Fluctuations In Ensemble Research Articles

The RCI server: rapid and accurate calculation of protein flexibility using chemical shifts

Protein motions play important roles in numerous biological processes such as enzyme catalysis, muscle contractions, antigen–antibody interactions, gene regulation and virus assembly. Knowledge of protein flexibility is also important in rational drug design, protein docking and protein engineering. However, the experimental measurement of protein motions is often difficult, requiring sophisticated experiments, complex data analysis and detailed information about the protein's tertiary structure. As a result, there is a considerable interest in developing simpler, more effective ways of quantifying protein flexibility. Recently, we described a method, called the random coil index (RCI), which is able to quantitatively estimate backbone root mean square fluctuations (RMSFs) of structural ensembles and order parameters using only chemical shifts. The RCI method is very fast (<5 s) and exceedingly robust. It also offers an excellent alternative to traditional methods of measuring protein flexibility. We have recently extended the RCI concept and implemented it as a web server. This server allows facile, accurate and fully automated predictions of MD RMSF values, NMR RMSF values and model-free order parameters (S2) directly from chemical shift assignments. It also performs automatic chemical shift re-referencing to ensure consistency and reproducibility. On average, the correlation between RCI predictions and experimentally obtained motional amplitudes is within the range from 0.77 to 0.82. The server is available at http://wishart.biology.ualberta.ca/rci.

Generalized Hawking–Page phase transition

The issue of radiant spherical black holes being in stable thermal equilibrium with their radiation bath is reconsidered. Using a simple equilibrium statistical mechanical analysis incorporating Gaussian thermal fluctuations in a canonical ensemble of isolated horizons, the heat capacity is shown to diverge at a critical value of the classical mass of the isolated horizon, given (in Planckian units) by the microcanonical entropy calculated using loop quantum gravity. The analysis reproduces the Hawking–Page phase transition discerned for anti-de Sitter black holes and generalizes it in the sense that nowhere is any classical metric made use of.

Spectral fluctuation characterization of random matrix ensembles through wavelets

A recently developed wavelet based approach is employed to characterize the scaling behavior of spectral fluctuations of random matrix ensembles, as well as complex atomic systems. Our study clearly reveals anti-persistent behavior and supports the Fourier power spectral analysis. It also finds evidence for multi-fractal nature in the atomic spectra. The multi-resolution and localization nature of the discrete wavelets ideally characterizes the fluctuations in these time series, some of which are not stationary.

Simulation-based equation of state of the hard disk fluid and prediction of higher-order virial coefficients

We present new molecular dynamics results for the pressure of the pure hard disk fluid up to the hexatic transition (reduced density about 0.9). The data, combined with the known virial coefficients (up to B 10), are used to build an equation of state, to estimate higher-order virial coefficients, and also to obtain a better value of B 10. Finite size effects are discussed in detail. The ‘van der Waals-like’ loop reported in the literature in the vicinity of the fluid/hexatic transition is explained by suppressed density fluctuations in the canonical ensemble. The inflection point on the pressure–density dependence is predicted by the equation of state even if the hexatic phase simulation data are not considered.

Particle number fluctuations in relativistic Bose and Fermi gases

Particle number fluctuations are studied in relativistic Bose and Fermi gases. The calculations are done within both the grand canonical and canonical ensemble. The fluctuations in the canonical ensemble are found to be different from those in the grand canonical one. Effects of quantum statistics strongly increase in the grand canonical ensemble for large chemical potential. This is, however, not the case in the canonical ensemble, and in the limit of large charge density a strongest difference between the grand canonical and canonical ensemble results is observed.

Fluctuations in the canonical ensemble

The particle number and energy fluctuations in the system of charged particles are studied in the canonical ensemble for non-zero net values of the conserved charge. In the thermodynamic limit the fluctuations in the canonical ensemble are different from the fluctuations in the grand canonical one. The system with several species of particles is considered. We calculate the quantum statistics effects which can be taken into account for the canonical ensemble fluctuations in the infinite volume limit. The fluctuations of the particle numbers in the pion-nucleon gas are considered in the canonical ensemble as an example of the system with two conserved charges - baryonic number and electric charge.

Particle number fluctuations in the microcanonical ensemble

Particle number fluctuations are studied in the microcanonical ensemble. For the Boltzmann statistics we deduce exact analytical formulae for the microcanonical partition functions in the case of non-interacting massless neutral particles and charged particles with zero net charge. The particle number fluctuations are calculated and we find that in the microcanonical ensemble they are suppressed in comparison to the fluctuations in the canonical and grand canonical ensembles. This remains valid in the thermodynamic limit too, so that the well-known equivalence of all statistical ensembles refers to average quantities, but does not apply to fluctuations. In the thermodynamic limit we are able to calculate the particle number fluctuations in the system of massive bosons and fermions when the exact conservation laws of both the energy and charge are taken into account.

Condensate fluctuations in the dilute Bose gas

The fluctuations of a number of particles in the Bose-Einstein condensate are studied in the grand canonical ensemble with an effective single-mode Hamiltonian, which is derived from an assumption that the mode corresponding to the Bose-Einstein condensate does not asymptotically correlate with other modes. The fluctuations are evaluated in the dilute limit with a proposed simple method, which is beyond the mean-field approximation. The accuracy of the latter is estimated; it is shown that the mean-field scheme does not work for the single-mode Hamiltonian, while for the Hartree Hamiltonian it allows us to estimate the condensate fluctuations up to a numerical factor. As a hypothesis, a formula is proposed that relates the fluctuations in the canonical ensemble with that of the grand canonical one.

Creating Order from Random Fluctuations in Small Spin Ensembles

We demonstrate the ability to create spin order by using a magnetic resonance force microscope to harness the naturally occurring statistical fluctuations in small ensembles of electron spins. In one method, we hyperpolarized the spin system by selectively capturing the transient spin order created by the statistical fluctuations. In a second method, we took a more active approach and rectified the spin fluctuations by applying real-time feedback to the entire spin ensemble. The created spin order can be stored in the laboratory frame for a period on the order of the longitudinal relaxation time of 30 seconds and then read out.

Fluctuations in the statistical ensembles

In this paper, we address multiplicity fluctuations of the ideal hadron-resonance gas in different ensembles: grand-canonical, canonical and microcanonical. Two different calculation methods are used: asymptotic expansions and full Monte Carlo simulations. The method based on asymptotic expansion allows a quick numerical calculation of dispersions in the hadron gas with three conserved charges at primary hadron level, while the Monte-Carlo simulation is suitable to study the effect of resonance decays. Even though mean multiplicities converge to the same values, major differences in fluctuations for these ensembles persist in the thermodynamic limit, as pointed out in recent studies. This difference is ultimately related to the non-additivity of the variances in the ensembles with exact conservation of extensive quantities.

Bayesian Ensemble Approach to Error Estimation of Interatomic Potentials

Using a Bayesian approach a general method is developed to assess error bars on predictions made by models fitted to data. The error bars are estimated from fluctuations in ensembles of models sampling the model-parameter space with a probability density set by the minimum cost. The method is applied to the development of interatomic potentials for molybdenum using various potential forms and databases based on atomic forces. The calculated error bars on elastic constants, gamma-surface energies, structural energies, and dislocation properties are shown to provide realistic estimates of the actual errors for the potentials.

Particle number fluctuations in a canonical ensemble

Fluctuations of charged particle number are studied in the canonical ensemble. In the infinite volume limit the fluctuations in the canonical ensemble are different from the fluctuations in the grand canonical one. Thus, the well-known equivalence of both ensembles for the average quantities does not extend for the fluctuations. In view of a possible relevance of the results for the analysis of fluctuations in nuclear collisions at high energies, a role of the limited kinematical acceptance is studied.

Stability of atomic clocks based on entangled atoms.

We analyze the effect of realistic noise sources for an atomic clock consisting of a local oscillator that is actively locked to a spin-squeezed (entangled) ensemble of N atoms. We show that the use of entangled states can lead to an improvement of the long-term stability of the clock when the measurement is limited by decoherence associated with instability of the local oscillator combined with fluctuations in the atomic ensemble's Bloch vector. Atomic states with a moderate degree of entanglement yield the maximal clock stability, resulting in an improvement that scales as N(1/6) compared to the atomic shot noise level.

Relativistic particle simulation of color diffusion in a SU(2) Yang–Mills plasma

We report on a relativistic particle-in-cell simulation study of color transport in a SU(2) Yang–Mills plasma. Simulation results reveal that the color distribution starting from a spiky form spreads to a flat one, through a diffusive process in color space. By numerically performing a “Perrin” like experiment, we explicitly show that the ensemble averaged color fluctuation square (〈(ΔI)2〉) grows linearly with time indicating that the color vector of a parton executes Brownian motion in color space. Depending on the choice of input parameters, color diffusion time scale can either be shorter or longer than the momentum diffusion time scale. Simulation results for a case where color diffuses faster than momentum are presented.

Ensemble fluctuations and the origin of quantum probabilistic rule

We demonstrate that the origin of the so-called quantum probabilistic rule (which differs from the classical Bayes’ formula by the presence of cos θ-factor) might be explained in the framework of ensemble fluctuations which are induced by preparation procedures. In particular, quantum rule for probabilities (with nontrivial cos θ-factor) could be simulated for macroscopic physical systems via preparation procedures producing ensemble fluctuations of a special form. We discuss preparation and measurement procedures which may produce probabilistic rules which are neither classical nor quantum; in particular, hyperbolic “quantum theory.”

Eigenvalue correlations on hyperelliptic Riemann surfaces

In this note we compute the functional derivative of the induced charge density, on a thin conductor, consisting of the union of g+1 disjoint intervals, $J:=\cup_{j=1}^{g+1}(a_j,b_j),$ with respect to an external potential. In the context of random matrix theory this object gives the eigenvalue fluctuations of Hermitian random matrix ensembles where the eigenvalue density is supported on J.

On the Distribution of Eigenvalues of Grand Canonical Density Matrices

Using physical arguments and partition theoretic methods, we demonstrate under general conditions, that the eigenvalues w(m) of the grand canonical density matrix decay rapidly with their index m, like w(m)∼exp[−βB−1(ln m)1+1/α], where B and α are positive constants, O(1), which may be computed from the spectrum of the Hamiltonian. We compute values of B and α for several physical models, and confirm our theoretical predictions with numerical experiments. Our results have implications in a variety of questions, including the behaviour of fluctuations in ensembles, and the convergence of numerical density matrix renormalization group techniques.

A new simulation method for the determination of phase equilibria in mixtures in the grand canonical ensemble

A grand canonical Monte Carlo (GCMC) simulation method is presented for the determination of the phase equilibria of mixtures. The coexistence is derived by expanding the pressure into a Taylor series as a function of the temperature and the chemical potentials that are the independent intensive variables of the grand canonical ensemble. The coefficients of the Taylor series can be calculated from ensemble averages and fluctuation formulae that are obtained from GCMC simulations in both phases. The method is able to produce the equilibrium data in a certain domain of the (T, p) plane from two GCMC simulations. The vapour-liquid equilibrium results obtained for a Lennard-Jones mixture agree well with the corresponding Gibbs ensemble Monte Carlo data.

The DWT power spectrum analysis of the large scale structure in the universe : Method and simulation tests

Based on the discrete wavelet transformation (DWT), we present a pixelized method of estimating the power spectra of galaxy samples. With local properties of wavelet both in physical and wavenumber spaces, DWT power spectrum is equal to the corresponding band average of Fourier power spectrum. The DWT estimator is optimized in the sense that the spatial resolution is adaptive automatically to the perturbation wavelength to be studied. Under the assumption of ergodicity, the spatial average of local DWT fluctuation modes provides a fair estimation of the ensemble average. We test DWT spectra of four typical cold dark matter (CDM) structure formation models with numerical simulations. To consider the infections of various observation effects to the DWT spectra, we introduce irregular survey geometries, a given sampling rate, radial selection effects and redshift distortion effects into our mock samples. The numerical results show that, owing to its local properties, DWT spectrum is less affected by the sampling rate, survey geometry, and statistical ensemble fluctuations. With fast wavelet decomposition algorithm, DWT can be used to analyze large survey samples, which is of direct significance in precise measurement of the cosmological parameters from the galaxy redshift surveys of next generation.

Frequency-dependent specific heat of viscous silica

We apply the Mori-Zwanzig projection operator formalism to obtain an expression for the frequency dependent specific heat $c(z)$ of a liquid. By using an exact transformation formula due to Lebowitz et al., we derive a relation between $c(z)$ and $K(t),$ the autocorrelation function of temperature fluctuations in the microcanonical ensemble. This connection thus allows to determine $c(z)$ from computer simulations in equilibrium, i.e., without an external perturbation. By considering the generalization of $K(t)$ to finite wavevectors, we derive an expression to determine the thermal conductivity $\ensuremath{\lambda}$ from such simulations. We present the results of extensive computer simulations in which we use the derived relations to determine $c(z)$ over eight decades in frequency, as well as $\ensuremath{\lambda}.$ The system investigated is a simple but realistic model for amorphous silica. We find that at high frequencies the real part of $c(z)$ has the value of an ideal gas. ${c}^{\ensuremath{'}}(\ensuremath{\omega})$ increases quickly at those frequencies which correspond to the vibrational excitations of the system. At low temperatures ${c}^{\ensuremath{'}}(\ensuremath{\omega})$ shows a second step. The frequency at which this step is observed is comparable to the one at which the \ensuremath{\alpha}-relaxation peak is observed in the intermediate scattering function. Also the temperature dependence of the location of this second step is the same as the one of the $\ensuremath{\alpha}$ peak, thus showing that these quantities are intimately connected to each other. From ${c}^{\ensuremath{'}}(\ensuremath{\omega})$ we estimate the temperature dependence of the vibrational and configurational part of the specific heat. We find that the static value of $c(z)$ as well as $\ensuremath{\lambda}$ are in good agreement with experimental data.