Yim Man Yi = 亞擴散雙穩分數福克-普朗克系統的線性響應及隨機共振和有色噪音對雙穩系統所引起的效應 / 嚴敏儀. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2006. / Includes bibliographical references (leaves 85-89). / Text in English; abstracts in English and Chinese. / Yim Man Yi = Ya kuo san shuang wen fen shu Fuke-Pulangke xi tong de xian xing xiang ying ji sui ji gong zhen he you se zao yin dui shuang wen xi tong suo yin qi de xiao ying / Yan Minyi. / Chapter 1 --- Introduction --- p.1 / Chapter 1.1 --- Brownian motion and anomalous dynamics --- p.1 / Chapter 1.2 --- White and colored noises --- p.3 / Chapter 2 --- Linear response theory of sub diffusive Fokker-Planck systems --- p.6 / Chapter 2.1 --- Introduction to subdiffusive Fokker-Planck systems --- p.6 / Chapter 2.2 --- Spectral density --- p.8 / Chapter 2.3 --- Linear response --- p.11 / Chapter 2.4 --- Signal-to-noise ratio (SNR) --- p.13 / Chapter 2.5 --- Stochastic energy --- p.14 / Chapter 3 --- Perturbation due to a sinusoidal signal --- p.16 / Chapter 3.1 --- Presence of an external sinusoidal forcing --- p.16 / Chapter 3.1.1 --- PDF and linear reponse --- p.16 / Chapter 3.1.2 --- Phase lag --- p.19 / Chapter 3.1.3 --- SNR --- p.19 / Chapter 3.1.4 --- Stochastic energy --- p.23 / Chapter 3.2 --- Presence of a sinusoidally time-varying diffusion coefficient --- p.23 / Chapter 4 --- Perturbation due to a rectangular signal --- p.26 / Chapter 4.1 --- Linear response to a rectangular pulse --- p.26 / Chapter 4.2 --- Perturbation due to a periodic rectangular signal --- p.28 / Chapter 4.2.1 --- Linear response --- p.29 / Chapter 4.2.2 --- SNR --- p.31 / Chapter 4.2.3 --- Stochastic energy --- p.33 / Chapter 4.3 --- Comparison with the response to a sinusoidal driving force --- p.35 / Chapter 5 --- The Effects of Colored Noise on the Stationary Probability Distribution --- p.38 / Chapter 5.1 --- Formulation of a general colored noises-driven system --- p.39 / Chapter 5.2 --- Approximation schemes for a colored noise --- p.42 / Chapter 5.2.1 --- Decoupling approximation --- p.42 / Chapter 5.2.2 --- UCNA --- p.45 / Chapter 5.2.3 --- Small r approximation . --- p.46 / Chapter 5.2.4 --- Presence of an additive noise: g(x) = 1 --- p.47 / Chapter 5.2.5 --- Presence of a multiplicative noise: g(x) = x --- p.52 / Chapter 5.3 --- Approximation scheme for two colored noises --- p.53 / Chapter 5.3.1 --- Presence of two additive noises --- p.55 / Chapter 5.3.2 --- Presence of an additive noise and a multiplicative noise --- p.55 / Chapter 5.4 --- Unimodal-bimodal transitions --- p.61 / Chapter 6 --- A stochastic genetic regulatory transcription model with colored noises --- p.68 / Chapter 6.1 --- Biological background --- p.69 / Chapter 6.2 --- Effect of a single noise --- p.72 / Chapter 6.3 --- Effects of two noises --- p.75 / Chapter 6.4 --- Biological implications of colored noises in the model --- p.81 / Chapter 7 --- Conclusions --- p.82 / Bibliography --- p.85 / Chapter A --- Mittag-Leffler function --- p.90 / Chapter B --- H-function (Fox function) --- p.92 / Chapter C --- Operator algebra --- p.94 / Chapter D --- Mean first passage time --- p.97

Chow Cheuk Wang = 含粒子阱之福克-普朗克方程及亞擴散分數福克-普朗克方程之研究 / 周卓宏. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2004. / Includes bibliographical references (leaves 70-72). / Text in English; abstracts in English and Chinese. / Chow Cheuk Wang = Han li zi jing zhi Fuke-Pulangke fang cheng ji ya kuo san fen shu Fuke-Pulangke fang cheng zhi yan jiu / Zhou Zhuohong. / Chapter 1 --- Introduction --- p.1 / Chapter 2 --- Derivation of the Fokker-Planck equation --- p.5 / Chapter 2.1 --- Diffusion equation --- p.5 / Chapter 2.2 --- Kramers-Moyal Equation --- p.7 / Chapter 2.3 --- Fokker-Planck Equation --- p.9 / Chapter 2.3.1 --- Eigenfunction Expansion --- p.10 / Chapter 2.3.2 --- Mapping FPE to a pseudo-Schrodinger equation --- p.11 / Chapter 3 --- Conventional Fokker-Planck equation with sinks --- p.15 / Chapter 3.1 --- Propagator with sinks --- p.16 / Chapter 3.1.1 --- One-sink propagator --- p.17 / Chapter 3.1.2 --- Two-sink propagator --- p.18 / Chapter 3.2 --- Survival probability --- p.19 / Chapter 3.3 --- Expectation value of the position --- p.22 / Chapter 3.4 --- Mean survival time --- p.25 / Chapter 4 --- Fractional Fokker-Planck equation with sinks --- p.38 / Chapter 4.1 --- Fractional diffusion equation --- p.39 / Chapter 4.2 --- Propagator of the subdiffusive system --- p.41 / Chapter 4.3 --- Survival probability and expectation value of the position --- p.43 / Chapter 4.4 --- Mean survival-time distribution --- p.47 / Chapter 5 --- Boundary value problems for diffusion and subdiffusion --- p.53 / Chapter 5.1 --- Diffusion in a linear potential U(x) = Fx --- p.54 / Chapter 5.2 --- Two absorbing boundaries --- p.55 / Chapter 5.3 --- One absorbing boundary and one reflecting boundary --- p.58 / Chapter 5.4 --- Two reflecting boundaries --- p.59 / Chapter 6 --- Summary --- p.68 / Bibliography --- p.70 / Chapter A --- Laplace transform and the method of Abate and Whitt --- p.73 / Chapter B --- Mittag-Leffler function and its two-point Pade approximant --- p.77

Firman So. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1999. / Includes bibliographical references (leaves [159]-160). / Text in English; abstracts in English and Chinese. / Firman So. / Abstract --- p.i / Acknowledgement --- p.iii / Contents --- p.iv / List of Figures --- p.vii / List of Tables --- p.xii / Chapter Chapter 1. --- Introduction --- p.1 / Chapter Chapter 2. --- Derivation of the Fokker-Planck Equation --- p.4 / Chapter 2.1 --- Brownian Motion --- p.4 / Chapter 2.2 --- Non-Linear Langevin Equation --- p.7 / Chapter 2.3 --- Conditional Probability Density --- p.9 / Chapter 2.4 --- Kramers-Moyal Expansion --- p.11 / Chapter 2.5 --- Fokker-Planck Equation --- p.13 / Chapter Chapter 3. --- Method & Solution of the One-variable Fokker-Planck Equation with Time-Independent Coefficients --- p.15 / Chapter 3.1 --- Stationary Solution --- p.16 / Chapter 3.2 --- Ornstein-Ulhenbeck Process: An Exactly Solvable Fokker-Planck Equation --- p.17 / Chapter 3.3 --- Eigenfunction Expansion --- p.19 / Chapter 3.4 --- Ornstein-Ulhenbeck process by Eigenfunction Expansion --- p.29 / Chapter 3.5 --- Eigenfunctions and Eigenvalues of Inverted Potentials --- p.30 / Chapter 3.6 --- Kramers' Escape Rate --- p.32 / Chapter Chapter 4. --- Diffusion in Potential Wells --- p.36 / Chapter 4.1 --- Symmetric Double-Well Potential --- p.36 / Chapter 4.2 --- Asymmetric Bistable Potential --- p.61 / Chapter Chapter 5. --- Stochastic Resonance --- p.100 / Chapter 5.1 --- Introduction --- p.100 / Chapter 5.2 --- Probability Density........................... --- p.101 / Chapter 5.3 --- Power Spectrum of the Autocorrelation Function of x --- p.113 / Chapter 5.4 --- Stochastic Resonance --- p.120 / Chapter Chapter 6. --- Colored Noise --- p.124 / Chapter 6.1 --- Introduction --- p.124 / Chapter 6.2 --- Approximation Schemes for the Colored Noise Problem --- p.125 / Chapter 6.3 --- Stationary Probability Density of the Colored Noise Driven Bistable System --- p.132 / Chapter 6.4 --- Escape Rate in the Presence of Colored Noise --- p.140 / Chapter Chapter 7. --- Conclusion --- p.146 / Appendix A --- p.149 / Chapter A.1 --- State-Dependent Diagonalization Method --- p.149 / Chapter A.2 --- Infinite-Square-Well Basis Diagnalization --- p.153 / Chapter A.3 --- Solving the Fokker-Planck equation --- p.156 / References

Thesis (Ph. D.)--University of Wisconsin--Madison, 1983. / Typescript. Vita. Description based on print version record. Includes bibliographical references.

In problems involving highly forward-peaked scattering, the Boltzmann transport equation can be simplified using the Fokker-Planck model. The purpose of this project was to develop an analytical solution to the resulting Fokker-Planck equation. This analytical solution can then be used to benchmark numerical transport codes. A numerical solution to the Fokker-Planck equation was also developed. The analytical solution found is a generalized function. It satisfies the purpose of the project with two limitations. The first limitation is that the solution can only be evaluated for certain sources. The second limitation is that the solution can only be evaluated for small times. The moments of the Fokker-Planck equation can be evaluated for any time. The numerical solution developed works for all sources and all times. The analytical solution, then, provides an accurate and precise benchmark under certain conditions. The numerical solution provides a less accurate benchmark under all conditions.

This thesis presents a detailed study of the composition dependence of the thermal and transient mobility of electrons in binary rare gas mixtures. The time independent electron real mobility in binary inert gas mixtures is calculated versus mole fraction for different electric field strengths. The deviations from the linear variation of the reciprocal of the mobility of the mixture with mole fraction, that is from Blanc's law, is determined and explained in detail. Very large deviations from the linear behavior were calculated for several binary mixtures at specific electric strengths, in particular for He-Xe mixtures. An interesting effect was observed whereby the electron mobility in He-Xe mixtures, for particular compositions and electron field strength could be greater than in pure He or less than in pure Xe. The time dependent electron real mobility and the corresponding relaxation time, in particular for He-Ar and He-Ne mixtures are reported for a wide range of concentrations, field strengths (d.c. electric field), and frequencies (microwave electric field). For a He-Ar mixture, the time dependent electron mobility is strongly influenced by the Ramsauer-Townsend minimum and leads to the occurrence of an overshoot and a negative mobility in the transient mobility. For He-Ne, a mixture without the Ramsauer-Townsend minimum, the transient mobility increases monotonically towards the thermal value. The energy thermal relaxation times 1/Pτ for He-Ne, and Ne-Xe mixtures are calculated so as to find out the validity of the linear relationship between the 1/Pτ of the mixture and mole fraction. A Quadrature Discretization Method of solution of the time dependent Boltzmann-Fokker-Planck equation for electrons in binary inert gas mixture is employed in the study of the time dependent electron real mobility. The solution of the Fokker-Planck equation is based on the expansion of the solution in the eigenfunctions of the Fokker-Planck operator. / Science, Faculty of / Chemistry, Department of / Graduate

The goal of this thesis is to contribute to the ambitious program of the foundation of developing statistical physics using chaos. We build a deterministic model of Brownian motion and provide a microscpoic derivation of the Fokker-Planck equation. Since the Brownian motion of a particle is the result of the competing processes of diffusion and dissipation, we create a model where both diffusion and dissipation originate from the same deterministic mechanism - the deterministic interaction of that particle with its environment. We show that standard diffusion which is the basis of the Fokker-Planck equation rests on the Central Limit Theorem, and, consequently, on the possibility of deriving it from a deterministic process with a quickly decaying correlation function. The sensitive dependence on initial conditions, one of the defining properties of chaos insures this rapid decay. We carefully address the problem of deriving dissipation from the interaction of a particle with a fully deterministic nonlinear bath, that we term the booster. We show that the solution of this problem essentially rests on the linear response of a booster to an external perturbation. This raises a long-standing problem concerned with Kubo's Linear Response Theory and the strong criticism against it by van Kampen. Kubo's theory is based on a perturbation treatment of the Liouville equation, which, in turn, is expected to be totally equivalent to a first-order perturbation treatment of single trajectories. Since the boosters are chaotic, and chaos is essential to generate diffusion, the single trajectories are highly unstable and do not respond linearly to weak external perturbation. We adopt chaotic maps as boosters of a Brownian particle, and therefore address the problem of the response of a chaotic booster to an external perturbation. We notice that a fully chaotic map is characterized by an invariant measure which is a continuous function of the control parameters of the map. Consequently if the external perturbation is made to act on a control parameter of the map, we show that the booster distribution undergoes slight modifications as an effect of the weak external perturbation, thereby leading to a linear response of the mean value of the perturbed variable of the booster. This approach to linear response completely bypasses the criticism of van Kampen. The joint use of these two phenomena, diffusion and friction stemming from the interaction of the Brownian particle with the same booster, makes the microscopic derivation of a Fokker-Planck equation and Brownian motion, possible.

A ferromagnetic particle with single domain, at some kinds of applied field (at some angle or strangth), the particle's free energy would be two state model. The rate of barrier crossing could be solve by Fokker-Planck equation .And use master equation to find out the Total rate between two potential well. In this thysis, we use the upper method to simulate particle's magnetic moment under time varying magnetic field at fixed angle or fixed magnetic applied rotate the particle. In numerical method, we use the back Euler method to prevent the divergence of the calculation.

The pseudospectral method is a family of numerical methods for the solution of differential equations based on the expansion of basis functions defined on a set of grid points. In this thesis, the relationship between the distribution of grid points and the accuracy and convergence of the solution is emphasized. The polynomial and sinc pseudospectral methods are extensively studied along with many applications to quantum and statistical mechanics involving the Fokker-Planck and Schroedinger equations. The grid points used in the polynomial methods coincide with the points of quadrature, which are defined by a set of polynomials orthogonal with respect to a weight function. The choice of the weight function plays an important role in the convergence of the solution. It is observed that rapid convergence is usually achieved when the weight function is chosen to be the square of the ground-state eigenfunction of the problem. The sinc method usually provides a slow convergence as the grid points are uniformly distributed regardless of the behaviour of the solution. For both polynomial and sinc methods, the convergence rate can be improved by redistributing the grid points to more appropriate positions through a transformation of coordinates. The transformation method discussed in this thesis preserves the orthogonality of the basis functions and provides simple expressions for the construction of discretized matrix operators. The convergence rate can be improved by several times in the evaluation of loosely bound eigenstates with an exponential or hyperbolic sine transformation. The transformation can be defined explicitly or implicitly. An explicit transformation is based on a predefined mapping function, while an implicit transformation is constructed by an appropriate set of grid points determined by the behaviour of the solution. The methodologies of these transformations are discussed with some applications to 1D and 2D problems. The implicit transformation is also used as a moving mesh method for the time-dependent Smoluchowski equation when a function with localized behaviour is used as the initial condition.

Stochastic models of biochemical reaction networks are used for understanding the properties of molecular regulatory circuits in living cells. The state of the cell is defined by the number of copies of each molecular species in the model. The chemical master equation (CME) governs the time evolution of the the probability density function of the often high-dimensional state space. The CME is approximated by a partial differential equation (PDE), the Fokker-Planck equation and solved numerically. Direct solution of the CME rapidly becomes computationally expensive for increasingly complex biological models, since the state space grows exponentially with the number of dimensions. Adaptive numerical methods can be applied in time and space in the PDE framework, and error estimates of the approximate solutions are derived. A method for splitting the CME operator in order to apply the PDE approximation in a subspace of the state space is also developed. The performance is compared to the most widely spread alternative computational method.