Time-dependent Probability Density Research Articles

Population growth as a nonlinear stochastic process

The evolution of the probability density of a biological population is described using nonlinear stochastic differential equations for the growth process and the related Fokker-Planck equations for the time-dependent probability densities. It is shown that the effect of the initial conditions disappears rapidly from the evolution of the mean of the process. But the behaviour of the variance depends on the initial condition. It may monotonically increase, reaching its maximum in the steady state, or have a rather complicated evolution reaching the maximum near the point where growth rates (not population size) is maximal. The variance then decreases to its steady-state value. This observation has implications for risk assessments associated with growing populations, such as microbial populations, which cause food poisoning if the population size reaches a critical level.

Neural representation of probabilistic information.

It has been proposed that populations of neurons process information in terms of probability density functions (PDFs) of analog variables. Such analog variables range, for example, from target luminance and depth on the sensory interface to eye position and joint angles on the motor output side. The requirement that analog variables must be processed leads inevitably to a probabilistic description, while the limited precision and lifetime of the neuronal processing units lead naturally to a population representation of information. We show how a time-dependent probability density rho(x; t) over variable x, residing in a specified function space of dimension D, may be decoded from the neuronal activities in a population as a linear combination of certain decoding functions phi(i)(x), with coefficients given by the N firing rates a(i)(t) (generally with D << N). We show how the neuronal encoding process may be described by projecting a set of complementary encoding functions phi;(i)(x) on the probability density rho(x; t), and passing the result through a rectifying nonlinear activation function. We show how both encoders phi;(i)(x) and decoders phi(i)(x) may be determined by minimizing cost functions that quantify the inaccuracy of the representation. Expressing a given computation in terms of manipulation and transformation of probabilities, we show how this representation leads to a neural circuit that can carry out the required computation within a consistent Bayesian framework, with the synaptic weights being explicitly generated in terms of encoders, decoders, conditional probabilities, and priors.

Calculation of ac Electric Field Effects on the Average Height of a Charged Colloid: Effects of Electrophoretic and Brownian Motions

Numerical solutions are presented for the time-dependent probability density of a charged colloidal particle near a planar wall, subjected to an alternating electric field, Brownian motion, and DLVO forces. These are the first time-dependent probability distribution solutions for a driven particle in a mobility gradient. The effects of electrophoretic particle motion on the probability distribution for finding a single, 6 μm diameter, polystyrene particle at a given height above an indium−tin oxide surface were studied via numerical solution of the complete convective−diffusion equation, ignoring inertia, in one dimension. Application of an ac electric field monotonically increased the average particle height over a complete cycle of the electric field; the strongest increase was at low frequency and large electric field strengths. These results agreed with observed behavior in potassium hydroxide solutions at high voltages but did not capture the depression of the average height at low voltages. The calculations agreed with the monotonic increase of average height observed in sodium bicarbonate solutions, but the slope of the increase was far less than experimentally observed. The calculated effects are present in the experimental results but do not encompass all the observations.

Buildup dynamics of transmission resonances in superlattices

A closed analytical expression for the time-dependent probability density of transmitted particles in superlattices is derived from a formal solution of the time-dependent Schr\"odinger equation. Such an expression consists of Breit-Wigner type resonance terms and interference contributions with explicit time dependence, which is applied to different superlattices to describe step-by-step how the transmission resonances are constructed as a function of time. In particular, it is found that for incidence at a resonance of position ${\ensuremath{\varepsilon}}_{n}$ and width ${\ensuremath{\Gamma}}_{n},$ the buildup of the transmission peaks is governed by a simple exponential law ${\mathcal{T}}_{n}({\ensuremath{\varepsilon}}_{n}{,t)=T}_{n}^{\mathrm{peak}}[1\ensuremath{-}\mathrm{exp}(\ensuremath{-}{\ensuremath{\Gamma}}_{n}t/2\ensuremath{\Elzxh}){]}^{2},$ where ${\mathcal{T}}_{n}({\ensuremath{\varepsilon}}_{n},t)$ is the probability density at the right edge of the superlattice and ${T}_{n}^{\mathrm{peak}}$ the height of the corresponding transmission peak. We show that our results are valid for periodic superlattices as well as for asymmetrical or even disordered potential profiles.

Optimal use of time-dependent probability density data to extract potential-energy surfaces

A novel algorithm was recently presented to utilize emerging time dependent probability density data to extract molecular potential energy surfaces. This paper builds on the previous work and seeks to enhance the capabilities of the extraction algorithm: An improved method of removing the generally ill-posed nature of the inverse problem is introduced via an extended Tikhonov regularization and methods for choosing the optimal regularization parameters are discussed. Several ways to incorporate multiple data sets are investigated, including the means to optimally combine data from many experiments exploring different portions of the potential. Results are presented on the stability of the inversion procedure, including the optimal combination scheme, under the influence of data noise. The method is applied to the simulated inversion of a double well system.

Multilevel Redfield description of the dissipative dynamics at conical intersections

Redfield theory is applied to investigate the photoinduced dynamics at a conical intersection (the so-called system) which is weakly coupled to a thermal environment (the so-called bath). The dynamics of the system is described by a two-state three-mode model Hamiltonian, chosen to represent the S1(nπ*)–S2(ππ*) conical intersection in pyrazine. Dissipative effects are introduced through a bilinear coupling of the system vibrational modes with a harmonic bath, which represents the remaining vibrational degrees of freedom of the molecule and/or interactions with a condensed-phase environment. The Redfield equations for the reduced density matrix are solved numerically without further approximations. From the reduced density matrix the time evolutions of electronic-state populations and vibrational coherences are obtained, as well as time-dependent probability densities of individual vibrational modes. The results provide a visualization of the essential features of the ultrafast (time scale of 10 fs) internal-conversion process at the conical intersection and the ensuing vibrational cooling process on the lower adiabatic potential-energy surface. The effect of vibrational damping on the linear optical absorption spectrum is also investigated.

Vibrational spectra and intramolecular vibrational redistribution in highly excited deuterobromochlorofluoromethane CDBrClF: Experiment and theory

The rovibrational spectra of deuterobromochlorofluoromethane (CDBrClF) were measured at intermediate (0.1 cm−1) and high resolution (0.0024 cm−1 full bandwidth, half-maximum) by interferometric Fourier transform infrared spectroscopy in the range from the far infrared at 200 cm−1 to the near infrared (12 000 cm−1) covering all the fundamentals and CD stretching overtones up to polyad N=5. The spectra are completely analyzed in terms of their vibrational assignments to fundamentals, combinations and overtones. At high excitation the analysis reveals the dominant anharmonic coupling between four high frequency vibrational modes; the CD stretching (ν1), two CD bending (ν2,ν3), and the CF stretching mode (ν4). The analysis is carried out using effective model Hamiltonians including three and four vibrational degrees of freedom. We also present vibrational variational calculations on a grid in a four-dimensional normal coordinate subspace. The potential energy and the dipole moment function are calculated ab initio on this grid using self-consistent field second order Møller–Plesset perturbation theory (MP2). Experimental and theoretical results for band positions and integrated intensities as well as effective spectroscopic parameters are found to be in good agreement. The important anharmonic coupling between the CD chromophore and the CF stretching vibration can be described by an effective cubic Fermi resonance coupling constant ksff′≈(50±10) cm−1, which leads to intramolecular vibrational redistribution between the CD and CF chromophores on the femtosecond time scale. Time dependent intramolecular vibrational redistribution processes in CDBrClF are derived in various representations, including time dependent probability densities (“wave packets”) in coordinate space and finally time dependent entropy.

Dynamical Properties of Bidirectional Ring Laser with Both the Additive and Multiplicative Noises and Saturation Effect**The project supported by National Natural Science Foundation of China and Natural Science Foundation of Jiangsu Province of China

The dynamical properties of a two-mode ring laser with both the additive and multiplicative noises are analyzed. Time-dependent probability density is obtained by the method of one-dimensional approximation. Some special behaviors of and are discussed when the multiplicative pump noise and saturation effect are included in the laser theory.

Molecular Dipole Function Inversion from Time Dependent Probability Density and Electric Field Data

A noniterative quantum mechanical algorithm is presented to extract the dipole function from time dependent probability density and external electric field data. The algorithm determines the dipole function as the solution of an exact linear integral equation without the need to solve the Schrodinger equation. The inversion in regular regions of the dipole is accurate and stable under perturbations from noisy data. The regular regions of the dipole are automatically identified by the algorithm, and Tikhonov regularization is employed. There is much freedom in the external electric field with the best choices generally producing broad excitations of many eigenstates. Field designs may be estimated from the Hamiltonian or through closed loop learning techniques in the laboratory. The inversion algorithm is tested in a simulation for O−H, which shows that the algorithm is very reliable. Since the inversion algorithm is fast, it is argued that closed loop laboratory learning techniques may be applied to optim...

Transient equations for multiple nucleation on solid electrodes: A stochastic description

We consider the phenomenon of heterogeneous nucleation in electrolytic cells. Through the proposition of a stochastic model, we derive an equation to describe the migration of solute ions and their reaction near the electrode surface. Defining particular boundary conditions, the stochastic equation is solved and the time dependent probability density is related to the flux of ions towards the growing nucleus. The flux of ions is identified as the current transient expression, which is adequate for a comparison with the experimental results. The theory is powerful enough to reproduce results obtained by electrodepositing cobalt on n-type silicon and to explain deviations from predictions of the standard model for three-dimensional nucleation diffusion limited growth.

Potential surfaces from the inversion of time dependent probability density data

This article presents an algorithm to assess the feasibility of inverting probability density data to extract potential surfaces. Such data admit the generation of a noniterative quantum inversion algorithm that does not require the solution of the Schrödinger equation. Tikhonov regularization is employed to manage the singular nature of the problem. The inversion in regular regions has excellent accuracy, and an error analysis also indicates that the potential in the regular regions is stable under perturbations from noisy data. The regular regions of the potential are identified by the algorithm. The algorithm does not require knowledge of the excitation process initiating the evolution of the system. Analysis indicates that the most detailed potential surface information will result from broadband excitation leaving the molecule with significant population in as many quantum states as possible. The inversion algorithm is tested in a simulation for the O–H potential, which shows that the algorithm is very fast and reliable.

PROBABILITY DENSITIES FOR NOISY DELAY BIFURCATIONS

The delay of a transition in a nonlinear system due to a slowly varying control parameter can be significantly reduced by very small noise. A new asymptotic approximation for the time-dependent probability density function gives a complete description of the process into the transition region, and is easily interpreted in terms of the noisy dynamics. It is also used to calculate mean transition times. The method is applied to two nonlinear systems with noise: a one-dimensional canonical model for a steady bifurcation and the noisy FitzHugh–Nagumo model.

A Dynamic Model for the Attachment of a Brownian Particle Mediated by Discrete Macromolecular Bonds

A model is presented for the attachment of a Brownian particle to a surface mediated by both the conservative colloidal forces and the formation of macromolecular bonds. By considering Brownian motion and bond formation as coupled stochastic processes, the model derives a governing equation for the time-dependent probability density of having a given number of bonds and separation distance from the surface. The model predicts the deposition rate of particles to a surface as a function of the physicochemical parameters of the binding molecules, including the density, interaction length, stiffness, and formation and dissociation kinetic rate constants. Furthermore, two limiting simplifications of the full model are explored which correspond to particle attachment rate limited by the rate of Brownian motion or by the rate of bond formation.

Spacetime structures in simple quantum systems

Recently W Kinzel [1995 Phys. Bl. 51 1190] has argued that even simple quantum systems can exhibit surprising phenomena. As an example he presented the formation of canals and ridges in the time-dependent probability density of a particle caught in a square well with infinitely high walls. We show how these structures emerge from the wavefunction and present a simple derivation of their location in the spacetime continuum.

Displaced and squeezed number states

From a background of displaced (coherent) and squeezed states, previous (often forgotten) work on displaced and squeezed number states is reviewed. We derive the most general of such states. The time-dependent expectation values, uncertainties, wave-functions, and probability densities are obtained. We comment on the possibility of experimentally observing these states.

Nonspreading wave packets

The spreading or nonspreading of wave packets in one-dimensional linear and quadratic potentials can readily be determined by solving Ehrenfest’s equations. The results for quadratic potentials can be understood in simple intuitive terms by using quantum distribution functions in phase space which, like classical distributions, rotate clocklike in time once per motional cycle. For more complicated situations, such as potentials with quantum energy spacings which are integral multiples of a basic unit, much can be learned instead by studying the quantum recurrence of their time-dependent wave functions or probability densities.

A criterion for stochastic resonance

A criterion for stochastic resonance in a bistable system which is modulated by a harmonic force can be developed from the study of time dependent probability densities. The amplitude with which the probability density oscillates is a measure of the total probability flow between the potential wells. Stochastic resonance can be observed in an increase of the amplitude for specific values of the noise strength.

Response and Reliability of Poisson-Driven Systems by Path Integration

The paper deals with the stochastic response and the reliability of a nonlinear and nonhysteretic single-degree-of-freedom (SDOF) oscillator subject to a stationary Poisson-driven train of impulses. The state vector made up of the displacement and the velocity components then becomes a Markov process. The applied solution method is based on path integration, which essentially implies that a mesh of discrete states of the Markov vector process is initially defined with a suitable distribution throughout the phase plane, next, the transition probability matrix related to the Markov chain originating from this discretization is calculated, assuming the transition time interval to be sufficiently small so that at most one impulse is likely to arrive during the interval. Obviously, this assumption is best fulfilled for processes with low pulse arrival rates. Consequently, the method is the most effective in such cases in contrast to all other approaches to the considered problems. The time-dependent joint probability density function (PDF) of the displacement and velocity is obtained by passing the system through a sequence of transient states. In the reliability problems, the probability mass is absorbed at the exit part of the boundary of the safe domain during transitions. The considered first passage time problem assumes time-invariant single or double barriers with deterministic or stochastic start in the safe domain. The method has been applied to a Duffing oscillator with linear viscous damping, and the computed results have been compared with those obtained from extensive Monte Carlo simulations.

The Liouville Equation and Its Potential Usefulness for the Prediction of Forecast Skill. Part II: Applications

The Liouville equation represents the consistent and comprehensive framework for the treatment of the uncertainty inherent in meteorological forecasts. By its very nature, consideration of the Liouville equation avoids problems that are inherent to commonly used methods for predicting forecast skill, such as the need for higher-moment closure within stochastic-dynamic prediction, or the need for the generation of large ensemble sizes within ensemble forecasting. The general analytical solution of the Liouville equation presented in the first part of this work is used here to find the solution of the Liouville equation relevant for two low-dimensional nonlinear dynamical systems and to investigate the potential usefulness of the Liouville equation in the context of predicting forecast skill. The analytical solution of the Liouville equation in these examples, namely, the time-dependent probability density function, is discussed and compared with corresponding results obtained by stochastic-dynamic prediction and ensemble forecasting. The negative effect of the unavoidable higher-moment discard within stochastic-dynamic prediction is quantitatively demonstrated. It is also shown that a large number of ensemble members is necessary to obtain accurate estimates of statistics, such as means and (co)variances, even in the low-dimensional situations considered. It is concluded that, due to its fundamental role in dealing with initial state uncertainty in dynamical models, the Liouville equation must be considered as a highly valuable and useful guideline during the process of developing a coherent methodology for forecasting forecast skill that minimizes deficiencies of currently used methods.

Dynamics of the infinite-ranged Potts model

We formulate a theory of single-spin-flip dynamics for the infinite-rangeq-state Potts model. We derive a Fokker-Planck equation, without diffusive term, from a phenomenological master equation. It describes the approach to equilibrium of the time-dependent probability density and thus generalizes Griffiths' (1966) result for the Ising model. We particularly compare the dynamic evolutions ofq=2 andq=3 systems when sinusoidal external fields are applied. In the caseq=2 we find evidence of a nonequilibrium phase transition and forq=3 period doubling bifurcations are observed, yielding a good estimate of Feigenbaum's universal exponent.