Generalized Fokker-Planck equation, Brownian motion, and ergodicity

Microscopic theory of Brownian motion of a particle of mass $M$ in a bath of molecules of mass $m\ll M$ is considered beyond lowest order in the mass ratio $m/M$. The corresponding Langevin equation contains nonlinear corrections to the dissipative force, and the generalized Fokker-Planck equation involves derivatives of order higher than two. These equations are derived from first principles with coefficients expressed in terms of correlation functions of microscopic force on the particle. The coefficients are evaluated explicitly for a generalized Rayleigh model with a finite time of molecule-particle collisions. In the limit of a low-density bath, we recover the results obtained previously for a model with instantaneous binary collisions. In general case, the equations contain additional corrections, quadratic in bath density, originating from a finite collision time. These corrections survive to order $(m/M)^2$ and are found to make the stationary distribution non-Maxwellian. Some relevant numerical simulations are also presented

Verkorting van de kweekcyclus bij ui (Allium cepa L.)

With an annual production of about 250 000 to 300 000 tons, onion (Allium cepa L.) is the most important outdoor vegetable crop in the Netherlands. The greater part, namely 70 to 80 %, has to be exported, and is meeting increasing competition. For this reason and in connection with changed growing and handling methods, including mechanization, there is a need to breed improved varieties.Breeding-work with the biennial onion, however, takes much time and shortening of the breeding cycle would be very profitable. This has been the purpose of the present research. By means of literature data, all phases of growth and development .have been investigated. Flower induction and initiation appeared to be dependent on temperature, but not on day length.At first we tried to promote rapid flowering of the bulbs by means of temperature treatments (chapter 3.1). This did not give satisfactory results. Development was very irregular and slow and only exceptionnally all the plants became generative (table 3). Nevertheless, we obtained more information about the factors required for development. We showed that the optimum temperature for flower induction for the cultivar "Rijnsburger" is 9° C rather than 13° C (graph 4). Moreover, short heat treatments before and after the storage period stimulate the regrowth (graph 2), but counteract the floral development (antivernalisation and devernalisation). This explains why the yield of seeds after storage of the mother bulbs under supra-optimal temperatures is lower than after storage under sub-optimal temperatures.Abortion of the young scapes (flower stalks) occurs exclusively under long days and to a greater extent at high temperatures (table 6). Our observations indicate that abortion is dependent on the distribution of assimilates between the scape and an axillary bud (photo 3). A "bulb/flower competition hypothesis" was formulated in the sense that, just as with onions grown for bulb production, in long days and at higher temperatures this axillary bud swells, which results in abortion of the scape (photo 4).Hitherto rarely different stages of development in onion in relation to optimal conditions have been distinguished. For a good development different conditions of day length and temperature appeared to be required. For that reason an outline was proposed that consists of three phases:Thermo-phase . Flower induction and initiation are dependent on the temperature, irrespective of day length.Competition-phase . Under long day and high temperatures, abortion may occur but at low temperatures (about 9° C) the flower scape develops, even in long days.Completion-phase . For a good further development long days are necessary and a high temperature is beneficial.Based on this information an annual breeding cycle has been realized by using onion plants (chapter 3.2). Early in September the seeds were sown in a greenhouse under natural short day conditions, in which bulbing and the associated rest period are prevented. With supplementary light during the daytime and day and night- temperatures of 16° C and 13° C respectively, in about 100 days strong plants were obtained.In the middle of December this plant material was transplanted and kept under long days at 9° C, under which circumstances flower induction and initiation were promoted and bulbing and abortion were completely inhibited (photo 7). In April the temperature was raised to 18° C and in May to at least 21° C. Under these temperature conditions and natural long days without supplementary light, all the plants developed into good seed plants, the seeds of which were harvested by mid July. In the beginning of August, 11 months after the initial sowing, these new seeds could be sown.An extra gain in time of at least one month is possible, by using unripe seeds with a maximum moisture content of 50% (chapter 3.3). By using a higher light intensity during the raising period and further improvements in the day length and temperature regime during the competition and completion phases, it is probably possible to obtain four generations in three years.Finally, some possibilities for application of the results in breeding projects have been discussed (chapter 4)

On the theory of electric dc-conductivity : linear and non-linear microscopic evolution and macroscopic behaviour

We consider the Schrodinger time evolution of charged particles subject to a static substrate potential and to a homogeneous, macroscopic electric field (a magnetic field may also be present). We investigate the microscopic velocities and the resulting macroscopic current. We show that the microscopic velocities are in general non-linear with respect to the electric field. One kind of non-linearity arises from the highly non-linear adiabatic evolution and (or) from an admixture of parts of it in so-called intermediate states, and the other kind from non-quadratic transition rates between adiabatic states. The resulting macroscopic dc-current may or may not be linear in the field. Three cases can be distinguished : (a) The microscopic non-linearities can be neglected. This is assumed to be the case in linear response theory (Kubo formalism, ...). We give arguments which make it plausible that often such an assumption is indeed justified, in particular for the current parallel to the field. (b) The microscopic non-linearitites lead to macroscopic non-linearities. An example is the onset of dissipation by increasing the electric field in the breakdown of the quantum Hall effect. (c) The macroscopic current is linear although the microscopic non-linearities constitute an essential part of it and cannot be neglected. We show that the Hall current of a quantized Hall plateau belongs to this case. This illustrates that macroscopic linearity does not necessarily result from microscopic linearity. In the second and third cases linear response theory is inadequate. We elucidate also some other problems related to linear response theory.Comment: 24 pages, 6 figures, some typing errors have been corrected. Remark : in eq. (1) of the printed article an obvious typing error remain

Enhanced quantum tunnelling induced by disorder

We reconsider the problem of the enhancement of tunnelling of a quantum particle induced by disorder of a one-dimensional tunnel barrier of length $L$, using two different approximate analytic solutions of the invariant imbedding equations of wave propagation for weak disorder. The two solutions are complementary for the detailed understanding of important aspects of numerical results on disorder-enhanced tunnelling obtained recently by Kim et al. (Phys. rev. B{\bf 77}, 024203 (2008)). In particular, we derive analytically the scaled wavenumber $(kL)$-threshold where disorder-enhanced tunnelling of an incident electron first occurs, as well as the rate of variation of the transmittance in the limit of vanishing disorder. Both quantities are in good agreement with the numerical results of Kim et al. Our non-perturbative solution of the invariant imbedding equations allows us to show that the disorder enhances both the mean conductance and the mean resistance of the barrier.Comment: 10 page

How accurate are the non-linear chemical Fokker-Planck and chemical Langevin equations?

The chemical Fokker-Planck equation and the corresponding chemical Langevin equation are commonly used approximations of the chemical master equation. These equations are derived from an uncontrolled, second-order truncation of the Kramers-Moyal expansion of the chemical master equation and hence their accuracy remains to be clarified. We use the system-size expansion to show that chemical Fokker-Planck estimates of the mean concentrations and of the variance of the concentration fluctuations about the mean are accurate to order $\Omega^{-3/2}$ for reaction systems which do not obey detailed balance and at least accurate to order $\Omega^{-2}$ for systems obeying detailed balance, where $\Omega$ is the characteristic size of the system. Hence the chemical Fokker-Planck equation turns out to be more accurate than the linear-noise approximation of the chemical master equation (the linear Fokker-Planck equation) which leads to mean concentration estimates accurate to order $\Omega^{-1/2}$ and variance estimates accurate to order $\Omega^{-3/2}$. This higher accuracy is particularly conspicuous for chemical systems realized in small volumes such as biochemical reactions inside cells. A formula is also obtained for the approximate size of the relative errors in the concentration and variance predictions of the chemical Fokker-Planck equation, where the relative error is defined as the difference between the predictions of the chemical Fokker-Planck equation and the master equation divided by the prediction of the master equation. For dimerization and enzyme-catalyzed reactions, the errors are typically less than few percent even when the steady-state is characterized by merely few tens of molecules.Comment: 39 pages, 3 figures, accepted for publication in J. Chem. Phy

Stochastic dynamics beyond the weak coupling limit: thermalization

We discuss the structure and asymptotic long-time properties of coupled equations for the moments of a Brownian particle's momentum derived microscopically beyond the lowest approximation in the weak coupling parameter. Generalized fluctuation-dissipation relations are derived and shown to ensure convergence to thermal equilibrium at any order of perturbation theory.Comment: 6+ page

Directed flow in non-adiabatic stochastic pumps

We analyze the operation of a molecular machine driven by the non-adiabatic variation of external parameters. We derive a formula for the integrated flow from one configuration to another, obtain a "no-pumping theorem" for cyclic processes with thermally activated transitions, and show that in the adiabatic limit the pumped current is given by a geometric expression.Comment: 5 pages, 2 figures, very minor change

Waiting time distribution for electron transport in a molecular junction with electron-vibration interaction

On the elementary level, electronic current consists of individual electron tunnelling events that are separated by random time intervals. The waiting time distribution is a probability to observe the electron transfer in the detector electrode at time $t+\tau$ given that an electron was detected in the same electrode at earlier time $t$. We study waiting time distribution for quantum transport in a vibrating molecular junction. By treating the electron-vibration interaction exactly and molecule-electrode coupling perturbatively, we obtain master equation and compute the distribution of waiting times for electron transport. The details of waiting time distributions are used to elucidate microscopic mechanism of electron transport and the role of electron-vibration interactions. We find that as nonequilibrium develops in molecular junction, the skewness and dispersion of the waiting time distribution experience stepwise drops with the increase of the electric current. These steps are associated with the excitations of vibrational states by tunnelling electrons. In the strong electron-vibration coupling regime, the dispersion decrease dominates over all other changes in the waiting time distribution as the molecular junction departs far away from the equilibrium