Ergodic Hypothesis Research Articles

Analyzing the dynamics of cell cycle processes from fixed samples through ergodic principles

Tools to analyze cyclical cellular processes, particularly the cell cycle, are of broad value for cell biology. Cell cycle synchronization and live-cell time-lapse observation are widely used to analyze these processes but are not available for many systems. Simple mathematical methods built on the ergodic principle are a well-established, widely applicable, and powerful alternative analysis approach, although they are less widely used. These methods extract data about the dynamics of a cyclical process from a single time-point “snapshot” of a population of cells progressing through the cycle asynchronously. Here, I demonstrate application of these simple mathematical methods to analysis of basic cyclical processes—cycles including a division event, cell populations undergoing unicellular aging, and cell cycles with multiple fission (schizogony)—as well as recent advances that allow detailed mapping of the cell cycle from continuously changing properties of the cell such as size and DNA content. This includes examples using existing data from mammalian, yeast, and unicellular eukaryotic parasite cell biology. Through the ongoing advances in high-throughput cell analysis by light microscopy, electron microscopy, and flow cytometry, these mathematical methods are becoming ever more important and are a powerful complementary method to traditional synchronization and time-lapse cell cycle analysis methods.

Ergodicity test of the eddy-covariance technique

Abstract. The ergodic hypothesis is a basic hypothesis typically invoked in atmospheric surface layer (ASL) experiments. The ergodic theorem of stationary random processes is introduced to analyse and verify the ergodicity of atmospheric turbulence measured using the eddy-covariance technique with two sets of field observational data. The results show that the ergodicity of atmospheric turbulence in atmospheric boundary layer (ABL) is relative not only to the atmospheric stratification but also to the eddy scale of atmospheric turbulence. The eddies of atmospheric turbulence, of which the scale is smaller than the scale of the ABL (i.e. the spatial scale is less than 1000 m and temporal scale is shorter than 10 min), effectively satisfy the ergodic theorems. Under these restrictions, a finite time average can be used as a substitute for the ensemble average of atmospheric turbulence, whereas eddies that are larger than ABL scale dissatisfy the mean ergodic theorem. Consequently, when a finite time average is used to substitute for the ensemble average, the eddy-covariance technique incurs large errors due to the loss of low-frequency information associated with larger eddies. A multi-station observation is compared with a single-station observation, and then the scope that satisfies the ergodic theorem is extended from scales smaller than the ABL, approximately 1000 m to scales greater than about 2000 m. Therefore, substituting the finite time average for the ensemble average of atmospheric turbulence is more faithfully approximate the actual values. Regardless of vertical velocity or temperature, the variance of eddies at different scales follows Monin–Obukhov similarity theory (MOST) better if the ergodic theorem can be satisfied; if not it deviates from MOST. The exploration of ergodicity in atmospheric turbulence is doubtlessly helpful in understanding the issues in atmospheric turbulent observations and provides a theoretical basis for overcoming related difficulties.

Estimating error in diffusion coefficients derived from molecular dynamics simulations.

The computationally expensive nature of molecular dynamics simulation limits the access to length (nanometer) and time scales (nanosecond) that are orders of magnitude smaller than the experiment it models. This limitation warrants a careful estimation of statistical uncertainty associated with the properties calculated from these simulations. The assumption that a simulation is long enough so that the ergodic hypothesis applies is often invoked in the literature for the computation of properties of interest from a single molecular dynamics simulation. Here, we demonstrate that making this assumption without validation results in poor estimates of the self-diffusion coefficient from a single molecular dynamics simulation of Lennard-Jones fluid. This problem is shown to be even more severe when the diffusion coefficient of macromolecules is calculated from a single molecular dynamics simulation. We have shown that conducting multiple independent simulations is necessary to obtain reliable estimates of diffusion coefficients and their associated statistical uncertainties. We show that even a “routine” calculation of the self-diffusion coefficient for a Lennard-Jones fluid, as determined from a linear fit of the mean squared displacement of particles as a function of time, violates the key assumptions of linear regression. A rigorous approach for addressing these issues is presented.

The Spatio-temporal Statistical Structure and Ergodic Behaviour of Scalar Turbulence Within a Rod Canopy

Connections between the spatial and temporal statistics of turbulent flow, and their possible convergence to ensemble statistics as assumed by the ergodic hypothesis, are explored for passive scalars within a rod canopy. While complete ergodicity is not expected to apply over all the spatial domain within such heterogeneous flows, the fact that canopy turbulence exhibits self-similar characteristics at a given depth within the canopy encourages a discussion on necessary conditions for an ‘operational’ ergodicity framework. Flows between roughness elements such as within canopies exhibit features that distinguish them from their well-studied classical boundary-layer counterparts. These differences are commonly attributed to short-circuiting of the energy cascade and the prevalence of intermittent von Karman vortex streets in the deeper layers of the canopy. Using laser-induced fluorescence measurements at two different depths within a rod canopy situated in a large flume, the spatio-temporal statistical properties and concomitant necessary conditions for ergodicity of passive scalar turbulence statistics are evaluated. First, the integral time and length scales are analyzed and their corresponding maximum values are used to guide the construction of an ensemble of independent realizations from repeated spatio-temporal concentration measurements. As a statistical analysis for an operational ergodicity check, a Kolmogorov–Smirnov test on the distributions of temporal and spatial concentration series against the ensemble was conducted. The outcome of this test reveals that ergodicity is reasonably valid over the entire domain except close to the rod elements where wake-induced inhomogeneities and damped turbulence prevail. The spatial concentration statistics within a grid-cell (square domain formed by four corner rods) appear to be less ergodic than their temporal counterparts, which is not surprising given the periodicity and persistence of von Karman vortices in the flow field. Also, a local advection velocity of dominant eddies is inferred using lagged cross-correlations of scalar concentration time series at different spatial locations. The computed probability density function of this advection velocity agrees well with the laser Doppler anemometry measurements for the same rod canopy.

A new approach to Boltzmann’s ergodic hypothesis

The worst (in the sense of convergence to an equilibrium) many-particle Hamiltonian system is considered, namely, a linear system of dimension 2N for which there is an N-parameter family of invariant tori of dimension N. We ask what minimal contact with the external world ensures convergence to the Liouville measure on the surface of constant energy from any initial state on this surface. The simplest example of such a contact is as follows: the velocity of a distinguished particle changes sign at discrete moments of time, and between these moments, particles move according to Hamiltonian dynamics. The only deviation from determinacy is that the intervals between the moments of sign changes are assumed to be independent identically distributed random variables.

Classical Ergodicity and Modern Portfolio Theory

What role have theoretical methods initially developed in mathematics and physics played in the progress of financial economics? What is the relationship between financial economics and econophysics? What is the relevance of the “classical ergodicity hypothesis” to modern portfolio theory? This paper addresses these questions by reviewing the etymology and history of the classical ergodicity hypothesis in 19th century statistical mechanics. An explanation of classical ergodicity is provided that establishes a connection to the fundamental empirical problem of using nonexperimental data to verify theoretical propositions in modern portfolio theory. The role of the ergodicity assumption in the ex post/ex ante quandary confronting modern portfolio theory is also examined.

A rejoinder to O'Donnell's critique of the ergodic/nonergodic explanation of Keynes's concept of uncertainty

:O’Donnell’s attack on the ergodic/nonergodic approach to Keynes’s uncertainty concept is erroneous. Keynes’s produced a general theory by imposing fewer restrictive axioms than classical theory. Keynes’s explicitly attacked the classical theory’s axiom that an actuarially certain knowledge regarding the future is available to decision makers; thus Keynes’s analysis rejects the ergodic hypothesis. Keynes’s theory does not substitute the assumption that all economic activity must be nonergodic. Keynes’s theory simply permits an analysis where the payout for some economic decisions (e.g., “animal spirits” investment spending) are made under uncertainty, i.e., without actuarial knowledge of the future payout of such spending decisions. O’Donnell’s epistemological approach implicitly supports the laissez faire view that there should not be any government interference with operation of free product and financial markets for government interference policies may only worsen future outcomes in a world where humans including Keynes, economists, and government officials do not have the capability to understand how the economy works.

Get Your kICS by Measuring Membrane Protein Dynamics

Akin to thepuckon an ice hockey rink,the lateral dynamics of a representativemembrane protein exhibits rich be-havior(1).Dependingontheinteractionwiththehockeyplayersandotherobsta-cles (membrane constituents), the puck(protein)candart aroundthe ice(mem-brane surface) or be stalled momen-tarily (transient confinement zone). Ifwe watched the motion of a singlepuck, we would need to watch manyhockey games beforewe might eventu-allyworkouttherulesofthegame.Butwhat if we could watch many games inparallel? This is where the suite of im-age correlation spectroscopy (ICS)techniques and its newest member, k-space image correlation spectroscopy(kICS) comes in Abu-Arish et al. (2).ICSandkICSaremembersofafam-ily of techniques which began withfluorescence correlation spectroscopy(3). Fluorescence fluctuation analysisprovides the statistical mechanicalfoundation with correlations betweenfluorescence fluctuations measuredrelative to a given lag vector. Correla-tions can be made as a function oftime (fixed space) called FCS (3), fluc-tuationsinspace(fixedtime)calledICS(4), and space-time fluctuations calledSTICS (5). The ergodic principle en-sures equivalence between occupancyfluctuations whether in time or space.In temporal ICS or TICS, one essen-tially images in time (using a laserscanning fluorescence confocal micro-scope) many molecules in parallel asthey change positions from one imageto the next image. If the molecules donot move between the first image andthe second image, the spatial correla-tion between image one and imagetwo will be high. However, if move-ment does occur between image oneand imagetwo,there isaloss of spatialcorrelation. By analyzing the correla-tion between images collected atdifferent times, information on proteinmotioncouldbeobtained.Thismethodprovidesinformation from manymole-cules without the need for tracking in-dividual particles. A problem endemicto all these methods is that fluctuationsin fluorescence are measured asopposed to particle fluctuations whichare the desired quantities in evaluatingtransport properties.In kICS (6), images collected from atime-seriesarefirstFouriertransformedinto k-space before image cross-corre-lation is performed. This has twodistinct advantages over (real space)ICS. First, fluctuations from photophy-sics (e.g., cis-trans isomerization, (de)protonation, reversible dark statequenching, reversible photobleaching)do not contribute to the determinationof transport coefficients (6). Second,determinationofthelateralpointspreadfunction dimension is not required, un-like conventional fluorescence fluctua-tion approaches (6).In this issueof the Biophysical Jour-nal, the pioneer of ICS and kICS, PaulWisemanhasteamedupwithAsmahanAbu-Arish,ElvisPandzic,JulieGoepp,Elizabeth Matthes, and John Hanrahan(2) to gain new insights into the dy-namics of a biomedically importantmembrane protein, the cystic fibrosistransmembrane conductance regulator.Using the kICS technique, the authorshaveprovidedevidencefortwopopula-tions of cystic fibrosis transmembraneconductance regulator (CFTR) mole-cules, which differed in degree ofconfinement and lateral motion on thecell surface. Impressively, they wereable to extract information on the dy-namics of CFTR inside domains,CFTR dynamicsoutside(andbetween)domains, fractional populations, anddegree of confinement (within do-mains). ICS analysis delivered clusterdensities and mean number of mole-cules per cluster.The authors examined the effect ofcholesterol on dynamics and clusteringbehavior of CFTR. Cholesterol has aremarkable effect on membrane pro-tein assembly, dynamics and function(7–9). Depletion of cholesterol causedthe confinedfractionand averagenum-ber of CFTR molecules per cluster todecrease, whereas increase in choles-terol were found to be associated withincrease in clustering and increasedconfined fraction. Interestingly, viralinfection was shown to increase clus-tering further into larger platformswith reduced CFTR mobility. Theseobservations and analyses suggestthat cholesterol-influenced membranedomains play an important role in thecell surface behavior and pathologyof CFTR.Aside from the important new in-sights into this anion channel, the re-sults of this study revealed howcomplex cell surface dynamics andclustering can be measured usingpowerful fluorescence microscopytechniques. With these new methodsin hand, biophysicists can sit back,relax and enjoy the hockey game.REFERENCES

New algorithm for simulation of 3D classical spin glasses under the influence of external electromagnetic fields

We study statistical properties of 3D classical spin-glass under the influence of external fields. It is proved that in the framework of the nearest-neighboring model 3D spin-glass problem at performing of Birkhoff’s ergodic hypothesis regarding to orientations of spins in the 3D space can be reduced to the problem of disordered 1D spatial spin-chains (SSC) ensemble where each spin-chain interacts with a random environment. The 1D SSC is defined as a periodic 1D lattice, where spins in nodes are randomly oriented in 3D space, in addition they all interact with each other randomly. For minimization of the Hamiltonian in an arbitrary node of the 1D lattice lattice, we obtained recurrent equations and corresponding Sylvester criterion, which allow to find energy local minimum. On the bases of these equations the high-performance parallel algorithm is developed which allows to calculate all statistical parameters of 3D spin glass, including distribution of a constant of spin-spin interaction, from the first principles of the classical mechanics.

A stochastic model of inward diffusion in magnetospheric plasmas

The inward diffusion of particles, often observed in magnetospheric plasmas (either naturally created stellar ones or laboratory devices), creates a spontaneous density gradient, which seemingly contradicts the entropy principle. We construct a theoretical model of diffusion that can explain the inward diffusion in a dipole magnetic field. The key is the identification of the proper coordinates on which an appropriate diffusion operator can be formulated. The effective phase space is foliated by the adiabatic invariants; on the symplectic leaf, the invariant measure (by which the entropy must be calculated) is distorted, by the inhomogeneous magnetic field, with respect to the conventional Lebesgue measure of the natural phase space. The collision operator is formulated to be consistent to the ergodic hypothesis on the symplectic leaf, i.e., the resultant diffusion must diminish gradients on the proper coordinates. The non-orthogonality of the cotangent vectors of the configuration space causes a coupling between the perpendicular and parallel diffusions, which is derived by applying Ito’s formula of changing variables. The model has been examined by numerical simulations. We observe the creation of a peaked density profile that mimics radiation belts in planetary magnetospheres as well as laboratory experiments.

Maximal coupling of empirical copulas for discrete vectors

For a vector X with a purely discrete multivariate distribution, we give simple short proofs of uniform a.s. convergence on their whole domain of two versions of genuine empirical copula functions, obtained either via probabilistic continuation, i.e. kernel smoothing, or via the distributional transform. These results give a positive answer to some delicate issues related to the convergence of copula functions in the discrete case. They are obtained under the very weak hypothesis of ergodicity of the sample, a framework which encompasses most types of serial dependence encountered in practice. Moreover, they allow to derive, as simple corollaries, almost sure consistency results for some recent extensions of concordance measures attached to discrete vectors. The proofs are based on a maximal coupling construction of the empirical cdf, a result of independent interest.

Ergodic theorem, ergodic theory, and statistical mechanics

This perspective highlights the mean ergodic theorem established by John von Neumann and the pointwise ergodic theorem established by George Birkhoff, proofs of which were published nearly simultaneously in PNAS in 1931 and 1932. These theorems were of great significance both in mathematics and in statistical mechanics. In statistical mechanics they provided a key insight into a 60-y-old fundamental problem of the subject—namely, the rationale for the hypothesis that time averages can be set equal to phase averages. The evolution of this problem is traced from the origins of statistical mechanics and Boltzman's ergodic hypothesis to the Ehrenfests' quasi-ergodic hypothesis, and then to the ergodic theorems. We discuss communications between von Neumann and Birkhoff in the Fall of 1931 leading up to the publication of these papers and related issues of priority. These ergodic theorems initiated a new field of mathematical-research called ergodic theory that has thrived ever since, and we discuss some of recent developments in ergodic theory that are relevant for statistical mechanics.

Brief Communication: Understanding disasters and early-warning systems

Abstract. This paper discusses some methodological questions on understanding disasters. Destructive earthquakes continue to claim thousands of lives. Tsunamis may be caused by recoil of the upper plate. Darwin's twin-epicenter hypothesis is applied to a theory of tsunamis. The ergodicity hypothesis may help to estimate the return periods of extremely rare events. A social science outline on the causation of the Tôhoku nuclear disaster is provided.

Collisionless electron heating in periodic arrays of inductively coupled plasmas

A novel mechanism of collisionless heating in large planar arrays of small inductive coils operated at radio frequencies is presented. In contrast to the well-known case of non-local heating related to the transversal conductivity, when the electrons move perpendicular to the planar coil, we investigate the problem of electrons moving in a plane parallel to the coils. Two types of periodic structures are studied. Resonance velocities where heating is efficient are calculated analytically by solving the Vlasov equation. Certain scaling parameters are identified. The concept is further investigated by a single particle simulation based on the ergodic principle and combined with a Monte Carlo code allowing for collisions with Argon atoms. Resonances, energy exchange, and distribution functions are obtained. The analytical results are confirmed by the numerical simulation. Pressure and electric field dependences are studied. Stochastic heating is found to be most efficient when the electron mean free path exceeds the size of a single coil cell. Then the mean energy increases approximately exponentially with the electric field amplitude.

Microscopic pressure-cooker model for studying molecules in confinement

A model for a system of a finite number of molecules in confinement is presented and expressions for determining the temperature, pressure, and volume of the system are derived. The present model is a generalisation of the Zwanzig–Langevin model because it includes pressure effects in the system. It also has general validity, preserves the ergodic hypothesis, and provides a formal framework for previous studies of hydrogen clusters in confinement. The application of the model is illustrated by an investigation of a set of prebiotic compounds exposed to varying pressure and temperature. The simulations performed within the model involve the use of a combination of molecular dynamics and density functional theory methods implemented on a computer system with a mixed CPU–GPU architecture.

Fractional Sampling Theorem for <formula formulatype="inline"> <tex Notation="TeX">$\alpha$</tex> </formula>-Bandlimited Random Signals and Its Relation to the von Neumann Ergodic Theorem

Considering that fractional correlation function and the fractional power spectral density, for α-stationary random signals, form a fractional Fourier transform pair. We present an interpolation formula to estimate a random signal from a temporal random series, based on the fractional sampling theorem for α-bandlimited random signals. Furthermore, by establishing the relationship between the sampling theorem and the von Neumann ergodic theorem, the estimation of the power spectral density of a random signal from one sample signal becomes a suitable approach. Thus, the validity of the sampling theorem for random signals is closely linked to an ergodic hypothesis in the mean sense.

Evidence for broken ergodicity due to chemical alloying from the dissociation kinetics of binary clusters.

The interplay between thermal relaxation and statistical dissociation in binary Morse clusters (AB)N has been investigated using numerical simulations and simple statistical approaches, for a variety of interaction parameters covering miscible and non-miscible regimes. While all clusters exhibit a core/shell phase separation pattern in their most stable, T = 0 structure, different melting mechanisms are identified depending on the ranges and their mismatch, including two-step melting of the surface and the core or premelting as alloying. The preference for emitting A or B particles upon evaporation has been evaluated assuming that the cluster is either thermally equilibrated or vibrationally excited in its ground state structure, and compared to the predictions of the Weisskopf theory. The variations of the dissociation rate constants with increasing energy and the branching ratio between the two channels show significant differences in both cases, especially when the clusters are miscible and bound by short-range forces, which indicates that the time scale for evaporation is much shorter than the equilibration time. Our results suggest that dissociation properties could be used to test the ergodic hypothesis in such compounds.

How Fast is Your Camera? Timescales for Molecular Motion and their Role in Restraining Molecular Dynamics

How Fast is Your Camera? Timescales for Molecular Motion and their Role in Restraining Molecular Dynamics

Enhancing retinal images by extracting structural information

High-resolution imaging of the retina has significant importance for science: physics and optics, biology, and medicine. The enhancement of images with poor contrast and the detection of faint structures require objective methods for assessing perceptual image quality. Under the assumption that human visual perception is highly adapted for extracting structural information from a scene, we introduce a framework for quality assessment based on the degradation of structural information. We implemented a new processing technique on a long sequence of retinal images of subjects with normal vision. We were able to perform a precise shift-and-add at the sub-pixel level in order to resolve the structures of the size of single cells in the living human retina. Last, we quantified the restoration reliability of the distorted images using an improved quality assessment. To that purpose, we used the single image restoration method based on the ergodic principle, which has originated in solar astronomy, to deconvolve aberrations after adaptive optics compensation.

Relationship between temporal and spatial averages in grid turbulence

AbstractA long-time direct numerical simulation (DNS) based on the lattice Boltzmann method is carried out for grid turbulence with the view to compare spatially averaged statistical properties in planes perpendicular to the mean flow with their temporal counterparts. The results show that the two averages become equal a short distance downstream of the grid. This equality indicates that the flow has become homogeneous in a plane perpendicular to the mean flow. This is an important result, since it confirms that hot-wire measurements are appropriate for testing theoretical results based on spatially averaged statistics. It is equally important in the context of DNS of grid turbulence, since it justifies the use of spatial averaging along a lateral direction and over several realizations for determining various statistical properties. Finally, the very good agreement between temporal and spatial averages validates the comparison between temporal (experiments) and spatial (DNS) statistical properties. The results are also interesting because, since the flow is stationary in time and spatially homogeneous along lateral directions, the equality between the two types of averaging provides strong support for the ergodic hypothesis in grid turbulence in planes perpendicular to the mean flow.