Crossover from anomalous to normal diffusion: truncated power-law noise correlations and applications to dynamics in lipid bilayers

Let us first consider an exponential tempering of the form

$$\begin{eqnarray}&&\langle {v}^{2}{\rangle }_{\tau }=\displaystyle \frac{{{ \mathcal D }}_{H}}{{\rm{\Gamma }}(2H-1)}{\tau }^{2H-2}{{\rm{e}}}^{-\tau /{\tau }_{\star }},\end{eqnarray} \tag{ 10 }$$

for τ > 0, where ${\tau }_{\star }\gt 0$ is a characteristic crossover time scale. For instance, in the case of moving cells the crossover time ${\tau }_{\star }$ would correspond to the time scale when the cell motion becomes uncorrelated, similar to the decorrelation of the lipid motion in the example of the lipid bilayer system discussed below.

Here we note that one should keep in mind that the autocorrelation function $\langle {v}^{2}{\rangle }_{\tau }$ cannot be chosen arbitrarily. Namely, its Fourier transform, the spectrum $\langle {\tilde{v}}^{2}(\omega )\rangle$ of the random process v(t) must be non-negative [41]. The positivity of $\langle {\tilde{v}}^{2}(\omega )\rangle$ for the case of exponential tempering in equation (10) is shown in appendix A. Note also that now ${\int }_{0}^{\infty }{\rm{d}}\tau \langle {v}^{2}{\rangle }_{\tau }={{ \mathcal D }}_{H}{\tau }_{\star }^{2H-1}$ is finite, thus we expect normal diffusion at long times.

With the use of expression (4) the MSD for the exponentially truncated fractional Gaussian noise takes on the exact form

$$\begin{eqnarray}&&\langle {x}^{2}(t)\rangle =\displaystyle \frac{2{{ \mathcal D }}_{H}{\tau }_{\star }^{2H}}{{\rm{\Gamma }}(2H-1)}\left[\displaystyle \frac{t}{{\tau }_{\star }}\gamma \left(2H-1,\displaystyle \frac{t}{{\tau }_{\star }}\right)-\gamma \left(2H,\displaystyle \frac{t}{{\tau }_{\star }}\right)\right],\end{eqnarray} \tag{ 11 }$$

where $\gamma (a,z)={\int }_{0}^{z}{t}^{a-1}{{\rm{e}}}^{-t}{\rm{d}}{t}$ is the incomplete γ-function. Using the asymptotic $\gamma (a,z)\sim {z}^{a}/a$ for z ≪ 1, and $\gamma (a,z)\sim {\rm{\Gamma }}(a)$ for z ≫ 1, we observe superdiffusive behaviour at short times, and normal diffusion at long times, namely,

$$\begin{eqnarray}\langle {x}^{2}(t)\rangle \sim \left\{\begin{array}{ll}\displaystyle \frac{2{{ \mathcal D }}_{H}}{{\rm{\Gamma }}(2H+1)}{t}^{2H}, & t\ll {\tau }_{\star }\\ 2{{ \mathcal D }}_{H}{\tau }_{\star }^{2H-1}t, & t\gg {\tau }_{\star }.\end{array}\right.\end{eqnarray} \tag{ 12 }$$

The emerging normal diffusion thus has the effective diffusivity ${{ \mathcal D }}_{H}{\tau }_{\star }^{2H-1}$. Note that the approximate formula at long times is in concordance with the simple estimate given by expression (5).

Figure 1 shows the crossover behaviour from superdiffusion to normal diffusion encoded in expression (11), along with the short and long time asymptotes given by result (12). As can be discerned from the plot, the crossover region is fairly short, spanning less than a decade in time for the chosen parameters.

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We now consider the softer power-law truncation of the form

$$\begin{eqnarray}&&\langle {v}^{2}{\rangle }_{\tau }=\displaystyle \frac{{{ \mathcal D }}_{H}}{{\rm{\Gamma }}(2H-1)}{\tau }^{2H-2}{\left(1+\displaystyle \frac{\tau }{{\tau }_{\star }}\right)}^{-\mu },\end{eqnarray} \tag{ 13 }$$

for τ > 0, μ > 0 and compare the resulting behaviour with the scenario of exponential tempering. Here, apart from the crossover time ${\tau }_{\star }$ the new power-law exponent μ is introduced which effects the exact nature of the dynamics at long times, as we are going to show below. We remark that the positivity of the spectrum for the power-law truncated form is discussed in appendix A. After plugging (13) into expression (4) we find for the MSD that

$$\begin{eqnarray}&&\langle {x}^{2}(t)\rangle =\displaystyle \frac{2{{ \mathcal D }}_{H}{\tau }_{\star }^{2H}}{{\rm{\Gamma }}(2H-1)}\left[\displaystyle \frac{t}{{\tau }_{\star }}f\left(\mu ,2H-1;\displaystyle \frac{t}{{\tau }_{\star }}\right)-f\left(\mu ,2H;\displaystyle \frac{t}{{\tau }_{\star }}\right)\right],\end{eqnarray} \tag{ 14 }$$

where we introduced the notation

$$\begin{eqnarray}&&f(\mu ,\alpha ;a)={\int }_{0}^{a}\displaystyle \frac{{y}^{\alpha -1}}{{(1+y)}^{\mu }}{\rm{d}}{y}.\end{eqnarray} \tag{ 15 }$$

Now, using the integral representation [42] of the hypergeometric function ${}_{2}{F}_{1}$ [43] we rewrite the integral in equation (15) as

$$\begin{eqnarray}&&f(\mu ,\alpha ;a)=\displaystyle \frac{{a}^{\alpha }}{\alpha }{}_{2}{F}_{1}(\mu ,\alpha ,\alpha +1;-a),\end{eqnarray} \tag{ 16 }$$

and thus rewrite the MSD (14) in the final form

$$\begin{eqnarray}\begin{array}{rcl}\langle {x}^{2}(t)\rangle & = & \displaystyle \frac{2{{ \mathcal D }}_{H}{t}^{2H}}{{\rm{\Gamma }}(2H-1)}\left[\displaystyle \frac{1}{2H-1}{}_{2}{F}_{1}\left(\mu ,2H-1;2H;-\displaystyle \frac{t}{{\tau }_{\star }}\right)\right.\\ & & \left.-\displaystyle \frac{1}{2H}{}_{2}{F}_{1}\left(\mu ,2H;2H+1;-\displaystyle \frac{t}{{\tau }_{\star }}\right)\right].\end{array}\end{eqnarray} \tag{ 17 }$$

In this notation the MSD can be directly evaluated by Wolfram Mathematica [44]. Note that ${}_{2}{F}_{1}(0,b;c;z)=1$, and thus result (17) reduces exactly to the MSD (9) for the untruncated case μ = 0. To obtain the limiting behaviours of the MSD (17) at short times $t\ll {\tau }_{\star }$ we use the Gauss hypergeometric series for the function ${}_{2}{F}_{1}$, see 15.1.1 in [42]. As result, to leading order we recover the MSD (9) of untruncated fractional Brownian motion.

At long times $t\gg {\tau }_{\star }$ the situation for power-law tempering is actually richer than for the case of exponential tempering. To see this, we first employ the linear transformation formula 15.3.7 in [42] and write expression (17) in the form

$$\begin{eqnarray}\begin{array}{c}\begin{array}{rcl} \langle {x}^{2}(t) \rangle & = & \displaystyle \frac{2{{ \mathcal D }}_{H}{\tau }_{\star }^{2H-1}t}{{\rm{\Gamma }}(2H-1)}\left[\displaystyle \frac{{\rm{\Gamma }}(2H-1){\rm{\Gamma }}(\mu +1-2H)}{{\rm{\Gamma }}(\mu )}-\displaystyle \frac{{\rm{\Gamma }}(2H+1){\rm{\Gamma }}(\mu -2H)}{2H{\rm{\Gamma }}(\mu )}\displaystyle \frac{{\tau }_{\star }}{t}\right.\\ & & +\,\displaystyle \frac{1}{2H-\mu -1}{\left(\displaystyle \frac{{\tau }_{\star }}{t}\right)}^{\mu +1-2H}{}_{2}{F}_{1}\left(\mu ,\mu +1-2H;\mu +2-2H;-\displaystyle \frac{{\tau }_{\star }}{t}\right)\\ & & \left.-\,\displaystyle \frac{1}{2H-\mu }{\left(\displaystyle \frac{{\tau }_{\star }}{t}\right)}^{\mu +1-2H}{}_{2}{F}_{1}\left(\mu ,\mu -2H;\mu +1-2H;-\displaystyle \frac{{\tau }_{\star }}{t}\right)\right].\end{array}\end{array}\end{eqnarray} \tag{ 18 }$$

We consider two possible cases:

2.2.1. Weak power-law truncation, $0\lt \mu \lt 2H-1\lt 1$

In this case the third and fourth terms in the square brackets of expression (18) are dominating and we find

$$\begin{eqnarray} &&\langle {x}^{2}(t) \rangle \sim \displaystyle \frac{2{{ \mathcal D }}_{H}{\tau }_{\star }^{\mu }}{(2H-\mu )(2H-1-\mu ){\rm{\Gamma }}(2H-1)}{t}^{2H-\mu }\end{eqnarray} \tag{ 19 }$$

for $t\gg {\tau }_{\star }$. Note that in the limit μ → 0 result (19) reduces to the untruncated formula (9). Thus, since we observe the inequality $2H-\mu \gt 1$ in the case of weak power-law truncation the dynamics is still superdiffusive, however, with a reduced anomalous diffusion exponent smaller than the value $2H$ in the short time limit.

2.2.2. Strong power-law truncation, $\mu \gt 2H-1\gt 0$

Note that in this case the integral of the velocity autocorrelation function (13) over the whole time domain converges, ${\int }_{0}^{\infty }{\rm{d}}\tau \langle {v}^{2}{\rangle }_{\tau }={{ \mathcal D }}_{H}{\tau }_{\star }^{2H-1}{\rm{\Gamma }}(\mu -2H+1)/{\rm{\Gamma }}(\mu )$, see 2.2.5.24 in [45]. Thus, with expression (5) we expect a linear time behaviour in the long time limit, whereas the term to next order in (4) gives ${\int }^{t}{\rm{d}}\tau \tau \langle {v}^{2}{\rangle }_{\tau }\simeq {\int }^{t}{\rm{d}}\tau {\tau }^{2H-1-\mu }\simeq {t}^{2H-\mu }$, a sublinear contribution since $2H-\mu \lt 1$. Alternatively, it follows from (18) that the main contribution comes from the first term in the square brackets. Thus, in full accordance with expression (5) we get

$$\begin{eqnarray}&&\langle {x}^{2}(t)\rangle \sim \displaystyle \frac{2{{ \mathcal D }}_{H}{\rm{\Gamma }}(\mu -2H+1){\tau }_{\star }^{2H-1}t}{{\rm{\Gamma }}(\mu )}\end{eqnarray} \tag{ 20 }$$

at $t\gg {\tau }_{\star }$.

Finally, for the borderline case $0\lt \mu =2H-1\lt 1$ it is in fact easier to consider equation (17). Making use of formula 7.3.1.81 in [46] we see that the leading contribution comes from the first hypergeometric function in the square brackets in expression (17), as ${}_{2}{F}_{1}(2H-1,2H-1;2H;z)\sim {\rm{\Gamma }}(2H){{\rm{\Gamma }}}^{-1}(2H-1){(-z)}^{-2H+1}\mathrm{ln}(-z)$. For the MSD we then finally obtain

$$\begin{eqnarray}&&\langle {x}^{2}(t)\rangle \sim \displaystyle \frac{2{D}_{H}{\tau }_{\star }^{2H-1}}{{\rm{\Gamma }}(2H-1)}t\mathrm{ln}\left(\displaystyle \frac{t}{{\tau }_{\star }}\right).\end{eqnarray} \tag{ 21 }$$

Thus, in this borderline limit between weak truncation (leading to reduced superdiffusion at long times) and strong truncation (normal long time diffusion) we here obtain normal diffusion with a logarithmic correction.

Figure 2 demonstrates that for the power-law tempering the crossover region is significantly enhanced, spanning several orders of magnitude, as compared to the much swifter crossover in the case of exponential tempering.

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The MSDs for both cases of exponential and power-law truncation are directly compared in figure 3, along with the time derivative of the MSD. As can be seen, the crossover for the exponential tempering occurs much more rapidly. Thus also the amplitude of the long time Brownian scaling is higher in the case of the power-law tempering for the same value of the crossover time scale ${\tau }_{\star }$.

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A graphical representation of the correlation functions (6), (10) and (13) is given in figure 4. The exponential cutoff appears more abrupt, as it should. However, this difference will obviously be reduced for larger values of the cutoff exponent μ. To fit data, the crossover shape can thus be adjusted by the choice of μ for the case of power-law tempering, thus having the possibility to effect a gradual adjustment from softer power-law to hard exponential tempering.

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Let us recall the derivation of the MSD from equations (22) and (23). With our choice x(0) = 0 we obtain for the Laplace transform of x(t), $\tilde{x}(s)={\int }_{0}^{\infty }x(t)\exp (-{st}){\rm{d}}{t}$ that

$$\begin{eqnarray}&&\tilde{x}(s)=\displaystyle \frac{\tilde{\xi }(s)}{{ms}{\tilde{\gamma }}_{H}(s)}.\end{eqnarray} \tag{ 24 }$$

Inverse Laplace transformation produces

$$\begin{eqnarray}&&x(t)=\displaystyle \frac{1}{m}{\int }_{0}^{t}\xi (t^{\prime} )H(t-t^{\prime} ){\rm{d}}{t}^{\prime} ,\end{eqnarray} \tag{ 25 }$$

where the kernel H(t) is the inverse Laplace transform of $\tilde{H}(s)=1/[s{\tilde{\gamma }}_{H}(s)]$. After some transformation we recover the MSD

$$\begin{eqnarray}\begin{array}{rcl}\langle {x}^{2}(t)\rangle & = & \displaystyle \frac{2}{{m}^{2}}{\displaystyle \int }_{0}^{t}{{\rm{d}}{t}}_{1}{\displaystyle \int }_{{t}_{1}}^{t}{{\rm{d}}{t}}_{2}H(t-{t}_{1})H(t-{t}_{2})\langle {\xi }^{2}{\rangle }_{{t}_{2}-{t}_{1}}\\ & = & \displaystyle \frac{2{k}_{B}T}{m}{\displaystyle \int }_{0}^{t}H(t^{\prime} )M(t^{\prime} ){\rm{d}}{t}^{\prime} ,\end{array}\end{eqnarray} \tag{ 26 }$$

where we introduced $M(t)={\int }_{0}^{t}{\gamma }_{H}(t^{\prime} )H(t-t^{\prime} ){\rm{d}}{t}^{\prime}$. Its Laplace transform is $\tilde{M}(s)={\tilde{\gamma }}_{H}(s)\tilde{H}(s)=1/s$, and thus simply M(t) = 1. We therefore arrive at

$$\begin{eqnarray}&&\langle {x}^{2}(t)\rangle =\displaystyle \frac{2{k}_{B}T}{m}{\int }_{0}^{t}H(t^{\prime} ){\rm{d}}{t}^{\prime} .\end{eqnarray} \tag{ 27 }$$

In Laplace space, this relation reads

$$\begin{eqnarray}&&\langle \tilde{{x}^{2}}(s)\rangle =\displaystyle \frac{2{k}_{B}T}{m}\displaystyle \frac{\tilde{H}(s)}{s}=\displaystyle \frac{2{k}_{B}T}{m}\displaystyle \frac{1}{{s}^{2}{\tilde{\gamma }}_{H}(s)}.\end{eqnarray} \tag{ 28 }$$

We stop to include a note on when exactly we expect asymptotically normal diffusion in the generalised Langevin equation model. The reasoning is similar to that presented at the beginning of section 2. Namely, from equation (28) it follows that diffusion is normal at long times if ${\tilde{\gamma }}_{H}(s)$ tends to a constant in the limit s → 0. This is equivalent to requiring that the average ${\int }_{0}^{\infty }{\gamma }_{H}(\tau ){\rm{d}}\tau$ is finite or, taking into account the fluctuation dissipation relation (23) that ${\int }_{0}^{\infty }\langle {\xi }^{2}{\rangle }_{\tau }{\rm{d}}\tau$ is finite (similar to the conclusion in section 2). Then, from expression (28) we infer the following behaviour in the long time limit (compare with equation (5))

$$\begin{eqnarray}&&\langle {x}^{2}(t)\rangle =\displaystyle \frac{2{k}_{B}T}{m{\int }_{0}^{\infty }{\gamma }_{H}(\tau ){\rm{d}}\tau }t.\end{eqnarray} \tag{ 29 }$$

According to this, anomalous diffusion is expected at long times whenever ${\int }_{0}^{\infty }{\gamma }_{H}(\tau ){\rm{d}}\tau$ is either infinite (subdiffusion) or zero (superdiffusion)¹³ .

In accordance with section 2 we choose the friction kernel in the power-law form

$$\begin{eqnarray}&&{\gamma }_{H}(\tau )=\displaystyle \frac{{{\rm{\Gamma }}}_{H}}{{\rm{\Gamma }}(2H-1)}{\tau }^{2H-2},\end{eqnarray} \tag{ 30 }$$

where the coefficient Γ_H is of dimension $[{{\rm{\Gamma }}}_{H}]={{\rm{s}}}^{-2H}$. The normal Brownian case is recovered from equation (22) for H = 1/2 since for H → 1/2+ we see that ${\gamma }_{H}(t)\to {{\rm{\Gamma }}}_{1/2}\delta (t)$ (note that in this Brownian limit, Γ_1/2 = η) and equation (22) assumes the form of the standard Langevin equation driven by white Gaussian noise obeying the regular fluctuation dissipation theorem. We note that the memory kernel for the power-law form (30) can be rewritten in terms of a fractional derivative, and the resulting version of equation (22) is then often referred to as the fractional Langevin equation [7, 48–50]. Power-law memory kernels of the form (30) are typical for many viscoelastic systems [8, 9, 14–17, 19, 20, 22, 50].

We now use the Laplace transform of equation (30), ${\tilde{\gamma }}_{H}(s)={{\rm{\Gamma }}}_{H}{s}^{1-2H}$, plug this into the above expression, and take an inverse Laplace transformation. This procedure leads to the final result

$$\begin{eqnarray}&&\langle {x}^{2}(t)\rangle =\displaystyle \frac{1}{{\rm{\Gamma }}(3-2H)}\displaystyle \frac{2{k}_{B}T}{m{{\rm{\Gamma }}}_{H}}{t}^{2-2H},\end{eqnarray} \tag{ 31 }$$

which reduces to the classical result $\langle {x}^{2}(t)\rangle =2({k}_{B}T/[m\eta ])t$ for normal Brownian motion in the limit H = 1/2. Therefore, due to the requirement that the system is thermalised and thus the Kubo–Zwanzig fluctuation theorem is fulfilled, the same noise leads to subdiffusion in this case with anomalous diffusion exponent $\alpha =2-2H$ and $0\lt 2-2H\lt 1$. Indeed, due to the coupling in relation (23) large noise values lead to large friction values, and therefore the persistence of the noise is turned into antipersistent diffusion dynamics [7, 48, 50].

We now derive the autocorrelation function of the displacements, following the procedure laid out by Pottier [51]. First, we note that the double Laplace transform of the correlation function of the random force can be written as

$$\begin{eqnarray}&&\langle \tilde{\xi }({s}_{1})\tilde{\xi }({s}_{2})\rangle ={k}_{B}{Tm}{\int }_{0}^{\infty }{{\rm{d}}{t}}_{1}{\int }_{0}^{\infty }{{\rm{d}}{t}}_{2}{{\rm{e}}}^{-{s}_{1}{t}_{1}-{s}_{2}{t}_{2}}{\gamma }_{H}(| {t}_{2}-{t}_{1}| ).\end{eqnarray} \tag{ 32 }$$

Then we split the domain of integration over t₂ into the two domains $0\leqslant {t}_{2}\leqslant {t}_{1}$ and ${t}_{1}\leqslant {t}_{2}\lt \infty$. After introducing $\tau ={t}_{1}-{t}_{2}$ and $\tau ={t}_{2}-{t}_{1}$ in each domain, respectively, we arrive at

$$\begin{eqnarray}&&\langle \tilde{\xi }({s}_{1})\tilde{\xi }({s}_{2})\rangle ={k}_{B}{Tm}\displaystyle \frac{{\tilde{\gamma }}_{H}({s}_{1})+{\tilde{\gamma }}_{H}({s}_{2})}{{s}_{1}+{s}_{2}}.\end{eqnarray} \tag{ 33 }$$

This expression represents the Laplace domain formulation of the fluctuation dissipation theorem (23). By help of equations (22) and (33) we then obtain the double Laplace transform of the displacement correlation function,

$$\begin{eqnarray}&&\langle \tilde{x}({s}_{1})\tilde{x}({s}_{2})\rangle =\displaystyle \frac{{k}_{B}T}{m}\left(\displaystyle \frac{1/{\tilde{\gamma }}_{H}({s}_{1})}{{s}_{1}{s}_{2}({s}_{1}+{s}_{2})}+\displaystyle \frac{1/{\tilde{\gamma }}_{H}({s}_{2})}{{s}_{1}{s}_{2}({s}_{1}+{s}_{2})}\right).\end{eqnarray} \tag{ 34 }$$

In the first term in the parentheses we first take the inverse Laplace transformation over s₂, going from $1/[{s}_{2}({s}_{1}+{s}_{2})]$ to $[1-\exp (-{s}_{1}{t}_{2})]/{s}_{1}$. Exchanging s₂ for s₁ we perform the same operation on the second term. Then we inverse Laplace transform the first term with respect to s₁ and make use of the translation formula ${{\mathscr{L}}}_{s}^{-1}\{\exp (-{bs}){{\mathscr{L}}}_{s}\{f(t)\}\}=f(t-b){\rm{\Theta }}(t-b)$, where b > 0 and Θ(t) is the Heaviside step function. As result yields

$$\begin{eqnarray}&&\langle x({t}_{1})x({t}_{2})\rangle =\displaystyle \frac{1}{{\rm{\Gamma }}(3-2H)}\displaystyle \frac{{k}_{B}T}{m{{\rm{\Gamma }}}_{H}}({t}_{1}^{2-2H}+{t}_{2}^{2-2H}-| {t}_{2}-{t}_{1}{| }^{2-2H}).\end{eqnarray} \tag{ 35 }$$

The velocity autocorrelation function is obtained by differentiation of this expression,

$$\begin{eqnarray}&&\langle v({t}_{1})v({t}_{2})\rangle =\langle {v}^{2}{\rangle }_{\tau }=-\displaystyle \frac{\sin (\pi [2H-1]){\rm{\Gamma }}(2H)}{\pi }\displaystyle \frac{{k}_{B}T}{m{{\rm{\Gamma }}}_{H}}| \tau {| }^{-2H},\end{eqnarray} \tag{ 36 }$$

where $\tau ={t}_{2}-{t}_{1}$. We see that in the relevant parameter range 1/2 < H < 1 the velocity autocorrelation is negative, $\langle {v}^{2}{\rangle }_{\tau }\lt 0$, reflecting the antipersistent character of the resulting motion.

For the exponentially truncated friction kernel and thus noise autocorrelation

$$\begin{eqnarray}&&{\gamma }_{H}(\tau )=\displaystyle \frac{\langle {\xi }^{2}{\rangle }_{\tau }}{{k}_{B}{Tm}}=\displaystyle \frac{{{\rm{\Gamma }}}_{H}}{{\rm{\Gamma }}(2H-1)}{\tau }^{2H-2}{{\rm{e}}}^{-\tau /{\tau }_{\star }}\end{eqnarray} \tag{ 37 }$$

we obtain the corresponding Laplace transform

$$\begin{eqnarray}&&{\tilde{\gamma }}_{H}(s)={{\rm{\Gamma }}}_{H}{(s+{\tau }_{\star }^{-1})}^{1-2H}.\end{eqnarray} \tag{ 38 }$$

After plugging this expression into relation (28) and taking the inverse Laplace transformation we obtain

$$\begin{eqnarray}&&\langle {x}^{2}(t)\rangle =\displaystyle \frac{2{k}_{B}T}{m{{\rm{\Gamma }}}_{H}}{t}^{2-2H}{E}_{1,3-2H}^{1-2H}\left(-\displaystyle \frac{t}{{\tau }_{\star }}\right)\end{eqnarray} \tag{ 39 }$$

in terms of the three parameter Mittag-Leffler function ${E}_{\alpha ,\beta }^{\delta }(z)$ (see appendix B for its definition and some relevant properties). When the crossover time ${\tau }_{\star }$ tends to infinity, ${E}_{\alpha ,\beta }^{\delta }(0)=1/{\rm{\Gamma }}(\beta )$, and we arrive at result (31) for the untruncated noise. In the limit H = 1/2 we have δ = 0 and ${E}_{1,2}^{0}(z)=1/{\rm{\Gamma }}(2)=1$, such that equation (39) reduces to the MSD of normal Brownian motion.

At short times $t\ll {\tau }_{\star }$ the MSD (39) reduces to the subdiffusive expression (31), whereas at long times $t\gg {\tau }_{\star }$ with the help of ${E}_{1,3-2H}^{1-2H}(-t/{\tau }_{\star })\sim {(t/{\tau }_{\star })}^{2H-1}$ (see appendix A), in accordance with relation (29) the MSD exhibits normal Brownian behaviour,

$$\begin{eqnarray}&&\langle {x}^{2}(t)\rangle \sim \displaystyle \frac{2{k}_{B}T}{m{{\rm{\Gamma }}}_{H}{\tau }_{\star }^{2H-1}}t.\end{eqnarray} \tag{ 40 }$$

We note that a similar crossover was observed in [52] where a modified three parameter Mittag-Leffler form for the kernel γ_H(τ) was considered.

The crossover from subdiffusion to normal diffusion in this exponentially tempered generalised Langevin equation picture is shown in figure 5. The crossover behaviour occurs over an interval of the order of a decade in time for the chosen parameters.

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Let us now turn to the autocorrelation functions. Using expression (38) in equation (34) we obtain

$$\begin{eqnarray}\begin{array}{rcl}\langle \tilde{x}({s}_{1})\tilde{x}({s}_{2})\rangle & = & \displaystyle \frac{{k}_{B}T}{m{{\rm{\Gamma }}}_{H}}\left(\displaystyle \frac{1}{{s}_{1}{s}_{2}({s}_{1}+{s}_{2}){({s}_{1}+{\tau }_{\star }^{-1})}^{1-2H}}\right.\\ & & \left.+\displaystyle \frac{1}{{s}_{1}{s}_{2}({s}_{1}+{s}_{2}){({s}_{2}+{\tau }_{\star }^{-1})}^{1-2H}}\right).\end{array}\end{eqnarray} \tag{ 41 }$$

As above, in the first term in the parentheses we take an inverse Laplace transformation with respect to s₂, and over s₁ in the second term. Then, with the translation formula and the Laplace transform (B.2) of the three parameter Mittag-Leffler function, we find

$$\begin{eqnarray}\begin{array}{rcl}\langle x({t}_{1})x({t}_{2})\rangle & = & \displaystyle \frac{{k}_{B}T}{m{{\rm{\Gamma }}}_{H}}\left({t}_{1}^{2-2H}{E}_{1,3-2H}^{1-2H}\left(-\displaystyle \frac{{t}_{1}}{{\tau }_{\star }}\right)+{t}_{2}^{2-2H}{E}_{1,3-2H}^{1-2H}\left(-\displaystyle \frac{{t}_{2}}{{\tau }_{\star }}\right)\right.\\ & & \left.-| {t}_{2}-{t}_{1}{| }^{2-2H}{E}_{1,3-2H}^{1-2H}\left(-\displaystyle \frac{| {t}_{2}-{t}_{1}| }{{\tau }_{\star }}\right)\right).\end{array}\end{eqnarray} \tag{ 42 }$$

Differentiation over t₁ and t₂ (with the help of equation (B.6)) then produces the velocity autocorrelation function,

$$\begin{eqnarray}&&\langle v({t}_{1})v({t}_{2})\rangle =\langle {v}^{2}{\rangle }_{\tau }=\displaystyle \frac{{k}_{B}T}{m{{\rm{\Gamma }}}_{H}{\tau }^{2H}}{E}_{1,1-2H}^{1-2H}\left(-\displaystyle \frac{\tau }{{\tau }_{\star }}\right)\end{eqnarray} \tag{ 43 }$$

with $\tau ={t}_{2}-{t}_{1}\gt 0$. Using the definition (B.1) of the three parameter Mittag-Leffler function it is easy to check that ${E}_{1,\delta }^{\delta }(z)=\exp (z)/{\rm{\Gamma }}(\delta )$. Thus, for the velocity autocorrelation function we find the result

$$\begin{eqnarray}&&\langle {v}^{2}{\rangle }_{\tau }=-\displaystyle \frac{\sin (\pi [2H-1]){\rm{\Gamma }}(2H)}{\pi }\displaystyle \frac{{k}_{B}T}{m{{\rm{\Gamma }}}_{H}}{\tau }^{-2H}{{\rm{e}}}^{-\tau /{\tau }_{\star }},\end{eqnarray} \tag{ 44 }$$

which is anticorrelated and reduces to the untruncated result (36) when the crossover time ${\tau }_{\star }$ tends to infinity.

For the power-law truncated friction kernel and noise autocorrelator,

$$\begin{eqnarray}&&{\gamma }_{H}(\tau )=\displaystyle \frac{\langle {\xi }^{2}{\rangle }_{\tau }}{{k}_{B}{Tm}}=\displaystyle \frac{{{\rm{\Gamma }}}_{H}}{{\rm{\Gamma }}(2H-1)}{\tau }^{2H-2}{\left(1+\displaystyle \frac{\tau }{{\tau }_{\star }}\right)}^{-\mu }\end{eqnarray} \tag{ 45 }$$

with τ > 0, μ > 0 the Laplace transform of the memory kernel can be performed by use of the integral representation of the Tricomi hypergeometric function U(a, b; z) (see 13.2.5 of [42]), leading to

$$\begin{eqnarray}&&{\tilde{\gamma }}_{H}(s)={{\rm{\Gamma }}}_{H}{\tau }_{\star }^{2H-1}U(2H-1,2H-\mu ;s{\tau }_{\star }).\end{eqnarray} \tag{ 46 }$$

With the general relation (28) we thus have

$$\begin{eqnarray}&&\langle {x}^{2}(t)\rangle =\displaystyle \frac{2{k}_{B}T}{m{{\rm{\Gamma }}}_{H}{\tau }_{\star }^{2H-1}}g(t)\end{eqnarray} \tag{ 47 }$$

with the abbreviation

$$\begin{eqnarray}&&g(t)={{\mathscr{L}}}_{s}^{-1}\left\{\displaystyle \frac{1}{{s}^{2}U(2H-1,2H-\mu ;s{\tau }_{\star })}\right\}.\end{eqnarray} \tag{ 48 }$$

The inverse Laplace transform of expression (47) cannot be performed analytically. However, we make use of the Tauberian theorems¹⁴ to find the MSD at short and long times.

At short times with $s{\tau }_{\star }\gg 1$ we use the large argument asymptotic of the Tricomi function, $U(2H-1,2H-\mu ;s{\tau }_{\star })\sim {(s{\tau }_{\star })}^{1-2H}$ (13.5.2 in [42]) and thus ${\tilde{\gamma }}_{H}(s)\sim {{\rm{\Gamma }}}_{H}{s}^{1-2H}$. From equation (28) (or, equivalently, equations (47) and (48)) we then get to result (31) by use of the Tauberian theorem.

Similar to the case considered in section 2 at long times corresponding to $s{\tau }_{\star }\ll 1$ the situation is actually richer than for the case of exponential tempering. To see this we first make use of (13.1.3) in [42] to express the Tricomi function via the Kummer function M(a, b; z) through

$$\begin{eqnarray}\begin{array}{rcl}U(2H-1,2H-\mu ;s{\tau }_{\star }) & = & \displaystyle \frac{\pi }{\sin (\pi [2H-\mu ])}\left[\displaystyle \frac{M(2H-1,2H-\mu ;s{\tau }_{\star })}{{\rm{\Gamma }}(\mu ){\rm{\Gamma }}(2H-\mu )}\right.\\ & & \left.-{(s{\tau }_{\star })}^{\mu +1-2H}\displaystyle \frac{M(\mu ,\mu +2-2H;s{\tau }_{\star })}{{\rm{\Gamma }}(2H-1){\rm{\Gamma }}(\mu +2-2H)}\right].\end{array}\end{eqnarray} \tag{ 49 }$$

Taking into account the series expansion of the Kummer function ((13.1.2) in [42]) we consider the following two possibilities:

3.4.1. Weak power-law truncation, $0\lt \mu \lt 2H-1\lt 1$

In this case the second term in (49) is dominant at small s and thus

$$\begin{eqnarray}&&U(2H-1,2H-\mu ;s{\tau }_{\star })\sim \displaystyle \frac{\pi {(s{\tau }_{\star })}^{1+\mu -2H}}{\sin (\pi [2H-\mu -1]){\rm{\Gamma }}(2H-1){\rm{\Gamma }}(\mu +2-2H)}.\end{eqnarray} \tag{ 50 }$$

Plugging this leading behaviour into expressions (47) and (48) and using the Tauberian theorem, after few transformations we obtain the long time behaviour of the MSD,

$$\begin{eqnarray}&&\langle {x}^{2}(t)\rangle \sim \displaystyle \frac{{\rm{\Gamma }}(2H-1)}{{\rm{\Gamma }}(2H-\mu -1){\rm{\Gamma }}(\mu +3-2H)}\displaystyle \frac{2{k}_{B}T}{m{{\rm{\Gamma }}}_{H}{\tau }_{\star }^{\mu }}{t}^{\mu +2-2H}.\end{eqnarray} \tag{ 51 }$$

Note that in the limit μ → 0 expression (51) reduces to the untruncated formula (31). Thus, since we observe the inequality $0\lt \mu +2-2H\lt 1$ in the present case of a weak power-law truncation, the dynamics is still subdiffusive, however, with an anomalous diffusion exponent larger than the value $2-2H$ in the short time limit.

3.4.2. Strong power-law truncation, $\mu \gt 2H-1\gt 0$

In this case the first term in the square brackets in equation (49) becomes dominant at small s and $U(2H-1,2H-\mu ;s{\tau }_{\star })\sim {\rm{\Gamma }}(\mu +1-2H)/{\rm{\Gamma }}(\mu )$, where we made us of the reflection formula for the Gamma function. From results (47) and (48) by use of the Tauberian theorem we obtain

$$\begin{eqnarray}&&\langle {x}^{2}(t)\rangle \sim \displaystyle \frac{{\rm{\Gamma }}(\mu )}{{\rm{\Gamma }}(\mu +1-2H)}\displaystyle \frac{2{k}_{B}T}{m{{\rm{\Gamma }}}_{H}{\tau }_{\star }^{2H-1}}t,\end{eqnarray} \tag{ 52 }$$

valid for $t\gg {\tau }_{\star }$. As expected, we find the desired crossover to the normal Brownian scaling of the MSD. Note that this result is in full accordance with equation (29). Indeed, from expression (45) we get (see 2.2.5.24 [45])

$$\begin{eqnarray}&&{\int }_{0}^{\infty }{\gamma }_{H}(\tau ){\rm{d}}\tau =\displaystyle \frac{{\rm{\Gamma }}(\mu +1-2H)}{{\rm{\Gamma }}(\mu )}{{\rm{\Gamma }}}_{H}{\tau }_{\star }^{2H-1}.\end{eqnarray} \tag{ 53 }$$

After plugging expression (53) into (29) we arrive at result (52). Note also that the condition of a strong power-law truncation is equivalent to the condition that integral (53) converges.

In the borderline case with $0\lt \mu =2H-1\lt 1$ we use 13.5.9 in [42] and find $U(2H-1,1;s{\tau }_{\star })\sim -\mathrm{ln}(s{\tau }_{\star })/{\rm{\Gamma }}(2H-1)$. With the use of the Tauberian theorem equations (47) and (48) yield

$$\begin{eqnarray}&&\langle {x}^{2}(t)\rangle \sim {\rm{\Gamma }}(2H-1)\displaystyle \frac{2{k}_{B}T}{m{{\rm{\Gamma }}}_{H}{\tau }_{\star }^{2H-1}}\displaystyle \frac{t}{\mathrm{ln}(t/{\tau }_{\star })}\end{eqnarray} \tag{ 54 }$$

at $t\gg {\tau }_{\star }$. Thus, in this borderline situation between the cases of weak truncation (leading to increased subdiffusion at long times) and strong truncation (normal long time diffusion) we observe a logarithmic correlation to normal diffusion.

Figure 6 shows the crossover dynamics for power-law tempering for the two possible cases: for weak power-law truncation with μ = 0.3 we observe the predicted crossover from slower to faster subdiffusion, while in the case of strong power-law truncation the subdiffusive dynamics crosses over to normal diffusion.

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Figure 7 shows a direct comparison between the cases of exponential and power-law truncation. As expected, the crossover is faster for the exponential tempering, and thus the resulting amplitude in this case exceeds the amplitude for the power-law tempering. Note that the latter observation contrasts the case of the truncated fractional Brownian motion in figure 3, for which the amplitude of the power-law tempering is higher.

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3.4.3. Velocity autocorrelation function

To gain some insight into the correlation behaviour we use equation (34) with ${\tilde{\gamma }}_{H}(s)$ from equation (46). Taking the inverse Laplace transformation over s₁ and s₂ in the same way as above we obtain the position autocorrelation function

$$\begin{eqnarray}&&\langle x({t}_{1})x({t}_{2})\rangle =\displaystyle \frac{{k}_{B}T}{m{{\rm{\Gamma }}}_{H}{\tau }_{\star }^{2H-1}}(g({t}_{1})+g({t}_{2})-g(| {t}_{2}-{t}_{1}| )),\end{eqnarray} \tag{ 55 }$$

where g(t) is given by relation (48). From here the velocity autocorrelation function is obtained as

$$\begin{eqnarray}&&\langle {v}^{2}{\rangle }_{\tau }=\displaystyle \frac{{k}_{B}T}{m{{\rm{\Gamma }}}_{H}{\tau }_{\star }^{2H-1}}\displaystyle \frac{{{\rm{d}}}^{2}}{{\rm{d}}{\tau }^{2}}g(\tau )\end{eqnarray} \tag{ 56 }$$

with τ > 0. We first note that expression (56) along with (48) may suggest that the Tauberian theorem may be directly applied to the expression ${U}^{-1}(2H-1,2H-\mu ;s{\tau }_{\star })$ in order to calculate the asymptotic behaviour of the velocity autocorrelation function $\langle {v}^{2}{\rangle }_{\tau }$. However, for short times corresponding to $s{\tau }_{\star }\gg 1$ the function ${U}^{-1}\sim {(s{\tau }_{\star })}^{2H-1}$, and since 1/2 < H < 1, the Tauberian theorem does not apply as $2H-1$ is positive. Instead, we should first obtain the asymptotic of g(τ) at short times $\tau \ll {\tau }_{\star }$ by use of the Tauberian theorem, and only then differentiate twice to get the asymptotic of the velocity autocorrelation function. This way we arrive at expression (36). At long times $\tau \gg {\tau }_{\star }$ we again consider the cases of weak and strong power-law truncations separately.

For the weak power-law truncation with $0\lt \mu \lt 2H-1\lt 1$ the situation is similar to the short time limit above. Indeed, ${U}^{-1}\sim {(s{\tau }_{\star })}^{2H-1-\mu }$, see result (50), and the Tauberian theorem does not apply. Instead we first plug relation (50) into expression (48) and then apply the Tauberian theorem. Following relation (56) we then find

$$\begin{eqnarray}&&\langle {v}^{2}{\rangle }_{\tau }\sim -C\displaystyle \frac{{k}_{B}T}{m{{\rm{\Gamma }}}_{H}{\tau }_{\star }^{\mu }}\displaystyle \frac{1}{{\tau }^{2H-\mu }},\end{eqnarray} \tag{ 57 }$$

where $C=(2H-\mu -1){\pi }^{-1}\sin (\pi [2H-\mu -1]){\rm{\Gamma }}(2H-1)$ is a positive constant. Note that for weak power-law truncation we have $1\lt 2H-\mu \lt 2$, and in the limit μ → 0 expression (57) reduces to the velocity autocorrelation function (36) in absence of truncation. From comparison of result (57) with (36) we see that the autocorrelation function in the truncated case decays slower than in the untruncated case. This may appear counterintuitive, however, it is in agreement with the antipersistent character of the fractional Langevin equation model in which the MSD scales like $\simeq {t}^{2-2H}$ and the velocity autocorrelation function at long times scales as $\simeq -{\tau }^{-2H}$ for 1/2 < H < 1. This means that a steeper decay of the velocity autocorrelation function corresponds to a more subdiffusive regime. In other words, when H is closer to 1/2 (the subdiffusive regime is closer to normal diffusion) then the decay of the autocorrelation function is slower. To see this better consider the effective Hurst index H_eff = H − μ/2. Then, for weak power-law truncation the MSD scales like $\simeq {t}^{2-2{H}_{\mathrm{eff}}}$ with $1/2\lt {H}_{\mathrm{eff}}\lt H\lt 1$, and the velocity autocorrelation function decays as $\simeq \,-\,{\tau }^{-}2{H}_{\mathrm{eff}}$. Thus, in the truncated case the diffusion becomes closer to normal, as it should be, while the velocity autocorrelation function decays slower than in the untruncated case, fully consistent with the antipersistent fractional Langevin equation model.

Now let us turn to the case of strong power-law truncation with $\mu \gt 2H-1\gt 0$ in which for simplicity we assume that $\mu +1-2H\ne n$ where $n\in {\mathbb{N}}$ is a positive integer. We are interested in the exponent of the power-law decay of the velocity autocorrelation function. Then expression (49) yields $U(2H-1,2H-\mu ;s{\tau }_{\star })\,\sim {a}_{0}+{a}_{1}s+{a}_{2}{s}^{2}+\ldots \,{a}_{k}{s}^{k}+{a}_{\mu }{s}^{\mu +1-2H}+{a}_{k+1}{s}^{k+1}+\ldots$, where a_i with $i=0,1,2,\,\ldots$ are constants that can be easily found from expansion 13.1.2 in [42] for the first Kummer function in the square brackets of expression (49) and $k=[\mu +1-2H]$ denotes the integer part of the corresponding argument in the Landau bracket $[\cdot ]$. Then ${U}^{-1}(2H-1,2H-\mu ;s{\tau }_{\star })\sim {b}_{0}+{b}_{1}s\,+\ldots +\,{b}_{k}{s}^{k}+{b}_{\mu }{s}^{\mu +1-2H}+\ldots$ where the b_i with i = 0, 1, 2, ...are again constant factors. From here and with equations (48) and (56) we find after application of the Tauberian theorem and subsequent double differentiation

$$\begin{eqnarray}&&\langle {v}^{2}{\rangle }_{\tau }\sim -C\displaystyle \frac{{k}_{B}T}{m{{\rm{\Gamma }}}_{H}{\tau }_{\star }^{2H-1}}\displaystyle \frac{1}{{\tau }^{\mu +2-2H}},\end{eqnarray} \tag{ 58 }$$

where C is a positive constant. Note that in the borderline case $1\gt \mu =2H-1\gt 0$ both expressions (57) and (58) tend to the same limit resulting in the logarithmic correction to normal diffusion in expression (54).

A graphical representation of the velocity autocorrelation function (36), (44) and (58) is shown in figure 8.

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We here demonstrate the usefulness of our tempered fractional Gaussian noise approach to a concrete physical system. The data we have in mind are from all atom molecular dynamics simulations of lipid bilayer membranes [30]. In their simplest form, these are double layered leaves made up of relatively short amphiphilic polymers called lipids. Immersed in water the double layer arrangement prevents the exposure of the hydrophobic tail groups to the ambient water, while the hydrophilic head groups are in contact with the water. At room temperature the lipid bilayer assumes a quite disordered liquid structure [30]. In this lipid matrix, comparatively large membrane proteins may be additionally embedded [30]. Natural biological membranes are composed of lipids of many different chemistries, and they are crowded with membrane proteins. Supercomputing studies have the task to reveal the dynamics of both proteins and lipids in such protein-decorated bilayer systems. This thermally driven diffusion of the constituents influence biological properties of the bilayer, such as diffusion limited aggregation, domain formation, or the membrane penetration by nanoparticles [30].

Figure 9 depicts the simulations results in a chemically uniform, liquid disordered lipid bilayer membrane as well as in the liquid ordered state in the presence of cholesterol molecules—the system is specified in detail in [22]. The motion of the lipids is Gaussian for all cases and best described as viscoelastic diffusion governed by the generalised Langevin equation (22) fuelled by power-law noise [22, 24, 25]¹⁵ . As can be seen in figure 9 the MSD of the liquid disordered lipid systems exhibits a clear crossover from subdiffusion to normal diffusion at roughly 10 ns, the typical crossover time scale discussed in literature, at which two nearest neighbour lipid molecules exchange their mutual positions and thus decorrelate their motion [22, 30, 31]. For the liquid ordered cases, one lipid chemistry also shows a subdiffusive–normal crossover, while the two other lipid chemistries lead to a crossover from slower to faster subdiffusion [22]. From fit of the parameters (see the summary in table 1) to the data we observe an excellent agreement with the short and long time scaling regimes and, remarkably, the model fully describes the crossover behaviours without further tuning for both liquid disordered and ordered situations. We note that subdiffusive–diffusive crossovers are also observed for protein-crowded membranes [23, 24, 53].

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Table 1.  Fit parameters for the model membrane simulations data shown in figure 9. The colours mentioned in the first column correspond to the colour coding in figure 9. Bold face stands for liquid ordered systems with embedded cholesterols.

	H	μ	${\tau }_{\star }$	${k}_{B}T/[m{{\rm{\Gamma }}}_{H}]$	${\alpha }_{\mathrm{short}}$	${K}_{{\alpha }_{\mathrm{short}}}$	αlong	${K}_{{\alpha }_{\mathrm{long}}}$
			$(\mathrm{ns})$	$({\mathrm{nm}}^{2}/{\mathrm{nsec}}^{2-2H})$		$({\mathrm{nm}}^{2}/{\mathrm{nsec}}^{2-2H})$		$({\mathrm{nm}}^{2}/{\mathrm{nsec}}^{{\alpha }_{\mathrm{long}}})$
DSPC (purple)	0.70	—	4.0	0.050	0.60	0.034	1.0	0.029
SOPC (pink)	0.67	—	2.5	0.88	0.66	0.064	1.0	0.064
DOPC (blue)	0.69	—	3.0	0.067	0.62	0.046	1.0	0.044
DSPC (grey)	0.76	0.41	0.60	0.019	0.48	0.010	0.89	0.003 5
SOPC (green)	0.75	0.44	0.22	0.025	0.50	0.014	0.94	0.002 6
DOPC (brown)	0.72	—	4.3	0.038	0.57	0.024	1.0	0.021

We note that from equation (31) and the effective diffusion coefficient

$$\begin{eqnarray}&&{K}_{\alpha }^{* }(t)=\displaystyle \frac{1}{2}\displaystyle \frac{{\rm{d}}}{{\rm{d}}{t}}\langle {x}^{2}(t)\rangle .\end{eqnarray} \tag{ 59 }$$

we find the short time limiting behaviour

$$\begin{eqnarray}&&{K}_{\alpha }^{* }(t)={K}_{\alpha ,\mathrm{short}}^{* }{t}^{1-2H}\end{eqnarray} \tag{ 60 }$$

with

$$\begin{eqnarray}&&{K}_{\alpha ,\mathrm{short}}^{* }=\displaystyle \frac{{k}_{B}T}{m{{\rm{\Gamma }}}_{H}}\displaystyle \frac{1}{{\rm{\Gamma }}(2-2H)}.\end{eqnarray} \tag{ 61 }$$

For the long time limit, from equation (40), it follows that

$$\begin{eqnarray}&&{K}_{\alpha ,\mathrm{long}}^{* }=\displaystyle \frac{{k}_{B}T}{m{{\rm{\Gamma }}}_{H}}\displaystyle \frac{1}{{\tau }_{\star }^{2H-1}}\end{eqnarray} \tag{ 62 }$$

for the exponential tempering, whereas the cases of DSPC and SOPC lipid chemistries the long time limit in the weak power-law truncation case is given by

$$\begin{eqnarray}&&{K}_{\alpha ,\mathrm{long}}^{* }=\displaystyle \frac{{\rm{\Gamma }}(2H-1)}{{\rm{\Gamma }}(2H-\mu +1){\rm{\Gamma }}(\mu +2-2H)}\displaystyle \frac{{k}_{B}T}{m{{\rm{\Gamma }}}_{H}{\tau }_{\star }^{\mu }}.\end{eqnarray} \tag{ 63 }$$

The fit values given in table 1 are in very good agreement with those obtained in the simulations study [22]. We note, however, that for the weak power-law tempering model fit the crossover time is somewhat underestimated.

Meerschaert and Sabzikar defined this extension of fractional Brownian motion by applying an exponential truncation in Mandelbrot's definition [36, 57] ¹⁶ ,

$$\begin{eqnarray}\begin{array}{rcl}{B}_{H,\lambda }(t) & = & {\displaystyle \int }_{-\infty }^{0}[{{\rm{e}}}^{-\lambda (t-t^{\prime} )}{(t-t^{\prime} )}^{H-\tfrac{1}{2}}-{{\rm{e}}}^{-\lambda (-t^{\prime} )}{(-t^{\prime} )}^{H-\tfrac{1}{2}}]B^{\prime} (t^{\prime} ){\rm{d}}{t}^{\prime} \\ & & +{\displaystyle \int }_{0}^{t}[{{\rm{e}}}^{-\lambda (t-t^{\prime} )}{(t-t^{\prime} )}^{H-\tfrac{1}{2}}]B^{\prime} (t^{\prime} ){\rm{d}}{t}^{\prime} ,\end{array}\end{eqnarray} \tag{ 64 }$$

where H, λ, t > 0. $B^{\prime} (t)$ is white Gaussian noise of δ-covariance $\langle B^{\prime} ({t}_{1})B^{\prime} ({t}_{2})\rangle ={\sigma }^{2}\delta ({t}_{1}-{t}_{2})$ and zero mean. The parameter λ stands for the truncation parameter, and classical fractional Brownian motion is then obtained in the limiting case λ → 0 when $H\in (0,1)$. It should be noted that the prefactor 1/Γ(H + 1/2) in Mandelbrot's original definition is dropped here in line with the procedure of [57]. The MSD encoded in equation (64) is (see appendix C for the derivation)

$$\begin{eqnarray}&&\langle {B}_{H,\lambda }^{2}(t)\rangle ={\sigma }^{2}{C}_{t}^{2}{t}^{2H},\end{eqnarray} \tag{ 65 }$$

where the prefactor is

$$\begin{eqnarray}&&{C}_{t}^{2}=\left[\displaystyle \frac{2{\rm{\Gamma }}(2H)}{{(2\lambda t)}^{2H}}-\displaystyle \frac{2{\rm{\Gamma }}(H+1/2)}{\sqrt{\pi }}\displaystyle \frac{{K}_{H}(| \lambda t| )}{{(2\lambda t)}^{H}}\right].\end{eqnarray} \tag{ 66 }$$

K_H(z) denotes the modified Bessel function of the second kind, which for small argument z behaves as [42]

$$\begin{eqnarray}&&{K}_{H}(z)\sim \displaystyle \frac{{\rm{\Gamma }}(H)}{{2}^{1-H}}{z}^{-H}+\displaystyle \frac{{\rm{\Gamma }}(-H)}{{2}^{1+H}}{z}^{H}+\displaystyle \frac{{\rm{\Gamma }}(H)}{{2}^{3-H}(1-H)}{z}^{2-H}\end{eqnarray} \tag{ 67 }$$

while for large z we have ${K}_{H}(z)\sim \sqrt{\pi /(2z)}{{\rm{e}}}^{-z}$. The fact that the prefactor C_t² is an explicit function of time contrasts the result of standard fractional Brownian motion, and we will readily see the ensuing consequences.

In the short time limit t ≪ λ⁻¹ expression (65) has the compound power-law form

$$\begin{eqnarray}&&\langle {B}_{H,\lambda }^{2}(t)\rangle \sim {\sigma }^{2}{{\rm{\Gamma }}}^{2}(H+1/2){V}_{H}{t}^{2H}+\displaystyle \frac{{\sigma }^{2}{\rm{\Gamma }}(2H)}{{2}^{1+2H}(H-1)}{\lambda }^{2-2H}{t}^{2}\end{eqnarray} \tag{ 68 }$$

with ${V}_{H}=1/[{\rm{\Gamma }}(2H+1)\sin (\pi H)]$. Thus, the limit λ → 0 indeed reduces to the expression for standard fractional Brownian motion. In the long time limit t ≫ λ⁻¹ the MSD of this tempered fractional Brownian motion, remarkably, converges exponentially towards a constant value,

$$\begin{eqnarray}&&\langle {B}_{H,\lambda }^{2}(t)\rangle \sim {\sigma }^{2}\left(\displaystyle \frac{2{\rm{\Gamma }}(2H)}{{(2\lambda )}^{2H}}-\displaystyle \frac{{2}^{1/2-H}{\rm{\Gamma }}(H+1/2)}{{\lambda }^{H+1/2}}{t}^{H-1/2}{{\rm{e}}}^{-\lambda t}\right),\end{eqnarray} \tag{ 69 }$$

a result which is at first surprising. This point will be discussed and compared to the fractional Ornstein–Uhlenbeck process below. The functional behaviour of result (69) is shown in figure 14. We note that if we consider the Langevin equation (2) in combination with the directly tempered noise ${B}_{H\lambda }^{{\prime} }(t)$, expression (65) and its limiting behaviours (68) and (69) exactly correspond to the dynamics of the MSD $\langle {x}^{2}(t)\rangle$.

As shown in [58] it is possible to define a tempered fractional Gaussian noise following Mandelbrot and van Ness' smoothening procedure involving a short time lag δ (see appendix C.2). The autocorrelation function of this tempered fractional Gaussian noise is given through

$$\begin{eqnarray}\begin{array}{rcl}\langle {B}_{H,\lambda }^{{\prime} }(t){B}_{H,\lambda }^{{\prime} }(t+\tau )\rangle & = & \displaystyle \frac{{\rm{\Gamma }}\left(H+\tfrac{1}{2}\right){\sigma }^{2}}{\sqrt{\pi }{(2\lambda )}^{H}{\delta }^{2}}[2{\tau }^{H}{K}_{H}(| \lambda \tau | )-{(\tau +\delta )}^{H}{K}_{H}(\lambda | \tau +\delta | )\\ & & -| \tau -\delta {| }^{H}{K}_{H}(\lambda | \tau -\delta | )].\end{array}\end{eqnarray} \tag{ 70 }$$

An important feature of the autocorrelation function (70) for tempered fractional Gaussian noise is its antipersistent behaviour over the whole range 0 < H < 1 for any finite λ, that is, the integral of expression (70) over the entire domain of τ vanishes:

$$\begin{eqnarray}&&{\int }_{0}^{\infty }\langle {B}_{H,\lambda }^{{\prime} }(t){B}_{H,\lambda }^{{\prime} }(t+\tau )\rangle {\rm{d}}\tau =0.\end{eqnarray} \tag{ 71 }$$

This is in sharp contrast to (conventional) fractional Gaussian noise. Indeed, in the limit $\lambda \to 0$ the noise autocorrelation function (70) approaches the one of fractional Gaussian noise [36, 57], as can be derived by using the small argument expansion (67) of the Bessel function. In this limit λ → 0 for any finite τ the autocorrelation function (70) converges to

$$\begin{eqnarray}&&\mathop{\mathrm{lim}}\limits_{\lambda \to 0}\langle {B}_{H,\lambda }^{{\prime} }(t){B}_{H,\lambda }^{{\prime} }(t+\tau )\rangle \sim \displaystyle \frac{{{\rm{\Gamma }}}^{2}\left(H+\tfrac{1}{2}\right){\sigma }^{2}{V}_{H}}{2{\delta }^{2}}[{(\tau +\delta )}^{2H}+| \tau -\delta {| }^{2H}-2{\tau }^{2H}]\end{eqnarray} \tag{ 72 }$$

and shows negative correlations for 0 < H < 1/2 and positive correlations for 1/2 < H < 1, see appendix C.3.

The autocorrelation function (70) and its limit for λ → 0 are shown in figures 10 and 11 for different values of the Hurst parameter. While for the tempered process it is antipersistent for the whole range of H, in the limit λ → 0 we clearly see the difference between the antipersistent case with the overshoot to negative values and a slow recovery back to zero. The autocorrelation function for the persistent case is always positive.

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It is easy to show that for τ ≪ 1/λ and δ → 0 the autocorrelation function (70) decays as a power-law, consistent with the behaviour of fractional Gaussian noise,

$$\begin{eqnarray}\begin{array}{rcl}\langle {B}_{H,\lambda }^{{\prime} }(t){B}_{H,\lambda }^{{\prime} }(t+\tau )\rangle & \sim & {\sigma }^{2}(2H-1)H{{\rm{\Gamma }}}^{2}(H+1/2){V}_{H}| \tau {| }^{2H-2}\\ & & -\displaystyle \frac{{\sigma }^{2}{\rm{\Gamma }}(2H){\lambda }^{2-2H}}{{2}^{2H+1}(1-H)},\end{array}\end{eqnarray} \tag{ 73 }$$

while the asymptotic behaviour at long observation times, τ ≫ λ⁻¹ ,

$$\begin{eqnarray}&&\langle {B}_{H,\lambda }^{{\prime} }(t){B}_{H,\lambda }^{{\prime} }(t+\tau )\rangle \sim \displaystyle \frac{{\tau }^{H-\tfrac{1}{2}}{{\rm{e}}}^{-\lambda \tau }{\sigma }^{2}}{{2}^{H-\tfrac{1}{2}}{\lambda }^{H+\tfrac{1}{2}}{\delta }^{2}}\left[1-\cosh (\lambda \delta )+\sinh (\lambda \delta )\displaystyle \frac{\left(H-\tfrac{1}{2}\right)\delta }{\tau }\right]\end{eqnarray} \tag{ 74 }$$

decays exponentially, in contrast to the non-tempered limit in equation (73). This different asymptotic behaviour of tempered versus non-tempered fractional Gaussian noise around the truncation time, is shown in figure 12.

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Considering the internal noise ξ(t) of the system as the tempered fractional Gaussian noise ${B}_{H,\lambda }^{{\prime} }(t)$ defined above, the overdamped tempered fractional Langevin equation reads [58]

$$\begin{eqnarray}&&{\int }_{0}^{t}{\gamma }_{H}(t-\tau )\displaystyle \frac{{\rm{d}}{x}}{{\rm{d}}\tau }{\rm{d}}\tau =\xi (t),\end{eqnarray} \tag{ 75 }$$

in which ${\gamma }_{H}(\tau )=2\langle {B}_{H,\lambda }^{{\prime} }(t){B}_{H,\lambda }^{{\prime} }(t+\tau )\rangle$. Similar to our derivation above, we obtain the Laplace transform of the MSD (28) in dimensionless units,

$$\begin{eqnarray}&&\langle {\tilde{x}}^{2}(s)\rangle =\displaystyle \frac{2}{{s}^{2}{\tilde{\gamma }}_{H}(s)},\end{eqnarray} \tag{ 76 }$$

in which we have to find the Laplace transformation of the autocorrelation function (70). We assume that σ² = 1 for simplicity from now on. To proceed, in the second and third terms we change the variables and split the resulting integrals,

$$\begin{eqnarray}&&{\tilde{\gamma }}_{H}(s)=\displaystyle \frac{2{\rm{\Gamma }}\left(H+\tfrac{1}{2}\right)}{\sqrt{\pi }{(2\lambda )}^{H}{\delta }^{2}}\left\{2[1-\cosh (\delta s)]{\int }_{0}^{\infty }{{\rm{d}}{t}{\rm{e}}}^{-{st}}{t}^{H}{K}_{H}(\lambda t)\right.\end{eqnarray} \tag{ 77 }$$

$$\begin{eqnarray}&&+\,\left.2{\int }_{0}^{\delta }{\rm{d}}{t}\sinh (s(\delta -t)){t}^{H}{K}_{H}(\lambda t)\right\}.\end{eqnarray} \tag{ 78 }$$

First, we expand the above functions up to second order in δ. Since in the second integral δ ≪ λ⁻¹ and t < δ the relevant regimes are δs ≪ 1 and λt ≪ 1. Therefore, to second order in δ, ${\tilde{\gamma }}_{H}(s)$ is

$$\begin{eqnarray}\begin{array}{rcl}{\tilde{\gamma }}_{H}(s) & \sim & \displaystyle \frac{2{\rm{\Gamma }}\left(H+\tfrac{1}{2}\right)}{\sqrt{\pi }{(2\lambda )}^{H}{\delta }^{2}}\left\{2\displaystyle \frac{-{(\delta s)}^{2}}{2}{\displaystyle \int }_{0}^{\infty }{{\rm{d}}{t}{\rm{e}}}^{-{st}}{t}^{H}{K}_{H}(\lambda t)\right.\\ & & \left.+2{\displaystyle \int }_{0}^{\delta }{\rm{d}}{t}(s(\delta -t)){t}^{H}{K}_{H}(\lambda t)\right\}.\end{array}\end{eqnarray} \tag{ 79 }$$

Using expansion (67) and keeping terms up to the second order of δ we find

$$\begin{eqnarray}&&2{\int }_{0}^{\delta }{\rm{d}}{t}(s(\delta -t)){t}^{H}{K}_{H}(\lambda t)\sim \displaystyle \frac{{2}^{H}s\pi }{\sin (\pi H){\rm{\Gamma }}(1-H){\lambda }^{H}}\displaystyle \frac{{\delta }^{2}}{2}.\end{eqnarray} \tag{ 80 }$$

Insertion of this result back to relation (79) yields

$$\begin{eqnarray}&&{\tilde{\gamma }}_{H}(s)\sim \displaystyle \frac{2{\rm{\Gamma }}\left(H+\tfrac{1}{2}\right)}{\sqrt{\pi }{(2\lambda )}^{H}{\delta }^{2}}\left\{\displaystyle \frac{\pi {2}^{H-1}s{\delta }^{2}}{\sin (\pi H){\rm{\Gamma }}(1-H){\lambda }^{H}}-{(\delta s)}^{2}{\int }_{0}^{\infty }{{\rm{d}}{t}{\rm{e}}}^{-{st}}{t}^{H}{K}_{H}(\lambda t)\right\}.\end{eqnarray} \tag{ 81 }$$

The integral in (81) is a Laplace transformation, for which we apply equation (2.16.6.3) of [59]. Hence we find the expression for the autocorrelation function in Laplace space,

$$\begin{eqnarray}\begin{array}{rcl}{\tilde{\gamma }}_{H}(s) & \sim & \displaystyle \frac{2{\rm{\Gamma }}\left(H+\tfrac{1}{2}\right)}{\sqrt{\pi }{(2\lambda )}^{H}{\delta }^{2}}\left\{\displaystyle \frac{\pi {2}^{H-1}s{\delta }^{2}}{\sin (\pi H){\rm{\Gamma }}(1-H){\lambda }^{H}}\right.\\ & & \left.-{(\delta s)}^{2}\displaystyle \frac{{s}^{-1}{\lambda }^{-H}}{{2}^{H+1}}\sqrt{\pi }\displaystyle \frac{{\rm{\Gamma }}(2H+1)}{{\rm{\Gamma }}(H+3/2)}{}_{2}{F}_{1}\left(\displaystyle \frac{1}{2},1;H+\displaystyle \frac{3}{2};1-\displaystyle \frac{{\lambda }^{2}}{{s}^{2}}\right)\right\}\end{array}\end{eqnarray} \tag{ 82 }$$

in terms of the hypergeometric function ${}_{2}{F}_{1}$ [42].

4.2.1. Short time behaviour of the MSD

For the regime of short observation times, δ ≪ t ≪ 1/λ we apply the linear transformations for hypergeometric functions (for more details see (appendix C.4)). Then, with the general definition for hypergeometric functions up to second order and some simplifications, we find the dominant term for the autocorrelation function,

$$\begin{eqnarray}&&{\tilde{\gamma }}_{H}(s)\sim \displaystyle \frac{2{{\rm{\Gamma }}}^{2}\left(H+\tfrac{1}{2}\right)}{2\sin (\pi H)}{s}^{1-2H}.\end{eqnarray} \tag{ 83 }$$

Substituting this into expression (76), we see that

$$\begin{eqnarray}&&\langle {\tilde{x}}^{2}(s)\rangle \sim \displaystyle \frac{2\sin (\pi H)}{2{{\rm{\Gamma }}}^{2}\left(H+\tfrac{1}{2}\right)}{s}^{2H-3}.\end{eqnarray} \tag{ 84 }$$

By inverse Laplace transformation we find the asymptotic MSD behaviour in time,

$$\begin{eqnarray}&&\langle {x}^{2}(t)\rangle \sim \displaystyle \frac{\sin (\pi H)}{{{\rm{\Gamma }}}^{2}\left(H+\tfrac{1}{2}\right)}\displaystyle \frac{{t}^{2-2H}}{{\rm{\Gamma }}(3-2H)}.\end{eqnarray} \tag{ 85 }$$

This result corresponds to subdiffusion for 1/2 < H < 1 in agreement with the findings in section 3. For 0 < H < 1/2 the behaviour is superdiffusive.

4.2.2. Long time behaviour of the MSD

For the long times regime t ≫ 1/λ or λ/s ≫ 1 we go back to expression (82) and use the same method as in the previous subsection (see also (appendix C.5)). It can be seen that the dominant term is a linear function of s,

$$\begin{eqnarray}&&\tilde{\gamma }(s)\sim \displaystyle \frac{\sqrt{\pi }}{2{\lambda }^{2H}}\displaystyle \frac{2{\rm{\Gamma }}\left(H+\tfrac{1}{2}\right)}{\sin (\pi H){\rm{\Gamma }}(1-H)}s.\end{eqnarray} \tag{ 86 }$$

Getting back to equation (76) for the MSD, this yields

$$\begin{eqnarray}&&\langle {\tilde{x}}^{2}(s)\rangle \sim \displaystyle \frac{2{\lambda }^{2H}\sin (\pi H){\rm{\Gamma }}(1-H)}{2{\rm{\Gamma }}\left(H+\tfrac{1}{2}\right)\sqrt{\pi }}{s}^{-3}.\end{eqnarray} \tag{ 87 }$$

After inverse Laplace transformation, we obtain

$$\begin{eqnarray}&&\langle {x}^{2}(t)\rangle \sim \displaystyle \frac{\sin (\pi H){\rm{\Gamma }}(1-H){\lambda }^{2H}}{{\rm{\Gamma }}\left(H+\tfrac{1}{2}\right)\sqrt{\pi }}{t}^{2}=\displaystyle \frac{\sqrt{\pi }{\lambda }^{2H}}{{\rm{\Gamma }}\left(H+\tfrac{1}{2}\right){\rm{\Gamma }}(H)}{t}^{2}.\end{eqnarray} \tag{ 88 }$$

Thus, at long times this process converges to ballistic diffusion, as already observed in [58].

The general behaviour of the MSD and its crossover from short time power-law behaviour to long time ballistic motion is shown in figure 13 for different Hurst exponents.

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To come back to the above observed finite limiting value at long times, encoded in expression (69), of the MSD in the tempered fractional Brownian process we briefly study the confined fractional Brownian motion in an harmonic potential. Experimentally, such a situation arises, for instance, when particle tracking is performed with an optical tweezers setup in a viscoelastic environment [9, 20]. We thus consider the Ornstein–Uhlenbeck process

$$\begin{eqnarray}&&\displaystyle \frac{{\rm{d}}{x}(t)}{{\rm{d}}{t}}=-\lambda x(t)+{B}_{H}^{{\prime} }(t),\end{eqnarray} \tag{ 89 }$$

for t > 0 and with x(0) = 0, where the noise ${B}_{H}^{{\prime} }(t)$ is again fractional Gaussian noise. The MSD reads (see appendix D)

$$\begin{eqnarray}&&\langle {x}^{2}(t)\rangle ={\sigma }^{2}{V}_{H}{t}^{2H}{{\rm{e}}}^{-\lambda t}\left[1+\displaystyle \frac{\lambda t}{4H+2}({{\rm{e}}}^{\lambda t}{f}_{H}(-\lambda t)-{{\rm{e}}}^{-\lambda t}{f}_{H}(\lambda t))\right],\end{eqnarray} \tag{ 90 }$$

where ${f}_{H}(x)\equiv M(2H+1;2H+2;x)$ is Kummer's confluent hypergeometric function. For $t\ll {\lambda }^{-1}$, the MSD of this fractional Ornstein–Uhlenbeck process assumes the form

$$\begin{eqnarray}&&\langle {x}^{2}(t)\rangle \sim {\sigma }^{2}{V}_{H}{t}^{2H}(1-\lambda t),\end{eqnarray} \tag{ 91 }$$

which corresponds to unconfined fractional Brownian motion with a correction proportional to λt. In the long time limit an exponentially fast convergence occurs to the stationary limit

$$\begin{eqnarray}&&\langle {x}^{2}(t)\rangle \sim \displaystyle \frac{{\sigma }^{2}}{2\sin (\pi H){\lambda }^{2H}}.\end{eqnarray} \tag{ 92 }$$

Figure 14 compares the MSDs of tempered fractional Brownian motion and of the fractional Ornstein–Uhlenbeck process. Both of them saturate at long times, where the plateau value depends on the value of H, compare also [60, 61]. Curiously, the plateau values of both processes become identical for the Hurst exponent H = 0.768 149.

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From the comparison with this fractional Ornstein–Uhlenbeck process we see that the direct tempering model of Meerschaert and Sabziker actually describes a confined motion, in contrast to the simple intuition of the tempering in equation (64). In that sense it is fundamentally different from the truncated models considered in the previous sections which show a crossover between two regimes of steadily increasing MSD.

The effect of direct tempering for the fractional Langevin equation model, a priori is even more surprising. Namely, as we saw from equations (85) and (88), this model demonstrates a crossover from a short time subdiffusive to a ballistic regime at long times. Such a behaviour appears counterintuitive. However, as we show now, it is actually a simple consequence of the two basic features of the directly tempered internal fractional Gaussian noise (75): (i) the integral of its autocorrelation function over the entire time domain from zero to infinity is identical to zero, see relation (71); (ii) at long times the autocorrelation function exhibits the exponential decay (74). To demonstrate that these two conditions indeed effect the ballistic long time behaviour, consider a toy model for the noise ξ(t) in the fractional Langevin equation (75), namely, we assume the autocorrelation function

$$\begin{eqnarray}&&\langle \xi (t)\xi (t+\tau )\rangle ={\gamma }_{H}(\tau )=\delta (\tau )-\lambda {{\rm{e}}}^{-\lambda \tau }.\end{eqnarray} \tag{ 93 }$$

Note that the spectral density of the noise is non-negative and the autocorrelation function (93) obeys conditions (i) and (ii). Now, the Laplace transform of the autocorrelation function (93) reads ${\tilde{\gamma }}_{H}(s)=s/(s+\lambda )$, and with relation (76) we thus find the MSD

$$\begin{eqnarray}&&\langle {\tilde{x}}^{2}(s)\rangle =\displaystyle \frac{2}{{s}^{2}{\tilde{\gamma }}_{H}(s)}=\displaystyle \frac{2(s+\lambda )}{{s}^{3}},\end{eqnarray} \tag{ 94 }$$

in Laplace space. As function of time, this indeed produced the ballistic long time behaviour $\langle {x}^{2}(t)\rangle \sim \lambda {t}^{2}$ for t ≫ 1/λ.

As we see the direct tempering approach leads to unexpected behaviours. Because of the stationary limit (69) the model by Meerschaert and Sabzikar may be more appropriate for modelling the velocity process rather than the position of a diffusing particle, compare [62]. Conversely, the emergence of the ballistic motion (88) at long times for the directly tempered fractional Langevin equation may find useful applications for active systems.

Due to the white Gaussian noise in equation (64) the MSD of tempered fractional Brownian motion (64) can be written as

$$\begin{eqnarray}\begin{array}{rcl}\langle {B}_{H,\lambda }^{2}(t)\rangle & = & {\sigma }^{2}\left[{\displaystyle \int }_{0}^{t}{{\rm{e}}}^{-2\lambda (t-u)}{(t-u)}^{2H-1}{\rm{d}}{u}\right.\\ & & \left.+{\displaystyle \int }_{-\infty }^{0}{({{\rm{e}}}^{-\lambda (t-u)}{(t-u)}^{H-1/2}-{{\rm{e}}}^{\lambda u}{(-u)}^{H-1/2})}^{2}{\rm{d}}{u}\right].\end{array}\end{eqnarray} \tag{ C.1 }$$

After expanding the square of the second integral and using the appropriate changes of variable, it becomes

$$\begin{eqnarray}&&\langle {B}_{H,\lambda }^{2}(t)\rangle ={\sigma }^{2}\left[{\int }_{0}^{\infty }{{\rm{e}}}^{-2\lambda {ts}}{s}^{2H-1}{\rm{d}}{s}-{{\rm{e}}}^{-\lambda t}{\int }_{0}^{\infty }{{\rm{e}}}^{-2\lambda {ts}}{(1+s)}^{H-1/2}{s}^{H-1/2}{\rm{d}}{s}\right].\end{eqnarray} \tag{ C.2 }$$

These integrals can be found, for instance, as equations (3.3814) and (3.3838) in [77]. This produces equation (65).

In the classical paper by Mandelbrot and van Ness [36] a smooth fractional Brownian motion is defined in terms of the small and positive parameter δ, through

$$\begin{eqnarray}&&{B}_{H}(t;\delta )=\displaystyle \frac{1}{\delta }{\int }_{t}^{t+\delta }{B}_{H}(u){\rm{d}}{u}.\end{eqnarray} \tag{ C.3 }$$

Its derivative is known as the fractional Gaussian noise

$$\begin{eqnarray}&&{B}_{H}^{\prime} (t;\delta )=\displaystyle \frac{1}{\delta }[{B}_{H}(t+\delta )-{B}_{H}(t)],\end{eqnarray} \tag{ C.4 }$$

where we omit the explicit dependence on δ in the main text. The autocorrelation function of equation (C.4) is given in expression (72).

The same procedure can be applied to tempered fractional Brownian motion to define the corresponding continuous fractional noise

$$\begin{eqnarray}&&{B}_{H,\lambda }^{{\prime} }(t;\delta )=\displaystyle \frac{1}{\delta }[{B}_{H,\lambda }(t+\delta )-{B}_{H,\lambda }(t)].\end{eqnarray} \tag{ C.5 }$$

With the identity

$$\begin{eqnarray}&&2(a-b)(c-d)={(a-d)}^{2}+{(b-c)}^{2}-{(a-c)}^{2}-{(b-d)}^{2},\end{eqnarray} \tag{ C.6 }$$

and the fact that tempered fractional Brownian motion has stationary increments, and ${B}_{H,\lambda }(0)=0$, we obtain

$$\begin{eqnarray}&&\langle {B}_{H,\lambda }^{{\prime} }(t;\delta ){B}_{H,\lambda }^{{\prime} }(t+\tau ;\delta )\rangle =\displaystyle \frac{1}{2{\delta }^{2}}[\langle {B}_{H,\lambda }^{2}(\tau -\delta )\rangle +\langle {B}_{H,\lambda }^{2}(\tau +\delta )\rangle -2\langle {B}_{H,\lambda }^{2}(\tau )\rangle ].\end{eqnarray} \tag{ C.7 }$$

By virtue of relation (65) the autocorrelation function of tempered fractional Gaussian noise becomes expression (70). The autocorrelation function of tempered fractional Gaussian noise (70) has a well defined limit when δλ → 0,

$$\begin{eqnarray}\begin{array}{rcl}\lt {B}_{H,\lambda }^{{\prime} }(t){B}_{H,\lambda }^{{\prime} }(t+\tau )\gt & = & \displaystyle \frac{{\sigma }^{2}{\rm{\Gamma }}(H+1/2){\lambda }^{2-2H}}{{2}^{H}\sqrt{\pi }}[{(\lambda | \tau | )}^{H-1}{K}_{1-H}(\lambda | \tau | )\\ & & -{(\lambda | \tau | )}^{H}{K}_{2-H}(\lambda | \tau | )].\end{array}\end{eqnarray} \tag{ C.8 }$$

Taking the integral over expression (72) and denoting

$$\begin{eqnarray}&&{W}_{H}=\displaystyle \frac{{{\rm{\Gamma }}}^{2}\left(H+\tfrac{1}{2}\right)}{2{\rm{\Gamma }}(2H+1)\sin (\pi H)}\end{eqnarray} \tag{ C.9 }$$

one gets

$$\begin{eqnarray}\begin{array}{rcl}{\mathbb{K}} & = & {\displaystyle \int }_{0}^{\infty }{\rm{d}}\tau \mathop{{\rm{lim}}}\limits_{\lambda \to 0}\langle {B}_{H,\lambda }^{{\prime} }(t){B}_{H,\lambda }^{{\prime} }(t+\tau )\rangle \\ & = & \displaystyle \frac{{W}_{H}}{{\delta }^{2}}\times \mathop{{\rm{lim}}}\limits_{A\to \infty }\left[{\displaystyle \int }_{0}^{A}{\rm{d}}\tau {(\tau +\delta )}^{2H}+{\displaystyle \int }_{0}^{A}{\rm{d}}\tau | \tau -\delta {| }^{2H}-2{\displaystyle \int }_{0}^{A}{\rm{d}}\tau {\tau }^{2H}\right]\\ & = & \displaystyle \frac{{W}_{H}}{{\delta }^{2}}\times \mathop{{\rm{lim}}}\limits_{A\to \infty }\left[{\displaystyle \int }_{\delta }^{A+\delta }{\rm{d}}\tau {\tau }^{2H}+{\displaystyle \int }_{-\delta }^{A-\delta }{\rm{d}}\tau {\tau }^{2H}-2{\displaystyle \int }_{0}^{A}{\rm{d}}\tau {\tau }^{2H}\right]\\ & = & \displaystyle \frac{{W}_{H}}{(2H+1){\delta }^{2}}\times \mathop{{\rm{lim}}}\limits_{A\to \infty }[{(A+\delta )}^{2H+1}-{\delta }^{2H+1}+{(A-\delta )}^{2H+1}-{\delta }^{2H+1}-2{A}^{2H+1}]\\ & = & \displaystyle \frac{{W}_{H}}{(2H+1){\delta }^{2}}\times \mathop{{\rm{lim}}}\limits_{A\to \infty }\left[{A}^{2H+1}\left(1+(2H+1)\displaystyle \frac{\delta }{A}+2H(2H+1)\displaystyle \frac{{\delta }^{2}}{2{A}^{2}}\right)-{\delta }^{2H+1}\right.\\ & & \left.+{A}^{2H+1}\left(1-(2H+1)\displaystyle \frac{\delta }{A}+2H(2H+1)\displaystyle \frac{{\delta }^{2}}{2{A}^{2}}\right)-{\delta }^{2H+1}-2{A}^{2H+1}\right]\\ & = & \displaystyle \frac{{W}_{H}}{(2H+1){\delta }^{2}}\times \mathop{{\rm{lim}}}\limits_{A\to \infty }[2H(2H+1){\delta }^{2}{A}^{2H-1}]\\ & = & \displaystyle \frac{{{\rm{\Gamma }}}^{2}\left(H+\tfrac{1}{2}\right)}{2{\rm{\Gamma }}(2H)\sin (\pi H)}\times \mathop{{\rm{lim}}}\limits_{A\to \infty }[{A}^{2H-1}]\\ & = & \left\{\begin{array}{ll}\infty , & H\gt \displaystyle \frac{1}{2}\\ 0, & H\lt \displaystyle \frac{1}{2}\end{array}\right..\end{array}\end{eqnarray} \tag{ C.10 }$$

For the regime of short observation times, $\delta \ll t\ll {\lambda }^{-1}$, we apply the linear transformation 15.3.6 from [42] for hypergeometric functions. In the resulted definition, the argument of the hypergeometric function is small,

$$\begin{eqnarray}\begin{array}{rcl}{\tilde{\gamma }}_{H}(s) & = & \displaystyle \frac{2{\rm{\Gamma }}\left(H+\tfrac{1}{2}\right)}{\sqrt{\pi }{(2\lambda )}^{H}}\left\{\displaystyle \frac{\pi {2}^{H-1}s}{\sin (\pi H){\rm{\Gamma }}(1-H){\lambda }^{H}}-{s}^{2}\displaystyle \frac{{s}^{-1}{\lambda }^{-H}}{{2}^{H+1}}\sqrt{\pi }{\rm{\Gamma }}\left[\begin{array}{l}1,2H+1\\ H+\displaystyle \frac{3}{2}\end{array}\right]\right.\\ & & \times \left[\displaystyle \frac{{\rm{\Gamma }}\left(H+\tfrac{3}{2}\right){\rm{\Gamma }}(H)}{{\rm{\Gamma }}(H+1){\rm{\Gamma }}\left(H+\tfrac{1}{2}\right)}{}_{2}{F}_{1}\left(\displaystyle \frac{1}{2},1;1-H;\displaystyle \frac{{\lambda }^{2}}{{s}^{2}}\right)\right.\\ & & +\left.\left.{\left(\displaystyle \frac{{\lambda }^{2}}{{s}^{2}}\right)}^{H}\displaystyle \frac{{\rm{\Gamma }}\left(H+\tfrac{3}{2}\right){\rm{\Gamma }}(-H)}{{\rm{\Gamma }}\left(\tfrac{1}{2}\right){\rm{\Gamma }}(1)}{}_{2}{F}_{1}\left(H+1,H+\displaystyle \frac{1}{2};H+1;\displaystyle \frac{{\lambda }^{2}}{{s}^{2}}\right)\right]\right\}.\end{array}\end{eqnarray} \tag{ C.11 }$$

For small arguments we use the general definition of hypergeometric functions, 15.1.1 in [42], up to the second order. Then

$$\begin{eqnarray}\begin{array}{rcl}{\tilde{\gamma }}_{H}(s) & = & \displaystyle \frac{2{\rm{\Gamma }}\left(H+\tfrac{1}{2}\right)}{\sqrt{\pi }{(2\lambda )}^{H}}\left\{\displaystyle \frac{\pi {2}^{H-1}s}{\sin (\pi H){\rm{\Gamma }}(1-H){\lambda }^{H}}-\displaystyle \frac{\sqrt{\pi }s}{{2}^{H+1}{\lambda }^{H}}{\rm{\Gamma }}\left[\begin{array}{l}1,2H+1\\ H+\displaystyle \frac{3}{2}\end{array}\right]\right.\\ & & \times \left[\displaystyle \frac{{\rm{\Gamma }}\left(H+\tfrac{3}{2}\right){\rm{\Gamma }}(H)}{{\rm{\Gamma }}(H+1){\rm{\Gamma }}\left(H+\tfrac{1}{2}\right)}\displaystyle \frac{{\rm{\Gamma }}(1-H)}{{\rm{\Gamma }}\left(\tfrac{1}{2}\right)}\left(\displaystyle \frac{{\rm{\Gamma }}(\tfrac{1}{2})}{{\rm{\Gamma }}(1-H)}+\displaystyle \frac{{\rm{\Gamma }}(1+\tfrac{1}{2}){\rm{\Gamma }}(2)}{{\rm{\Gamma }}(2-H)}\displaystyle \frac{{\lambda }^{2}}{{s}^{2}}\right)\right.\\ & & +{\left(\displaystyle \frac{{\lambda }^{2}}{{s}^{2}}\right)}^{H}\displaystyle \frac{{\rm{\Gamma }}\left(H+\tfrac{3}{2}\right){\rm{\Gamma }}(-H)}{{\rm{\Gamma }}\left(\tfrac{1}{2}\right)}\displaystyle \frac{{\rm{\Gamma }}(H+1)}{{\rm{\Gamma }}\left(H+\tfrac{1}{2}\right){\rm{\Gamma }}(H+1)}\left({\rm{\Gamma }}\left(H+\displaystyle \frac{1}{2}\right)\right.\\ & & \left.\left.\left.+{\rm{\Gamma }}\left(H+\displaystyle \frac{3}{2}\right)\displaystyle \frac{{\lambda }^{2}}{{s}^{2}}\right)\right]\right\}.\end{array}\end{eqnarray} \tag{ C.12 }$$

Now, we simplify the Gamma functions using the duplication formula 6.1.18 in [42],

$$\begin{eqnarray}\begin{array}{rcl}{\tilde{\gamma }}_{H}(s) & = & \displaystyle \frac{2{\rm{\Gamma }}\left(H+\tfrac{1}{2}\right)}{\sqrt{\pi }{(2\lambda )}^{H}}\left\{\displaystyle \frac{\pi {2}^{H-1}s}{\sin (\pi H){\rm{\Gamma }}(1-H){\lambda }^{H}}-\displaystyle \frac{{2}^{H-1}{\rm{\Gamma }}(H)s}{{\lambda }^{H}}-\displaystyle \frac{{2}^{H-1}{\rm{\Gamma }}(H)s}{{\lambda }^{H}2(1-H)}\displaystyle \frac{{\lambda }^{2}}{{s}^{2}}\right.\\ & & -\displaystyle \frac{{2}^{H-1}s}{{\lambda }^{H}}\displaystyle \frac{H}{\left(H,+,\tfrac{1}{2}\right)}\displaystyle \frac{{\rm{\Gamma }}(H){\rm{\Gamma }}(-H){\rm{\Gamma }}\left(H+\tfrac{3}{2}\right)}{{\rm{\Gamma }}\left(\tfrac{1}{2}\right)}{\left(\displaystyle \frac{{\lambda }^{2}}{{s}^{2}}\right)}^{H}\\ & & \left.-\displaystyle \frac{{2}^{H-1}s}{{\lambda }^{H}}\displaystyle \frac{H}{\left(H,+,\tfrac{1}{2}\right)}\displaystyle \frac{{\rm{\Gamma }}(H){\rm{\Gamma }}(-H){\rm{\Gamma }}\left(H+\tfrac{3}{2}\right)}{{\rm{\Gamma }}\left(\tfrac{1}{2}\right)}\left(H+\displaystyle \frac{1}{2}\right){\left(\displaystyle \frac{{\lambda }^{2}}{{s}^{2}}\right)}^{H+1}\right\}.\end{array}\end{eqnarray} \tag{ C.13 }$$

Using Euler's reflection formula,

$$\begin{eqnarray}&&{\rm{\Gamma }}(z){\rm{\Gamma }}(1-z)=\displaystyle \frac{\pi }{\sin (\pi z)}\end{eqnarray} \tag{ C.14 }$$

the first two terms cancel each other and it can be seen that the dominant term in the autocorrelation function scales as ${s}^{1-2H}$,

$$\begin{eqnarray}\begin{array}{rcl}{\tilde{\gamma }}_{H}(s) & = & \displaystyle \frac{2{\rm{\Gamma }}\left(H+\tfrac{1}{2}\right)}{\sqrt{\pi }{(2\lambda )}^{H}}\left\{\displaystyle \frac{{2}^{H-1}}{{\lambda }^{H-1}\left(H+\tfrac{1}{2}\right)}\displaystyle \frac{\pi }{\sin (\pi H)}\displaystyle \frac{{\rm{\Gamma }}\left(H+\tfrac{3}{2}\right)}{\sqrt{\pi }}{\left(\displaystyle \frac{\lambda }{s}\right)}^{2H-1}\right.\\ & & \left.-\displaystyle \frac{{2}^{H-2}{\rm{\Gamma }}(H)}{{\lambda }^{H-1}(1-H)}\left(\displaystyle \frac{\lambda }{s}\right)+\displaystyle \frac{{2}^{H-1}}{{\lambda }^{H-1}}\displaystyle \frac{\pi {\rm{\Gamma }}\left(H+\tfrac{3}{2}\right)}{\sin (\pi H)\sqrt{\pi }}{\left(\displaystyle \frac{\lambda }{s}\right)}^{2H+1}\right\}.\end{array}\end{eqnarray} \tag{ C.15 }$$

For the regime of long observation time or $\tfrac{\lambda }{s}\gg 1$, we go back to equation (82) and use relation (15.3.8) from [42] for hypergeometric functions with small arguments. Then, by applying the expansion of hypergeometric functions up to the second order for small argument, s/λ ≪ 1,

$$\begin{eqnarray}&&\begin{array}{l}{}_{2}{F}_{1}\left(\displaystyle \frac{1}{2},1;H+\displaystyle \frac{3}{2};1-\displaystyle \frac{{\lambda }^{2}}{{s}^{2}}\right)=\left(\displaystyle \frac{s}{\lambda }\right)\displaystyle \frac{{\rm{\Gamma }}\left(H+\tfrac{3}{2}\right){\rm{\Gamma }}\left(\tfrac{1}{2}\right)}{{\rm{\Gamma }}(1){\rm{\Gamma }}(H+1)}{}_{2}{F}_{1}\left(\displaystyle \frac{1}{2},H+\displaystyle \frac{1}{2};\displaystyle \frac{1}{2};\displaystyle \frac{{s}^{2}}{{\lambda }^{2}}\right)\\ \quad +\,\left(\displaystyle \frac{{s}^{2}}{{\lambda }^{2}}\right)\displaystyle \frac{{\rm{\Gamma }}\left(H+\tfrac{3}{2}\right){\rm{\Gamma }}\left(-\tfrac{1}{2}\right)}{{\rm{\Gamma }}\left(\tfrac{1}{2}\right){\rm{\Gamma }}\left(H+\tfrac{1}{2}\right)}{}_{2}{F}_{1}\left(1,H+1;\displaystyle \frac{3}{2};\displaystyle \frac{{s}^{2}}{{\lambda }^{2}}\right)\\ \quad =\,\left(\displaystyle \frac{s}{\lambda }\right)\displaystyle \frac{{\rm{\Gamma }}\left(H+\tfrac{3}{2}\right)\sqrt{\pi }}{{\rm{\Gamma }}(H+1)}\left[\displaystyle \frac{{\rm{\Gamma }}\left(\tfrac{1}{2}\right)}{{\rm{\Gamma }}\left(\tfrac{1}{2}\right){\rm{\Gamma }}\left(H+\tfrac{1}{2}\right)}\displaystyle \sum _{k=0}^{\infty }\displaystyle \frac{{\rm{\Gamma }}\left(k+\tfrac{1}{2}\right){\rm{\Gamma }}\left(k+H+\tfrac{1}{2}\right)}{{\rm{\Gamma }}\left(k+\tfrac{1}{2}\right)}\displaystyle \frac{1}{k!}{\left(\displaystyle \frac{{s}^{2}}{{\lambda }^{2}}\right)}^{k}\right]\\ \quad +\,\left(\displaystyle \frac{{s}^{2}}{{\lambda }^{2}}\right)\left(H+\displaystyle \frac{1}{2}\right)(-2)\left[\displaystyle \frac{{\rm{\Gamma }}\left(\tfrac{3}{2}\right)}{{\rm{\Gamma }}(1){\rm{\Gamma }}(H+1)}\displaystyle \sum _{k=0}^{\infty }\displaystyle \frac{{\rm{\Gamma }}(k+1){\rm{\Gamma }}(k+H+1)}{{\rm{\Gamma }}\left(k+\tfrac{3}{2}\right)}\displaystyle \frac{1}{k!}{\left(\displaystyle \frac{{s}^{2}}{{\lambda }^{2}}\right)}^{k}\right]\\ \quad =\,\displaystyle \frac{s}{\lambda }\displaystyle \frac{\left(H+\tfrac{1}{2}\right)\sqrt{\pi }}{{\rm{\Gamma }}(H+1)}\left[{\rm{\Gamma }}\left(H+\displaystyle \frac{1}{2}\right)+{\rm{\Gamma }}\left(H+\displaystyle \frac{3}{2}\right)\displaystyle \frac{{s}^{2}}{{\lambda }^{2}}\right]\\ \quad -\,\displaystyle \frac{{s}^{2}}{{\lambda }^{2}}\displaystyle \frac{\left(H+\tfrac{1}{2}\right)\sqrt{\pi }}{{\rm{\Gamma }}(H+1)}\left[\displaystyle \frac{{\rm{\Gamma }}(H+1)}{{\rm{\Gamma }}\left(\tfrac{3}{2}\right)}+\displaystyle \frac{{\rm{\Gamma }}(2){\rm{\Gamma }}(H+2)}{{\rm{\Gamma }}\left(\tfrac{3}{2}+1\right)}\displaystyle \frac{{s}^{2}}{{\lambda }^{2}}\right].\end{array}\end{eqnarray} \tag{ C.16 }$$

As a result, the integral in expression (82) is approximated as

$$\begin{eqnarray}&&\begin{array}{l}{\displaystyle \int }_{0}^{\infty }{{\rm{d}}{t}{\rm{e}}}^{-{st}}{t}^{H}{K}_{H}(\lambda t)\sim {s}^{-1}{2}^{H-1}{\lambda }^{-H}\left\{\sqrt{\pi }{\rm{\Gamma }}\left(H+\displaystyle \frac{1}{2}\right)\displaystyle \frac{s}{\lambda }\right.\\ \quad \left.+\sqrt{\pi }{\rm{\Gamma }}\left(H+\displaystyle \frac{3}{2}\right)\displaystyle \frac{{s}^{3}}{{\lambda }^{3}}-2{\rm{\Gamma }}(H+1)\displaystyle \frac{{s}^{2}}{{\lambda }^{2}}-\displaystyle \frac{4}{3}(H+1){\rm{\Gamma }}(H+1)\displaystyle \frac{{s}^{4}}{{\lambda }^{4}}\right\}.\end{array}\end{eqnarray} \tag{ C.17 }$$

Applying these approximations, the resulting expression for the autocorrelation function in the Laplace domain is

$$\begin{eqnarray}\begin{array}{rcl}\tilde{\gamma }(s) & = & \displaystyle \frac{2{\rm{\Gamma }}\left(H+\tfrac{1}{2}\right)}{\sqrt{\pi }{(2\lambda )}^{H}}\left\{\displaystyle \frac{\pi {2}^{H-1}s}{\sin (\pi H){\rm{\Gamma }}(1-H){\lambda }^{H}}-{s}^{2}{s}^{-1}{2}^{H-1}{\lambda }^{-H}\right.\\ & & \times \left[\sqrt{\pi }{\rm{\Gamma }}\left(H+\displaystyle \frac{1}{2}\right)\displaystyle \frac{s}{\lambda }+\sqrt{\pi }{\rm{\Gamma }}\left(H+\displaystyle \frac{3}{2}\right)\displaystyle \frac{{s}^{3}}{{\lambda }^{3}}-2{\rm{\Gamma }}(H+1)\displaystyle \frac{{s}^{2}}{{\lambda }^{2}}\right.\\ & & \left.\left.-\displaystyle \frac{4}{3}(H+1){\rm{\Gamma }}(H+1)\displaystyle \frac{{s}^{4}}{{\lambda }^{4}}\right]\right\}.\end{array}\end{eqnarray} \tag{ C.18 }$$

It can be seen that the dominant term is a linear function of s,

$$\begin{eqnarray}&&\tilde{\gamma }(s)=\displaystyle \frac{2{\rm{\Gamma }}\left(H+\tfrac{1}{2}\right)}{\sqrt{\pi }{(2\lambda )}^{H}}\displaystyle \frac{\pi {2}^{H-1}s}{\sin (\pi H){\rm{\Gamma }}(1-H){\lambda }^{H}}.\end{eqnarray} \tag{ C.19 }$$