State-dependent Diffusion Coefficients Research Articles

EPR-Net: constructing a non-equilibrium potential landscape via a variational force projection formulation.

We present EPR-Net, a novel and effective deep learning approach that tackles a crucial challenge in biophysics: constructing potential landscapes for high-dimensional non-equilibrium steady-state systems. EPR-Net leverages a nice mathematical fact that the desired negative potential gradient is simply the orthogonal projection of the driving force of the underlying dynamics in a weighted inner-product space. Remarkably, our loss function has an intimate connection with the steady entropy production rate (EPR), enabling simultaneous landscape construction and EPR estimation. We introduce an enhanced learning strategy for systems with small noise, and extend our framework to include dimensionality reduction and the state-dependent diffusion coefficient case in a unified fashion. Comparative evaluations on benchmark problems demonstrate the superior accuracy, effectiveness and robustness of EPR-Net compared to existing methods. We apply our approach to challenging biophysical problems, such as an eight-dimensional (8D) limit cycle and a 52D multi-stability problem, which provide accurate solutions and interesting insights on constructed landscapes. With its versatility and power, EPR-Net offers a promising solution for diverse landscape construction problems in biophysics.

Stochastic resonance and hysteresis in climate with state-dependent fluctuations.

We consider two aspects in climatic science where bistability between the two stable states of the systems is observed. One is the transition between the glacial and the interglacial period in Earth's glacial cycle. Another is the thermohaline circulation in the North Atlantic ocean. Both of these phenomena can be modeled by the overdamped dynamics of a Brownian particle in a double-well potential subject to periodic forcing. For the former case, the two wells represent two different climates and the periodic forcing is sufficiently weak not to cause a transition between the two states without the effect of the noise. Whereas in case of the latter phenomenon, the two states correspond to the two different conditions of the flow and the strength of the periodic forcing is high enough to give rise to hysteresis in the system. We propose that one important component of the dynamics, short-term fluctuations related to weather, in both of these cases, depends on the current climatic state of the system. This leads to introduction of the state-dependent diffusion coefficients in the dynamics because the diffusion coefficient represents the strength of the fluctuations. We justify our argument by analyzing the δ^{18}O record for the glacial cycle model. We have shown that this consideration can produce certain features in the dynamics which agree with the real observations in case of both glacial cycles and thermohaline flow.

Open Problem—Weakly Interacting Particle Systems on Dense Random Graphs

We first briefly introduce the following classic mean-field particle system given as a collection of [Formula: see text]-valued stochastic processes {[Formula: see text](t) : t ∈ [0, T][Formula: see text].[Formula: see text]Here, {Wi} are independent and identically distributed standard d-dimensional Brownian motions, and [Formula: see text] × [Formula: see text] → [Formula: see text] is assumed to be bounded and Lipschitz, but one could also allow for more general initial states, drift coefficient b, and a state-dependent diffusion coefficient.

State-dependent diffusion coefficients and free energies for nucleation processes from Bayesian trajectory analysis

ABSTRACTThe rate of nucleation processes such as the freezing of a supercooled liquid or the condensation of supersaturated vapour is mainly determined by the height of the nucleation barrier and the diffusion coefficient for the motion across it. Here, we use a Bayesian inference algorithm for Markovian dynamics to extract simultaneously the free energy profile and the diffusion coefficient in the nucleation barrier region from short molecular dynamics trajectories. The specific example we study is the nucleation of vapour bubbles in liquid water under strongly negative pressures, for which we use the volume of the largest bubble as a reaction coordinate. Particular attention is paid to the effects of discretisation, the implementation of appropriate boundary conditions and the optimal selection of parameters. We find that the diffusivity is a linear function of the bubble volume over wide ranges of volumes and pressures, and is mainly determined by the viscosity of the liquid, as expected from the Rayleigh–Plesset theory for macroscopic bubble dynamics. The method is generally applicable to nucleation processes and yields important quantities for the estimation of nucleation rates in classical nucleation theory.

Interplay of fast and slow dynamics in rare transition pathways: The disk-to-slab transition in the 2d Ising model.

Rare transitions between long-lived stable states are often analyzed in terms of free energy landscapes computed as functions of a few collective variables. Here, using transitions between geometric phases as example, we demonstrate that the effective dynamics of a system along these variables are an essential ingredient in the description of rare events and that the static perspective provided by the free energy alone may be misleading. In particular, we investigate the disk-to-slab transition in the two-dimensional Ising model starting with a calculation of a two-dimensional free energy landscape and the distribution of committor probabilities. While at first sight it appears that the committor is incompatible with the free energy, they can be reconciled with each other using a two-dimensional Smoluchowski equation that combines the free energy landscape with state dependent diffusion coefficients. These results illustrate that dynamical information is not only required to calculate rate constants but that neglecting dynamics may also lead to an inaccurate understanding of the mechanism of a given process.

Time delay can facilitate coherence in self-driven interacting-particle systems.

Directional switching in a self-propelled particle model with delayed interactions is investigated. It is shown that the average switching time is an increasing function of time delay. The presented results are applied to studying collective animal behavior. It is argued that self-propelled particle models with time delays can explain the state-dependent diffusion coefficient measured in experiments with locust groups. The theory is further generalized to heterogeneous groups where each individual can respond to its environment with a different time delay.

First-passage and escape problems in the Feller process

The Feller process is an one-dimensional diffusion process with linear drift and state-dependent diffusion coefficient vanishing at the origin. The process is positive definite and it is this property along with its linear character that have made Feller process a convenient candidate for the modeling of a number of phenomena ranging from single-neuron firing to volatility of financial assets. While general properties of the process have long been well known, less known are properties related to level crossing such as the first-passage and the escape problems. In this work we thoroughly address these questions.

Nonlinear diffusion effects on biological population spatial patterns

Motivated by the observation that anomalous diffusion is a realistic feature in the dynamics of biological populations, we investigate its implications in a paradigmatic model for the evolution of a single species density u(x,t). The standard model includes growth and competition in a logistic expression, and spreading is modeled through normal diffusion. Moreover, the competition term is nonlocal, which has been shown to give rise to spatial patterns. We generalize the diffusion term through the nonlinear form ∂tu(x,t)=D∂xxu(x,t)ν (with D,ν>0), encompassing the cases where the state-dependent diffusion coefficient either increases (ν>1) or decreases (ν<1) with the density, yielding subdiffusion or superdiffusion, respectively. By means of numerical simulations and analytical considerations, we display how that nonlinearity alters the phase diagram. The type of diffusion imposes critical values of the model parameters for the onset of patterns and strongly influences their shape, inducing fragmentation in the subdiffusive case. The detection of the main persistent mode allows analytical prediction of the critical thresholds.

Reaction–diffusion models of development with state-dependent chemical diffusion coefficients

Reaction–diffusion models are widely used to model developmental processes. The great majority of current models invoke constant diffusion coefficients. However, the diffusion of metabolites or signals through tissues is frequently such that this assumption may reasonably be questioned. We consider several different physical mechanisms leading to effective diffusion coefficients in biological tissues which vary with the local conditions, including models in which juxtacrine signaling results in the diffusion of a signal in the absence of material transport. We develop a mathematical formalism for transforming local transport laws into diffusive terms. This procedure is appropriate when the typical length scale over which the concentrations change significantly is much greater than the dimensions of a cell. We review previous developmental models which considered the possibility of state-dependent diffusion coefficients. We also provide a few new motivating examples.

The Bismut-Elworthy formula for backward SDE's and applications to nonlinear Kolmogorov equations and control in infinite dimensional spaces

We consider a forward-backward system of stochastic evolution equations in a Hilbert space. Under nondegeneracy assumptions on the diffusion coefficient (that may be nonconstant) we prove an analogue of the well-known Bismut-Elworthy formula. Next, we consider a nonlinear version of the Kolmogorov equation, i.e. a deterministic quasilinear equation associated to the system according to Pardoux, E and Peng, S. (1992). "Backward stochastic differential equations and quasilinear parabolic partial differential equations". In: Rozowskii, B.L., Sowers, R.B. (Eds.), Stochastic Partial Differential Equations and Their Applications , Lecture Notes in Control Inf. Sci., Vol. 176, pp. 200-217. Springer: Berlin. The Bismut-Elworthy formula is applied to prove smoothing effect, i.e. to prove existence and uniqueness of a solution which is differentiable with respect to the space variable, even if the initial datum and (some) coefficients of the equation are not. The results are then applied to the Hamilton-Jacobi-Bellman equation of stochastic optimal control. This way we are able to characterize optimal controls by feedback laws for a class of infinite-dimensional control systems, including in particular the stochastic heat equation with state-dependent diffusion coefficient.

Prior mapping for nonlinear flows in random environments.

We analyze nonlinear flows in randomly heterogeneous environments, which are characterized by state-dependent diffusion coefficients with spatially correlated structures. The prior Kirchhoff mapping is used to describe such systems by linear stochastic partial differential equations with multiplicative noise. These are solved through moment equations which are closed, alternatively, either by perturbation expansions, or by a posterior linear mapping closure. The latter relies on the assumption that the state variable is a spatially distributed Gaussian field. We demonstrate that the former approach is more robust.

Bistability driven by Gaussian colored noise: First-passage times.

Herein we present a calculation of the mean first-passage time for a bistable one-dimensional system driven by Gaussian colored noise of strength D and correlation time ${\ensuremath{\tau}}_{c}$. We obtain quantitative agreement with experimental analog-computer simulations of this system. We disagree with some of the conclusions reached by previous investigators. In particular, we demonstrate that all available approximations that lead to a state-dependent diffusion coefficient lead to the same result for small D${\ensuremath{\tau}}_{c}$.

A systematic elimination procedure for Ito stochastic differential equations and the adiabatic approximation

We study a class of nonlinear Ito stochastic differential equations (with possibly state dependent diffusion coefficients), in which the variables can be divided into linearly damped (slaved) variables s and linearly undamped variablesu (order parameters). We devise a systematic and constructive procedure to eliminate the slaved variables. We take explicit time and chance dependence of the slaved variables into account, the latter via a family of diffusion processesZ t (v) . These act as fluctuating coefficients of the Center Manifoldst=s(ut, t,Z t (v) (v=2, 3, ...)) and appear explicitly in the elimination procedure. We show how in the Ito calculus fluctuating and deterministic coefficients of the Center Manifold are more completely separated than in the previously treated Stratonovich case [1]. The adiabatic approximation is defined as a partial summation of the elimination expansion and the stochastic generalization ofs=0 is derived. We show how thus ambiguity of stochastic calculi is removed. Closed form summations are given in two examples. We briefly indicate the potential use of perturbation theory techniques in the systematic elimination procedure.