Inertial Manifolds for a Smoluchowski Equation on the Unit Sphere

The existence of inertial manifolds for a Smoluchowski equation—a nonlinear Fokker-Planck equation on the unit sphere which arises in modeling of colloidal suspensions—is investigated. A nonlinear and nonlocal transformation is used to eliminate the gradient from the nonlinear term.     

Bounds for Kac's Master Equation

Mark Kac considered a Markov Chain on the n-sphere based on random rotations in randomly chosen coordinate planes. This same walk was used by Hastings on the orthogonal group. We show that the walk has spectral gap bounded below by c/n ³. This and curvature information are used to bound the rate of convergence to stationarity. Received: 26 April 1999 / Accepted: 30 August 1999     

Fractional Moment Bounds and Disorder Relevance for Pinning Models

We study the critical point of directed pinning/wetting models with quenched disorder. The distribution K(·) of the location of the first contact of the (free) polymer with the defect line is assumed to be of the form K(n) = n ^−α-1 L(n), with α ≥ 0 and L(·) slowly varying. The model undergoes a (de)-localization phase transition: the free energy (per unit length) is zero in the delocalized phase and positive in the localized phase. For α < 1/2 disorder is irrelevant: quenched and annealed critical points coincide for small disorder, as well as quenched and annealed critical exponents [3,28]. The same has been proven also for α = 1/2, but under the assumption that L(·) diverges sufficiently fast at infinity, a hypothesis that is not satisfied in the (1 + 1)-dimensional wetting model considered in [12,17], where L(·) is asymptotically constant. Here we prove that, if 1/2 < α < 1 or α > 1, then quenched and annealed critical points differ whenever disorder is present, and we give the scaling form of their difference for small disorder. In agreement with the so-called Harris criterion, disorder is therefore relevant in this case. In the marginal case α = 1/2, under the assumption that L(·) vanishes sufficiently fast at infinity, we prove that the difference between quenched and annealed critical points, which is smaller than any power of the disorder strength, is positive: disorder is marginally relevant. Again, the case considered in [12,17] is out of our analysis and remains open. The results are achieved by setting the parameters of the model so that the annealed system is localized, but close to criticality, and by first considering a quenched system of size that does not exceed the correlation length of the annealed model. In such a regime we can show that the expectation of the partition function raised to a suitably chosen power is small. We then exploit such an information to prove that the expectation of the same fractional power of the partition function goes to zero with the size of the system, a fact that immediately entails that the quenched system is delocalized.     

Analytical Solution of Smoluchowski Equation in Harmonic Oscillator Potential

Non-equilibrium fission has been described by diffusion model. In order to describe the diffusion process analytically, the analytical solution of Smoluchowski equation in harmonic oscillator potential is obtained. This analytical solution is able to describe the probability distribution and the diffusive current with the variable x and t. The results indicate that the probability distribution and the diffusive current are relevant to the initial distribution shape, initial position, and the nuclear temperature T; the time to reach the quasi-stationary state is proportional to friction coefficient β, but is independent of the initial distribution status and the nuclear temperature T. The prerequisites of negative diffusive current are justified. This method provides an approach to describe the diffusion process for fissile process in complicated potentials analytically.     

Smoluchowski Equation for Networks: Merger Induced Intermittent Giant Node Formation and Degree Gap

The dynamical phase diagram of a network undergoing annihilation, creation, and coagulation of nodes is found to exhibit two regimes controlled by the combined effect of preferential attachment for initiator and target nodes during coagulation and for link assignment to new nodes. The first regime exhibits smooth dynamics and power law degree distributions. In the second regime, giant degree nodes and gaps in the degree distribution are formed intermittently. Data for the Japanese firm network in 1994 and 2014 suggests that this network is moving towards the intermittent switching region.     

Global Well-Posedness for a Smoluchowski Equation Coupled with Navier-Stokes Equations in 2D

We prove global existence for a nonlinear Smoluchowski equation (a nonlinear Fokker-Planck equation) coupled with Navier-Stokes equations in 2d. The proof uses a deteriorating regularity estimate in the spirit of [5] (see also [1]).     

Sharp Entropy Dissipation Bounds and Explicit Rate of Trend to Equilibrium for the Spatially Homogeneous Boltzmann Equation

We derive a new lower bound for the entropy dissipation associated with the spatially homogeneous Boltzmann equation. This bound is expressed in terms of the relative entropy with respect to the equilibrium, and thus yields a differential inequality which proves convergence towards equilibrium in relative entropy, with an explicit rate. Our result gives a considerable refinement of the analogous estimate by Carlen and Carvalho [9, 10], under very little additional assumptions. Our proof takes advantage of the structure of Boltzmann's collision operator with respect to the tensor product, and its links with Fokker–Planck and Landau equations. Several variants are discussed. Received: 24 June 1998 / Accepted: 23 December 1998     

Stationary probability flow and vortices for the Smoluchowski–Feynman ratchet

Smoluchowski–Feynman ratchet is in contact with two heat reservoirs with different temperatures. We study the non-equilibrium stationary state with a ratchet rotating unidirectionally, based on the probability flow field, which are obtained either by solving the Fokker–Planck equation (with no inertial effect) or by computer simulations (with an inertial effect). Vortex pattern in the probability flow is found, whose sense of rotation, on the one hand, is determined by a simple thermodynamic argument and, on the other hand, determines rotational direction of the ratchet. Small efficiency of the ratchet is also discussed within this vortex framework.     

The Smoluchowski limit for a simple mechanical model

We consider a vertical stick constantly accelerated along thex-axis by a forceF and which elastically collides with point particles of the same mass (atoms). The atoms are initially Poisson distributed and are allowed to have four velocities only. It is shown that under suitable scaling of the system the displacementQ(t) of the stick satisfies a nontrivial CLT:Q(t)=vFt+D ^1/2 W(t) (Smoluchowski equation), where the values ofv andD depend on the fact that one atom may collide several times.     

Globally Hyperbolic Moment Model of Arbitrary Order for One-Dimensional Special Relativistic Boltzmann Equation

This paper extends the model reduction method by the operator projection to the one-dimensional special relativistic Boltzmann equation. The derivation of arbitrary order globally hyperbolic moment system is built on our careful study of two families of the complicate Grad type orthogonal polynomials depending on a parameter. We derive their recurrence relations, calculate their derivatives with respect to the independent variable and parameter respectively, and study their zeros and coefficient matrices in the recurrence formulas. Some properties of the moment system are also proved. They include the eigenvalues and their bound as well as eigenvectors, hyperbolicity, characteristic fields, linear stability, and Lorentz covariance. A semi-implicit numerical scheme is presented to solve a Cauchy problem of our hyperbolic moment system in order to verify the convergence behavior of the moment method. The results show that the solutions of our hyperbolic moment system converge to the solution of the special relativistic Boltzmann equation as the order of the hyperbolic moment system increases.     

Eigenvalues of the one-dimensional Smoluchowski equation

The eigenvalues and eigenfunctions of the Smoluchowski equation are investigated for the case of potentials withN deep wells. The small parameter =kT/V, which measures the ratio of the thermal energy to a typical well depth, is used in connection with the method of matched asymptotic expansion to obtained asymptotic approximations to all the eigenvalues and eigenfunctions. It is found that the eigensolutions fall into two classes, namely (i) the top-of-the-well and (ii) the bottom-of-the-well eigensolutions. The eigenvalues for both classes of solutions are integer multiples of the squqres of the frequencies at the top or bottom of the various wells. The eigenfunctions are, in general, localized to the top or bottom of the corresponding well. The very small eigenvalues require special consideration because the asymptotic analysis is incapable of distinguishing them from the zero eigenvalue with multiplicityN. Another approximation reveals that, in addition to the true zero eigenvalue, there areN-1 eigenvalues of order exp(–). The case of other possible multiple eigenvalues is also examined.     

The diffusion in the quantum Smoluchowski equation

A novel quantum Smoluchowski dynamics in an external, nonlinear potential has been derived recently. In its original form, this overdamped quantum dynamics is not compatible with the second law of thermodynamics if applied to periodic, but asymmetric ratchet potentials. An improved version of the quantum Smoluchowski equation with a modified diffusion function has been put forward in L. Machura et al. (Phys. Rev. E 70 (2004) 031107) and applied to study quantum Brownian motors in overdamped, arbitrarily shaped ratchet potentials. With this work we prove that the proposed diffusion function, which is assumed to depend (in the limit of strong friction) on the second-order derivative of the potential, is uniquely determined from the validity of the second law of thermodynamics in thermal, undriven equilibrium. Put differently, no approximation-induced quantum Maxwell demon is operating in thermal equilibrium. Furthermore, the leading quantum corrections correctly render the dissipative quantum equilibrium state, which distinctly differs from the corresponding Gibbs state that characterizes the weak (vanishing) coupling limit.     

Derivation of Smoluchowski equations with corrections for Fokker-Planck and BGK collision models

The time evolution of the phase space distribution function for a classical particle in contact with a heat bath and in an external force field can be described by a kinetic equation. From this starting point, for either Fokker-Planck or BGK (Bhatnagar-Gross-Krook) collision models, we derive, with a projection operator technique, Smoluchowski equations for the configuration space density with corrections in reciprocal powers of the friction constant. For the Fokker-Planck model our results in Laplace space agree with Brinkman, and in the time domain, with Wilemski and Titulaer. For the BGK model, we find that the leading term is the familiar Smoluchowski equation, but the first correction term differs from the Fokker-Planck case primarily by the inclusion of a fourth order space derivative or super Burnett term. Finally, from the corrected Smoluchowski equations for both collision models, in the spirit of Kramers, we calculate the escape rate over a barrier to fifth order in the reciprocal friction constant, for a particle initially in a potential well.     

A comment on early-time solutions of the Smoluchowski equation

We present a simple derivation of classes of early-time solutions of the Smoluchowski equation in the presence of boundaries, simplifying and generalizing an analysis by van Kampen.     

Joint probability distribution beyond the Smoluchowski limit for brownian motion in position space

We obtain a time convolutionless partial differential equation for the joint probability distribution in position space of a non-markovian brownian particle under the influence of some potential. We discuss the corrections to the Smoluchowski limit in this context.     

Bounds for the anisotropic Ising model

Rigorous upper bounds are found for the magnetisation, susceptibility, critical temperature and crossover exponent in an anisotropic Ising system.     

Charged Brownian particles: Kramers and Smoluchowski equations and the hydrothermodynamical picture

We consider a charged Brownian gas under the influence of external and non-uniform electric, magnetic and mechanical fields, immersed in a non-uniform bath temperature. With the collision time as an expansion parameter, we study the solution to the associated Kramers equation, including a linear reactive term. To the first order we obtain the asymptotic (overdamped) regime, governed by transport equations, namely: for the particle density, a Smoluchowski-reactive like equation; for the particle’s momentum density, a generalized Ohm’s-like equation; and for the particle’s energy density, a Maxwell-Cattaneo-like equation. Defining a nonequilibrium temperature as the mean kinetic energy density, and introducing Boltzmann’s entropy density via the one particle distribution function, we present a complete thermohydrodynamical picture for a charged Brownian gas. We probe the validity of the local equilibrium approximation, Onsager relations, variational principles associated to the entropy production, and apply our results to: carrier transport in semiconductors, hot carriers and Brownian motors. Finally, we outline a method to incorporate non-linear reactive kinetics and a mean field approach to interacting Brownian particles.