Non-Gaussian Brownian Diffusion in Dynamically Disordered Thermal

Article pubs.acs.org/JPCB

Non-Gaussian Brownian Diffusion in Dynamically Disordered Thermal Environments Neha Tyagi and Binny J. Cherayil* Dept. of Inorganic and Physical Chemistry, Indian Institute of Science, Bangalore 560012, India ABSTRACT: In this article, we suggest simple alternatives to the methods recently used by Jain and Sebastian [J. Phys. Chem. B 2016, 120, 3988] and Chechkin et al. [Phys. Rev. X 2017, 7, 021002] to treat a model of non-Gaussian Brownian diffusion based on the dynamics of a particle governed by Ornstein− Uhlenbeck modulated white noise. In addition to substantiating these authors’ earlier findings (which show that a particle can execute a simple random walk even when the distribution of its displacements deviates from Gaussianity), our approach identifies another process, two-state white noise, that exhibits the same “anomalous” Brownian behavior. Indeed, we find that the modulation of white noise by any stochastic process whose time correlation function decays exponentially is likely to behave similarly, suggesting that the occurrence of such behavior can be widespread and commonplace.

1. INTRODUCTION As single-particle tracking methods have improved over the years, so also has our understanding of the nature of thermally driven molecular motion in complex fluid environments. We now know, for instance, that in these systems ordinary Brownian diffusion, characterized chiefly by the linear dependence on time t of the particle’s mean-square displacement (MSD), is often not the rule but the exception. More common is the occurrence of subdiffusive or superdiffusive motion, in which the MSD of the particle varies as tα, with α < 1 for subdiffusion and α > 1 for superdiffusion.1−5 Such behavior usually stems from the absence of a clear separation between the relaxation times of the particle and the relaxation times of the molecules in its vicinity, which can lead to particle displacements that are no longer independent and identically distributed. But there is considerable evidence to show that even in the sluggish surroundings that typically lead to anomalous behavior, it is possible for a particle to both execute a simple random walk and be governed (in certain time intervals, anyway) by a non-Gaussian distribution of displacements.6−8 Explanations of this phenomenon (which is not confined to particle motion, but can encompass other kinds of stochastic fluctuations, including those that characterize stocks and shares9,10) tend to be based on one or other version of the idea that in a crowded heterogeneous medium a particle’s diffusivity can vary from instant to instant. This idea, referred to by some authors as “diffusing diffusivity”,11 though more properly described, in our view, as Brownian motion under dynamic disorder,12 has recently been implemented analytically by Jain and Sebastian13−15 and by Chechkin et al.16 using a model in which white noise forces acting on a particle are modulated by an Ornstein−Uhlenbeck process.17 The model © XXXX American Chemical Society

predicts much the same kind of non-Gaussian Brownian diffusion that has been widely observed in simulations and experiments. In this article, we would like to draw attention to certain overlooked features of the above model that when more fully explored lead to new results and suggest new directions in which to extend its findings. In particular, based on the model’s defining equations for particle motion, we demonstrate that it is possible to (1) derive the expression for the particle’s MSD more simply, (2) provide an alternate, less elaborate route to the calculation of the probability distribution function of the particle’s displacements, and (3) identify another process (or class of processes) that can also serve as a paradigm of nonGaussian Brownian diffusion. After recapitulating details of the model in Section 2 below, we show how it may be used to calculate a particle’s mean square displacement. In Section 3 we describe the calculation of the time-dependent distribution of particle displacements, while in Section 4, we calculate these same quantities for another stochastic process that we find is “anomalously” Brownian. We discuss these results in Section 5 and present some general conclusions.

2. DYNAMICALLY DISORDERED BROWNIAN MOTION Consider a particle moving through a fluid under the action of thermal fluctuations; if the amplitude of these fluctuations is itself assumed to vary stochastically, the evolution of the Received: April 25, 2017 Revised: June 7, 2017

A

DOI: 10.1021/acs.jpcb.7b03870 J. Phys. Chem. B XXXX, XXX, XXX−XXX

Article

The Journal of Physical Chemistry B

(which itself is governed by η(t)), so that in fact P(x,t) must really be understood as the marginal density distribution of the function P(x,D,t) = ⟨δ(x − x(t))δ(D − D(t))⟩; in other words, P(x,t) = ∫ ∞ −∞dDP(x,D,t).) To calculate P(x,t), one can (among other options) either solve the Fokker−Planck equation (FPE) that is satisfied by P(x,D,t) and then integrate over D, or directly evaluate the noise averages in eq 6 using path integral methods. Checkin et al.16 adopt a variant of the first approach that invokes the notion of “subordination”, while Jain and Sebastian13 adopt a phase space version of the second. Both approaches lead to essentially the same results, though Chechkin et al. suggest that the FPE approach is both simpler to implement and of potentially wider applicability, given its roots in the subordination process. Our own calculation of P(x,t), which we now present, is something of a hybrid of these two approaches, and is somewhat more transparent than either, in our opinion. It begins with the construction of an FPE from eq 6 that initially assumes D(t) to be a definite function of time, and not subject to fluctuations. This FPE is easily found (using the functional calculus formalism,18 for instance), and can be shown to be

particle’s position x(t) at time t can be described by the equation dx(t ) = D(t )θ(t ) dt

(1)

Here θ(t) and D(t) are random variables, and for the present purposes they are chosen to correspond, respectively, to white noise and to the Ornstein−Uhlenbeck (O−U) process.17 White noise is taken to be defined by the correlations ⟨θ(t)⟩ = 0 and ⟨θ(t)θ(t′)⟩ = δ(t − t′), while the O−U process is described by the equation dD(t ) = −aD(t ) + η(t ) dt

(2)

where a is a constant and η(t) is another white noise variable, but now defined by the correlations ⟨η(t)⟩ = 0 and ⟨η(t)η(t′)⟩ = bδ(t − t′), with b a second constant. Aside from minor differences of notation and convention, eqs 1 and 2 are equivalent to the equations of the diffusing diffusivity model formulated in ref 16 (cf. eq 19a−c of that reference.) It should be emphasized, however, that D(t) in the present calculations is the analogue not of the variable D(t) in ref 16 (though sharing the same symbol) but of the variable Y(t), and can therefore assume both positive and negative values. So in our calculations it is the quantity D2(t) that is to be identified with a diffusion coefficient; indeed, as can be verified, this quantity has the right units for such an identification. It may also be verified that a the units of a rate, and b the units of an area per square time. Eq 1 is formally a first order differential equation in time that under the initial condition x(0) = 0 can be integrated at once to t x(t) = ∫0 dt′D(t′)θ(t′). This expression, if squared and then averaged over the distribution of the two noise terms, assumed to be independent, yields 2

⟨x (t )⟩ =

∫0

t

dt1

∫0

∂ 2P(x , t ) ∂P(x , t ) 1 = D 2 (t ) 2 ∂t ∂x 2

(When D(t) does fluctuate, and its fluctuations are defined by eq 2, the corresponding FPE is ∂P(x,D,t)/∂t = [(D2/2)∂2/∂x2 + a(∂/∂D)D + (b/2)(∂2/∂D2)]P(x,D,t).) As is easily verified by direct substitution, the solution of eq 7, under the initial condition P(x,0) = δ(x), is

b −a |t1− t2| [e − e−a(t1+ t2)] 2a

P ̅ (x , t ) =

The substitution of eq 4 along with the correlation function of white noise into eq 3, followed by integration, leads to the following exact result bt 1⎛ 2 b ⎞ −2at ⎜D − ⎟(1 − e ) + 0 2a 2a ⎝ 2a ⎠

⎡ x2 dt ′D2(t ′))−1/2 exp⎢− ( ⎣ 2

∫0

t

⎤ dt ′D2(t ′))−1⎥ ⎦

(8a) (8b)

∫ +[D]P(x , t|[D])P[D]

(9)

where P[D] is the weight functional for a given D(t) trajectory, and ∫ +[D]represents the functional (or path) integral over all such trajectories. The evaluation of the average in eq 9 is clearly the key step in the determination of P̅(x,t), and for this we need the functional P[D], which can be obtained from eq 2 following the methods described in, for instance, ref 19 and summarized in papers from this lab.20−22 Without repeating the details, we simply note that P[D] is given by

(5)

where D0 is the initial value of D(t). Eq 5 is the first of our main findings. It is easily shown that eq 5 reduces to ⟨x2(t)⟩ = D20t + O(t2) as t → 0, and to ⟨x2(t)⟩ → bt/2a + O(1) as t → ∞.

3. DISTRIBUTION FUNCTION OF PARTICLE DISPLACEMENTS In general, the probability density P(x,t) that a random variable x(t) has the value x at time t is given formally by P(x , t ) = ⟨δ(x − x(t ))⟩

t

where the definition introduced in eq 8b is intended to highlight the fact that the distribution function of x is also a functional of D(t). D(t) itself is, of course, actually a random variable, so the distribution of x that is relevant to the present discussion is the average of P(x,t|[D]) over the trajectories of D(t), a quantity we shall denote P̅(x,t) and define as

(3)

(4)

⟨x 2(t )⟩ =

∫0

≡P(x , t |[D])

The two-time correlation function of the O−U process in this relation can be derived from eq 2; it is given by ⟨D(t1)D(t 2)⟩ = D02e−a(t1+ t2) +

1 ( 2π

P(x , t ) =

t

dt 2⟨D(t1)D(t 2)⟩⟨θ(t1)θ(t 2)⟩

(7)

⎡ a 1 P[D] = 5e at /2exp⎢ − (Df2 − D02) − 2b ⎣ 2b −

(6)

a2 2b

∫0

t

⎤ dt ′D2(t ′)⎥ ⎦

∫0

t

dt ′Ḋ (t ′)2

(10)

where Df and D0 are the final and initial values of D(t), Ḋ (t′) stands for dD(t′)/dt′and 5 is a proportionality constant that will be chosen to ensure that ∫ ∞ −∞dxP̅ (x,t) = 1.

where the angular brackets denote an average over realizations of the noise that drives the dynamics of x(t). In the present model, these dynamics are generated both by θ(t) and D(t) B


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derived by Jain and Sebastian13 (cf. eq 25 in their paper in the limit n = 1.)

To proceed further, P(x,t|[D]) is now rewritten identically as P(x , t |[D]) =

1 2π

⎡

∞

∫−∞ dyexp⎢⎣ixy −

y2 2

∫0

t

⎤ dt ′D2(t ′)⎥ ⎦

4. TWO-STATE PROCESS (11)

It is clear from the structure of eqs 1 and 3 that the meansquare displacement of any particle whose dynamics are driven by randomly modulated white noise will vary linearly with time (in some interval) if the time-correlation function of the modulating variable decays exponentially. This means that there is at least one other stochastic process that could replace D(t) in eq 1 and (potentially) still produce non-Gaussian Brownian diffusion, viz., dichotomic or two-state noise. In this section, we will demonstrate that this is in fact the case. A dichotomic variable ξ(t) that fluctuates randomly between states with the values c1 and c2 is defined, quite generally, by the 1 following properties:24,25 (i) ⟨ξ(t )⟩ ≡ ξ ̅ = 2ν (ν12c 2 + ν21c1), ̅ where νij is the transition rate from i to j, and v ̅  (ν12 + ν21)/2 is the mean transition rate, (ii)

and then substituted back into eq 9. The result can be put in the form P ̅ (x , t ) =

1 2π

∞

∞

∞

∫−∞ dyeixy ∫−∞ dDf ∫−∞ dD0Gy(Df , t|D0)

× Peq(D0)

(12)

where 2

2

Gy (Df , t |D0) ≡ e at /2 − a(Df − D0 )/2b

D(t ) = Df

∫D(0)=D

0

×

∫0

t

dt ′Ḋ (t ′)2 −

ky 2

∫0

t

⎡ 1 +[D]exp⎢ − ⎣ 2b ⎤ dt ′D2(t ′)⎥ ⎦ (13)

⟨ξ(t1)ξ(t 2)⟩ = ⟨ξ(t1)⟩2 +

Here, ky  y + a /b, and Peq(D0) is the distribution of initial D values, which is taken to be the steady state distribution of the FPE corresponding to the O−U process of eq 2, and is given by 2

Peq(D0) =

2

⎛ aD 2 ⎞ a exp⎜ − 0 ⎟ πb ⎝ b ⎠

(14)

exp( −2ν ̅ |t1 − t 2|),

dx(t ) = σ (t )θ (t ) dt

and

(17)

where θ(t) is the white noise variable introduced in eq 1, the same steps leading to eq 5 can be applied to eq 17 to show that

⎛ ⎞ k yb 2 2 ⎟ Gy (Df , t |D0) ∝ e at /2 − a(Df − D0 )/2b⎜⎜ ⎟ ⎝ bsinh( k yb t ) ⎠ ⎡ k yb × exp⎢− {(Df2 + D02)cosh( k yb t ) ⎢ 2bsinh( k yb t ) ⎣ 1/2

⟨x 2(t )⟩ = μ2 t

(18)

at all times, the angular brackets here referring to averages over θ(t) and σ(t). Eq 18 is the third of our main findings. To evaluate the corresponding noise-averaged density distribution of x [which for consistency, but at the risk of causing some confusion with the quantity defined in eq 9, we denote P̅(x,t)] we repeat the steps from eq 6 to eq 8a, replacing D(t) with σ(t), and construct the functional P(x,t|[σ]), which we then rewrite as

(15)

where the unspecified proportionality constant in this relation can be combined with the factor of 5 in eq 10, and the combination of the two (denoted 5′) fixed by normalization. After substituting the expressions for Gy(Df,t|D0) and Peq(D0) (eqs 14 and 15) into eq 12, the evaluation of P̅(x,t) reduces to the evaluation of two Gaussian integrals and a Fourier transform. When the Gaussian integrals are carried out, the result, after setting 5′ to 1/ 2π , is

P(x , t |[σ ]) =

1 2π

∞

⎡

∫−∞ dyexp⎢⎣ixy −

y2 2

∫0

t

⎤ dt ′σ 2(t ′)⎥ ⎦ (19)

If property (iii) of σ(t) is substituted into this expression, and the result averaged over σ(t), we find that

⎛ ⎞1/2 k yb 1 at /2 ixy⎜ ⎟ 2a e P ̅ (x , t ) = dye ⎜ ⎟ −∞ 2π sinh( ) k b t y ⎝ ⎠ 1 × 2 (a + 2a k yb coth( k yb t ) + k yb)1/2 (16)

∫

4ν ̅ 2

(iii) ξ (t) = (c1 + c2)ξ(t) − c1c2. For our purposes, it proves convenient to introduce a new two-state variable σ(t) defined as σ(t) = ξ(t) − ξ̅. For this variable, (i) ⟨σ(t)⟩  σ̅ = 0, (ii) ⟨σ(t1)σ(t2)⟩ = μ2e−γ|t1−t2|, where μ2 = ν12ν21(c1 − c2)2/4ν2̅ = (c1 + c2)ξ̅ − c1c2 − ξ̅2 and γ = 2ν,̅ and (iii) σ2(t) = μ1σ(t) + μ2, where μ1 = (c1 + c2 − 2ξ̅). If the dynamics of x(t) is now assumed to be given by 2

The path integral of eq 13 is of known form (being related to the propagator of the classical harmonic oscillator), and its evaluation is readily carried out. Omitting details of the calculation (which may be found elsewhere19−23), it can be shown that

⎤ − 2Df D0}⎥ ⎥ ⎦

ν12ν21(c1 − c 2)2

∞

P ̅ (x , t ) =

1 2π

∞

∫−∞ dyeixyΦ(y , t )

(20)

where Φ(y,t) = e−at⟨exp[−β∫ t0dt′σ(t′)]⟩, with α = μ2y2/2 and β = μ1y2/2.We next take the Laplace transform of Φ(y,t)(the ̂ of a function f(t) being defined as f(s) ̂ = Laplace transform f(s) ∞ −st ∫0 dte f(t)), and expand the exponential involving σ(t) to all orders, producing

ixy which, using the result ∫ ∞ = 2πδ(y), may be verified to −∞dxe ∞ satisfy the condition ∫ −∞dxP̅(x,t) = 1. Eq 16 is the second of our main findings, and it can be shown to have the same analytic structure as the corresponding distribution function

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∫0

Φ̂(y , s) =

⎡ dt e−s ′ t ⎢1 − β ⎣

∞

+

β2 2!

∫0

t

dt1

∫0

t

∫0

show, the same short time form also emerges from the distribution function derived by Jain and Sebastian.13 In this t → 0 limit, but referring now to eqs 26 and 27, the function Φ(y,t) simplifies to

t

dt1σ(t1)

⎤ dt 2σ(t1)σ(t 2) + ···⎥ ⎦

Φ(y , t → 0) ≈ 1 −

(21)

where s′ = s + α. Following Morita,26 we assume that σ(t) is Poissonian; it then follows that in nonoverlapping time intervals, the correlations of σ(t) satisfy

(22)

P ̅ (x , t ) =

This means that

∫0

∞

dte−s′t

∫0

t

dt1

∫0

t

dt 2 ...

∫0

t

dt 2m − 1

... σ(t 2m − 1)σ(t 2m)⟩ = (2m) !

∫0

t

dt 2m⟨σ(t1)σ(t 2)

(23)

The substitution of the above expression into eq 21 leads to Φ̂(y , s) =

1 s′[1 − μ2 β /s′(s′ + γ )] 2

P ̅ (x , t ) =

(24)

s+P s + sQ + R 2

(25)

where P = μ2y /2 + γ, Q = 2P − γ and R = γμ2y /2 + (μ22 − μ2μ21)y4/4. The inverse Laplace transform of this function is easily found; it is given by 2

Φ(y , t ) =

2

1 −Qt/2 e [γ sinh(Λt /2) + Λcosh(Λt /2)] Λ

(26)

Q 2 − 4R . P̅(x,t) is now given exactly by ∞ 1 1 P ̅ (x , t ) = dy eixy − Qt/2[γ sinh(Λt /2) 2π −∞ Λ

where Λ =

(27)

=

1 π

∞

ixy

∫−∞ dy [2a + t(2ae2 + by2 )]1/2

2a ⎛ 2a ⎞ |x|⎟ K 0⎜ bt ⎝ bt ⎠

(32)

Eq 32, along with eq 18, yields the following limiting behaviors for α2(t): α2(t ) ∼

5. DISCUSSION To show that a particle can both execute a random walk and yet be governed by a non-Gaussian distribution of displacements, it would have been helpful if the integral representations of P̅(x,t) in eqs 16 and 27 were known in closed form. And though they are not, they can be determined exactly in certain limits. In particular, as t → 0, eq 16 is easily shown to simplify to27 1 2a 2π

(31)

4

which, like eq 16, is unlikely to yield a closed form expression, though it can presumably be evaluated numerically. Eq 27 is the fourth of our main findings.

P ̅(x , t ) ≈

⎛ x2 ⎞ 1 ⎟⎟ exp⎜⎜ − 2πμ2 t ⎝ 2μ2 t ⎠

⎞ 6μ2 μ12 ⎛ 1 −γt 2 2 ⟨x (t )⟩ = ⎜t − [1 − e ]⎟ + 3μ2 t γ ⎝ γ ⎠

∫

+ Λcosh(Λt /2)]

(30)

a Gaussian in x. There is no obvious criterion by which to choose between the possibilities represented by eqs 30 and 31, but additional information about the shape of the distribution function can be obtained from the so-called non-Gaussianity parameter α2(t), which is defined for one-dimensional systems as3 α2(t) = −1 + < x4(t) > /(3 < x2(t) > 2), and is zero for Gaussian distributions and nonzero for non-Gaussian distributions. To calculate α2(t) for our model involving two-state noise, we need an expression for ⟨x4(t)⟩ [in addition to ⟨x2(t)⟩, which has already been calculated in eq 18]; this quantity can be obtained the same way as ⟨x2(t)⟩, i.e., by integrating eq 17, taking the fourth power of the result, and averaging over the two sources of noise. This leads to the exact expression

which after collecting terms and simplifying assumes the form Φ̂(y , s) =

⎛ |x | ⎞ 1 ⎟ K 0⎜⎜ π μ2 t ⎝ μ2 t ⎟⎠

which is entirely equivalent to eq 28. The only caveat to be noted here is that eq 29 could equally well have been resummed to the form exp(−μ2y2t/2), and this form, when substituted into eq 27 and the integral evaluated, yields

(μ2 β 2)m (s′)m + 1(s′ + γ )m

(29)

Now to this order of t, it is possible to write eq 29 in several equivalent resummed forms that allow for the Fourier inversion of eq 27, one of them being (1 + μ2y2t)−1/2. If this form is substituted into eq 27 and the integral over y carried out, the result is

⟨σ(t1)σ(t 2)... σ(t 2m − 1)σ(t 2m)⟩ = ⟨σ(t1)σ(t 2)⟩⟨σ(t3)σ(t4)⟩ ...⟨σ(t 2m − 1)σ(t 2m)⟩, t1 ≥ t 2 ≥ t3 ≥ ... t 2m − 1 ≥ t 2m

1 2 μ y t + O(t 2) 2 2

μ12 μ2

,t→0

α2(t ) ∼ 0, t → ∞

(33a) (33b)

This means that P̅ (x,t) starts off as a non-Gaussian distribution, but eventually becomes Gaussian. The same qualitative behavior is observed for particle motion governed by O−U modulated white noise [for which ⟨x4(t)⟩ = 3b2(a2t2 + 2at − 1 + e−2at)/(4a4)], and for both this and the former process, the long-time decay of α2(t) to 0 is monotonic. But in certain anomalously diffusing systems, such as phospholipids on biomembrane surfaces,28 finite-sized tracers in heterogeneously crowded 2-d environments,29 and RNA-protein aggregates in viscoelastic media,30 experiments and simulations suggest that α2(t) can actually exhibit a rise in time (from small but finite initial values) before eventually falling to 0 at long times. That neither of the theoretical models in the present work is able to capture this trend suggests that at intermediate times there may be other stochastic processes, with characteristic time constants

+ O(t 2)

(28)

where K0(z) is a modified Bessel function of order zero, a function that is clearly non-Gaussian. Barring notational differences, eq 28 has exactly the same functional form as the short time limit of the distribution function derived by Chechkin et al.16 (cf. eq 63 in their paper). As these authors D



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comparable to those of D(t) or σ(t), that regulate or influence particle motion. Tests of this possibility are presently being explored. Among the other conclusions that can be drawn from the various results in the foregoing sections are the following: (i) It is not necessary to determine the density distribution of x in order to calculate its mean-square displacement; this quantity can be found by straightforward integration of the Langevin equation, in conjunction with the correlation function of the variable that modulates white noise. When this variable describes the Ornstein−Uhlenbeck process, the MSD is given exactly by eq 5, and when it describes the two-state process, the MSD is given exactly by eq 18. In contrast, the expressions for the MSD in refs 13 and16 (in the notation and convention of those references), are of the form ct, where c = 4 in ref 13 and c = 1 in ref 16, and they were obtained from the calculated expressions for the distribution function of x. We have shown that eq 5 reduces to this form in both the short and long-time limits. (ii) The calculation of the distribution function of x can be carried out without appeal to the subordination process that underlies the approach in ref 16 or to the elaborate phase space path integral method that underlies the approach in ref 13. In the present approach, the Langevin equation of dynamically disordered Brownian motion is first expressed in its equivalent Fokker−Planck form. The solution of this FPE, which contains the time dependent diffusion coefficient D2(t), is next represented as a Fourier integral. This expression is then averaged over the stochastic trajectories of D(t) (which are governed by the O−U equation) using path integration. The path integration step (which in other calculations, such as those in ref 13, can become quite involved) is particularly easy to implement in this instance because it reduces immediately to the evaluation of the path integral of a classical harmonic oscillator, a standard and well-known textbook procedure. (Jain and Sebastian adopt this simpler approach in their recent treatment of a Lévy walk model of diffusing diffusivity,15 but start their analysis from an FPE whose time-dependent diffusion coefficient is D(t); the subsequent steps, viz., representation of the solution of this FPE as a Fourier integral and then path integration over the Lévy trajectories of D(t), remain the same.) (iii) A still simpler process than the Ornstein−Uhlenbeck process (i.e., two-state noise) can also lead to non-Gaussian Brownian diffusion when it modulates white noise. Indeed, modulation of white noise by any stochastic variable whose two-point correlation function decays exponentially is likely to lead to such behavior (at least in certain time regimes), so its widespread occurrence is not especially surprising, though it is still not fully understood.31 At a fairly basic level, however, it can be seen as an illustration of the phenomenon of dynamic disorder, in which the overlap between the characteristic relaxation time scales of two or more competing rate processes produces complex dynamics.

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ACKNOWLEDGMENTS The authors are grateful to K. L. Sebastian and R. Jain for their critical reading of the manuscript.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. ORCID

Neha Tyagi: 0000-0003-0593-0349 Binny J. Cherayil: 0000-0003-3656-5030 Notes

The authors declare no competing financial interest. E


Article

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Non-Gaussian Brownian Diffusion in Dynamically Disordered Thermal

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