Breaking the Brownian barrier: models and manifestations of molecular diffusion in complex fluids

Harish Srinivasan *^ab, Veerendra K. Sharma *^ab and Subhankur Mitra ^ab
aSolid State Physics Division, Bhabha Atomic Research Centre, Mumbai, 400085, India. E-mail: harishs@barc.gov.in; sharmavk@barc.gov.in
^bHomi Bhabha National Institute, Mumbai, 400094, India

First published on 12th November 2024

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Harish Srinivasan

Dr Harish Srinivasan has been a research scientist at the Solid State Physics Division of the Bhabha Atomic Research Centre, Mumbai, since 2015. He earned his doctoral degree from the Homi Bhabha National Institute, with a focus on the non-Gaussian and non-Markovian behaviors of molecular diffusion in complex fluids. His expertise lies in non-equilibrium statistical mechanics, quasielastic neutron scattering, and molecular dynamics (MD) simulations. By integrating these disciplines, he has significantly advanced the understanding of diffusion mechanisms in various complex fluids, including supercooled liquids, deep eutectic solvents, lipid bilayers, and micelles.

Veerendra K. Sharma

Prof. Veerendra K. Sharma is a scientist currently serving at the Solid State Physics Division of the Bhabha Atomic Research Centre (BARC), India. He also holds the position of Associate Professor at the Homi Bhabha National Institute (HBNI), Mumbai. His research encompasses a broad range of fields, including membrane biophysics, soft condensed matter, complex fluids, and energy materials. He has expertise in neutron scattering and molecular dynamics (MD) simulations to investigate these systems, providing valuable insights into their structure and dynamics. Prof. Sharma earned his PhD from HBNI, Mumbai, where his doctoral work was recognized with the HBNI Outstanding Doctoral Thesis Award. He completed his postdoctoral research at the Oak Ridge National Laboratory (ORNL), USA. As a testament to his exceptional contributions to science, Prof. Sharma has been inducted into all three of India's premier national science academies in their respective young scientist leagues. He has been elected as an Associate of the Indian Academy of Sciences (IASc) and holds memberships in the Indian National Young Academy of Sciences (INYAS), the Indian National Science Academy (INSA), and The National Academy of Sciences, India (NASI). Prof. Sharma has received numerous prestigious accolades, including the INSA Distinguished Lecturer Fellowship, NASI Young Scientist Award, IPA-Buti Foundation Award, SMC Bronze Medal, Best Young Physicist Award at the Young Physicists' Colloquium, and DAE Young Scientist Award. His contributions extend to leadership roles, having served as a national core committee member of INYAS from 2022 to 2024.

Subhankur Mitra

Prof. Subhankur Mitra is presently working at the Solid State Physics Division, Bhabha Atomic Research Centre, Mumbai, India and as a Professor at the Homi Bhabha National Institute, Mumbai, India. He earned his MSc in physics from the University of North Bengal and his doctoral degree from the University of Mumbai. He has carried out his postdoctoral research at the Laboratoire Leon Brillouin, France. He has 30 years of research experience in neutron scattering and molecular dynamics simulation techniques involving the dynamic behavior of guest molecules such as water and hydrocarbons inside a variety of host materials including clays and zeolitic materials. He has also extensively studied the dynamics in micelles, vesicles, and lipids using neutron scattering and molecular dynamics simulation.

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1 Historical overview of Brownian diffusion

The notion of random motion has been a topic of great philosophical and scientific discourse throughout the history of humanity. As one of the earliest proponents of this idea, a Roman poet and philosopher Lucretius in his scientific poem “On the Nature of Things”¹ wrote the following about the behaviour of dust particles:

“And thus they flit around in all directions,

At random all and every way, and fill

The hidden nooks of things with a seething whirl.”

An idea he proposed to reinforce the atomistic models in Roman philosophy bears a gross resemblance to the nature of the random motion of particles suspended in a fluid. However, the first true discovery of random motion of particles in a fluid is credited to the botanist Robert Brown. In 1827, during a study on the suspension of pollen grains in water, he observed numerous particles ejected by pollen grains performing zig-zag motion. Through a series of observations on different materials, Brown noted that this motion was not related to life, but rather a physical phenomenon that had no biological origin.² After Brown's notable work, very little progress was made in understanding the physical principles of this process.

In the second half of the 19th century, significant advancements were made in the development of molecular kinetic theory, spearheaded by scientists such as Maxwell,³ Boltzmann,⁴ and Gibbs.⁵ These pioneers laid the groundwork for understanding the behavior of gases at a molecular level, describing how macroscopic properties arise from microscopic interactions. This progress in kinetic theory prompted the French physicist Gouy⁶ to propose that the erratic motion observed in Brownian particles could be explained by the random collisions of molecules in a fluid.

However, it was not until 1905 that a more rigorous and quantitative framework for Brownian motion was established by Albert Einstein.⁷ Einstein's theoretical work provided a statistical description of the random movements of particles, linking the observable diffusion coefficient to fundamental properties such as temperature and viscosity. This breakthrough was pivotal in demonstrating that Brownian motion could serve as direct evidence for the existence of atoms and molecules, bridging the gap between macroscopic observations and microscopic theory.

Simultaneously, other scientists were arriving at similar conclusions. Smoluchowski⁸ developed a parallel theory, independently corroborating Einstein's findings and extending the mathematical treatment of Brownian motion. Additionally, the Australian physicist Sutherland⁹ contributed to the theoretical understanding of this phenomenon, reinforcing the emerging consensus about the molecular origins of Brownian motion.

A crucial experimental validation of these theoretical advancements was provided by Jean Baptiste Perrin. In a series of meticulous experiments, Perrin observed the Brownian motion of colloidal particles suspended in a liquid and quantitatively verified the predictions made by Einstein's theory. By measuring the distribution and movement of these particles, Perrin was able to determine Avogadro's number, thereby providing the first direct evidence for the existence of atoms and molecules. His work not only confirmed the theoretical models of Brownian motion but also played a critical role in establishing the atomic theory of matter.¹⁰

1.1 Einstein's model

In his groundbreaking paper published in 1905,⁷ Einstein built upon the progress in molecular kinetic theory, primarily pioneered by the efforts of Boltzmann, Maxwell, and Gibbs. By employing these fundamental principles, Einstein addressed the phenomenon of Brownian motion, elucidating the molecular or atomic underpinnings of the persistent and irregular motion displayed by microscopic particles suspended in a stationary fluid.

Owing to the probabilistic nature of particle dynamics, a probability distribution function (PDF), P(x, t), characterizing the probability of finding the particle at some position x at a time t, is considered along with certain initial conditions P(x, 0). In Einstein's model, the Brownian motion of particles is propagated by making small displacements of length Δ. These displacements essentially occur due to random molecular collisions on the particle and are characterized by a probability distribution, f(Δ). An equation governing the time-evolution of the PDF P(x, t + τ) can be given as,⁷

	(1)

The above equation quantitatively relates the probability of detecting particles at position x at time t + τ to two factors: (i) the likelihood of encountering the particle at x − Δ at time t and (ii) the probability of the particle undergoing a displacement of length Δ. It is crucial to emphasize several key assumptions that form the foundation of the aforementioned equation. Foremost among these assumptions is that it considers the occurrence of a single displacement within the time interval τ. Additionally, the equation assumes that the probability of the jump's magnitude remains unaffected by the displacements at an earlier time. These assumptions align naturally with the meaningful scales inherent to the time interval τ. This interval is considerably small when compared to the macroscopic time frame of experimental measurements, a factor that allows it to be treated as a single displacement. However, it retains a reasonably large magnitude when compared to average time-interval between molecular collisions, thereby maintaining the assumption of independence between consecutive jumps. In order to obtain a rate equation governing the probability density P(x, t), we consider the Taylor expansion of (1) with respect to τ on the right hand side and Δ on the left hand side, yielding,

	(2)

The first term on the sides cancel out, owing to the normalization of f(Δ) (as each displacement for the particle to go from x to x + Δ is mutually exclusive, the sum of the probability of all these mutually exclusive processes will be equal to one under normalization). Furthermore, considering that the suspended particles experience isotropic collisions, it is enforced that the mean jump-length is zero (〈Δ〉 = 0). Therefore we have,

	(3)

where we have retained terms of only lowest order with non-zero contribution in the Taylor expansion. This truncation is equivalent to the physical assumption that the observation length x ≫ Δ and time scales t ≫ τ. It is easy to recognize the structure of eqn (3) as the diffusion equation, with the diffusivity, D, given by 〈Δ²〉/(2τ). The solution to eqn (3) is given by a standard Gaussian distribution, for an initial condition P(x, 0) = δ(x), as shown in ref. 7 and 11,

	(4)

Furthermore, an immediate consequence (eqn (4)) is the nature of average particle displacements. Due to the isotropic nature of the motion, mean displacement 〈x〉 is zero. However, the mean-squared displacement (MSD), 〈x²〉 = 2Dt, exhibits a linear relationship with time. The square-root time-dependence of displacement is a distinctive feature of Brownian motion.

1.2 Langevin's model – treatment of velocity

While Einstein's model provided a breakthrough in understanding of molecular kinetic theory it primarily lacked the description of velocity of the Brownian particles in the system. In 1908, Langevin provided a model¹² to describe the velocity of the diffusing particle. At its core, Langevin's model aims to describe how a single particle moves through a viscous medium (bath), like a small particle suspended in a liquid. The equation governing the velocity of the particle is given by,

	(5)

The equation describes the motion of a particle in a fluid, influenced by viscous drag and random thermal fluctuations. The first term represents the viscous drag characterized by frictional constant, γ. The second term, ξ(t), is the random force experienced by the Brownian particles due to collision atoms/molecules of the fluid. The random force, ξ(t), is referred to as a stochastic process, which represents a random variable that varies in time¹³ and is defined through its statistical measures or ensemble averages (denoted by 〈〉). The stochastic process employed in the Langevin equation (eqn (5)) is called the Gaussian white noise. Notably, it has zero average, 〈ξ(t)〉 = 0, with an underlying Gaussian distribution, reflecting the isotropic nature of collisions on the Brownian particle. Secondly, the variance of ξ(t) is given by a delta correlation, indicating that no two collisions happening at two different instants of time are correlated. Mathematically, this is given as, 〈ξ(t)ξ(t′)〉 = 2γk_BTδ(t − t′), where T is the temperature of the bath and k_B is the Boltzmann constant. This relationship between the variance of ξ(t) and frictional constant γ is fixed by the conditions of thermal equilibrium at a temperature T and is referred to as the fluctuation dissipation theorem.

The Langevin's model treats particle velocity as a stochastic process, driven by another stochastic process, ξ(t), which is a Gaussian white noise. Notably, the velocity also exhibits a Gaussian distribution owing to the underlying Gaussian nature of the white noise. Moreover, the velocity of particles exhibits Markovian behavior, meaning it has no history dependence and only depends on its current state. This Markovian property arises from the delta-correlation characteristic of the white noise term in the Langevin equation. Physically, this indicates that the size of Brownian particles is sufficiently large, such that subsequent collisions are completely uncorrelated. Later, it shall be discussed how this model breaks down, when molecular collisions become correlated and Markovianity is no more preserved.

The behaviour of velocity of Brownian particles can be obtained from eqn (5) by calculating the velocity autocorrelation function (VACF), C_v(t), given by,

	(6)

The VACF decays exponentially with the decay rate being dictated by two physical attributes, frictional constant, γ, and mass of the particle, m. The decay of the VACF is enhanced for a higher frictional constant and lower mass of the particle. In the case when the frictional force is higher in the liquid, the VACF decays faster owing to a larger number of collisions by molecules of the fluid. On the other hand, when particles are more massive, they are less likely to change their initial velocity due to stronger inertial effects. The VACF can also be used to calculate the mean-squared displacement (MSD) of the diffusing particle from the Green–Kubo relationship

	(7)

At sufficiently long times (particularly for t ≫ m/γ), it is evident that the MSD attains the linear time dependence, 〈x²(t)〉 ∼ 2(k_BT/γ)t, which resembles the behaviour of MSD from the Einstein's formalism based on the diffusion equation (eqn (3)) considering D = k_BT/γ.

The diffusion constant, D, is a crucial physical parameter of the diffusion process. In Einstein's prescription, we found that it was linked to the ratio between mean-squared displacement and waiting time between jumps. On the other hand, in the case of Langevin's model, it is linked to the frictional constant, γ. Through Stokes equation, the frictional force on a particle is the linked to viscosity of the liquid, η, according γ = 6πηr, where r is the radius of the diffusing particle. This provides the Stokes–Einstein relationship – link between diffusivity D and viscosity η,

	(8)

1.3 Equivalence of Einstein and Langevin models

It is clear that at the long-time limit, both Einstein and Langevin models accurately reproduce the linear time-dependence of MSD and the Gaussian nature of probability distributions. Both the underlying physical motivation and mathematical structure of these models are fundamentally different and complementary in approach. The Einstein model directly deals with probability density of the position of the diffusing particle and provides a basis for the Fokker–Planck equation (FPE). On the other hand, Langevin's model essentially sets the stage for directly solving differentials involving stochastic processes leading to the formulation of stochastic differential equations (SDEs). It is possible to obtain a one-to-one correspondence between FPEs and SDEs under certain circumstances. We will explore this in trying to establish the link between these two seminal models of diffusion by considering the long-time limiting behaviour of the Langevin equation (eqn (5)). This limit can also be achieved equivalently by considering a massive particle, referred to as the inertial limit. In either case the contribution of the acceleration to the Langevin equation (eqn (5)) is ignored, leading to,

	(9)

where is a dimensionless Gaussian white noise of unit variance. In this limit, the velocity of the particle behaves as a Gaussian white noise and the position of the particle is obtained as an integral over the white noise,

	(10)

The integral over the dimensionless white-noise, χ(t), is also called Wiener's process (W(t)) and holds a special position in calculus of stochastic processes.¹³ If P(x, t) is the probability density associated with the stochastic process x(t) in eqn (10), the FPE equivalent equation for the SDE in eqn (9) is,¹³

	(11)

Clearly, this equation resembles eqn (3) derived in the formalism provided by Einstein, when treating Brownian motion using a random walk model. The relationship between eqn (9) and (11) is a canonical example of FPE ↔ SDE correspondence.

Two essential features emerge in the solution of Brownian motion in either of these frameworks. Firstly, the mean-squared displacement, 〈x²(t)〉, varies linearly with respect to time, t, and this behaviour is commonly referred to as Fickianity. Secondly, the Gaussianity exhibited by distribution of displacement arises from solutions of eqn (11). As a concluding remark, we note the two fundamental physical assumptions that have been used in the description of the Brownian motion model.

• Markovianity refers to lack of history dependence in the diffusion process. This assumption is grounded in the physical concept that, for particles with size or mass greater than the molecules in the fluid, collisions upon the particle are uncorrelated.

• Gaussianity pertains to the character of displacement distribution being Gaussian in nature. This behaviour is achieved as a limit of a large number of small displacements, as a manifestation of the central limit theorem.

As we shall see in the subsequent sections, both these assumptions tend to breakdown when considering the description of diffusion of molecules within complex fluids. Under such circumstances, we shall employ more general models that can describe the diffusion based on mathematically consistent models.

This perspective article focuses on probing molecular diffusion using advanced techniques such as quasielastic neutron scattering (QENS) and molecular dynamics (MD) simulations. By examining the results of these experiments and simulations, we seek to explain the observed behaviors within the framework of non-Gaussian and non-Markovian dynamics. Through detailed case studies and methodological insights, we aim to provide a comprehensive understanding of how modern techniques and models have revolutionized our approach to studying diffusion in complex fluids.

In Section 2, we describe the developments in the area of modelling dynamics beyond the Brownian regime, with a particular emphasis on building models for non-Markovian (2.1) and non-Gaussian (2.2) dynamics. In the final Section 2.3, we also explore the newly introduced model of non-Gaussian fractional Brownian motion (nGfBm). Section 3 presents a brief introduction to the techniques of quasielastic neutron scattering (QENS) and molecular dynamics (MD) simulations. Section 4 presents various case studies based on the combined power QENS and MD simulations for investigating the dynamics of molecules in complex media. Finally, Section 5 provides a brief summary and future challenges in the area of studying molecular diffusion mechanisms.

2 Beyond the Brownian regime

As our understanding of molecular diffusion has evolved, it has become increasingly evident that the classical assumptions of Gaussianity and Markovianity often fail to adequately describe the complex dynamics observed in many systems.^14–27 In particular, the simplistic view of Brownian motion as a purely random process is insufficient for capturing the intricacies of diffusion in heterogeneous environments, where factors such as molecular interactions, memory effects, and spatial heterogeneities play significant roles. This has led to the development of a variety of advanced models that extend beyond the Brownian framework to better characterize these phenomena.

One notable model is the continuous time random walk (CTRW),²⁸ which incorporates the concept of waiting times and allows for both subdiffusive and superdiffusive behaviors, highlighting the heterogeneous features of diffusion.^29,30 However, fundamentally CTRW models that exhibit subdiffusion because of divergent mean waiting times tend to exhibit weak ergodicity breaking^31,32 and may therefore not be suitable for equilibrium systems. On the other hand, subdiffusion behaviour described using fractional Brownian motion (fBm), which introduces non-Markovianity into diffusion processes,³³ exhibits ergodicity.^34,35 While under special circumstances of confined superdiffusion³⁶ and massive particles,³⁷ fBm tends to exhibit ergodicity breaking, they are not mainly relevant for the systems explored in this perspective. It has been observed that fBm could be linked to viscoelasticity of the medium, which eventually reflects as a memory driven stochastic process.³⁸ It is notable that fBm processes can be described naturally as an integral of fractional Gaussian noise (fGn)^34,35,39 in the framework of the generalized Langevin equation (GLE). This has emerged as a powerful tool for modeling systems with memory effects.¹¹ In Section 2.1, we shall explore in more detail about the GLE with specific choice memory kernels that can lead to different results. It is notable nevertheless that these models while non-Markovian are entirely driven by Gaussian noises and hence don't reproduce non-Gaussian diffusion processes.

Recent studies have also explored non-Gaussian dynamics in a variety of soft matter systems.^40–45 A number of theoretical techniques have been utilized to describe non-Gaussian diffusion processes. In the framework of CTRW strongly non-Gaussian scale-free Levy processes have been shown.^29,46 On the other hand more modern theories have focused on explaining non-Gaussian diffusion through superstatistical Langevin equations.^47,48 These models also have some strong connections with ideas from subordinated stochastic processes,⁴⁷ which are described through diffusing-diffusivity models.^49,50 However, in this perspective we build the non-Gaussian models with an aim to explain the typical jump-diffusion process that is observed in molecular diffusion in strongly interacting liquids.^51–55 In these systems, it is particularly observed that the non-Gaussian dynamics is a consquence of the cage-jump mechanism of diffusion.^52,56,57

In the following subsections, we will delve deeper into the specific aspects of non-Markovian dynamics (Section 2.1) and the characteristics of non-Gaussian dynamics (Section 2.2), and also show the development of non-Gaussian fractional Brownian motion (Section 2.3), each of which illustrates their extension beyond the traditional models and provides a framework for understanding molecular diffusion in complex fluids. It should be noted that the models discussed in this section will consider only 1D systems. The 3D generalization that will be applied in experiments is quite straightforward, except in the case of general conditions of the jump kernel used to describe non-Gaussian diffusion. The 3D extension of the generalized jump kernel is treated in Section 4.2.3 explicitly.

2.1 Non-Markovian diffusion processes

Although Markovian approximation serves good in various physical theories, its breakdown is sometimes inevitable when exploring dynamics involving crowding and strong interactions. In such circumstances, non-Markovian diffusion phenomena have been observed in numerous soft matter systems including lateral diffusion in membranes,^20,23,58,59 protein dynamics,^{30,58,60–62} polymer diffusion,^63–65 and crowded fluids.^16,66–68 The origin of this non-Markovian nature has been linked to various physical origins such as viscoelasticity,^38,69 crowding of molecules^19,23,59,70etc. While a general prescription to handle non-Markovian processes is notoriously difficult, a generally preferred prescription is using the generalized Langevin equation (GLE).^11,71–73 Typically the GLE is given by incorporating a memory kernel, M(t), in the Langevin equation (eqn (5)),

	(12)

where ζ(t) is the random force, which has an underlying Gaussian distribution. In order to satisfy the conditions of thermodynamic equilibrium, fluctuation-dissipation^11,74 relation for the GLE sets a constraint between the memory kernel M(t) and autocorrelation of the random force ζ(t),

〈ζ(t)ζ(t′)〉 = kBTM(t − t′)	(13)

It is easy to verify that the Langevin equation (eqn (5)) is recovered by considering the memory kernel of the form M(t) = 2γδ(t). This essentially corresponds to no memory in the random force effectively leading to the Markovian behaviour in velocity.

A general expression for the velocity autocorrelation (VACF) for particles obeying GLE can be obtained from eqn (12). Multiply on both sides of eqn (12) by v(0) and take the ensemble average. By noting that 〈v(0)ζ(t)〉 = 0 due to causality, we find that

	(14)

The calculation of the VACF is a convenient way to characterize the diffusion process. Furthermore, through the Green–Kubo relation (eqn (7)) it provides a way to calculate the mean-squared displacement (MSD) as well. Considering that the Laplace transform of the above equation (eqn (14)) can serve as a mathematically convenient way to calculate the VACF for any given memory function, we can define

	(15)

where _v(u) and (u) are the Laplace transforms of C_v(t) and M(t) respectively. We shall consider two particular two cases of memory kernel – exponential and inverse power-law.

2.2 Non-Gaussian diffusion processes – jump diffusion

While the Generalized Langevin Equation (GLE) provides a robust framework to capture the inherent non-Markovian nature of diffusion processes, it fundamentally relies on Gaussian noise. Typically, Gaussian behavior in the limiting distribution of displacement is considered a consequence of the Central Limit Theorem (CLT). Violations of the CLT can occur under two main circumstances: (i) the distribution of individual displacements has a divergent variance, or (ii) the number of displacements is insufficient to reach the asymptotic limit.

In case (i), where the distribution of displacements has a divergent variance, a new class of processes known as Lévy flights is introduced. Lévy flights exhibit significant deviations from Gaussian behavior across all length and time scales due to their heavy-tailed distributions. Case (ii) is particularly relevant when studying diffusion at molecular length and time scales, especially in the presence of jump-diffusion resulting from strong molecular interactions. Here, the observation window might limit the number of displacive jumps observed, naturally leading to non-Gaussian effects. These effects are notably evident when measuring time-correlation functions at different length scales, as observed in neutron scattering through the Q-dependence.

In this section, we develop a formalism to highlight the non-Gaussian effects of jump-diffusion processes prevalent in complex molecular liquids. Specifically, we aim to create a framework that captures the transition from non-Gaussian behavior at small length scales to Gaussian behavior at larger length scales. In order to mathematically describe such processes, we propose a non-local diffusion (NLD) equation given by,

	(26)

where Λ_h(x) is the jump kernel that incorporates non-local displacements into the original diffusion equation. Considering only infinitesimally small local displacements based on a Dirac delta jump kernel, i.e. Λ_h(x) = Dδ(x), leads to the case of Brownian motion described by the original diffusion equation (eqn (11)). This implies that if the condition of non-local displacements is relaxed we obtain the original diffusion equation. Meanwhile, choosing a power-law kernel, Λ_h(x) ∼ x^−α, leads to the description of Levy flights,^77–79 which are strongly non-Gaussian at all length scales. In our study we aim to develop jump kernels, which exhibit non-Gaussianity over small length scales but tend to become Gaussian at large spatial distances, reverting to the hydrodynamic regime.

Before the development of the general jump kernel, we note that the general solution to eqn (26) is given easily in Fourier space, in terms of SISF, I(k, t). Considering the Fourier transform of eqn (26) leads to an ordinary differential equation in time whose solution is given by,

I(k, t) = I(k, 0)exp[−k2h(k)t]	(27)

where _h(k) is the Fourier transform of the jump kernel Λ_h(x). Therefore, the solutions of the NLD equation can in principle be obtained by calculating the inverse Fourier transform of eqn (27).

2.3 Non-Gaussian fractional Brownian motion

In the last section, we have discussed models that exhibit non-Markovian and non-Gaussian behaviours independently. While in the former, we chose the mathematical framework of SDEs to achieve non-Markovianity, the latter was accomplished by modifying the FPE for diffusion by incorporating non-locality. In this section, we aim to bring these two aspects together, by constructing a model that is simultaneously non-Markovian and non-Gaussian. In order to achieve this, we firstly look at the definition of the fractional Brownian motion (fBm) process as defined by Mandelbrot and van Ness.³³ Furthermore, we show the derivation of a Fokker–Planck equation for fBm in free space. Lastly, we incoporate the non-local jump kernel to obtain the Fokker–Planck equation for nGfBm.

3 Methods

In this section, we provide a brief overview of the quasielastic neutron scattering (QENS) experiments⁸⁴ and molecular dynamics (MD) simulations⁸⁵ used to investigate molecular diffusion mechanisms. Although other techniques, such as pulsed-field gradient nuclear magnetic resonance (PFG-NMR) and fluorescence spectroscopy, can also probe molecular-level diffusion, QENS offers a distinct advantage by simultaneously profiling both the spatial and temporal aspects of the diffusion process. This capability allows QENS to deliver not only relaxation timescales but also direct insights into the geometry and length scales of the dynamics. When combined with MD simulations, which cover similar time and length scales to QENS, additional atomistic details can be obtained. These simulations complement the QENS experiments, providing a comprehensive and detailed understanding of the molecular diffusion mechanisms observed.

3.1 Quasielastic neutron scattering

The advent of spectroscopy using thermal and cold neutrons has revolutionized the study of structure and dynamics at molecular scales. With neutron wavelengths in the range of Å and the corresponding energies in the millielectron volt range, neutron scattering techniques offer a unique window into processes occurring on timescales of nanoseconds to picoseconds and length scales from angstroms to nanometers. In quasielastic neutron scattering (QENS) experiments, the stochastic dynamics of molecules are captured through the analysis of the lineshape of the broadened elastic peak. This broadening arises from the motion of atoms and molecules within the sample, providing crucial information about the diffusion mechanisms at play. QENS is particularly powerful because it can simultaneously probe both the temporal and spatial characteristics of these dynamics.

By examining the energy transfer spectrum near the elastic scattering line, QENS can distinguish between different types of motions, such as translational and rotational diffusion. The resulting quasielastic broadening is directly related to the timescales and geometries of molecular motions, allowing for detailed insights into how molecules move and interact within various environments. This makes QENS an invaluable tool in the study of molecular diffusion in a number of complex fluids including polymers^41,86–88 proteins,^89–93 lipid bilayers,^94–97 micelles,^98–100 deep eutectic solvents,^25,52,53 ionic liquids^55,101,102 and biological systems^94,103–105

In 1954, Léon van Hove¹⁰⁶ introduced a comprehensive formalism for analyzing neutron scattering experimental spectra using correlation functions, now known as van Hove correlation functions. This formalism laid the groundwork for interpreting neutron scattering data in terms of molecular dynamics. van Hove's approach defines two distinct types of correlation functions: the van Hove distinct-correlation function and the van Hove self-correlation function. These functions provide detailed insights into the spatial and temporal correlations of particles within a system. In this section, we particularly focus on the van Hove self-correlation, which is relevant for describing the self-diffusion process.^{69,84,96,107,108} Notably, the van-Hove self-correlation function is linked to the incoherent scattering function of neutron scattering, which is the dominant contribution in hydrogeneous systems.^69,84,108 Furthermore, we also describe the typical features of QENS spectra of a particle executing Brownian motion.

The van-Hove self-correlation function, G_s(r, t), is defined as,^84,106

	(47)

where r_i(t) is the position of the ith particle at any instant of time t and N is the total number of particles in the system. The angular brackets 〈〉 indicate ensemble average. Intuitively, G_s(r, t) provides probability that the displacement of the particle is equal to r at any given time t, provided that the particle started at the origin at t = 0.

The QENS spectra measured in neutron scattering experiments are linked to the van-Hove self-correlation function through a double Fourier-transform in space and time,

	(48)

where Q is the momentum-transfer and ω is the energy transfer in the neutron scattering experiment. Here I_s(Q, t) is the self-intermediate scattering function (SISF), which is often convenient to describe the dynamics in complex fluids. The SISF, I_s(Q, t), is defined according to,

	(49)

It is notable that, while working with liquids, the obtained spectra are considered as an isotropic average, essentially reducing to the QENS spectra and SISF to be function of the magnitude of momentum transfer |Q| = Q. We also note that the SISF is a quantity that is conveniently calculated from MD simulations, as described in Section 3.2, and therefore often serves as a direct link between QENS and MD simulations.

In the case of a particle undergoing Brownian motion, it is easy to show that G_s(r, t) follows eqn (11). Therefore, the solutions in the Fourier-space can be given as, I_s(Q, t) = exp[−DQ²t], wherein D is the diffusivity of the particle. The IQENS spectra, therefore, can be given by,

	(50)

where Γ(Q) is the half-width at half-maximum (HWHM) of the Lorentzian and varies quadratically with Q, given by Γ(Q) = DQ² (in natural units, where ħ = 1). Hence, in systems demonstrating Brownian motion, fitting their QENS spectra with a Lorentzian and characterizing the Q-dependence of the HWHM of Lorentzian with quadratic variation can yield estimations of the diffusivity, D, of the particles within the medium. The Lorentzian lineshape of S_inc(Q, E) and the quadratic Q-dependence of the HWHM are the hallmark signatures of Brownian motion within the lens of QENS experiments. Therefore, any violations in either or both of these behaviours indicate a deviation from the model of Brownian motion, as we shall discuss in various examples in the subsequent sections.

While this perspective article mainly focuses on violations from the Brownian regime, it is expedient to briefly discuss the typical examples that exhibit Brownian motion as observed through QENS. Ideally, weakly interacting systems with monatomic diffusing units have become the best candidates to exhibit Brownian motion. For instance, among the earliest studies on liquid argon, Dasanacharya and Rao¹⁰⁹ observed an evident Brownian diffusion. In more recent times, studies exploring diffusion in liquid metals^110,111 have also shown the mechanism of diffusion to be robust under the Brownian consideration.

While these models work well with monatomic systems, the molecular liquids essentially have multiple dynamical degrees of freedom. Therefore, we elucidate the behavior of the SISF and the QENS spectra of systems where particles undergo two distinct independent dynamical processes. These processes typically include localized diffusion (rotational, cage-rattling, etc.) and long-range diffusion, characterized by their respective SISFs, I_loc(Q, t) and I_long(Q, t). The overall SISF in such cases can be expressed as a product of the individual components:

Is(Q, t) = Iloc(Q, t)Ilong(Q, t)	(51)

In the energy domain, this product relationship translates to a convolution such that the incoherent scattering function S_inc(Q, ω) is given by:

Sinc(Q, ω) = Sloc(Q, ω) ⊗ Slong(Q, ω)	(52)

This convolution indicates that the resultant QENS spectra embody contributions from both localized and long-range diffusion processes, effectively combining their individual dynamic signatures into a comprehensive spectral profile.

3.2 Molecular dynamics (MD) simulations

In the context of studying molecular diffusion, MD simulations offer a complementary approach to experimental techniques such as quasielastic neutron scattering (QENS). Neutron scattering spectra, which result from neutron interactions with an ensemble of atoms, typically require phenomenological models to interpret the mechanism of diffusion. In contrast, classical molecular dynamics (MD) simulations provide atomistic trajectories, allowing for the extraction of detailed dynamical information about particular groups of atoms and their individual degrees of freedom. This specificity in information is crucial for validating and enhancing the phenomenological models used to analyze QENS spectra, offering a complementary approach to understanding the system's dynamics. This synergy between MD and QENS is particularly valuable for interpreting complex diffusion processes in heterogeneous systems.

Using trajectories from molecular dynamics (MD) simulations, an extensive array of dynamical parameters can be calculated, including mean-squared displacement (MSD), velocity autocorrelation function (VACF), and self-intermediate scattering function (SISF). These functions provide critical insights into the dynamic behavior of the system under investigation.

Generally, MSD provides a first peek into the behaviour of the diffusion process occurring in the system. It is calculated from the simulation trajectories using the formula⁸⁵

	(53)

where N is the total number of particles in the system and t₀ is an arbitrary time-origin. It is notable that in eqn (53) a time-origin averaging over t₀ is considered to obtain the MSD as a thermodynamic quantity under the assumption of ergodicity.

While MSD and VACF provide basic information about the dynamics of the system, the SISF I_s(Q, t) provides deeper insight into the nature of the dynamical process as it possesses information about all the higher moments of displacement. Furthermore, the SISF is also related to the experimental spectra measured in QENS experiments through a time Fourier transform of the incoherent scattering function measured in neutron scattering. From MD simulation trajectories, it can be conveniently calculated using the formula,

	(54)

where Q is the scattering vector or momentum transfer. The overline denotes averaging over all Q-orientations to obtain isotropic averaging. From eqn (48), it is evident that the calculated SISF I_s(Q, t) is related to the time-Fourier transform of the incoherent neutron scattering law, S_inc(Q, ω).

4 Case studies

In this section, we present case studies based on QENS experiments and MD simulations across various systems to establish the occurrence of non-Markovian and non-Gaussian dynamics, either independently or together. In Section 4.1, we examine the lateral subdiffusion of lipids within a bilayer system.^23,59 Through detailed simulation analyses corroborated by QENS measurements, we demonstrate that while the lateral motion of lipids is Gaussian, it exhibits non-Markovian characteristics. The asymptotic behaviour of lateral subdiffusion is described using the power-law memory kernel in the GLE, as described in Section 2.1. In Section 4.2, we discuss a recent study on the cage-jump diffusion mechanism in deep eutectic solvents (DESs).^25,52 This system displays Markovian but non-Gaussian diffusion dynamics. To interpret the findings from simulations and experiments, we utilize the Non-Linear Diffusion (NLD) model discussed in Section 2.2. After exploring systems that exhibit either non-Gaussian or non-Markovian dynamics independently, we also show the recent findings on universal subdiffusion crossover observed in glass-forming liquids. This phenomenon is effectively explained using the non-Gaussian fractional Brownian motion (nGfBm) model developed in Section 2.3, which necessarily combines the non-Gaussian and non-Markovian effects.

4.1 Lateral subdiffusion of lipids

The lateral motion of the lipid within the leaflet is of key interest since it plays an important role in various physiologically relevant membrane processes including cell signalling, membrane trafficking, and cell recognition. However, the model description of the lateral motion of the lipids is not fully agreed upon in the literature. Different models such as Fickian diffusion,^95,112–115 ballistic flow like motion,^116,117 localized translational motion of lipids in a cylindrical volume,¹¹⁸ and sub-diffusive motions^{23,59,119,120} have been used to describe the lateral motion of the lipids. In this section, we show how the lateral motion of lipids can be described within the framework of the generalized Langevin equation (GLE) with a power-law memory kernel, for a lipid bilayer system made of cationic lipids dioctadecyl dimethyl ammonium bromide (DODAB).^23,120,121 The results of the DODAB bilayer are presented at two different temperatures, 298 K and 350 K, corresponding to ordered and the fluid phases of the DODAB membrane system respectively.

4.2 Jump diffusion in deep eutectic solvents (DESs)

The early years of this century witnessed the emergence of a new class of solvents, known as deep eutectic solvents (DESs),^123–126 which exhibited physicochemical properties very similar to ILs. DESs are created by mixing two or more compounds at a specific molar ratio corresponding to their eutectic point.^123–128 Generally, these mixtures have a considerably lower freezing point compared to the parent compounds.^125,126 These solvents have found extensive applications in various industrial processes, including electrodeposition,^129–133 catalysis,¹³⁴ nanoparticle¹³⁴ and nanotube^135,136 synthesis, drug transport,¹³⁷ CO₂ capture,¹³⁸etc.

Recently, it has been shown that a mixture of acetamide (CH₃CONH₂) and a group of lithium salts (LiX, X = ClO₄, Br, NO₃) in a molar ratio of 78:22 form DES having freezing points below room temperature.^127,139 It has been suggested that depression of freezing points in these systems can be ascribed to the ability of these salts to break the inter-amide hydrogen bonding. Among the three salts, LiClO₄ forms the least viscous DES at room temperature, indicating the larger affinity of perchlorate ions for the amide group.¹²⁷ While there are plenty of studies involving the macroscopic transport of DES systems, a comprehensive model of microscopic dynamics is not well established. The diffusion mechanism of acetamide in these DESs^25,52,140 is unraveled through MD simulations, and these results are utilized to model the QENS experimental data. The observed behaviour establishes the existence of caging and jumps in the diffusion process, which leads to the manifestation of dynamical heterogeneity (DH) and the non-Gaussian diffusion process.

4.3 Subdiffusion crossover in glass formers

Molecular and polymeric glass formers commonly exhibit a crossover from non-Gaussian to Gaussian diffusion while still being non-Fickian (subdiffusive).^{24,40,86,144–147} Although extensively studied via simulations^40,41,144 and experiments,^{40,41,144–147} a comprehensive understanding of the underlying basis for non-Gaussianity across these crossovers remains elusive due to the absence of a first-principles model. Glass formers typically exhibit an exponential decay in their displacement distribution,^42,49,148 a characteristic observed in various complex fluids such as colloidal suspensions,^42,44,45 Si atoms in a silica melt,^148,149 and Lennard-Jones particles.^148,150 This behavior is attributed to large deviations and randomization of the number of jumps in particle displacement.¹⁵¹ However, the precise nature of the displacement distribution in glass formers undergoing subdiffusion crossover remains unexplored.

In glass-formers the SISF is typically found to follow a stretched exponential function, I_s(Q, t) = exp[−(t/τ_s(Q))^β]. Here, β represents the stretching parameter, reflecting the deviation from an exponential relaxation profile, while τ_s denotes the characteristic relaxation time. Numerous experimental^{40,41,86,145,147} and computational^41,144,147 investigations reveal a crossover in τ_svs. Q relationship near the first maximum, Q₀, of the structure factor. Essentially, for Q < Q₀, τ(Q) ∼ Q^−2/β, meanwhile for Q > Q₀, τ(Q) ∼ Q⁻². At low-Q values (<Q₀), juxtaposing the relationship τ_s ∼ Q^−2/β with the stretched exponential decay inevitably leads to a Gaussian sub-diffusion with a MSD behaviour, 〈δr²(t)〉 ∼ t^β. But, in the high Q regime (>Q₀), where τ_s ∼ Q⁻², the diffusion mechanism cannot be described within the Gaussian approximation.^40,86,144 Therefore, this crossover in the behaviour of relaxation time has been attributed to a transition from Gaussian (for Q < Q₀) to a non-Gaussian (for Q > Q₀) sub-diffusion in these media.^40,144,145 In the subsequent discussion, we delve into characterizing this sub-diffusion crossover by formulating a Fokker–Planck equation for the nGfBm model. Our findings underscore that the emergence of this crossover is exclusively a result of the non-locality induced by the jump kernel in the model.

5 Summary and future outlook

The Brownian motion model has played a crucial role in advancing our understanding across diverse disciplines such as biology, materials science, finance, and environmental science. By providing insights into stochastic processes, it has facilitated numerous discoveries and applications. Yet, as experimental and simulation techniques evolve, we increasingly uncover instances where the assumptions of Gaussianity and Markovianity inherent in the Brownian motion model are inadequate. This realization underscores the necessity for alternative models that can capture the complexities of real-world systems without relying on these assumptions. This perspective article highlights recent advancements in addressing challenges encountered in modeling the diffusion mechanism of molecules in complex fluids. Initially, the approach focuses on studying systems that manifest either non-Gaussian or non-Markovian behavior exclusively. We consider mathematical models that extend beyond the traditional Brownian regime, offering a robust framework for characterizing diffusion in such systems. These models are strongly supported by experimental studies utilizing neutron scattering and computational studies employing MD simulations.

The Generalized Langevin equation (GLE) provides a framework for describing systems with non-Markovian dynamics driven by colored noise, such as power-law noise. This extension allows for modeling fractional Brownian motion (fBm), which deviates notably from traditional Brownian motion. In studies of lipid lateral motion, power-law noise has been identified as a driving force, characterized by an associated power-law memory kernel within the GLE framework. Molecular dynamics (MD) simulations further confirm the lateral diffusion as a Gaussian process, supporting the use of underlying Gaussianity in power-law noise. Experimental findings from Quasi-Elastic Neutron Scattering (QENS) experiments also exhibit stretched exponential relaxation behavior, aligning with this description. This study describes the dynamics of a system that exhibits exclusively non-Markovian dynamics.

A significant challenge in studying lipid dynamics is reconciling the disparate dynamical parameters obtained from various techniques such as Fluorescence Correlation Spectroscopy (FCS),¹⁵⁵ Nuclear Magnetic Resonance (NMR),¹⁵⁶ and Quasielastic Neutron Scattering (QENS).⁹⁵ Long-time molecular dynamics (MD) simulation studies have shown that while lateral motion exhibits subdiffusion, it eventually transitions to the Brownian regime after a few nanoseconds.^30,59 This transition can explain the apparent discrepancies in the dynamical parameters measured by different techniques, as each method probes a narrow time domain where the dynamics follow distinct mechanisms. Therefore, the next step is to develop a model that captures the transition from non-Brownian to Brownian regimes, making it applicable across various experimental observations.

Unlike the GLE, the non-local diffusion (NLD) model, with its non-local jump kernels, inherently exhibits non-Gaussianity, resulting in spatially heterogeneous diffusion processes. This trait is validated through the calculation of the non-Gaussian parameter of the model, which exhibits a very slow decay. It's therefore established that the cage-jump diffusion mechanism inherently introduces dynamical heterogeneity in the system. This article provides evidence of applicability of NLD in explaining the diffusion mechanism of molecules in deep eutectic solvents (DESs), paving the way for explaining the origin of dynamical heterogeneity in these complex media. Employing this model of NLD, the QENS data on deep eutectic solvents (DESs) were explained, particularly describing the cage-jump diffusion mechanism.

A number of studies have noted that the cage-jump diffusion mechanism can be an underlying origin of Stokes–Einstein breakdown.^157–159 Therefore a detailed analysis is required to understand the ramifications of NLD on Stokes–Einstein breakdown. It is possible that the scale-dependent diffusion process in the NLD model could lead to a scale dependent Stokes–Einstein breakdown, which has been observed in ionic liquids¹⁴¹ and deep eutectic solvents,^25,160,161 which also manifest the cage-jump diffusion process. In fact, some recent theoretical studies do indicate that distributed medium viscosity is linked to quasi-exponential tails in displacement distribution.¹⁶² Therefore, exploring this aspect will also provide important insights into the nature of relationship between Stokes–Einstein breakdown and dynamical heterogeneity.^25,160

Another notable mechanism that can be explored using the NLD model is the Fickian yet non-Gaussian diffusion (FnGD), which has recently been observed in a variety of systems.^42,44,45,163 As discussed in Section 2.2, the NLD model exhibits strongly non-Gaussian behavior while maintaining a mean-squared displacement that is linear in time. Several theoretical models, such as diffusing diffusivity,^49,164 subordinated diffusion,^47,50,151 and superstatistics,^47,48 have been proposed to describe FnGD. It is becoming evident that the cage-jump mechanism of diffusion could provide a novel and physically intuitive explanation for FnGD. However, further research is needed to establish the connection between the cage-jump model and the existing theoretical models. Such connections can possibly be explored based on theory of subordinated processes.^{47,151,165–167}

The previous section delves into systems exhibiting simultaneous non-Markovianity and non-Gaussianity. In such cases, traditional Markovian continuous time random walk (CTRW) models fall short due to jumps with inherent long time history dependence. Introducing the non-Gaussian fractional Brownian motion (nGfBm) model addresses this limitation, offering a modified version of NLD for systems driven by colored noise. The nGfBm model provides a valuable framework for naturally invoking the emergence of sub-diffusion crossover in glass-formers. This model also provides a method to show the origin of exponential tails in such dynamically heterogeneous systems.

While the nGfBm sufficiently describes the crossover from non-Gaussian to Gaussian subdiffusion, it doesn't provide more insights into the nature of viscosity–diffusion decoupling in the glass formers. Developing a microscopic theory on subordinated stochastic processes might provide a link to bulk relaxation processes in the medium giving additional insights into the possibility of Stokes–Einstein breakdown as a consequence of the inherent non-Gaussianity in the model. Further research on the microscopic foundations of the nGfBm is neccessary to understand these aspects.

In summary, this perspective article provides a comprehensive overview of molecular diffusion, which exhibits violation of Brownian motion. These violations have been described using frameworks, which essentially incorporate non-Markovianity and non-Gaussianity into the diffusion phenomena. Additionally, these models have also been applied and validated on data obtained on various systems using QENS experiments and MD simulations.

Data availability

No primary research results, software or code have been included and no new data were generated or analysed as part of this review.

Conflicts of interest

There are no conflicts to declare.

Notes and references

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