynamic kind

Distribution and Dynamic Properties of Xenon Dissolved in the Ionic Smectic Phase of [C16mim][NO3]: MD Simulation and Theoretical Model

We have investigated the structural and dynamic properties of Xe dissolved in the ionic liquid crystal (ILC) phase of 1-hexadecyl-3-methylimidazolium nitrate using classical molecular dynamics (MD) simulations. Xe is found to be preferentially dissolved within the hydrophobic environment of the alkyl chains rather than in the ionic layers of the smectic phase. The structural parameters and the estimated local diffusion coefficients concerning the short-time motion of Xe are used to parametrize a theoretical model based on the Smoluchowski equation for the macroscopic dynamics across the smectic layers, a feature which cannot be directly obtained from the relatively short MD simulations. This protocol represents an efficient combination of computational and theoretical tools to obtain information on slow processes concerning the permeability and diffusivity of the xenon in smectic ILCs