Jun Ohkubo

The stochastic pump current and the non-adiabatic geometrical phase

We calculate a pump current in a classical two-state stochastic chemical kinetics by means of the non-adiabatic geometrical phase interpretation. The two-state system is attached to two particle reservoirs, and under a periodic perturbation of the kinetic rates, it gives rise to a pump current between the two-state system and the absorbing states. In order to calculate the pump current, the Floquet theory for the non-adiabatic geometrical phase is extended from a Hermitian case to a non-Hermitian case. The dependence of the pump current on the frequency of the perturbative kinetic rates is explicitly derived, and a stochastic resonance-like behavior is obtained.

Molecular alignment in a liquid induced by a nonresonant laser field: Molecular dynamics simulation.

We carried out molecular dynamics (MD) simulations for a dilute aqueous solution of pyrimidine in order to investigate the mechanisms of field-induced molecular alignment in a liquid phase. An anisotopically polarizable molecule can be aligned in a liquid phase by the interaction with a nonresonant intense laser field. We derived the effective forces induced by a nonresonant field on the basis of the concept of the average of the total potential over one optical cycle. The results of MD simulations show that a pyrimidine molecule is aligned in an aqueous solution by a linearly polarized field of light intensity I approximately 10(13) W/cm2 and wavelength lambda = 800 nm. The temporal behavior of field-induced alignment is adequately reproduced by the solution of the Fokker-Planck equation for a model system in which environmental fluctuations are represented by Gaussian white noise. From this analysis, we have revealed that the time required for alignment in a liquid phase is in the order of the reciprocals of rotational diffusion coefficients of a solute molecule. The degree of alignment is determined by the anisotropy of the polarizability of a molecule, light intensity, and temperature. We also discuss differences between the mechanisms of optical alignment in a gas phase and a liquid phase.

Transition Phenomena Induced by Internal Noise and Quasi-Absorbing State

We study a simple chemical reaction system and effects of the internal noise. The chemical reaction system causes the same transition phenomenon discussed by Togashi and Kaneko [Phys. Rev. Lett. 86 (2001) 2459; J. Phys. Soc. Jpn. 72 (2003) 62]. By using the simpler model than Togashi-Kaneko’s one, we discuss the transition phenomenon by means of a random walk model and an effective model. The discussion makes it clear that quasi-absorbing states, which are produced by the change of the strength of the internal noise, play an important role in the transition phenomenon. Stabilizing the quasi-absorbing states causes bifurcation of the peaks in the stationary probability distribution discontinuously.

Current and fluctuation in a two-state stochastic system under nonadiabatic periodic perturbation.

We calculate a current and its fluctuation in a two-state stochastic system under a periodic perturbation. The system could be interpreted as a channel on a cell surface or a single Michaelis-Menten catalyzing enzyme. It has been shown that the periodic perturbation induces a so-called pump current, and the pump current and its fluctuation are calculated with the aid of the geometrical phase interpretation. We give a simple calculation recipe for the statistics of the current, especially in a nonadiabatic case. The calculation scheme is based on the nonadiabatic geometrical phase interpretation. Using the Floquet theory, the total current and its fluctuation are calculated, and it is revealed that the average of the current shows a stochastic-resonance-like behavior. In contrast, the fluctuation of the current does not show such behavior.

Replica Analysis of Preferential Urn Model

We analyze a preferential urn model with randomness using the replica method. The preferential urn model is a stochastic model based on the concept “the rich get richer.” In the replica analysis, we clarify that the preferential urn model with randomness shows a fat-tailed occupation distribution. We discuss the intuitive picture of the preferential urn model, a relationship between quenched and annealed systems, and a trial for the application of the preferential urn model to econophysics. The analytical treatments and results would be useful for various research fields such as complex networks, stochastic models, and econophysics.

Digital Image Inpainting based on Markov Random Field

Digital image inpainting means reconstruction of small damaged portions of images. In the present paper, we propose a novel algorithm for digital image inpainting based on the Markov random field model. The proposed algorithm requires only specifying damaged regions to be inpainted, and works automatically, i.e., it does not require any user intervention. In addition, our method is very simple to be implemented and fast

Duality in Interacting Particle Systems and Boson Representation

In the context of Markov processes, we show a new scheme to derive dual processes and a duality function based on a boson representation. This scheme is applicable to a case in which a generator is expressed by boson creation and annihilation operators. For some stochastic processes, duality relations have been known, which connect continuous time Markov processes with discrete state space and those with continuous state space. We clarify that using a generating function approach and the Doi-Peliti method, a birth-death process (or discrete random walk model) is naturally connected to a differential equation with continuous variables, which would be interpreted as a dual Markov process. The key point in the derivation is to use bosonic coherent states as a bra state, instead of a conventional projection state. As examples, we apply the scheme to a simple birth-coagulation process and a Brownian momentum process. The generator of the Brownian momentum process is written by elements of the SU(1,1) algebra, and using a boson realization of SU(1,1) we show that the same scheme is available.

Approximation scheme for master equations: Variational approach to multivariate case

We study an approximation scheme based on a second quantization method for a chemical master equation. Small systems, such as cells, could not be studied by the traditional rate equation approach because fluctuation effects are very large in such small systems. Although a Fokker-Planck equation obtained by the system size expansion includes the fluctuation effects, it needs large computational costs for complicated chemical reaction systems. In addition, discrete characteristics of the original master equation are neglected in the system size expansion scheme. It has been shown that the use of the second quantization description and a variational method achieves tremendous reduction in the dimensionality of the master equation approximately, without loss of the discrete characteristics. Here, we propose a new scheme for the choice of variational functions, which is applicable to multivariate cases. It is revealed that the new scheme gives better numerical results than old ones and the computational cost increases only slightly.

Algebraic Probability, Classical Stochastic Processes, and Counting Statistics

We study a connection between the algebraic probability and classical stochastic processes described by master equations. Introducing a definition of a state which has not been used for quantum cases, the classical stochastic processes can be reformulated in terms of the algebraic probability. This reformulation immediately gives the Doi–Peliti formalism, which has been frequently used in nonequilibrium physics. As an application of the reformulation, we give a derivation of basic equations for counting statistics, which plays an important role in nonequilibrium physics.

Power-law behavior and condensation phenomena in disordered urn models

We investigate the equilibrium statistical properties of urn models with disorder. Two urn models are proposed; one belongs to the Ehrenfest class, and the other corresponds to the Monkey class. These models are introduced from the view point of the power-law behavior and randomness; it is clarified that quenched random parameters play an important role in generating the power-law behavior. We evaluate the occupation probability P(k) with which an urn has k balls by using the concept of statistical physics of disordered systems. In the disordered urn model belonging to the Monkey class, we find that above critical density ?c for a given temperature, condensation phenomenon occurs and the occupation probability changes its scaling behavior from an exponential law to a heavy tailed power law in large-k regime. We also discuss an interpretation of our results for explaining macro-economy, in particular the emergence of wealth differentials.