LEFD: Langevin Equation with Fluctuating Diffusivity

• LEFD is a stochastic framework where diffusivity is a time- or state-dependent random process that captures environmental heterogeneity and anomalous diffusion.
• It employs multiplicative-noise dynamics and renewal process models to explain phenomena such as ageing, weak ergodicity breaking, and non-Gaussian particle displacements.
• Applications span from protein dynamics and polymer motion to diffusion in heterogeneous membranes, linking trajectory statistics with hidden mobility fluctuations.

The Langevin equation with fluctuating diffusivity (LEFD) is a stochastic framework in which the diffusion coefficient is itself a random process in time, and in a broader usage may also be a state-dependent scalar or tensor field on configuration space. In its most basic overdamped form, the position obeys

$\frac{dx(t)}{dt}=\sqrt{2D(t)}\,\xi(t),$

with Gaussian white noise $\xi(t)$, while $D(t)$ encodes heterogeneous mobility generated by internal conformational dynamics, environmental rearrangements, or coarse-grained hidden variables. In the contemporary literature, LEFD serves as a common language for Brownian yet non-Gaussian diffusion, anomalous diffusion, ageing, weak ergodicity breaking, and state-dependent Langevin dynamics with multiplicative noise (Akimoto et al., 16 Sep 2025, Yasuda et al., 11 Jun 2026).

1. Definition and scope

In the scalar temporal formulation used in several LEFD studies, the particle position satisfies

$$\frac{dx(t)}{dt}=\sqrt{2D(t)}\,\xi(t), \qquad \langle \xi(t)\rangle=0,\quad \langle \xi(t)\xi(t')\rangle=\delta(t-t'),$$

with the explicit assumption that $D(t)$ and $\xi(t)$ are statistically independent. In this setting, $\xi(t)$ represents fast, memoryless thermal noise, whereas $D(t)$ represents slow fluctuations of mobility, such as conformational changes or environmental rearrangements (Shirataki et al., 5 Dec 2025).

A broader formulation writes the dynamics in $n$ dimensions as

$$\frac{d\mathbf{r}(t)}{dt}=\sqrt{2}\,\mathbf{B}(t)\cdot \mathbf{w}(t),$$

with instantaneous diffusion matrix

$\xi(t)$0

where $\xi(t)$1 is Gaussian white noise and $\xi(t)$2 is a stationary stochastic process independent of $\xi(t)$3 and $\xi(t)$4. In this form, scalar LEFD is the isotropic special case $\xi(t)$5 (Uneyama et al., 2014).

A second, mathematically distinct usage treats fluctuating diffusivity as state-dependent diffusion. In that setting one writes, for mesoscopic coordinates $\xi(t)$6,

$\xi(t)$7

and interprets $\xi(t)$8 as a contravariant diffusion tensor that varies with configuration. Yasuda et al. make this coordinate-covariant by identifying the diffusion tensor with the inverse of a Riemannian metric on configuration space (Yasuda et al., 11 Jun 2026).

The scope of LEFD is therefore broader than a single equation. It includes temporally heterogeneous scalar diffusivities, tensorial and configuration-dependent diffusivities, renewal-type switching models, diffusing-diffusivity models, and generalized Langevin equations with fluctuating diffusivity. A plausible implication is that LEFD is best understood as a class of multiplicative-noise stochastic dynamics whose defining feature is that diffusivity is not fixed.

2. Canonical stochastic constructions

A central canonical model is the two-state LEFD, in which the diffusivity alternates between two values,

$\xi(t)$9

with sojourn-time PDFs $D(t)$0 and $D(t)$1. In the power-law regime,

$D(t)$2

the process becomes non-Markovian, and the asymptotic behavior depends on whether $D(t)$3 are smaller or larger than $D(t)$4. This two-state renewal model was developed to describe anomalous subdiffusion, ageing, scatter of the diffusion coefficient, and weak ergodicity breaking in single-particle tracking (Miyaguchi et al., 2016, Akimoto et al., 2016).

A second widely studied construction is diffusing diffusivity. One representative model introduces an auxiliary Ornstein–Uhlenbeck process $D(t)$5 and sets

$D(t)$6

where $D(t)$7 is symmetric dichotomous noise switching at rate $D(t)$8. Because $D(t)$9, the diffusivity is positive and bounded,

$$\frac{dx(t)}{dt}=\sqrt{2D(t)}\,\xi(t), \qquad \langle \xi(t)\rangle=0,\quad \langle \xi(t)\xi(t')\rangle=\delta(t-t'),$$0

This bounded diffusing-diffusivity model yields a stationary diffusivity distribution on a finite interval and provides explicit short-time displacement PDFs and long-time Gaussian diffusion (Lee et al., 13 Apr 2026).

A third construction ties diffusivity to hidden conformational coordinates. In the Double-Well-Controlled Diffusing Diffusivity (DWCDD) model, the diffusivity is linked to a conformational coordinate $$\frac{dx(t)}{dt}=\sqrt{2D(t)}\,\xi(t), \qquad \langle \xi(t)\rangle=0,\quad \langle \xi(t)\xi(t')\rangle=\delta(t-t'),$$1 by

$$\frac{dx(t)}{dt}=\sqrt{2D(t)}\,\xi(t), \qquad \langle \xi(t)\rangle=0,\quad \langle \xi(t)\xi(t')\rangle=\delta(t-t'),$$2

while $$\frac{dx(t)}{dt}=\sqrt{2D(t)}\,\xi(t), \qquad \langle \xi(t)\rangle=0,\quad \langle \xi(t)\xi(t')\rangle=\delta(t-t'),$$3 evolves in a double-well potential

$$\frac{dx(t)}{dt}=\sqrt{2D(t)}\,\xi(t), \qquad \langle \xi(t)\rangle=0,\quad \langle \xi(t)\xi(t')\rangle=\delta(t-t'),$$4

with two minima at $$\frac{dx(t)}{dt}=\sqrt{2D(t)}\,\xi(t), \qquad \langle \xi(t)\rangle=0,\quad \langle \xi(t)\xi(t')\rangle=\delta(t-t'),$$5 and $$\frac{dx(t)}{dt}=\sqrt{2D(t)}\,\xi(t), \qquad \langle \xi(t)\rangle=0,\quad \langle \xi(t)\xi(t')\rangle=\delta(t-t'),$$6, a barrier at $$\frac{dx(t)}{dt}=\sqrt{2D(t)}\,\xi(t), \qquad \langle \xi(t)\rangle=0,\quad \langle \xi(t)\xi(t')\rangle=\delta(t-t'),$$7, and reflecting boundaries at $$\frac{dx(t)}{dt}=\sqrt{2D(t)}\,\xi(t), \qquad \langle \xi(t)\rangle=0,\quad \langle \xi(t)\xi(t')\rangle=\delta(t-t'),$$8 and $$\frac{dx(t)}{dt}=\sqrt{2D(t)}\,\xi(t), \qquad \langle \xi(t)\rangle=0,\quad \langle \xi(t)\xi(t')\rangle=\delta(t-t'),$$9. At low enough temperature, the model coarse-grains to a telegraph process between

$D(t)$0

with exponentially distributed sojourn times and Arrhenius-type mean waiting times (Shirataki et al., 5 Dec 2025).

LEFD has also been extended to viscoelastic settings. In the generalized Langevin equation with fluctuating diffusivity (GLEFD), the position obeys

$D(t)$1

with a generalized fluctuation-dissipation relation encoded in the stochastic memory kernel $D(t)$2. Power-law kernels yield anomalous subdiffusion, non-Gaussianity, and stretched-exponential relaxation; single-exponential kernels yield analytically tractable plateau structures in the MSD and the self-intermediate-scattering function (Miyaguchi, 2022).

Finally, LEFD with a linear restoring force defines the Ornstein–Uhlenbeck process with fluctuating diffusivity,

$D(t)$3

for which relaxation functions can be written in terms of eigenvalues and eigenfunctions of a transfer operator associated with the diffusivity dynamics (Uneyama et al., 2019).

3. Statistical observables and characteristic signatures

For the overdamped scalar LEFD with $D(t)$4 independent of the white noise, the mean-squared displacement satisfies

$D(t)$5

If $D(t)$6 is stationary, the ensemble-averaged MSD is normal diffusive. In the isotropic matrix formulation,

$D(t)$7

provided $D(t)$8 is stationary. A common misconception is therefore that fluctuating diffusivity necessarily implies anomalous MSD scaling; stationary LEFD shows that this is false (Uneyama et al., 2014).

The nontrivial structure appears in higher-order statistics and finite-time observables. At short times, when diffusivity is effectively frozen over the observation window, the displacement PDF is a superposition of Gaussians with random variance. In the bounded dichotomous-driven OU model, the stationary diffusivity density is

$D(t)$9

which implies a logarithmic divergence of the displacement PDF at the origin and Gaussian tails modulated by a power law,

$\xi(t)$0

at short times. At long times the same model converges to ordinary Gaussian diffusion with effective diffusivity

$\xi(t)$1

The variance of the time-averaged stochastic diffusivity decays as $\xi(t)$2, demonstrating self-averaging (Lee et al., 13 Apr 2026).

The time-averaged MSD,

$\xi(t)$3

is central in LEFD because its mean can remain deceptively simple while its fluctuations encode diffusivity correlations. Uneyama, Miyaguchi, and Akimoto derived a general expression for the relative standard deviation

$\xi(t)$4

and showed that, for $\xi(t)$5 and diffusivity correlation time $\xi(t)$6,

$\xi(t)$7

where $\xi(t)$8 is the normalized diffusivity correlation function. If $\xi(t)$9 decays sufficiently fast,

$\xi(t)$0

The short-time plateau reflects static heterogeneity, while the long-time $\xi(t)$1 decay of $\xi(t)$2 reflects ergodic averaging. The crossover time is a weighted average relaxation time of the diffusivity (Uneyama et al., 2014).

In the two-state LEFD, the excess part of the RSD is directly related to the non-Gaussian parameter of the propagator, and in equilibrium it can be used to reconstruct the diffusivity autocorrelation function. This makes RSD a diagnostic observable when MSDs alone are blind to fluctuating diffusivity (Miyaguchi et al., 2016).

4. Nonequilibrium, memory, and observation-time-induced crossover

Nonequilibrium initial conditions are often decisive in LEFD. In the two-state renewal model with heavy-tailed sojourn times, the ensemble-averaged MSD can remain normal while time-averaged observables are anomalously broad, and the time-averaged diffusion coefficient can remain intrinsically random when the mean sojourn time for one state diverges. Occupation-time statistics then determine the distributional behavior of time-averaged diffusivity and produce Mittag-Leffler or Lamperti generalized arcsine laws, depending on the sojourn-time exponents (Akimoto et al., 2016).

Memory in the hidden conformational dynamics can change the long-time diffusion coefficient even when the position process remains normally diffusive. Kimura and Akimoto model the gyration radius $\xi(t)$3 by a generalized Langevin equation in a double-well potential with either exponential or power-law memory kernel, and set

$\xi(t)$4

For the exponential kernel, the stationary distribution of $\xi(t)$5 is Boltzmann and the global diffusion coefficient is independent of the memory time. For the power-law kernel, the stationary distribution depends on the initial condition, the stationary measure is not Boltzmann, and the global diffusion coefficient increases as the memory exponent decreases. The paper attributes this to an everlasting effect of the initial condition due to long-term memory (Kimura et al., 2023).

A distinct nonequilibrium phenomenon is the observation-time-induced crossover recently analyzed in the DWCDD model. Starting from the low-diffusivity basin, the effective diffusion coefficient measured over an observation window $\xi(t)$6,

$\xi(t)$7

crosses over from $\xi(t)$8 to the relaxed value when the observation time becomes comparable to the internal relaxation time. For the symmetric double-well,

$\xi(t)$9

with

$D(t)$0

The crossover temperature $D(t)$1 is defined by

$D(t)$2

which gives

$D(t)$3

Longer observation windows therefore shift the crossover to lower temperature. The paper identifies three necessary conditions for this observation-time-induced crossover: fluctuating diffusivity, a temperature-dependent relaxation time, and a nonequilibrium initial condition. It further argues that the protein dynamical transition in hydrated proteins can be interpreted as an instance of this general crossover mechanism rather than a genuine thermodynamic phase transition (Shirataki et al., 5 Dec 2025).

This suggests a useful distinction inside LEFD theory. Some anomalous phenomena arise from stationary heterogeneity and disappear in long measurements; others arise from nonequilibrium preparation and memory, so finite-time observations probe relaxation rather than equilibrium transport.

5. Geometric and thermodynamic structure

When diffusivity depends on the state, LEFD becomes a multiplicative-noise problem with a nontrivial drift structure. Yasuda et al. formulate state-dependent diffusion geometrically by defining the diffusion metric

$D(t)$4

so that configuration space becomes a Riemannian manifold. They introduce a scalar probability density

$D(t)$5

and a scalar free energy

$D(t)$6

The covariant Fokker–Planck equation then reads

$D(t)$7

and the covariant Langevin equation in Itô form is

$D(t)$8

where

$D(t)$9

and $n$0 are the Christoffel symbols of the diffusion metric. In this formulation, the spurious drift is the Christoffel contribution $n$1, and coordinate covariance is explicit (Yasuda et al., 11 Jun 2026).

The same paper shows that the conventional Lau–Lubensky form with spurious drift $n$2 is mathematically equivalent to the covariant form once the relation between $n$3 and $n$4 is used. The geometric interpretation is therefore not a different stochastic dynamics but a different organization of the same thermodynamic content.

This has direct consequences for modeling. In polar coordinates for Brownian motion in a two-dimensional harmonic potential, the correct radial equation is

$n$5

which yields the Rayleigh equilibrium radial density. Omitting the Christoffel contribution and using

$n$6

instead produces a Gaussian radial density and therefore an incorrect equilibrium measure. The same issue appears for Brownian motion on a sphere, where the correct equation for $n$7 contains the drift $n$8, and for diffusion projected from a curved surface to a flat plane (Yasuda et al., 11 Jun 2026).

A related lesson emerges from the OU process with fluctuating diffusivity. Although its primary relaxation function can be matched by a suitably chosen generalized Langevin equation or a multi-mode OU process, the second relaxation function generally satisfies

$n$9

whereas Gaussian linear models enforce $$\frac{d\mathbf{r}(t)}{dt}=\sqrt{2}\,\mathbf{B}(t)\cdot \mathbf{w}(t),$$0. This establishes that LEFD relaxation can be qualitatively different from that of conventional Gaussian memory models even when $$\frac{d\mathbf{r}(t)}{dt}=\sqrt{2}\,\mathbf{B}(t)\cdot \mathbf{w}(t),$$1 looks similar (Uneyama et al., 2019).

6. Physical realizations and research uses

LEFD has been used as a coarse-grained description of transport in proteins, supercooled liquids, polymers, membranes, and other heterogeneous media. In the review literature it is presented as a framework that captures Brownian yet non-Gaussian diffusion, anomalous diffusion, ageing, and weak ergodicity breaking, and that can connect trajectory statistics to hidden mobility fluctuations (Akimoto et al., 16 Sep 2025).

A concrete biophysical implementation appears in heterogeneous biological membranes. For a single protein diffusing on a two-dimensional membrane, the dynamics is written as

$$\frac{d\mathbf{r}(t)}{dt}=\sqrt{2}\,\mathbf{B}(t)\cdot \mathbf{w}(t),$$2

where $$\frac{d\mathbf{r}(t)}{dt}=\sqrt{2}\,\mathbf{B}(t)\cdot \mathbf{w}(t),$$3 is the normalized phase-field order parameter distinguishing ordered $$\frac{d\mathbf{r}(t)}{dt}=\sqrt{2}\,\mathbf{B}(t)\cdot \mathbf{w}(t),$$4 and disordered $$\frac{d\mathbf{r}(t)}{dt}=\sqrt{2}\,\mathbf{B}(t)\cdot \mathbf{w}(t),$$5 domains. In multi-particle systems, a drift term from a Lennard–Jones interaction is added,

$$\frac{d\mathbf{r}(t)}{dt}=\sqrt{2}\,\mathbf{B}(t)\cdot \mathbf{w}(t),$$6

with

$$\frac{d\mathbf{r}(t)}{dt}=\sqrt{2}\,\mathbf{B}(t)\cdot \mathbf{w}(t),$$7

In this setting, diffusivity fluctuations arise because the particle samples a heterogeneous membrane landscape; the residence-time distributions in $$\frac{d\mathbf{r}(t)}{dt}=\sqrt{2}\,\mathbf{B}(t)\cdot \mathbf{w}(t),$$8 and $$\frac{d\mathbf{r}(t)}{dt}=\sqrt{2}\,\mathbf{B}(t)\cdot \mathbf{w}(t),$$9 are empirically of the form

$\xi(t)$00

and the RSD of the TAMSD exhibits an intermediate plateau and $\xi(t)$01 behavior at short and long times. The paper interprets crowding and domain preference as mechanisms that regulate subdiffusion, effective diffusivity, and protein partitioning (Sakamoto et al., 2023).

Entangled polymers and supercooled liquids supply further archetypes. In the reptation model, the center-of-mass motion can be cast as an LEFD whose diffusivity matrix is controlled by the end-to-end vector; in supercooled-liquid-inspired two-state models, fast and slow diffusivity states represent dynamic heterogeneity. In both cases the ensemble MSD may remain normal, while RSD and related time-averaged observables resolve the hidden relaxation timescale of diffusivity fluctuations (Uneyama et al., 2014).

Protein dynamics is another recurring context. The non-Markovian GLE-based conformational model with $\xi(t)$02 and the DWCDD model with $\xi(t)$03 both use conformational variables to generate fluctuating mobility. One result is that finite-time effective diffusivities can depend strongly on memory, initial conditions, and observation protocol; another is that apparently sharp temperature-dependent changes can be explained as nonequilibrium crossovers controlled by internal relaxation times rather than equilibrium phase transitions (Kimura et al., 2023, Shirataki et al., 5 Dec 2025).

Across these applications, LEFD repeatedly separates two questions that are often conflated. One concerns the scaling of the MSD. The other concerns the statistics of diffusivity itself—its fluctuations, correlation time, sojourn-time law, and dependence on preparation. LEFD is useful precisely because it makes the second question explicit.