A generalization of the cumulant expansion. Application to a scale-invariant probabilistic model

Rodríguez, Ana Alonso; Tsallis, Constantino

doi:10.1063/1.3448944

Cited by 18 publications

(14 citation statements)

References 31 publications

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“…Similarly the cumulants determine the moments. In some cases theoretical treatments of problems in terms of cumulants are simpler than those of moments [25,26]. The nth moment of a probability density function f (x) of a variable x is the mean value of x n and is mathematically defined by…”

Section: Definitions Of the Moments Central Moments And Cumulantsmentioning

confidence: 99%

Finite-volume cumulant expansion in QCD-colorless plasma

et al. 2015

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Due to the finite-size effects, the localization of the phase transition in finite systems and the determination of its order, become an extremely difficult task, even in the simplest known cases. In order to identify and locate the finite-volume transition point T 0 (V ) of the QCD deconfinement phase transition to a colorless QGP, we have developed a new approach using the finite-size cumulant expansion of the order parameter and the L mn -method. The first six cumulants C 1,2,3,4,5,6 with the corresponding under-normalized ratios (skewness , kurtosis κ, pentosis Π ± , and hexosis H 1,2,3 ) and three unnormalized combinations of them, (O = σ 2 κ −1 , U = σ −2 −1 , N = σ 2 κ) are calculated and studied as functions of (T, V ). A new approach, unifying in a clear and consistent way the definitions of cumulant ratios, is proposed. A numerical FSS analysis of the obtained results has allowed us to locate accurately the finite-volume transition point. The extracted transition temperature value T 0 (V ) agrees with that expected T N 0 (V ) from the order parameter and the thermal susceptibility χ T (T, V ), according to the standard procedure of localization to within about 2 %. In addition to this, a very good correlation factor is obtained proving the validity of our cumulants method. The agreement of our results with those obtained by means of other models is remarkable.

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Section: Definitions Of the Moments Central Moments And Cumulantsmentioning

confidence: 99%

Finite-volume cumulant expansion in QCD-colorless plasma

et al. 2015

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“…The q-generalization of the inverse Fourier transform appears then to be a quite subtle mathematical problem if q = 1. It has nevertheless been solved recently [38,39], and further considerations are coming [40] related to q-moments [41][42][43]. Work is in progress attempting to transform the existence proof in [33] into a uniqueness one.…”

Section: Probability Distributions That Are Attractors In the Sense Omentioning

confidence: 99%

The Nonadditive Entropy Sq and Its Applications in Physics and Elsewhere: Some Remarks

Tsallis

2011

Entropy

179

184

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The nonadditive entropy S q has been introduced in 1988 focusing on a generalization of Boltzmann-Gibbs (BG) statistical mechanics. The aim was to cover a (possibly wide) class of systems among those very many which violate hypothesis such as ergodicity, under which the BG theory is expected to be valid. It is now known that S q has a large applicability; more specifically speaking, even outside Hamiltonian systems and their thermodynamical approach. In the present paper we review and comment some relevant aspects of this entropy, namely (i) Additivity versus extensivity; (ii) Probability distributions that constitute attractors in the sense of Central Limit Theorems; (iii) The analysis of paradigmatic low-dimensional nonlinear dynamical systems near the edge of chaos; and (iv) The analysis of paradigmatic long-range-interacting many-body classical Hamiltonian systems. Finally, we exhibit recent as well as typical predictions, verifications and applications of these concepts in natural, artificial, and social systems, as shown through theoretical, experimental, observational and computational results.

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“…The expanded components are on the exponent and the free energy difference could be expressed as a linear combination of the cumulants. [104][105][106] The distribution of the nonequilibrium works in near-equilibrium pulling is close to Gaussian according to the central limit theory. For normally distributed data, only the first two terms in cumulant expansion survive, as higher-order cumulants are all zeros.…”

Section: Methodsmentioning

confidence: 77%

Seeding the Multi-dimensional Nonequilibrium Pulling for Hamiltonian Variation: Indirect QM/MM Free Energy Simulations

Sun

He²

2021

Preprint

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<p>The combination of free energy simulations in the alchemical and configurational spaces provides a feasible route to access the thermodynamic profiles under a computationally demanding target Hamiltonian. Normally, due to the significant differences between the computational cost of ab initio quantum mechanics (QM) calculations and those of semi-empirical quantum mechanics (SQM) and molecular mechanics (MM), this indirect method is often applied to obtain the QM thermodynamics by combining the SQM or MM thermodynamics and the SQM-to-QM or MM-to-QM corrections. In our previous works, a multi-dimensional nonequilibrium pulling framework for Hamiltonian variations has been introduced based on bidirectional pulling and bidirectional reweighting. The method performs nonequilibrium free energy simulations in the configurational space to obtain the thermodynamic profile along the conformational change pathway under a selected computationally efficient Hamiltonian, and uses the nonequilibrium alchemical method to correct or perturb the thermodynamic profile to that under the target Hamiltonian. The BAR-based method is designed to achieve the best generality and transferability and thus leads to modest (~20 fold) speedup. In this work, we explore the possibility of further accelerating the nonequilibrium free energy simulation by employing unidirectional pulling and using the selection criterion to obtain the initial configurations used to initiate nonequilibrium trajectories following the idea of adaptive steered molecular dynamics (ASMD). A single initial condition is used to seed the whole multi-dimensional nonequilibrium free energy simulation and the sampling is performed fully in the nonequilibrium ensemble. The ASMD scheme estimates the free energy difference with the unidirectional exponential average (EXP), but it does not follow exactly the requirements of the EXP estimator. Another consequence of the seeding simulation is the inherently sequential or serial pulling due to the inter-segment dependency, which triggers some problems in the parallelizability of the simulation. Therefore, some tests are required to grasp some insights and guidelines for using this selection-criterion-based ASMD scheme. The ASMD method is tested thoroughly on a dihedral flipping model system and encouraging numerical results are obtained. The selection-criterion-based multi-dimensional ASMD framework follows the same perturbation framework of the BAR-based method, and thus could be used in various Hamiltonian-variation cases.</p>

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A generalization of the cumulant expansion. Application to a scale-invariant probabilistic model

Abstract: As well known, cumulant expansion is an alternative way to moment expansion to fully characterize probability distributions provided all the moments exist. If this is not the case, the so called

Cited by 18 publications

References 31 publications

Finite-volume cumulant expansion in QCD-colorless plasma

Finite-volume cumulant expansion in QCD-colorless plasma

The Nonadditive Entropy Sq and Its Applications in Physics and Elsewhere: Some Remarks

Seeding the Multi-dimensional Nonequilibrium Pulling for Hamiltonian Variation: Indirect QM/MM Free Energy Simulations

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