Fokker Planck Kramers Equation Research Articles

Quantum Langevin equation

We propose a Langevin equation to describe the quantum Brownian motion of bounded particles based on a distinctive formulation concerning both the fluctuation and dissipation forces. The fluctuation force is similar to that employed in the classical case. It is a white noise with a variance proportional to the temperature. The dissipation force is not restricted to be proportional to the velocity and is determined in a way as to guarantee that the stationary state is given by a density operator of the Gibbs canonical type. To this end we derived an equation that gives the time evolution of the density operator, which turns out to be a quantum Fokker–Planck–Kramers equation. The approach is applied to the harmonic oscillator in which case the dissipation force is found to be non Hermitian and proportional to the velocity and position.

Molecular dynamics simulation of partially diffusion-controlled reaction between mono- and diatomic molecules

Molecular dynamics simulations were performed for model partially diffusion-controlled reactions between mono- and diatomic molecules at liquid density with energetical and geometrical restrictions on the reactivity. Since the reorientational motion of the diatomic molecule was fast, a spherical reaction surface approximation could be applied in the analysis of the short-time transient effect on the rate constant. In this approximation, the two types of restrictions could be treated as different profiles of reactant velocity distributions at the reaction surface. The transient rate constants for the two types of restrictions obtained by the simulations were virtually identical if the initial rate constants were the same, though the reaction probability per collision and the spatial population distribution of reactants were different. The simulation results were compared with three theories: two based on the Fokker–Planck–Kramers equation and one on the diffusion equation. The three theories, which assumed different functional forms of velocity distribution, reasonably well explained the time dependences of the rate constants obtained by the simulations and their insensitivity on the detailed profile of reactant velocity distribution.

Analysis of Short-Time Transient Dynamics of a Diffusion-Controlled Reaction in a Hard-Sphere Fluid Based on Fokker–Planck–Kramers Equation

Abstract Molecular dynamics (MD) simulations are performed for a simple diffusion-controlled reaction in a hard-sphere fluid. The short-time transient rate constants are compared with the Markovian Langevin dynamics (LD), two theories based on the Fokker–Planck–Kramers equation (FPKE), and the Smoluchowski–Collins–Kimball (SCK) theory based on the diffusion equation. At n* = 0.7856, where n* is the total number density reduced by the diameter of the molecules, molecular motions are diffusive and Markovian, and the MD results agree well with the LD results. At n* = 0.9428, the MD results are much larger than the LD ones at short times, and the discrepancy can be explained by non-Markovian effects. Also at n* = 0.2000, where inertia effects are important, the time profile of the MD rate constant is satisfactorily explained by LD, while the MD rate constant shows a smaller decay than at higher densities. FPKE theory with a continuous velocity distribution reproduces the LD results under all conditions indicating that the theory is valid over a wide density range. FPKE theory with a discontinuous velocity distribution cannot explain the simulation results at the lowest density despite taking into account inertia effects. Although SCK theory is expected to be valid only at high densities, the predicted rate constant is close to that from the simulation even at the lowest density, if the intrinsic rate constant is chosen to reproduce the exact value of the initial rate constant.

Application of Fokker-Planck-Kramers equation treatment for short-time dynamics of diffusion-controlled reaction in supercritical Lennard-Jones fluids over a wide density range

The validity of a Fokker-Planck-Kramers equation (FPKE) treatment of the rate of diffusion-controlled reaction at short times [K. Ibuki and M. Ueno, J. Chem. Phys. 119, 7054 (2003)] is tested in a supercritical Lennard-Jones fluid over a wide density range by comparing it with the Langevin dynamics and molecular dynamics simulations and other theories. The density n range studied is 0.323n(c)< or =n< or =2.58n(c) and the temperature 1.52T(c), where n(c) and T(c) are the critical density and temperature, respectively. For the rate of bimolecular reactions, the transition between the collision-limited and diffusion-limited regimes is expected to take place in this density range. The simulations show that the rate constant decays with time extensively at high densities, and that the magnitude of decay decreases gradually with decreasing density. The decay profiles of the rate constants obtained by the simulations are reproduced reasonably well by the FPKE treatment in the whole density range studied if a continuous velocity distribution is used in solving the FPKE approximately. If a discontinuous velocity distribution is used instead of the continuous one, the FPKE treatment leads to a rate constant much larger than the simulation results at medium and low densities. The rate constants calculated from the Smoluchowski-Collins-Kimball (SCK) theory based on the diffusion equation are somewhat smaller than the simulation results in medium and low densities when the intrinsic rate constant is chosen to adjust the steady state rate constant in the low density limit to that derived by the kinetic collision theory. The discrepancy is relatively small, so that the SCK theory provides a useful guideline for a qualitative discussion of the density effect on the rate constant.

Fokker–Planck–Kramers equation treatment of dynamics of diffusion-controlled reactions using continuous velocity distribution in three dimensions

A theory has been developed for the short-time dynamics of diffusion-controlled reactions based on the Fokker–Planck–Kramers equation (FPKE) in three dimensions. A continuous velocity distribution function has been proposed to solve the FPKE approximately. The present theory agrees better with the Langevin dynamics results than the earlier theory using a discontinuous velocity distribution. This indicates the validity of the present theory in three dimensions, because the Langevin dynamics results can be assumed to be the exact solutions to the FPKE. The theory is compared with molecular dynamics (MD) simulations in Lennard-Jones fluids to examine the applicability for realistic systems. The present theory predicts a somewhat smaller rate constant than the MD simulation in the time range of a few picoseconds. The discrepancies can be explained qualitatively in terms of the non-Markovian effect on the molecular motions.

Improved Velocity Distribution Applied to Fokker–Planck–Kramers Equation Treatment for Dynamics of Diffusion-Controlled Reactions in Two Dimensions

Abstract The validity of theoretical treatments of short-time dynamics of diffusion-controlled reaction based on the Fokker–Planck–Kramers’ equation (FPKE) has been examined using molecular dynamics (MD) simulation in two-dimensional Lennard–Jones fluids. First, we made Langevin dynamics (LD) simulation assuming that the relative diffusion coefficient is given by the sum of the self-diffusion coefficients and that the potential between the reactants is given by the potential of mean force. The LD and the MD simulation results for the time dependence of the survival probabilities of the reactants agree well. This indicates that the FPKE derived from the Langevin equation can be applied to the problem of diffusion-controlled reaction. The time-dependent Harris theory based on the FPKE and the approximation of the half-range Maxwellian velocity distribution showed disagreements with the MD simulation. Such a limitation of the theory has not been recognized in three-dimensional fluids. In the approximation used in the time-dependent Harris theory, the velocity distribution function is not continuous. We have shown that the agreement between the MD simulation and the theory can be improved when we use a new approximation based on a continuous velocity distribution function.

Effect of potential of mean force on short-time dynamics of a diffusion-controlled reaction in a hard-sphere fluid

We calculated the time-dependent rate constant of a diffusion-controlled reaction between hard-spheres in a hard-sphere fluid at short times starting from the Fokker–Planck–Kramers equation combined with the approximation of half-range Maxwellian velocity distributions. For the potential function, we employed the potential of mean force (PMF) obtained from the equilibrium radial distribution function. The rate constant at short times was much larger than that neglecting the PMF effect, though the steady state rate constant did not sensitively depend on the PMF effect. This indicates that the effect of the initial distribution of the reactants is important in determining the rate constant at short times. The results were compared with a computer simulation. The dependences of the survival probability of a target on the time, the transmission coefficient, and the reactant concentration were examined, and satisfactory agreements between the calculation and the simulation were obtained at a relatively low density. At a high density, the non-Markovian effect should be taken into account to explain the simulation result.

Numerical test of Kramers reaction rate theory in two dimensions

The Fokker–Planck–Kramers equation for a system composed by a reactive coordinate x coupled to a solvent coordinate y is employed to study the effect of additional degrees of freedom on the dynamics of reactive events. The system is studied numerically in the diffusional regimes of both coordinates, for different topologies of the bistable potential function and anisotropies of friction. The eigenvalue spectrum is evaluated by representing the time evolution operator over a basis set of orthonormal functions. A detailed analysis of the effect of the explicit consideration of the slow nonreactive mode is carried on to show that a variation of qualitative picture (scenario) of the reaction dynamics occurs when friction along different directions is strongly anisotropic, depending also on the structure of the two-dimensional potential surface. The numerical study supports both the qualitative picture of the reaction dynamics and the rate constant expressions obtained analytically. For those cases where the Langer theory has a restricted range of applicability because of the change in the reaction dynamics scenario, this fact has been numerically demonstrated. Here the Langer expression for the rate constant is replaced by the one obtained as a result of the consideration of the effective one-dimensional problem along the solvent coordinate, characterized by a smaller activation energy than that in the initial problem. All of these facts were confirmed by the numerical test, which shows a satisfactory agreement with the analytical results.

Theory of the rotational brownian motion of a linear molecule in 3D: a statistical mechanics study

The Fokker-Planck-Kramers (FPK) equation for the rotational Brownian motion of a linear rigid rotator in 3D is derived from the basic principles of statistical mechanics applied to the two classical model Hamiltonians: 1. 1) The model of a rigid fixed rotor interacting with a bath of harmonic oscillators. 2. 2) The model of a linear rigid rodlike polymer diluted in a monoatomic solvent. An entropy formula is established for the rotator system allowing to formulate the second law of thermodynamics and to analyze its implication on the form of the dielectric and Kerr functions. Finally, a general scheme is proposed to solve the FPK equation in the form of series expansion. Approximative analytical expressions for the dielectric and Kerr functions, are obtained up to the third order in the electric field, considering a steady state regime of a dc field on which is superimposed an alternating field. These response functions generalize and extend all the results recently published on the topic. The particular case of the Debye-Smoluchowski diffusion model is straightforwardly recovered.

Improved rate theories of chemical reactions

A mathematical difficulty in Kramers’ theory on reaction rate is pointed out and his treatment is improved by considering the Brownian motion near the bottom of potential minimum explicitly. We used the Fokker–Planck–Kramers equation (FPK) and derived a modified Smoluchowski’s equation by eliminating the velocity dependence. Moreover, we assumed the stationary state and defined the reaction coefficient by the flux divided by the concentration of the reactant. Finally, we solved the modified Smoluchowski by using computer assisted symbolic manipulation programs and obtained the expression for the rate coefficient by a single theory. It is found that there exists a lower limit for the friction value which is corresponding to the critical damping motion for an ordinary harmonic damped oscillator. It is also shown that we can have Kramers’ turnover for the case of the steep decline of the potential after passing the maximum. The Boltzmann transport equation for the strong collision case with the collision term by Bhatnager, Gross, and Krook (BBGK) is also solved exactly by the symbolic manipulation to find the rate coefficient which does not lead to the Kramers turnover.

The rotational Brownian motion of a linear molecule and its application to the theory of Kerr effect relaxation

Exact expressions in the low-field limit for the Laplace transform of the Kerr effect response (KER), including inertial effects, of an assembly of needlelike dipolar anisotropically polarizable molecules are obtained using the Fokker–Planck–Kramers equation for the probability density function in configuration-angular velocity space. The spatial part of the solution of this equation is expanded as a series of associated Legendre functions. The coefficients of Legendre functions of the first and second order may be solved separately, using an expansion in Laguerre polynomials. The coefficients of the Legendre polynomials of order 1 and 2 are evaluated to yield the dielectric and KER after-effect solutions. The dielectric response is in full agreement with the result of Sack. The investigation shows clearly that the KER cannot be obtained by any simple transformation of the dielectric response unlike rotation in two dimensions.

Nonlinear Kerr effect relaxation including inertial effects in alternating fields

The complex birefringence of a dielectric medium is found for the nonlinear stationary response arising from the application of a unidirectional field superposed on an alternating field. Effects arising from dipolar inertia are included. The starting point is the Fokker–Planck–Kramers equation which is solved by assuming that the molecules are polar and anisotropically polarizable and are compelled to rotate in two dimensional space (plane rotator). Kerr effect relaxation is characterized by the ensemble average 〈cos 2θ〉(t). Time evolution of this gives rise to harmonic components in ω and 2ω. They are given to second order in the driving field. Cole–Cole diagrams for each component are presented for various values of P, the ratio of the induced dipole moment to the permanent dipole moment, and τI/τ, the inertial parameter (τI and τ being the friction time and the Debye relaxation time, respectively). It is shown how these diagrams behave in the high-frequency region, and a method of extracting information about molecular parameters such as τI, τ, and P from them is proposed.

The rotational Brownian motion of a nonspherical body and its application to the theory of dielectric relaxation

The conditional probability density function in angular velocity space for a symmetrical body is obtained exactly both from the stochastic differential equations governing the rotational Brownian motion and from the Fokker–Planck equation. Furthermore, it is shown how the Laplace transform of the angular velocity correlation function for an asymmetrical body may be calculated from the Fokker–Planck equation. At the same time, the Laplace transform of the dipole correlation function for the symmetrical body is calculated both from a stochastic integrodifferential equation and Fokker–Planck–Kramers equation. Also, the Laplace transform of the dipole correlation function for the asymmetrical body is obtained using the Fokker–Planck–Kramers equation. Also, the dipole correlation function as an explicit function of time is calculated numerically for a symmetrical body based on the Fokker–Plank–Kramers equation, and is compared to that obtained from the free rotational model and to a single exponential decay function with the Debye relaxation time.