Grand Canonical Distribution Research Articles

Efficient estimation of transition rates as functions of pH

AbstractExtracting the kinetic properties of a system whose dynamics depend on the pH of the environment with which it exchanges energy and atoms requires sampling the grand canonical ensemble. As an alternative, we present a novel strategy that requires simulating only the most recurrent canonical ensembles that compose the grand canonical ensemble. The simulations are used to estimate the grand canonical distribution for a specific pH value by reweighting and to construct the transition rate matrix by discretizing the Fokker–Planck equation by square root approximation and robust Perron cluster cluster analysis. As an application, we have studied the tripeptide Ala‐Asp‐Ala.

On the asymptotic behavior of solutions to a class of grand canonical master equations

In this article, we investigate the long-time behavior of solutions to a class of infinitely many master equations defined from transition rates that are suitable for the description of a quantum system approaching thermodynamical equilibrium with a heat bath at fixed temperature and a reservoir consisting of one species of particles characterized by a fixed chemical potential.We do so by proving a result which pertains to the spectral resolution of the semigroup generated by the equations, whose infinitesimal generator is realized as a trace-class self-adjoint operator defined in a suitably weighted sequence space. This allows us to prove the existence of global solutions which all stabilize toward the grand canonical equilibrium probability distribution as the time variable becomes large, some of them doing so exponentially rapidly but not all. When we set the chemical potential equal to zero, the stability statements continue to hold in the sense that all solutions converge toward the Gibbs probability distribution of the canonical ensemble which characterizes the equilibrium of the given system with a heat bath at fixed temperature.

Assessing transition rates as functions of environmental variables.

We present a method to estimate the transition rates of molecular systems under different environmental conditions that cause the formation or the breaking of bonds and require the sampling of the Grand Canonical Ensemble. For this purpose, we model the molecular system in terms of probable "scenarios," governed by different potential energy functions, which are separately sampled by classical MD simulations. Reweighting the canonical distribution of each scenario according to specific environmental variables, we estimate the grand canonical distribution, then use the Square Root Approximation method to discretize the Fokker-Planck operator into a rate matrix and the robust Perron Cluster Cluster Analysis method to coarse-grain the kinetic model. This permits efficiently estimating the transition rates of conformational states as functions of environmental variables, for example, the local pH at a cell membrane. In this work, we formalize the theoretical framework of the procedure, and we present a numerical experiment comparing the results with those provided by a constant-pH method based on non-equilibrium Molecular Dynamics Monte Carlo simulations. The method is relevant for the development of new drug design strategies that take into account how the cellular environment influences biochemical processes.

Accelerated simulation method for charge regulation effects.

The net charge of solvated entities, ranging from polyelectrolytes and biomolecules to charged nanoparticles and membranes, depends on the local dissociation equilibrium of individual ionizable groups. Incorporation of this phenomenon, charge regulation (CR), in theoretical and computational models requires dynamic, configuration-dependent recalculation of surface charges and is therefore typically approximated by assuming constant net charge on particles. Various computational methods exist that address this. We present an alternative, particularly efficient CR Monte Carlo method (CR-MC), which explicitly models the redistribution of individual charges and accurately samples the correct grand-canonical charge distribution. In addition, we provide an open-source implementation in the large-scale Atomic/Molecular Massively Parallel Simulator molecular dynamics (MD) simulation package, resulting in a hybrid MD/CR-MC simulation method. This implementation is designed to handle a wide range of implicit-solvent systems that model discreet ionizable groups or surface sites. The computational cost of the method scales linearly with the number of ionizable groups, thereby allowing accurate simulations of systems containing thousands of individual ionizable sites. By matter of illustration, we use the CR-MC method to quantify the effects of CR on the nature of the polyelectrolyte coil-globule transition and on the effective interaction between oppositely charged nanoparticles.

Nonequilibrium grand-canonical ensemble built from a physical particle reservoir

We introduce a nonequilibrium grand-canonical ensemble defined by considering the stationary state of a driven system of particles put in contact with a particle reservoir. When an additivity assumption holds for the large deviation function of density, a chemical potential of the reservoir can be defined. The grand-canonical distribution then takes a form similar to the equilibrium one. At variance with equilibrium, though, the probability weight is "renormalized" by a contribution coming from the contact, with respect to the canonical probability weight of the isolated system. A formal grand-canonical potential can be introduced in terms of a scaled cumulant generating function, defined as the Legendre-Fenchel transform of the large deviation function of density. The role of the formal Legendre parameter can be played, physically, by the chemical potential of the reservoir when the latter can be defined, or by a potential energy difference applied between the system and the reservoir. Static fluctuation-response relations naturally follow from the large deviation structure. Some of the results are illustrated on two different explicit examples, a gas of noninteracting active particles and a lattice model of interacting particles.

Thermodynamic properties of the solid and metal electrons in the nonextensive quantum statistics with a nonextensive parameter transformation

We study the thermodynamic properties of solid and metal electrons in the nonextensive quantum statistics with a nonextensive parameter transformation. First we study the nonextensive grand canonical distribution function and the nonextensive quantum statistics with a parameter transformation. Then we derive the generalized Boson distribution and Fermi distribution in the nonextensive quantum statistics. Further we study the thermodynamic properties of solid and metal electrons in the nonextensive quantum system, including the generalized Debye models, the generalized internal energies, the generalized capacities and chemical potential. We derive new expressions of these thermodynamic quantities, and we show that they all depend significantly on the nonextensive parameter and in the limit they recover to the forms in the classical quantum statistics. These new expressions may be applied to study the new characteristics in some nonextensive quantum systems where the long-range interactions and/or long-range correlations play a role.

The nonextensive Bose-Einstein condensation and photon gas with parameter transformation

The nonextensive boson system is revisited with the statistical ensemble theory, where the fundamental grand canonical distribution is derived from the equiprobability principle. By recourse to the approach of parameter transformation, one familiar and concise statistical formula for bosons is deduced, in an accurate fashion. Then the Bose-Einstein condensation phenomenon is re-discussed. The results show that the critical temperature is dependent on the nonextensive parameter $ \nu$ , and in the generalized expression of the heat capacity for the condensated phase of boson systems there exists one additional term, obeying the T3 law. By use of the statistical formula, the nonextensive photon gas is also researched. The internal energy and heat capacity for the nonextensive photon gas exhibit a similar dependence on the temperature to the classical photon gas, apart from a coefficient correction dependent on the parameter $ \nu$ . The Gibbs function for the nonextensive photon gas is still zero, showing that the photon field is also at thermal equilibrium, like the situation of classical photon field. The entropy of photon field can be calculated through the integral of the nonextensive quantum statistics formula and can also be derived from the original definition of Tsallis entropy, by recourse to the direct parameter transformation. This seems to indicate the validity of the treatment technique for the nonextensive quantum systems.

On Entropy Minimization and Convergence

We examine the minimization of information entropy for measures on the phase space of bounded domains, subject to constraints that are averages of grand canonical distributions. We describe the set of all such constraints and show that it equals the set of averages of all probability measures absolutely continuous with respect to the standard measure on the phase space (with the exception of the measure concentrated on the empty configuration). We also investigate how the set of constrains relates to the domain of the microcanonical thermodynamic limit entropy. We then show that, for fixed constraints, the parameters of the corresponding grand canonical distribution converge, as volume increases, to the corresponding parameters (derivatives, when they exist) of the thermodynamic limit entropy. The results hold when the energy is the sum of any stable, tempered interaction potential that satisfies the Gibbs variational principle (e.g.~Lennard-Jones) and the kinetic energy. The same tools and the strict convexity of the thermodynamic limit pressure for continuous systems (valid whenever the Gibbs variational principle holds) give solid foundation to the folklore local homeomorphism between thermodynamic and macroscopic quantities.

The nonextensive statistical ensembles with dual thermodynamic interpretations

The nonextensive statistical ensembles are revisited for the complex systems with long-range interactions and long-range correlations. An approximation, the value of nonextensive parameter (1-q) is assumed to be very tiny, is adopted for the limit of large particle number for most normal systems. In this case, Tsallis entropy can be expanded as a function of energy and particle number fluctuation, and thus the power-law forms of the generalized Gibbs distribution and grand canonical distribution can be derived. These new distribution functions can be applied to derive the free energy and grand thermodynamic potential in nonextensive thermodynamics. In order to establish appropriate nonextensive thermodynamic formalism, the dual thermodynamic interpretations are necessary for thermodynamic relations and thermodynamic quantities. By using a new technique of parameter transformation, the single-particle distribution can be deduced from the power-law Gibbs distribution. This technique produces a link between the statistical ensemble and the quasi-independent system with two kinds of nonextensive parameter having quite different physical explanations. Furthermore, the technique is used to construct nonextensive quantum statistics and effectively to avoid the factorization difficulty in the power-law grand canonical distribution.

Weakly Nonergodic Dynamics in the Gross-Pitaevskii Lattice.

The microcanonical Gross-Pitaevskii (also known as the semiclassical Bose-Hubbard) lattice model dynamics is characterized by a pair of energy and norm densities. The grand canonical Gibbs distribution fails to describe a part of the density space, due to the boundedness of its kinetic energy spectrum. We define Poincaré equilibrium manifolds and compute the statistics of microcanonical excursion times off them. The tails of the distribution functions quantify the proximity of the many-body dynamics to a weakly nonergodic phase, which occurs when the average excursion time is infinite. We find that a crossover to weakly nonergodic dynamics takes place inside the non-Gibbs phase, being unnoticed by the largest Lyapunov exponent. In the ergodic part of the non-Gibbs phase, the Gibbs distribution should be replaced by an unknown modified one. We relate our findings to the corresponding integrable limit, close to which the actions are interacting through a short range coupling network.

Statistical analogues of thermodynamic extremum principles

As shown by Jaynes, the canonical and grand canonical probability distributions of equilibrium statistical mechanics can be simply derived from the principle of maximum entropy, in which the statistical entropy is maximised subject to constraints on the mean values of the energy E and/or number of particles N in a system of fixed volume V. The Lagrange multipliers associated with those constraints are then found to be simply related to the temperature T and chemical potential μ. Here we show that the constrained maximisation of S is equivalent to, and can therefore be replaced by, the essentially unconstrained minimisation of the obvious statistical analogues of the Helmholtz free energy F = E − TS and the grand potential J = F − μN. Those minimisations are more easily performed than the maximisation of S because they formally eliminate the constraints on the mean values of E and N and their associated Lagrange multipliers. This procedure significantly simplifies the derivation of the canonical and grand canonical probability distributions, and shows that the well known extremum principles for the various thermodynamic potentials possess natural statistical analogues which are equivalent to the constrained maximisation of S.

Extension of Kirkwood-Buff theory to the canonical ensemble.

Kirkwood-Buff (KB) integrals are notoriously difficult to converge from a canonical simulation because they require estimating the grand-canonical radial distribution. The same essential difficulty is encountered when attempting to estimate the direct correlation function of Ornstein-Zernike theory by inverting the pair correlation functions. We present a new theory that applies to the entire, finite, simulation volume, so that no cutoff issues arise at all. The theory gives the direct correlation function for closed systems, while smoothness of the direct correlation function in reciprocal space allows calculating canonical KB integrals via a well-posed extrapolation to the origin. The present analysis method represents an improvement over previous work because it makes use of the entire simulation volume and its convergence can be accelerated using known properties of the direct correlation function. Using known interaction energy functions can make this extrapolation near perfect accuracy in the low-density case. Because finite size effects are stronger in the canonical than in the grand-canonical ensemble, we state ensemble correction formulas for the chemical potential and the KB coefficients. The new theory is illustrated with both analytical and simulation results on the 1D Ising model and a supercritical Lennard-Jones fluid. For the latter, the finite-size corrections are shown to be small.

Multicomponent Conduction and Selectivity of Biological Channels

We consider multispecies conduction and selectivity of narrow biological filters such as KcsA [1] and NaChBac [2]. We use equilibrium statistical theory to find filter occupation and conductivities when coupled to the bath solutions with mixed types of conducting ions. The analysis of the filter conduction and selectivity conduction and selectivity is thus reduced to the analysis of the effective grand canonical distribution of ions within the filter [3], energy spectra, and density of the states within the filter [4]. The effects of energy quantization due to discrete number of ions and due to dehydration are taken into account. To account for the interaction between fluxes of different species within the channel we introduce Maxwell-Stefan theory [5] of the ion channel conduction. The fluxes of various species through the filter are obtained analytically in the limit of weak coupling. The theory is applied to derive Eisenman relations of the filter selectivity from the condition of nearly diffusion limited conduction in KscA channel. The model predictions are compared with experimental results using self-consistent kinetic equations. The transition rates in kinetic model are derived using effective grand canonical ensemble. A good agreement with the experiment is demonstrated. An application of the theory to analysis of the channel mutation experiments are discussed.

Thermodynamics and galactic clustering with a modified gravitational potential

Based on thermodynamics, we study the galactic clustering of an expanding Universe by considering the logarithmic and volume (quantum) corrections to Newton's law along with the repulsive effect of a harmonic force induced by the cosmological constant ($\mathrm{\ensuremath{\Lambda}}$) in the formation of the large scale structure of the Universe. We derive the $N$-body partition function for extended-mass galaxies (galaxies with halos) analytically. For this partition function, we compute the exact equations of states, which exhibit the logarithmic, volume, and cosmological constant corrections. In this setting, a modified correlation (clustering) parameter (due to these corrections) emerges naturally from the exact equations of state. We compute a corrected grand canonical distribution function for this system. Furthermore, we obtain a deviation in differential forms of the two-point correlation functions for both the point-mass and extended-mass cases. The consequences of these deviations on the correlation function's power law are also discussed.

Stochastic boundary approaches to many-particle systems coupled to a particle reservoir.

Stochastic boundary conditions for interactions with a particle reservoir are discussed in many-particle systems. We introduce the boundary conditions with the injection rate and the momentum distribution of particles coming from a particle reservoir in terms of the pressure and the temperature of the reservoir. It is shown that equilibrium ideal gases and hard-disk systems with these boundary conditions reproduce statistical-mechanical properties based on the corresponding grand canonical distributions. We also apply the stochastic boundary conditions to a hard-disk model with a steady particle current escaping from a particle reservoir in an open tube, and discuss its nonequilibrium properties such as a chemical potential dependence of the current and deviations from the local equilibrium hypothesis.

Molecular simulation of methane adsorption in shale based on grand canonical Monte Carlo method and pore size distribution

To explore the influence of the pore size distribution on the methane adsorption capacity of shale, methane adsorption amount for different pore sizes has been calculated by using the simulation results and pore size distribution data. In the study, excess adsorption per unit area of different pore sizes has been simulated by the grand canonical Monte Carlo (GCMC) method. And the pore size distribution of Longmaxi Formation shale samples was characterized by high pressure mercury injection (HPMI) experiment, low pressure nitrogen adsorption (LP-N2-GA) experiment, and low pressure CO2 adsorption (LP-CO2-GA) experiment. The results indicate that in the range of 0.5–1.5 nm, the excess adsorption per unit area shows a stepped increase with the pore size increasing. When the pore size is larger than 1.5 nm, the excess adsorption per unit area keep unchanged with the increase in pore size. According to the methane adsorption capacity and pore surface area, pores in the shale samples can be divided to three parts: <1 nm, 1–4 nm, >4 nm. Pores in the first part (width <1 nm) have a large specific surface area, but the excess adsorption per unit area is much smaller than other pores. They account for about 65% of the total specific surface area, while accounting for only 48% of the total adsorption amount. Pores in the second part (1–4 nm) have a large specific pore area and large methane adsorption capacity. They provide 30% of the total specific surface area, while providing 44% of the total adsorption amount. Pores in the third part (width >4 nm) contribute only 8% of total adsorption amount as the specific surface area of these pores is notably small.

Permittivity and one-particle distribution functions in the thermodynamics of a coulomb system

Using the grand canonical distribution and the virial theorem, we show that the Gibbs thermodynamic potential of a nonrelativistic system of charged particles is uniquely determined by its permittivity and the distribution functions of electrons and nuclei without using perturbation theory. This means that consistent approximations for the permittivity and one-particle distribution functions of electrons and nuclei must be used to calculate thermodynamic functions of the Coulomb system. To construct such selfconsistent approximations, we propose using a decoupling procedure based on separating the “connected” and “regular” parts of the temperature Green’s functions in the equations of motion. We consider the self-consistent Hartree-Fock approximation corresponding to this procedure.

Metropolis–Hastings thermal state sampling for numerical simulations of Bose–Einstein condensates

We demonstrate the application of the Metropolis–Hastings algorithm to sampling of classical thermal states of one-dimensional Bose–Einstein quasicondensates in the classical fields approximation, both in untrapped and harmonically trapped case. The presented algorithm can be easily generalized to higher dimensions and arbitrary trap geometry. For truncated Wigner simulations the quantum noise can be added with conventional methods (half a quantum of energy in every mode). The advantage of the presented method over the usual analytical and stochastic ones lies in its ability to sample not only from canonical and grand canonical distributions, but also from the generalized Gibbs ensemble, which can help to shed new light on thermodynamics of integrable systems.

Counting statistics for electron capture in a dynamic quantum dot.

We report noninvasive single-charge detection of the full probability distribution P(n) of the initialization of a quantum dot with n electrons for rapid decoupling from an electron reservoir. We analyze the data in the context of a model for sequential tunneling pinch-off, which has generic solutions corresponding to two opposing mechanisms. One limit considers sequential "freeze-out" of an adiabatically evolving grand canonical distribution, the other one is an athermal limit equivalent to the solution of a generalized decay cascade model. We identify the athermal capturing mechanism in our sample, testifying to the high precision of our combined theoretical and experimental methods. The distinction between the capturing mechanisms allows us to derive efficient experimental strategies for improving the initialization.

Statistical Distribution of a Completely Open System Based on Incomplete Shannon Entropy

Micro canonical, canonical and grand canonical systems are special cases of completely open systems. Within the framework of non-extensive and incomplete statistics, we derive the statistical distribution for a completely open system on the basis of incomplete Shannon entropy, using the maximum entropy principle. We calculate the physical properties of a linear filament using this distribution. The results are the same as those calculated using the incomplete E-V distribution, except that average values of the thermodynamic variables replace the corresponding constant values. However, the relative fluctuations in thermodynamic variables are completely different. In the canonical and grand canonical distributions, relative fluctuations are proportional to . In the incomplete statistical distribution for a completely open system, the relative fluctuations are proportional to 1. This result can explain some phenomenon that traditional and current statistical mechanics cannot explain, such as the large fluctuations of critical, super-cooled and overheated states.