Diffusion in a bistable potential

The problem of diffusion of a particle in a bistable potential is studied on the basis of the one-dimensional Smoluchowski equation for the space- and time-dependent probability distribution. The potential is modeled as two parabolic wells separated by a parabolic barrier. For the model potential the Smoluchowski equation is solved exactly by a Laplace transform with respect to time for the initial condition that at time zero the probability distribution is given by a thermal equilibrium distribution in one of the wells. In the limit of a high barrier the rate of transition to the other well is given by an asymptotic result due to Kramers. For a potential barrier of moderate height there are significant corrections to the asymptotic result.     

Diffusion in a symmetric bistable potential: A variational approach

An approximation procedure for the solution of stochastic nonlinear equations, which was derived from a variational principle in a previous paper, is applied to the problem of a particle that diffuses in a symmetric bistable potential starting from the point of unstable equilibrium. The second moment and variance for the particle's position are calculated as functions of the timet. Good agreement is found with results recently obtained by Baibuzet al. from an approximate evaluation of a path integral expression for the probability density.     

Diffusion in a bistable potential: A systematic WKB treatment

We study the distributionP of a single stochastic variable, the evolution of which is described by a Fokker-Planck equation with a first moment deriving from a bistable potential, in the limit of constant and small diffusion coefficient. A systematic WKB analysis of the lowest eigenmodes of the equivalent Schrödinger-like equation yields the following results: the final approach to equilibrium is governed by the Kramers high-viscosity rate, which is shown to be exact in this limit; for intermediate times, we show that Suzuki's scaling statement does give the correct behavior for the transition between the one-peak and the two-peak structure forP. However, the intermediate time domain also contains a second half, whereP enters the diffusive equilibrium regions, characterized by a time scale of the same order as Suzuki's time.     

Diffusion in a bistable potential: The functional integral approach

We study, with the help of the Onsager-Machlup functional integral approach, the distributionP of a single stochastic variable, the evolution of which is described by a Fokker-Planck equation with a first moment deriving from a bistable potential. We set up the approximation scheme appropriate, in this approach, to the limit of constant and small diffusion coefficient. Two regimes are to be distinguished: Very long times (Kramers regime) are treated within the frame of a free-instanton-molecule gas approximation, and at intermediate times (Suzuki regime) a standard semiclassical calculation is legitimate. We thus rederive exactly the results obtained from the mode expansion and WKB method.We dedicate this work to our colleagues Yuri Orlov and Robert Nazarian.     

Diffusion in a bistable potential at intermediate and high friction

We study the motion of a Brownian particle in a bistable potential for intermediate and high-friction . Following ideas of Titulaer we perform a high-friction expansion of the distribution functionP(v,x,t) in velocity and space. We show (for arbitrary potential) that the expansion coefficients obey simple recursion relations, which allow them to be calculated easily. When terms of order ^–5 are neglected the resulting differential equations can be transformed into Hermitian Schrödinger-type equations. Using the WKB technique we solve these equations analytically for the case of the bistable potential and discuss the various time regimes involved in the system, in particular we show that the final approach to equilibrium is governed by the Kramers rate. Our results become exact in the limit of low temperatures.     

Diffusion in a bistable potential. General inverse friction expansion

The derivation of the characteristic times and of the density probability distribution for the motion of a Brownian particle in a bistable potential at intermediate friction was, until now, essentially limited to low orders in the inverse frictionγ ^?1. On the other hand, at least for temperatures low with respect to the barrier height, the Kramers time, which is the lowest nonzero eigenvalue in the bistable potential problem, is known exactly. This paper presents a systematic approach for the determination of the solution of the Fokker-Planck equation in an arbitrary potential in the overdamped regime. This calculation includes anharmonicity corrections up to orderγ ^?5. One feature of this paper is to show that the problem is equivalent to replacing the original potentialφ(x) by a free energy which, for a velocity distribution at equilibrium, simply is \(\widetilde\phi \) =φ(x) ?k _BT ln[g(x)], where $$g(x) = \left\{ {{{m\gamma } \mathord{\left/ {\vphantom {{m\gamma } {[2\phi ''(x)]}}} \right. \kern-\nulldelimiterspace} {[2\phi ''(x)]}}} \right\}\left\{ {1 - [{{1 - 4\phi ''(x)} \mathord{\left/ {\vphantom {{1 - 4\phi ''(x)} {m\gamma ^2 }}} \right. \kern-\nulldelimiterspace} {m\gamma ^2 }}]^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} } \right\}$$ For out-of-equilibrium velocity distribution an effective potential is also explicitly given. In every case the function g(x) plays a crucial role. This approach is then applied to the exact determination, in the low-temperature limit, of all the characteristic times and of the probability distribution in bistable potentials. Moreover, from the knowledge of the characteristic times and probability density distribution, it would be easy to determine the general and exact Suzuki scaling law for the relaxation from the instability point at intermediate friction.     

A WKB treatment of diffusion in a multidimensional bistable potential

We extend to the case of a finite set of stochastic variables whose distributionP obeys a nonlinear Fokker-Planck equation our previous treatment of diffusion in a bistable potentialU, in the limit of small, constant diffusion coefficient. This is done with the help of an extended WKB approximation due to Gervais and Sakita. The treatment is valid if there exists a well-defined most probable path connecting the minima ofU, and if the valley ofU along that path has a slowly varying width, and weak curvature and twisting. We find that: (i) the final approach to equilibrium is governed by Eyring's generalization of the Kramers high-viscosity rate, which we rederive; (ii) for intermediate times, if the initial distribution is concentrated in the region of instability (close vicinity of the saddle point ofU),P has, along the most probable path, the behavior described by Suzuki's scaling statement for a one-dimensional system. In a second part of this time domain,P enters the diffusive regions around the minima ofU and relaxes toward local longitudinal equilibrium on a time comparable with Suzuki's time scale. The time for relaxation toward transverse local equilibrium may, depending on the initial conditions, compete with these longitudinal times.We dedicate this work to our colleague, Yuri Orlov.     

A comparative study of solution based CIGS thin film growth on different glass substrates

CuIn_xGa_1−xSe_yS_2−y (CIGS) thin films were synthesized on glass substrates by a paste coating of Cu, In, and Ga precursor solution with a three-step heat treatment process: oxidation, sulfurization, and selenization. In particular, morphological changes of CIGS films for each heat treatment step were investigated with respect to the kinds of glass substrates: bare, Mo-coated, and F-doped SnO₂ (FTO) soda-lime glasses. Very high quality CIGS film with large grains and low degree of porosity was obtained on the bare glass substrate. Similar morphology of CIGS film was also acquired on the Mo-coated glass except the formation of an undesired Mo oxide interfacial layer due to the partial oxidation of Mo layer during the first heat treatment under ambient conditions. On the other hand, CIGS film with much smaller grains and higher degree of porosity was gained when FTO glass was used as a substrate, resulting in slight solar to electricity conversion behavior (0.20%). Higher power conversion efficiency (1.32%) was attained by the device with the CIGS film grown on Mo-coated glass in spite of the presence of a Mo oxide impurity layer.     

Particle Size Analysis: A comparative study of various methods

The particle size distribution of component materials can be important parameter governing the proper function of many industrial products. At present, accurate determination of particle size by currently available methods is difficult for both theoretical and practical reasons. In this paper, some elementary notions regarding particle sizing are reviewed as is the relative performance of commercially available particle sizing instrumentation which could be used for particle size analysis of dry powders. Our results indicate that various particle size instruments yield significantly (even vastly) different particle size distributions for the same material. Without great care on the part of the investigator, the results obtained from any given instrument may not accurately describe the powder in question and thus may cause incorrect technical recommendations. The use of Fraunhofer diffraction instruments on samples containing irregularly shaped small ( < 10 μm) particles of low refractive index requires considerable caution.     

Opinion dynamics as a movement in a bistable potential

In this paper, we investigate the Sznajd model of opinion dynamics with anticonformity on a complete graph. We show that below some threshold value of anticonformal behavior spontaneous reorientations occur between two stable states. Dealing with a complete graph allows us also for an analytical treatment. We provide analytical calculations both for the infinite and finite systems. We show that opinion dynamics can be understood as a movement of a public opinion in a symmetric bistable effective potential. We focus also on the spontaneous transitions between stable states in the case of the finite system and show that a typical waiting time can be observed.     

Different methods to calculate heat transfer coefficient in a double-tube heat exchanger: A comparative study

An experimental study performed to compare the results of different methods used in the literature for the calculation of heat transfer coefficient in double-tube heat exchangers. Then, a new fundamental approach was proposed which used the temperature profile and the local heat transfer coefficients. In this method, the heat transfer coefficient has been calculated for the total length of the heat exchanger including developing and fully developed regions. Numerous experimentations have been conducted in a double-tube heat exchanger. A significant difference observed between the results obtained from the suggested approach and those of the previous methods.     

Xyloglucan degradation using different radiation sources: a comparative study

Tamarind seed xyloglucan was subjected to different radiation sources-ultrasound, gamma-radiation, and microwave heating, and the effects of these energies upon its molecular and structural properties were characterised by gel permeation chromatography, viscometry, sugar analysis, FT-IR and NMR spectroscopic techniques. In dependence on the degradation methods and experimental conditions used, the decrease of the relative molecular mass (RMM) was accompanied with alteration of the primary structure. Depolymerisation by ultrasound at a frequency of 20 kHz yielded after 120 min products with RMM of about 131 x 10(3) without significant alteration of the primary structure of the polysaccharide. Intense degradation of XG started by microwave heating at pH 1.5 yielding polymers with RMM in the range of higher oligosaccharides, however, with changed sugar composition due to cleavage of the glycosyl side chains. At gamma-irradiation doses >40 kGy, next to chain cleavage, very high-molecular mass components exhibiting UV(254)-absorption were formed, and the RMM decreased to about 50 x 10(3) at the highest applied dose (100 kGy). The results of the comparative study suggest that ultrasonication was the most convenient procedure to decrease the RMM of xyloglucan to 130 x 10(3) and preserve the primary structure of the polysaccharide.     

A study of stochastic resonance in a bistable system with double stochasticity

In a simple stochastic system—an overdamped Kramers oscillator with two noise sources (sources of white and dichotomic noise)—stochastic resonance is investigated theoretically and by means of analog simulation as a function of the asymmetry of the potential and the amplitude and the correlation time of the dichotomic noise. It is found that stochastic resonance is observed under slow (compared with the Kramers switching time) dichotomic noise and the signal-to-noise ratio (SNR) has a maximum for a noise amplitude equal to static bias.     

Investigations of a nonlinear energy harvester with a bistable potential well

This paper investigates a nonlinear energy harvester that uses magnetic interactions to create an inertial generator with a bistable potential well. The motivating hypothesis for this work was that nonlinear behavior could be used to improve the performance of an energy harvester by broadening its frequency response. Theoretical investigations study the harvester's response when directly powering an electrical load. Both theoretical and experimental tests show that the potential well escape phenomenon can be used to broaden the frequency response of an energy harvester.     

A comparative study of the influence of different post-treatment methods on the properties of HfO2 single layers

Heat treatment, laser conditioning and ion post-treatment were used, respectively, on HfO₂ single layers deposited by electron beam evaporation (EBE). Optical, structural and laser-induced damage threshold (LIDT) properties of the films had been studied. It was found that all of the post-treatment methods are effective for improving the LIDT, but the mechanism for the improvement differs. Heat treatment in atmosphere was found to be effective for improving film stoichiometry. Laser conditioning was useful to decrease the defects by stress release. While in the ion post-treatment, the defects were removed by ion sputtering and the absorption was reduced at the effect of compression induced by ion bombardment. The most suitable method should be chosen for the required characteristics of the coatings.     

Exact solution for the diffusion in bistable potentials

We solve analytically the Fokker-Planck equation for a one-parameter family of symmetric, attractive, nonharmonic potentials which include double-well situations. The exact knowledge of the eigenfunctions and eigenvalues allows us to fully discuss the transient behavior of the probability density. In particular, for the bistable potentials, we can give analytical expressions for the probability current over the working barrier and for the onset time which characterizes the transition from uni- to bimodal probability densities.On leave from the Department of Theoretical Physics, Université de Genève, CH-1211, Genève 4, Switzerland.Supported by the Swiss National Fund for Scientific Research.On leave from the Institute of Theoretical Physics, Academia Sinica, Beijing, China.Supported in part by the Robert A. Welch Foundation.