Statistical physics of inhomogeneous transport: Unification of diffusion laws and inference from first-passage statistics

Characterization of composite materials, whose properties vary in space over microscopic scales, has become a problem of broad interdisciplinary interest. In particular, estimation of the inhomogeneous transport coefficients, e.g., the diffusion coefficient or the heat conductivity, which shape important processes in biology and engineering, is a challenging task. The analysis of such systems is further complicated because two alternative formulations of the inhomogeneous transport equations exist in the literature—the Smoluchowski and Fokker-Planck equations, which are also related to the so-called Ito-Stratonovich dilemma. Using the theory of statistical physics, we show that the two formulations, usually regarded as distinct models, are physically equivalent. From this result we develop efficient estimates for the transverse space-dependent diffusion coefficient in fluids near a phase boundary. Our method requires only measurements of escape probabilities and mean exit times of molecules leaving a narrow spatial region. We test our estimates in three case studies: (i) a Langevin model of a Büttikker-Landauer ratchet; atomistic molecular-dynamics simulations of liquid-water molecules in contact with (ii) vapor, and (iii) soap (surfactant) film which has promising applications in physical chemistry. Our analysis reveals that near the surfactant monolayer the mobility of water molecules is slowed down almost twice with respect to the bulk liquid. Moreover, the diffusion coefficient of water correlates with the transition from hydrophilic to hydrophobic parts of the film.

Figure 1

Illustration of first passage events in time series of four noninteracting particles. Initially all the four particles are located at $x=0\mathrm{nm}$. The time series of their positions are truncated at the respective first-passage events—when the particles' trajectories cross one of the boundaries at $x-L=-0.1\mathrm{nm}$ or at $x+L=0.1\mathrm{nm}$ ($L=0.1\mathrm{nm}$) for the first time. The red-colored series end with the negative first-passage events at $x-L$, whereas the blue-colored ones—with the positive event at $x+L$. The first-passage time can be read out from the vertical time axis at the final point of the trajectories.

Figure 2

Estimation of the diffusion coefficient in a Büttiker-like ratchet. Space-dependent diffusion coefficient, plotted together with the Smoluchowski and Fokker-Planck potentials, is compared with the estimates made from our simulations using Eq. (60). Error bars are given by three standard deviations. Simulation parameter values are as follows: $U_0=3k_{B}T$, $D_0=3{\mathrm{nm}}^2/\mathrm{ns}$, $\alpha =0.5$, $\varphi =0.2\mathrm{nm}$. Checks for first-passage events with $L=0.1\mathrm{nm}$ were performed at each simulation time step—every $2\mathrm{fs}$. A total of 1000 such events was acquired for each value of the initial position $x$.

Figure 3

Estimation of the gradient of the Fokker-Planck potential in a Büttiker-like ratchet. The slope of the Fokker-Planck potential is compared with the approximate Eq. (58) applied to the simulation data. Error bars and simulation parameters are the same as those in Fig. 2.

Figure 4

Water-vapor and water-surfactant interfaces in molecular-dynamics simulations. Panel (a): A snapshot of the ${\mathrm{H}}_2\mathrm{O}$ molecules in the water-vapor system with red-colored oxygen atoms and white-colored hydrogen atoms. Panel (b): A snapshot with the surfactant's carbon chains (green colored) and ${\mathrm{H}}_2\mathrm{O}$ molecules in the water-surfactant system. Panel (c): Equilibrium number density of water molecules at the center of the simulation box for the water-vapor (red) and water-surfactant (green) interfaces. The red dotted vertical line indicates a position of the Gibbs-dividing interface only in the right part of the simulation box for the water-vapor system. Likewise the green dotted vertical line delimits one of the two phase boundaries between water and surfactant in the left part of the simulated system. Red circles and green triangles indicate locations inspected in Fig. 6 for water-vapor and water-surfactant interfaces, respectively. The equilibrium density of molecules is reconstructed from histograms with bin size $0.2\mathrm{nm}$ extracted from a simulation of total duration $0.5\mathrm{ns}$ with time step $10\mathrm{fs}$.

Figure 5

First-passage statistics of ${\mathrm{H}}_2\mathrm{O}$ molecules in the water-vapor (red) and water-surfactant (green) systems. (a) Conditional probability $P_{+}\left(x\right)$ for water molecules near $x$ to first escape the interval $[x-L,x+L]$ from $x+L$. A first-passage probability $P_{+}\left(x\right)>1/2[P_{+}\left(x\right)<1/2]$ indicates an effective force acting on the molecules at a given point $x$ in the positive (negative) direction of the axis. (b) Mean escape time $\langle \tau \left(x\right)\rangle$ from the interval $[x-L,x+L]$ quantifies the molecules' mobility. Larger values of $\langle \tau \left(x\right)\rangle$ correspond to a slower diffusive kinetics. Error bars are given by three standard deviations. Red and green dotted lines indicate the Gibbs dividing interfaces as described in Fig. 4. See Sec. 3b for further details on the first-passage-time analysis.

Figure 6

Fitting local mean squared displacements (MSD) of water molecules observed in our molecular-dynamics simulations for selected locations, which are also indicated in the density profile in Fig. 4: (a) water-vapor system and (b) water-surfactant system. The legend shows the inferred values of the diffusion coefficient at different positions $x$ (in nanometers) as obtained by a linear regression of the MSDs vs time within selected intervals (solid lines).

Figure 7

Estimating the heterogeneous diffusion coefficient of ${\mathrm{H}}_2\mathrm{O}$ molecules in the water-vapor (red) and water-surfactant (green) systems. Panel (a) compares estimations of the transverse diffusion coefficient $D\left(x\right)$ as a function of the distance $x$ from the center of the simulation box obtained by three inference methods: (I) The first-passage method [Eq. (60)], (II) the method of Ref. [25], (III) the method of local mean-squared displacements (Appendix pp5). Panel (b) shows the density of hydrophilic and hydrophobic residues of the ${\mathrm{C}}_6{\mathrm{H}}_{12}$ molecules in the water-surfactant interface. Error bars in (a) are given by three standard deviations. Red and green dotted lines indicate the Gibbs dividing interfaces as described in Fig. 4.