Smoluchowski Dynamics Research Articles

Transient probability currents provide upper and lower bounds on non-equilibrium steady-state currents in the Smoluchowski picture.

Probability currents are fundamental in characterizing the kinetics of nonequilibrium processes. Notably, the steady-state current Jss for a source-sink system can provide the exact mean-first-passage time (MFPT) for the transition from the source to sink. Because transient nonequilibrium behavior is quantified in some modern path sampling approaches, such as the "weighted ensemble" strategy, there is strong motivation to determine bounds on Jss-and hence on the MFPT-as the system evolves in time. Here, we show that Jss is bounded from above and below by the maximum and minimum, respectively, of the current as a function of the spatial coordinate at any time t for one-dimensional systems undergoing overdamped Langevin (i.e., Smoluchowski) dynamics and for higher-dimensional Smoluchowski systems satisfying certain assumptions when projected onto a single dimension. These bounds become tighter with time, making them of potential practical utility in a scheme for estimating Jss and the long time scale kinetics of complex systems. Conceptually, the bounds result from the fact that extrema of the transient currents relax toward the steady-state current.

From the Liouville to the Smoluchowski equation for a colloidal solute particle in a solvent

We show how the Smoluchowski dynamics of a colloidal Brownian particle suspended in a molecular solvent can be reached starting from the microscopic Liouvillian evolution of the full classical model in the high friction limit. The integration of the solvent degrees of freedom goes through a multiple time scale perturbation expansion which removes the secular divergences. A simple dynamical Monte Carlo scheme is then proposed to solve the resulting evolution equation for the colloid solute particle. In particular we study the approach to the equilibrium Boltzmann distribution at late times and its resilience behavior at shorter times as influenced by the steepness of the external potential and the friction coefficient around their respective minima. This is very important to understand the fate of the Brownian particle’s random walk and its evolution history.

Numerical study of long-time dynamics and ergodic-nonergodic transitions in dense simple fluids.

Since the mid-1980s, mode-coupling theory (MCT) has been the de facto theoretic description of dense fluids and the transition from the fluid state to the glassy state. MCT, however, is limited by the approximations used in its construction and lacks an unambiguous mechanism to institute corrections. We use recent results from a new theoretical framework--developed from first principles via a self-consistent perturbation expansion in terms of an effective two-body potential--to numerically explore the kinetics of systems of classical particles, specifically hard spheres governed by Smoluchowski dynamics. We present here a full solution for such a system to the kinetic equation governing the density-density time correlation function and show that the function exhibits the characteristic two-step decay of supercooled fluids and an ergodic-nonergodic transition to a dynamically arrested state. Unlike many previous numerical studies--and in stark contrast to experiment--we have access to the full time and wave-number range of the correlation function with great precision and are able to track the solution unprecedentedly close to the transition, covering nearly 15 decades in scaled time. Using asymptotic approximation techniques analogous to those developed for MCT, we fit the solution to predicted forms and extract critical parameters. We find complete qualitative agreement with known glassy behavior (e.g. power-law divergence of the α-relaxation time scale in the ergodic phase and square-root growth of the glass form factors in the nonergodic phase), as well as some limited quantitative agreement [e.g. the transition at packing fraction η*=0.60149761(10)], consistent with previous static solutions under this theory and with comparable colloidal suspension experiments. However, most importantly, we establish that this new theory is able to reproduce the salient features seen in other theories, experiments, and simulations but has the advantages of being derived from first principles and possessing a clear mechanism for making systematic corrections.

Limit theorems for Smoluchowski dynamics associated with critical continuous-state branching processes

We investigate the well-posedness and asymptotic self-similarity of solutions to a generalized Smoluchowski coagulation equation recently introduced by Bertoin and Le Gall in the context of continuous-state branching theory. In particular, this equation governs the evolution of the L\'{e}vy measure of a critical continuous-state branching process which becomes extinct (i.e., is absorbed at zero) almost surely. We show that a nondegenerate scaling limit of the L\'{e}vy measure (and the process) exists if and only if the branching mechanism is regularly varying at 0. When the branching mechanism is regularly varying, we characterize nondegenerate scaling limits of arbitrary finite-measure solutions in terms of generalized Mittag-Leffler series.

On extremals of the entropy production by ‘Langevin–Kramers’ dynamics

We refer as ‘Langevin–Kramers’ dynamics to a class of stochastic differential systems exhibiting a degenerate ‘metriplectic’ structure. This means that the drift field can be decomposed into a symplectic and a gradient-like component with respect to a pseudo-metric tensor associated with random fluctuations affecting increments of only a sub-set of the degrees of freedom. Systems in this class are often encountered in applications as elementary models of Hamiltonian dynamics in a heat bath eventually relaxing to a Boltzmann steady state. Entropy production control in Langevin–Kramers models differs from the now well-understood case of Langevin–Smoluchowski dynamics for two reasons. First, the definition of entropy production stemming from fluctuation theorems specifies a cost functional which does not act coercively on all degrees of freedom of control protocols. Second, the presence of a symplectic structure imposes a non-local constraint on the class of admissible controls. Using Pontryagin control theory and restricting the attention to additive noise, we show that smooth protocols attaining extremal values of the entropy production appear generically in continuous parametric families as a consequence of a trade-off between smoothness of the admissible protocols and non-coercivity of the cost functional. Uniqueness is, however, always recovered in the over-damped limit as extremal equations reduce at leading order to the Monge–Ampère–Kantorovich optimal mass-transport equations.

Universal long-time dynamics in dense simple fluids.

Dense simple fluids appear to have a long-time, very slowly evolving state which--for high-enough density--approaches a glassy, nonergodic phase. In this high-density regime, the equilibrating system decays via a three-step process as identified in mode-coupling theory (MCT). MCT, however, is an inherently phenomenological theory without prospects for improvement, and so a systematic theory has been recently developed which naturally allows one to calculate cumulants between the fundamental particle density and response fields self-consistently in a perturbation expansion in a pseudopotential. This paper details the extension of this theory and specifically addresses the universal nature of this erogodic-nonergodic transition. We find that the nonergodic state can be characterized by the associated static equilibrium state, and--nontrivially--that these states are identical regardless of whether a system is driven by Smoluchowski or Newtonian dynamics. In addition, though the specific response fields differ between Smoluchowski and Newtonian dynamics, the two share identical linear fluctuation-dissipation relations connecting the ρ-B cumulants. While we show this universality of nonergodic states within perturbation theory, we expect it to be true more generally. At leading order in the perturbation expansion, one obtains a kinetic kernel quadratic in the density analogous to the "one-loop" theory of MCT. At this one-loop level one finds vertex corrections which depend on the three-point equilibrium cumulants. Here we assume these vertex-corrections can be ignored and instead focus on computing the contributions of higher-order loops. We show that it is possible to sum up all of the loop contributions and find that these higher-order loops do not change the nonergodic state parameters substantially.

Dynamic correlations in Brownian many-body systems.

For classical Brownian systems driven out of equilibrium, we derive inhomogeneous two-time correlation functions from functional differentiation of the one-body density and current with respect to external fields. In order to allow for appropriate freedom upon building the derivatives, we formally supplement the Smoluchowski dynamics by a source term, which vanishes at the physical solution. These techniques are applied to obtain a complete set of dynamic Ornstein-Zernike equations, which serve for the development of approximation schemes. The rules of functional calculus lead naturally to non-Markovian equations of motion for the two-time correlators. Memory functions are identified as functional derivatives of a unique space- and time-nonlocal dissipation power functional.

Lattice Fokker Planck for dilute polymer dynamics

We show that the actual diffusive dynamics, governing the momentum relaxation of a polymer molecule, and described by a Fokker-Planck equation, may be replaced by a BGK-type relaxation dynamics without affecting the slow (Smoluchowski) dynamics in configuration space. Based on the BGK-type description, we present a lattice-Boltzmann (LB) based direct discretization approach for the phase-space description of inertial polymer dynamics. We benchmark this formulation by determining the bulk rheological properties for both steady and time-dependent shear and extensional flows at moderate to large Weissenberg numbers. Finally, we compare the usefulness of the different discrete velocity models, typically used in the LB framework, for solving diffusive dynamics based on the Fokker-Planck equation.

Coarse-grained Monte Carlo simulations of non-equilibrium systems

We extend the scope of a recent method for generating coarse-grained lattice Metropolis Monte Carlo simulations [X. Liu, W. D. Seider, and T. Sinno, Phys. Rev. E 86, 026708 (2012); and J. Chem. Phys. 138, 114104 (2013)] from continuous interaction potentials to non-equilibrium situations. The original method has been shown to satisfy detailed balance at the coarse scale and to provide a good representation of various equilibrium properties in both atomic and molecular systems. However, we show here that the original method is inconsistent with non-equilibrium trajectories generated by full-resolution Monte Carlo simulations, which, under certain conditions, have been shown to correspond to Langevin dynamics. The modified coarse-grained method is generated by simultaneously biasing the forward and backward transition probability for every possible move, thereby preserving the detailed balance of the original method. The resulting coarse-grained Monte Carlo simulations are shown to provide trajectories that are consistent with overdamped Langevin (Smoluchowski) dynamics using a sequence of simple non-equilibrium examples. We first consider the purely diffusional spreading of a Gaussian pulse of ideal-gas particles and then include an external potential to study the influence of drift. Finally, we validate the method using a more general situation in which the particles interact via a Lennard-Jones interparticle potential.

Newtonian Kinetic Theory and the Ergodic-Nonergodic Transition

In a recent work we have discussed how kinetic theory, the statistics of classical particles obeying Newtonian dynamics, can be formulated as a field theory. The field theory can be organized to produce a self-consistent perturbation theory expansion in an effective interaction potential. In the present work we use this development for investigating ergodic-nonergodic (ENE) transitions in dense fluids. The theory is developed in terms of a core problem spanned by the variables $\rho$, the number density, and $B$, a response density. We set up the perturbation theory expansion for studying the self-consistent model which gives rise to a ENE transition. Our main result is that the low-frequency dynamics near the ENE transition is the same for Smoluchowski and Newtonian dynamics. This is true despite the fact that term by term in a density expansion the results for the two dynamics are fundamentally different.

Kinetic equations governing Smoluchowski dynamics in equilibrium

We continue our study of the statistical properties of particles in equilibrium obeying Smoluchowski dynamics. We show that the system is governed by a kinetic equation of the memory function form and that the memory function is given by one of the self-energies available via perturbation theory as introduced in previous work. We determine the memory function explicitly to second order in an expansion in a pseudopotential. The method we use allows for a straightforward computation of corrections via a formal expansion and we therefore view it as an improvement over the conventional mode-coupling theory (MCT) formalism where it is not clear how to make systematic corrections. In addition, the formalism we have introduced is flexible enough to allow for a wide array of different approximation schemes, including density expansions. The convergence criteria for our formal series are not worked out here, but the second-order equation that we derive is promising in the sense that it leads to analytic and numerical results consistent with expectations from computer simulations of the hard-sphere system in addition to replicating the desired features from conventional MCT (e.g., a two-step decay). These particular solutions will be discussed in forthcoming work.

Smoluchowski dynamics and the ergodic-nonergodic transition

We use the recently introduced theory for the kinetics of systems of classical particles to investigate systems driven by Smoluchowski dynamics. We investigate the existence of ergodic-nonergodic (ENE) transitions near the liquid-glass transition. We develop a self-consistent perturbation theory in terms of an effective two-body potential and work to second order in this potential. At second order, we have an explicit relationship between the static structure factor and the effective potential and choose the static structure factor in the case of hard spheres to be given by the solution of the Percus-Yevick approximation for hard spheres. Then, using the analytically determined ENE equation for the ergodicity function, we find an ENE transition for packing fraction η greater than a critical value η(*)=0.76, which is physically unaccessible. The existence of a linear fluctuation-dissipation theorem in the problem is shown and used to great advantage.

From equilibrium to steady-state dynamics after switch-on of shear

A relation between equilibrium, steady state, and waiting-time-dependent dynamical two-time correlation functions in dense glass-forming liquids subject to homogeneous steady shear flow is discussed. The systems under study show pronounced shear thinning, i.e., a significant speedup in their steady-state slow relaxation as compared to equilibrium. An approximate relation that recovers the exact limit for small waiting times is derived following the integration through transients (ITT) approach for the nonequilibrium Smoluchowski dynamics, and is exemplified within a schematic model in the framework of the mode-coupling theory of the glass transition (MCT). Computer simulation results for the tagged-particle density correlation functions corresponding to wave vectors in the shear-gradient directions from both event-driven stochastic dynamics of a two-dimensional hard-disk system and from previously published Newtonian-dynamics simulations of a three-dimensional soft-sphere mixture are analyzed and compared with the predictions of the ITT-based approximation. Good qualitative and semiquantitative agreement is found. Furthermore, for short waiting times, the theoretical description of the waiting time dependence shows excellent quantitative agreement to the simulations. This confirms the accuracy of the central approximation used earlier to derive fluctuation dissipation ratios [M. Krüger and M. Fuchs, Phys. Rev. Lett. 102, 135701 (2009)]. For intermediate waiting times, the correlation functions decay faster at long times than the stationary ones. This behavior is predicted by our theory and observed in simulations.

A natural missing link between activated and downhill protein folding scenarios.

We propose protein PTB1 : 4W as a good candidate for engineering into a downhill folder. PTB1 : 4W has a probe-dependent thermal unfolding curve and sub-millisecond T-jump relaxation kinetics on more than one time scale. Its refolding rate in denaturant is a non-linear function of denaturant concentration (curved chevron plot). Yet at high denaturant concentration its unfolding is probe-independent, and the folding kinetics can be fitted to a single exponential decay. The domain appears to fold via a mechanism between downhill folding and activated folding over several small barriers, and when denaturant is added, one of these barriers greatly increases and simplifies the observed folding to apparent two-state kinetics. We predict the simplest free energy function consistent with the thermal denaturation and kinetics experiments by using the singular value Smoluchowski dynamics (SVSD) model. PTB1 : 4W is a natural 'missing link' between downhill and activated folding. We suggest mutations that could move the protein into the downhill folding limit.

Aspects of lipid membrane bio-responses to subnanosecond, ultrahigh voltage pulsing

Membrane electroporation is probably one of the best-known effects of applying external voltages to biological cells. Reports in the literature have focused on relatively long voltage pulse durations (100 ns-1 ms). Here we probe the very short (< 1 ns), but intense electric field (> 500 kV/cm) regime that is made possible by advances in pulsed power technology. Our analyses based on continuum Smoluchowski and molecular dynamics (MD) approaches, predict two new aspects. First, it is shown that pore formation rates would be dramatically lower than predicted by conventional theory due to their dependence on local pore area. Second, such high fields are predicted to affect membrane proteins and ion-channels, without causing electroporation in regions between the proteins. Hence, such high voltage, short duration pulsing should not be associated with electroporation alone, but rather be viewed as a novel vehicle that opens possibilities for a range of new electrically-driven bio-response phenomena.

The Scaling Attractor and Ultimate Dynamics for Smoluchowski’s Coagulation Equations

We describe a basic framework for studying dynamic scaling that has roots in dynamical systems and probability theory. Within this framework, we study Smoluchowski’s coagulation equation for the three simplest rate kernels K(x,y)=2, x+y and xy. In another work, we classified all self-similar solutions and all universality classes (domains of attraction) for scaling limits under weak convergence (Menon and Pego in Commun. Pure Appl. Math. 57, 1197–1232, [2004]). Here we add to this a complete description of the set of all limit points of solutions modulo scaling (the scaling attractor) and the dynamics on this limit set (the ultimate dynamics). A key tool is Bertoin’s Lévy-Khintchine representation formula for eternal solutions of Smoluchowski’s equation (Bertoin in Ann. Appl. Probab. 12, 547–564, [2002a]). This representation linearizes the dynamics on the scaling attractor, revealing these dynamics to be conjugate to a continuous dilation, and chaotic in a classical sense. Furthermore, our study of scaling limits explains how Smoluchowski dynamics “compactifies” in a natural way that accounts for clusters of zero and infinite size (dust and gel).

Modeling and simulation of Li-ion conduction in poly(ethylene oxide)

Polyethylene oxide (PEO) containing a lithium salt (e.g., LiI) serves as a solid polymer electrolyte (SPE) in thin-film batteries and its ionic conductivity is a key parameter of their performance. We model and simulate Li + ion conduction in a single PEO molecule. Our simplified stochastic model of ionic motion is based on an analogy between protein channels of biological membranes that conduct Na +, K +, and other ions, and the PEO helical chain that conducts Li + ions. In contrast with protein channels and salt solutions, the PEO is both the channel and the solvent for the lithium salt (e.g., LiI). The mobile ions are treated as charged spherical Brownian particles. We simulate Smoluchowski dynamics in channels with a radius of ca. 0.1 nm and study the effect of stretching and temperature on ion conductivity. We assume that each helix (molecule) forms a random angle with the axis between these electrodes and the polymeric film is composed of many uniformly distributed oriented boxes that include molecules with the same direction. We further assume that mechanical stretching aligns the molecular structures in each box along the axis of stretching (intra-box alignment). Our model thus predicts the PEO conductivity as a function of the stretching, the salt concentration and the temperature. The computed enhancement of the ionic conductivity in the stretch direction is in good agreement with experimental results. The simulation results are also in qualitative agreement with recent theoretical and experimental results.

Thermostats for “Slow” Configurational Modes

Thermostats are dynamical equations used to model thermodynamic variables such as temperature and pressure in molecular simulations. For computationally intensive problems such as the simulation of biomolecules, we propose to average over fast momentum degrees of freedom and construct thermostat equations in configuration space. The equations of motion are deterministic analogues of the Smoluchowski dynamics in the method of stochastic differential equations.

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.

Stochastic Mode Reduction for Particle-Based Simulation Methods for Complex Microfluid Systems

We illustrate the stochastic mode reduction procedure as formulated recently by Majda, Timofeyev, and Vanden-Eijnden [Comm. Pure Appl. Math., 54 (2001), pp. 891--974] (MTV) on the equations of motion underlying various particle-based simulation approaches (such as Stokesian dynamics and Brownian dynamics) and the conceptually distinct dissipative particle dynamics (DPD) simulation approaches for complex microfluid systems. The resulting coarse-grained dynamics are compared and contrasted with each other. We show that the stochastic mode reduction procedure provides a way to recover the Smoluchowski dynamics for a standard model of multiple interacting particles in a fluid. The DPD, however, has some subtle aspects which obstruct the application of the stochastic mode reduction procedure. We discuss the mathematical and physical properties of the DPD method that underlie thisdifficulty.