Ensemble Formalism Research Articles

Effect of Savings on a Gas-Like Model Economy with Credit and Debt.

In kinetic exchange models, agents make transactions based on well-established microscopic rules that give rise to macroscopic variables in analogy to statistical physics. These models have been applied to study processes such as income and wealth distribution, economic inequality sources, economic growth, etc., recovering well-known concepts in the economic literature. In this work, we apply ensemble formalism to a geometric agents model to study the effect of saving propensity in a system with money, credit, and debt. We calculate the partition function to obtain the total money of the system, with which we give an interpretation of the economic temperature in terms of the different payment methods available to the agents. We observe an interplay between the fraction of money that agents can save and their maximum debt. The system’s entropy increases as a function of the saved proportion, and increases even more when there is debt.

Nonequilibrium ensemble derivation of hydrodynamic heat transport and higher-order generalizations

Thermal transport in classical fluids is analyzed through higher-order generalized hydrodynamics (or mesoscopic hydrothermodynamics) depending on the evolution of the energy density and its fluxes of all orders. It is derived by a kinetic theory based on the nonequilibrium statistical ensemble formalism. A general system of coupled evolution equations is derived. Maxwell times, which are of significance to determine the character of the motion, are derived. They also have an important role in the choice of the contraction of description (limitation in the number of fluxes to be retained) in the studies on hydrodynamic motions. In a description of order 1, an analysis of the technological process of thermal prototyping is presented.

Schrödinger group and non-linear generalizations of quantum mechanics

A free particle may be characterized abstractly in terms of its invariance group. Both the Hamilton-Jacobi and Schrödinger equations for classical and quantum free particles are invariant under the same group, the Schrödinger group.The aim of this paper is to discuss possible restrictions on non-linear generalizations of quantum mechanics. The basic idea is to search for those non-linear generalization of quantum mechanics for which a description of the non-relativistic free particle includes a representation of the Schrödinger group, as in classical and quantum mechanics. This condition imposes strong constraints on possible models of free particles.To carry out such an investigation it is necessary to select an appropriate mathematical formalism, one which can describe physical systems of non-relativistic particles and allows for the incorporation of non-linearities in a natural way. Here the formalism of ensembles on configuration space is used, which is an approach that is capable of describing both classical and quantum systems and, in addition, allows for more general theories that describe mixed classical-quantum systems. Modifications of the equations involving derivatives of up to fourth order are considered

Ensemble Green's function theory for interacting electrons with degenerate ground states

An ensemble Green's function formalism, based on the von Neumann density matrix approach, to calculate one-electron excitation spectra of a many-electron system with degenerate ground states is proposed. A set of iterative equations for the ensemble Green's function and self-energy is derived and a simplest approximation corresponding to an ensemble $GW$ approximation is naturally obtained. The derivation is based on the Schwinger functional derivative technique and does not assume any adiabatic connection between a noninteracting and an interacting ground state.

Magnetocaloric properties of V6 molecular magnet

The paper presents a theoretical study of magnetocaloric properties of polyoxovanadate molecular magnet V6 containing 6 vanadium ions carrying quantum spins S=1/2. The characteristic property of such structure is the presence of two weakly interacting spin triangles with all-antiferromagnetic couplings. The properties of the system are described using the exact numerical diagonalization approach applied to the quantum Heisenberg model and utilizing a field ensemble formalism. The dependence of the magnetic entropy and magnetic specific heat on the temperature and external magnetic field is calculated and extensively discussed. The magnetocaloric properties are quantified by isothermal entropy change and entropy derivative over the magnetic field. An interesting behaviour of isothermal entropy change is found, with high degree of tunability of the magnetocaloric effect with the initial and final magnetic field values.

Entangling quantum fields via a classical gravitational interaction

We consider the coupling of quantum fields to classical gravity in the formalism of ensembles on configuration space, a model that allows a consistent formulation of interacting classical and quantum systems. Explicit calculations show that there are solutions for which two initially independent quantum fields evolve into an entangled state, even though their interaction occurs solely via a common classical gravitational field. Thus in contrast to recent suggestions, an observed generation of entanglement would not provide a definitive test of the nonclassical nature of gravity.

An electronic temperature definition for the reactive electronic species: Conciliating practical approaches in conceptual chemical reactivity theory with a rigorous ensemble formulation.

By working under the framework of the Helmholtz potential as a functional of the equilibrium density matrix, in this contribution, we provide theoretical evidence about a particular thermodynamic situation, where electronic species display their highest susceptibility to exchange electrons to or from surroundings. This situation is denominated as the electronic temperature condition. Neutral chemical species display their lowest possible hardness value at the electronic temperature condition, and remarkably, under this circumstance, the exchange of any amount of electronic charge will necessarily be translated into a net increase in the corresponding chemical hardness. Chemical response functions defined as partial derivatives of the Helmholtz potential with respect to the (average) number of electrons and evaluated at the electronic temperature condition provide comparable results than those obtained from the coarse quadratic approximation to the exact dependence of the electronic energy vs the number of electrons, including composite quantities as the electrophilicity index. In this context, we show that the exact Helmholtz potential dependence with respect to the number of electrons can accurately be approximated by "temperature dependent" polynomial fits (up to fourth order), evaluated at the electronic temperature condition.

Nonadiabatic molecular dynamics with decoherence and detailed balance under a density matrix ensemble formalism

Author(s): Kang, J; Wang, LW | Abstract: The mixed quantum-classical nonadiabatic molecular dynamics (NAMD) is a powerful tool to study many phenomena, especially ultrafast carrier transport and cooling. Carrier decoherence and detailed balance are two major issues in NAMD. So far, there is no computationally inexpensive approach to incorporate both effects. While the decoherence effect can be easily included in the state density matrix formalism and the detailed balance can be included in surface hopping or the wave function collapse approach, it is difficult to include both of them in a unified formalism. In this work we introduce a state density matrix formalism (referred to as P-matrix) including both the decoherence and detailed balance effects for NAMD. This method is able to explicitly treat the decoherence between different pairs of adiabatic states. Moreover, the off-diagonal density matrix elements are divided into two parts, corresponding to energy-increasing and energy-decreasing transitions. The detailed balance is then enforced by a Boltzmann factor applied to the energy-increasing transition part. The P-matrix formalism is applied to study hot-hole cooling and transfer processes in Si quantum dot (QD) systems. The calculated hot-carrier relaxation time is consistent with experiments. In a QD-pair system, the hot-hole cooling time shows weak dependence on the QD spacing. However, the hot-carrier transfer rate from one QD to another is found to decrease exponentially with the QD-QD distance. When the QD spacing is small (∼1 nm), the hot-carrier transfer can be very efficient. It is also shown that the explicit treatment of decoherence time is important in order to treat this hot-carrier transfer correctly.

A Hierarchical Approach to Predict Conformation-Dependent Histidine Protonation States in Stable and Flexible Proteins.

Solution acidity measured by pH is an important environmental factor that affects protein structure. It influences the protonation state of protein residues, which in turn may be coupled to protein conformational changes, unfolding, and ligand binding. It remains difficult to compute and measure the p Ka of individual residues, as well as to relate them to pH-dependent protein transitions. This paper presents a hierarchical approach to compute the p Ka of individual protonatable residues, specifically histidines, coupled with underlying structural changes of a protein. A fast and efficient free energy perturbation (FEP) algorithm has also been developed utilizing a fast implementation of standard molecular dynamics (MD) algorithms. Specifically, a CUDA version of the AMBER MD engine is used in this paper. Eight histidine p Ka's are computed in a diverse set of pH stable proteins to demonstrate the proposed approach's utility and assess the predictive quality of the AMBER FF99SB force field. A reference molecule is carefully selected and tested for convergence. A hierarchical approach is used to model p Ka's of the six histidine residues of the diphtheria toxin translocation domain (DTT), which exhibits a diverse ensemble of individual conformations and pH-dependent unfolding. The hierarchical approach consists of first sampling equilibrium conformational ensembles of a protein with protonated and neutral histidine residues via long equilibrium MD simulations (Flores-Canales, J. C.; et al. bioRxiv, 2019, 572040). A clustering method is then used to identify sampled protein conformations, and p Ka's of histidines in each protein conformation are computed. Finally, an ensemble averaging formalism is developed to compute weighted average histidine p Ka's. These can be compared with an apparent experimentally measured p Ka of the DTT protein and thus allows us to propose a mechanism of pH-dependent unfolding of the DTT protein.

Study of Electron Transport in 4H-SiC by Using Nonequilibrium Statistical Ensemble Formalism

A theoretical study, by using Nonequilibrium Statistical Ensemble Formalism (NESEF), on the nonlinear transport of electrons in the transient and steady state of n-doped 4H-SiC under the influence of high electric fields is presented. The electron drift velocity and the nonequilibrium temperature are obtained, and their dependence on the electric field applied in the orientation parallel or perpendicular to the c-axis is derived and analyzed.

Ensemble Controllability of Cellular Oscillators

Many diseases including cancer, Parkinson’s disease and heart diseases are caused by loss or malfunction of regulatory mechanism of an oscillatory system. Successful treatment of these diseases might involve recovering the healthy behavior of the oscillators in the system, i.e., achieving synchrony or a desired distribution of the oscillators on their periodic orbit. In this letter, we consider the problem of controlling the distribution of a population of cellular oscillators described in terms of phase models. Different practical limitations on the observability and controllability of cellular states naturally lead to an ensemble control formulation in which a population-level feedback law for achieving a desired distribution is sought. A systems theoretic approach to this problem leads to Lyapunov- and LaSalle-like arguments, from which we develop our main contribution, novel necessary and sufficient conditions for the controllability of phase distributions in terms of the Fourier coefficients of the phase response curve. Since our treatment is based on a rather universal formulation of phase models, the results and methods proposed in this letter are readily applicable to the control of a wide range of other types of oscillating populations, such as circadian clocks, and spiking neurons.

Signatures of a universal jump in the superfluid density of a two-dimensional Bose gas with a finite number of particles

We study, within the classical fields approximation, a two-dimensional weakly interacting uniform Bose gas of a finite number of atoms. By using a grand canonical ensemble formalism we show that such systems exhibit, in addition to the Berezinskii-Kosterlitz-Thouless (BKT) and thermal phases, the intermediate region. This region is characterized by a decay of current-current correlations at low momenta and by an algebraic decay of the first-order correlations with an exponent being larger than the critical value predicted by the BKT theory. The density of the superfluid fraction at the temperature which separates the BKT phase from the intermediate region approaches the one found by Nelson and Kosterlitz for two-dimensional superfluids while the number of atoms is increased.

Density fields for branching, stiff networks in rigid confining regions.

We develop a formalism to describe the equilibrium distributions for segments of confined branched networks consisting of stiff filaments. This is applicable to certain situations of cytoskeleton in cells, such as for example actin filaments with branching due to the Arp2/3 complex. We develop a grand ensemble formalism that enables the computation of segment density and polarisation profiles within the confines of the cell. This is expressed in terms of the solution to nonlinear integral equations for auxiliary functions. We find three specific classes of behaviour depending on filament length, degree of branching and the ratio of persistence length to the dimensions of the geometry. Our method allows a numerical approach for semi-flexible filaments that are networked.

Experimental results on fluctuations of conserved charges confronted with predictions from canonical thermodynamics

The study of multiplicity distributions of identified particles in terms of their higher moments is at the focus of contemporary experimental and theoretical studies. In a thermalized system combinations of these moments are directly related to the Equation of State (EoS) through variations of the thermodynamic pressure with respect to changes in the chemical potential μQ, associated with a conserved charge Q. The ultimate goal of the experimental measurements in relativistic nuclear collisions is, by systematics comparison to the ab initio theoretical calculations, to probe the dynamics of genuine phase transitions between a hadron gas and the quark-gluon plasma. However, the comparison between experiment and theory is far from trivial, because several non-dynamical fluctuations need to be controlled prior to a meaningful comparison to theoretical predictions. In this report we present quantitative estimates for these non-dynamical contributions using the Canonical Ensemble (CE) formulation of the statistical mechanics. Together with the analytical formulas we provide also results from Monte Carlo simulations within the CE and compare our predictions with the corresponding measurements from the STAR experiment.

Finite-temperature auxiliary-field quantum Monte Carlo: Self-consistent constraint and systematic approach to low temperatures

We describe an approach for many-body calculations with a finite-temperature, grand canonical ensemble formalism using auxiliary-field quantum Monte Carlo (AFQMC) with a self-consistent constraint to control the sign problem. The usual AFQMC formalism of Blankenbecler, Scalapino, and Sugar suffers from the sign problem with most physical Hamiltonians, as is well known. Building on earlier ideas to constrain the paths in auxiliary-field space [Phys. Rev. Lett. \textbf{83}, 2777 (1999)] and incorporating recent developments in zero-temperature, canonical-ensemble methods, we discuss how a self-consistent constraint can be introduced in the finite-temperature, grand-canonical-ensemble framework. This together with several other algorithmic improvements discussed here leads to a more accurate, more efficient, and numerically more stable approach for finite-temperature calculations. We carry out a systematic benchmark study in the two-dimensional repulsive Hubbard model at $1/8$ doping. Temperatures as low as $T=1/80$ (in units of hopping) are reached. The finite-temperature method is exact at very high temperatures, and approaches the result of the zero-temperature constrained-path AFQMC as temperature is lowered. The benchmark shows that systematically accurate results are obtained for thermodynamic properties.

A Boltzmann-Tsallis approach towards cell survival curve in Radiobiology

This paper outlines a phenomenological approach towards cell survival curve at low dose using tools of extensive Statistical Mechanics and nonextensive Statistical Mechanics. An Ising chain model is developed for the cell survival curve and the canonical ensemble formalism based on Boltzmann Gibbs statistic and Tsallis statistic is presented. The resulting cell survival curve shows excellent agreement with the experimental data and the physical parameters from our Tsallis model (N’, q) can be shown to provide clear classification between healthy and cancerous cells. In this paper, we also provides possible biophysical interpretation to the (N’, q) parameters where N’ is representative of the amount of repairable DNA content in the nucleus and q represents the degree of correlation in DNA damage. Overall, this is the first time a Statistical Mechanics approach is used in Radiobiology, and could present a new perspective.

Estimation of physical parameters under location uncertainty using an ensemble2–expectation–maximization algorithm

Estimating the parameters of geophysical dynamic models is an important task in the data assimilation (DA) techniques used to forecast initialization and reanalysis. In the past, most parameter estimation strategies were derived by state augmentation, yielding algorithms that are easy to implement but may exhibit convergence difficulties. The expectation–maximization (EM) algorithm is considered advantageous because it employs two iterative steps to estimate the model state and the model parameter separately. In this work, we propose a novel ensemble formulation of the maximization step in EM that allows a direct optimal estimation of the physical parameters using iterative methods for linear systems. This departs from current EM formulations that are only capable of dealing with additive model error structures. This contribution shows how the EM technique can be used for dynamics identification problems with the model error parameterized as an arbitrary complex form. The proposed technique is used here for the identification of stochastic subgrid terms that account for processes unresolved by a geophysical fluid model. This method, together with the augmented state technique, is evaluated to estimate such subgrid terms through high‐resolution data. As compared to the augmented state technique, our method is shown to yield considerably more accurate parameters. In addition, in terms of prediction capacity, it leads to a smaller generalization error as a result of the overfitting of the trained model on the presented data and eventually better forecasts.

Monte Carlo approach to the excluded-volume hadron resonance gas in grand canonical and canonical ensembles

The Monte Carlo (MC) procedure for sampling the hadron yields within hadron resonance gas (HRG) model is presented. The effects of excluded-volume due to the finite hadron eigenvolumes and of exact charge conservation within the canonical ensemble (CE) formulation are simultaneously taken into account with the help of the importance sampling technique combined with the rejection sampling. The MC procedure allows one to calculate arbitrary moments of the event-by-event hadron yields. Note that the CE formulation for the excluded-volume HRG has not been considered before. The MC simulations coincide with the known analytic results in the thermodynamic limit for the excluded-volume HRG in the grand canonical ensemble and for the CE of non-interacting particles. The MC procedure is applied to study the simultaneous excluded-volume and CE effects. These effects are considered within the full HRG to calculate the particle number fluctuations and to estimate the finite size effects.

Statistical hadron-gas treatment of systems created in proton-proton interactions at energies available at the CERN Super Proton Synchrotron

We analyze the newest data from the NA61/SHINE collaboration which, in addition to previous results on pions and kaons, include mean multiplicities of $p$, $\Lambda$, and $\phi$-mesons produced in inelastic proton-proton (p+p) interactions at $\sqrt{s_{NN}}=6.3-17.3$~GeV. The canonical ensemble formulation of the ideal hadron resonance gas (HRG) model is used with exact conservation of net baryon number $B=2$, electric charge $Q=2$, and strangeness $S=0$. The chemical freeze-out parameters in p+p interactions are obtained and compared to those in central nucleus-nucleus collisions. Several features of p+p interactions at the CERN SPS within a statistical model are studied: 1) the inclusion of the $\phi$-meson yields in thermal fits worsens significantly the fit quality; 2) the data show large event-by-event multiplicity fluctuations in inelastic p+p interactions which can not be explained by a single fireball described by a statistical model; 3) the fits within the canonical ensemble formulation of HRG do not give any improvement over the fits within the grand canonical ensemble formulation, i.e., there are no indications for existence of a single statistical system in p+p inelastic interactions in the considered energy range.

On local representations of rotations on discrete configuration spaces

A quantum mechanical spin-half system may be characterized abstractly as having a set of two-valued observables which generate infinitesimal rotations in three dimensions. We consider a concrete realization of such a two-level system within the formalism of ensembles on configuration space, an approach which is not only capable of describing quantum mechanical systems but allows also for theories that are generalizations of quantum theory. Such a spin-half system may be called a rotational bit or robit, to distinguish it from the standard quantum qubit. After reviewing ensembles on configuration space and examining the example of constructing representations of the Galilean Lie algebra for the free particle, we construct probabilistic models for ensembles that consist of one and two spin-half systems. In the case of a single spin-half system, there are two main requirements: the configuration space must be a discrete set, labeling the outcomes of two-valued spin observables, and these observables must provide an algebraic representation of so(3). The case of a pair of spin-half systems is more complicated, in that additional physical requirements concerning locality and subsystem independence must also be taken into account and now the observables must provide an algebraic representation of so(3) ⊕ so(3). We compare the resulting theories to the corresponding quantum mechanical systems.