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Distributional Behaviors of Time-averaged Observables in Langevin Equation with Fluctuating Diffusivity: Normal Diffusion but Anomalous Fluctuations

We consider Langevin equation with dichotomously fluctuating diffusivity, where the diffusion coefficient changes dichotomously in time, in order to study fluctuations of time-averaged observables in temporary heterogeneous diffusion process. We find that occupation time statistics is a powerful tool for calculating the time-averaged mean square displacement in the model. We show that the time-averaged diffusion coefficients are intrinsically random when the mean sojourn time for one of the states diverges. Our model provides anomalous fluctuations of time-averaged diffusivity, which have relevance to large fluctuations of the diffusion coefficient in single-particle-tracking experiments.

We consider Langevin equation with dichotomously fluctuating diffusivity, where the diffusion coefficient changes dichotomously in time, in order to study fluctuations of time-averaged observables in temporary heterogeneous diffusion process. We find that occupation time statistics is a powerful tool for calculating the time-averaged mean square displacement in the model. We show that the time-averaged diffusion coefficients are intrinsically random when the mean sojourn time for one of the states diverges. Our model provides anomalous fluctuations of time-averaged diffusivity, which have relevance to large fluctuations of the diffusion coefficient in single-particle-tracking experiments.

I. INTRODUCTION
Law of large numbers plays an important role in statistical physics. In stationary stochastic processes X t , law of large numbers or the central limit theorem tells us that time-averaged observables such as diffusivity and the ratio of occupation time converge to a constant when the measurement time goes to infinity: where the observable O(·) is a function of the stochastic process X t . In experiments, time-averaged observables are not constant because of finite measurement times. However, in some stochastic processes describing nonequilibrium phenomena, time-averaged observables are intrinsically random because of the breakdown of law of large numbers or the central limit theorem [1,2]. In other words, they do not converge to a constant even when the measurement time goes to infinity and the fluctuations never disappear. Such anomalous behavior has been studied by infinite ergodic theory in dynamical systems [3]. Infinite ergodic theory states that time-averaged observables converge in distribution, and the distribution function depends on the invariant measure as well as a class of the observation function [4][5][6][7]. Continuous-time random walk (CTRW) is a model of anomalous diffusion, where the mean square displacement (MSD) increases sub-linearly with time, and is extensively studied in disorder materials [8] as well as biophysics [9,10]. In CTRW, a random walker waits for the next jump and the waiting time is a random variable whose probability density function (PDF) ρ(τ ) follows a power-law distribution: where c 0 is a scale factor. When α ≤ 1, the mean waiting time diverges, thereby causing a breakdown of law * akimoto@keio.jp of large numbers and the central limit theorem. In this case, it was shown that the time-averaged MSD (TMSD) for a fixed lag time ∆ ≪ t, defined as does not converge to a constant but converges in distribution as t → ∞ [9,11,12]. Moreover, the PDF of the normalized TMSD, i.e., δ 2 (∆; t)/ δ 2 (∆; t) , follows a universal distribution called the Mittag-Leffler distribution, which is one of distributional limit theorems in infinite ergodic theory [12]. This distributional property for a time-averaged observable is called distributional ergodicity in stochastic processes [13,14]. Other distributional behaviors have been found in other diffusion processes such as a quenched trap model [15,16] and stored-energy-driven Levy flight (SEDLF) [14,17], where the PDF of the normalized TMSDs (timeaveraged diffusion coefficients) follows other distributions depending on the power-law exponent in the waiting time distribution, the spatial dimension as well as parameters controlling jumps of a random walker. It is important to clarify whether fluctuations of time-averaged observables are intrinsic or not, because diffusion coefficients obtained by single-particle-tracking experiments in living cells exhibit large fluctuations [10,[18][19][20][21]. Such large fluctuations will have relevance to distributional behaviors in stochastic models of anomalous diffusion.

II. LANGEVIN EQUATION WITH DICHOTOMOUSLY FLUCTUATING DIFFUSIVITY
To investigate ergodic properties in heterogeneous diffusion processes, we consider the following Langevin equation with fluctuating diffusivity (LEFD), where w(t) is the n-dimensional white Gaussian noise with w(t) = 0, and w i (t)w j (t ′ ) = δ ij δ(t − t ′ ). On the other hand, the diffusion coefficient D(t) can be a non-Markovian stochastic process. We assume that D(t) and w(t) are statistically independent. Because the diffusion coefficient is determined by shape of the particle or surrounding environment, the LEFD can describe the dynamics of a particle with inner degree of freedom. In fact, this model can be utilized in the equation of motion for the center-of-mass of entangled polymer in reptation model [22] and is related to dynamic heterogeneity in supercooled liquids [23][24][25][26]. Moreover, because the stochastic process D(t) is generic, this system includes temporally heterogeneous diffusion models induced by spacial heterogeneity such as the ones studied in [27][28][29] In our previous study [30], we have obtained the relative standard deviation (RSD) of the TMSD as a function of measurement time t in LEFD when the stochastic process D(t) is in equilibrium, where the RSD is defined by In equilibrium processes, the RSD becomes where ψ 1 (t) is the normalized correlation function of diffusion coefficients, i.e., Therefore, information on the underlying diffusion coefficient D(t) can be extracted by the RSD analysis [30,31]. Here, we investigate ergodic properties of LEFD especially in non-equilibrium cases. In particular, we consider two-state models for the stochastic process D(t). When the mean sojourn time of a state in D(t) diverges, the stochastic process becomes non-stationary, which implies that the system is intrinsically in nonequilibrium. We show normal diffusion yet anomalous fluctuations of TMSD.
Here, we consider dichotomous processes for diffusivity D(t) (see Fig. 1), i.e., D(t) = D + if the state is + and D(t) = D − otherwise (− state). Sojourn times for + and − states are random variables following different probability density functions (PDFs), ρ + (τ ) and ρ − (τ ) for + and − states, respectively. We assume that the one of the PDFs ρ + (τ ) follows either a narrow distribution where all moments are finite or a broad distribution of power-law form [Eq. (2)], and that the other PDF follows a power-law distribution, whose Laplace transform is given byρ − (s) = 1 − a − s α− + o(s) (α − < 1). In particular, we consider three cases for ρ + (x): where m k is the kth moment of sojourn times of the state +. In what follows, we set α − = α. This kind of powerlaw behavior is observed in supercooled liquids [26].

III. REPRESENTATION OF TIME-AVERAGED MEAN SQUARE DISPLACEMENT
For ∆ ≪ t, TMSD is represented by where δr(∆; t ′ ) ≡ r(t ′ + ∆) − r(t ′ ), t i is the ith transition time from one state to the other state with t 0 = 0, N t is the number of transitions up to time t. Since a particle undergoes Brownian motion in each state, is the occupation time of the state ± up to time t [Thus, T + (t) + T − (t) = t], and τ 0 is a characteristic time for the transitions of D(t). The condition of ∆ ≪ τ 0 validates the approximation that the state in [t i , t i + ∆] does not change. We have where we define a time-averaged diffusion coefficient of each state as D ± (t) ≡ T±(t) 0 δr 2 ± (∆; t ′ )dt ′ /2nT ± (t). Therefore, TMSDs always show normal diffusion and the time-averaged diffusion coefficient defined as D(t) ≡ δ 2 (∆; t)/(2n∆) is given by Using Eqs. (9) and (10), we have the RSD [Eq. (5)]: In Eq. (10), the time-averaged diffusion coefficient D(t) is controlled by three stochastic variables, D ± (t), and T + (t). As shown below, the RSD of T + (t) decays slowly t −β with β < 1/2 in the limit t → ∞, while those of D ± (t) decay as t −0.5 . Therefore, in the long time limit, the fluctuations of T + (t) is dominant over those of D ± (t), and thus we can approximate as D ± (t) ≃ D ± . Under this approximation, we have an asymptotic behavior of the RSD: This is another representation of the RSD by the occupation time in LEFD with two-state diffusivity. We confirmed that the asymptotic behavior is the same as the RSD (6) in equilibrium processes. Since we neglect fluctuations of D ± (t), this expression for the RSD is valid only when the right-hand side of Eq. (11) decays slower than t −0.5 . Otherwise, the asymptotic behavior of the RSD is the same as that in Brownian motion (see Appendix. A): IV.

OCCUPATION TIME STATICS
Here, we consider the occupation time statics for three cases. We define the joint probability distribution, g ± n (y; t), of the occupation time T + (t) = y and the number of renewal N t = n up to time t under the condition that the initial state is ±, given by g ± n (y; t) = δ (y − T + (t)) I (t n ≤ t < t n+1 ) ± .
The Laplace transform of g ± n (y; t) with respect to y and t is given bŷ where n = 1, 2, . . . . For example, if the initial state is + and n = 2k or 2k + 1, it can be represented aŝ where τ k is the kth sojourn time, and thus t k = k i=1 τ i . Integrating the above equations, and using interindepen-dence of τ k and τ l (k = l), we havê The cases in which the system starts from − state can be calculated in the similar way. Then, the the PDF of T + (t) is obtained by summing up g ± n (y; t) in terms of n: g ± (y; t) = ∞ n=0 g ± n (y; t), and thus we havê where ρ(s, u) ≡ 1 −ρ + (s + u)ρ − (s). In the small s and u limit,ĝ Case (1) From Eq. (21), the Laplace transform of the PDF of T + (t) for the case (1) is given bŷ . (22) Using the relation between the moments of T + (t) and g ± (u; s), we have the asymptotic behavior of the nth moment of T + (t) where µ = m 1 . It follows that the ETMSD shows normal diffusion: where we used D ± (t) ∼ D ± and Eq. (23). Because TMSD converges to 2nD − ∆ as t → ∞, this process seems to be normal diffusion. In Brownian motion, D(t) converges to a constant and the distribution follows Gaussian. Thus, deviation from Gaussian detects anomaly of the process. Since D(t) is given by Eq. (10) and D(t) → D − , we consider the deviation, i.e., δD t ≡ D(t) − D − . By Eq. (10), we have Here, the first term in the right-hand side can be ne- Note that this condition is satisfied when α > 0.5 [see Eq. (23)]. By Eq. (23), moments of the normalized occupation time defined by T α (t) ≡ T + (t)/ T + (t) becomes When the PDF of a random variable M α follows the Mittag-Leffler distribution of order α, the Laplace transform is given by e −zMα = ∞ k=0 .Therefore, the distribution of δD t / δD t is not Gaussian but converges to the Mittag-Leffler distribution when α > 0.5 (see Fig. 2a). For α < 0.5, the first term in Eq. (25) becomes the leading term and the distribution of δD t / δD t becomes Gaussian with the mean 0 and the variance For α > 0.5, using Eq. (11) yields where A(α) = 2Γ(α+1) 2 Γ(2α+1) − 1. For D − = 0, the result is exactly the same as that in CTRW [9]: Σ(t; ∆) ∼ A(α).
As in the previous case, TMSD converges to 2nD − ∆ as t → ∞. By Eq. (11), the RSD decays as Although we do not have the limit distribution of T + (t)/ T + (t) , the tail should be a heavy tail (power-law distribution) because the second moment of T + (t)/ T + (t) diverges. By the relation between δD t and T + (t), i.e., Eq. (25), we find that the deviations of time-averaged diffusion coefficient, δD t / δD t , are random and the distribution is a non-trivial distribution characterized by a power law (see Fig. 2b). This is a similar situation for the PDF of time-averaged diffusion coefficients in some parameter region of SEDLF [17].

C.
Case (3) Contrary to the previous two cases, TMSDs do not converge to a constant in the case (3), whereas TMSD shows normal diffusion [see Eq. (9)]. Eq. (21) yields the Laplace transform of the PDF of T + (t): By Appendix B in [32], Eq. (34) implies that the limit distribution of T + (t)/t exists: and the distribution is given by where g T + /t (x) is the PDF of T + (t)/t, a = a − /a + and β ≡ 1/(1 + a). This is the Lamperti's generalized arcsine law [2], which is observed for time-averaged drift in superdiffusion [33]. By Eq. (10), the distribution of the time-averaged diffusion coefficient is given by that of T + (t)/t: Because the PDF of T + (t)/t follows the Lamperti's generalized arcsine law, Eq. (36), the PDF of D(t) is given by P D (x) = g α,β Because the mean and second moment of T + (t)/t are given by T + (t)/t = β and (T + /t) 2 = m(α, β) ≡ β(αβ + 1 − α), respectively [32], we have the RSD As shown in Fig. 3, theory is in good agreement with numerical results.

VI. CONCLUSION
We have shown three distributional limit theorems for time-averaged observables related to diffusivity in the Langevin equation with dichotomously fluctuating diffusivity. When one of the states is zero (D − = 0) in the case (1), statistical properties of TMSD are exactly the same as those in CTRW. Therefore, this model is a generalization of CTRW. When both diffusion coefficients are not zero, the TMSD asymptotically show normal diffusion in all cases, whereas fluctuations of TMSD (deviations of time-averaged diffusion coefficients) are intrinsically random. Especially in case (3), time-averaged diffusion coefficients are intrinsically random and the distribution follows the generalized arcsine law. As a result, we have found anomalous fluctuations in apparently normal diffusion processes.