On the Fractal Langevin Equation

In this paper, fractal stochastic Langevin equations are suggested, providing a mathematical model for random walks on the middle-τ Cantor set. The fractal mean square displacement of different random walks on the middle-τ Cantor set are presented. Fractal under-damped and over-damped Langevin equations, fractal scaled Brownian motion, and ultra-slow fractal scaled Brownian motion are suggested and the corresponding fractal mean square displacements are obtained. The results are plotted to show the details.

In a seminal paper, generalized standard calculus is formulated to define derivatives and integrals on totally disconnected fractal sets and fractal curves [20][21][22][23].Recently, an extension of fractal calculus for the fractals embedding in 2D is formulated [24].
Mean square displacements of random walks having power law are modeled utilizing F α -calculus to provide applications in statistical mechanics [23,25].The over-damped Langevin equation is investigated, which describes dynamics of Brownian particles in the long time limit.The anomalous diffusion of particles in free cooling granular gases is modeled in [26].
In this paper, we suggest fractal under-damped and over-damped Langevin equations, fractal scaled Brownian motion, and ultra-slow fractal scaled Brownian motion.Using stochastic fractal differential equations, the fractal mean square displacement is derived, which leads to a new hierarchy of random walks.
The outline of the paper is as follows: In Section 2, we review basic tools.We define the fractal Langevin equation with different coefficients and work out the mean square displacement for under-damped and over-damped Langevin equations in Section 3. In Section 4, we present fractal ultra-slow and scaled Brownian motion and their fractal mean displacements.Finally, we conclude our results in Section 5.

Basic Tools
In this section, we give a short review of local generalized Riemman calculus on fractal middle-τ Cantor set.

Middle-τ Cantor Set
The middle-τ Cantor set created by following stages: (II) Remove disjoint open intervals of length τ from the remaining sections of step I.
(III) Pick up disjoint open intervals of length τ from the remaining sections of previous step, and so on ad infinitum.
The Lebesgue measures of middle-τ Cantor sets are zero and their Hausdorff dimensions are given by where H(C τ ) is the Hausdorff measure [27].

Local Fractal Calculus
If C τ is middle-τ Cantor set, then the flag function is defined by [20,21,23], where [20,21,23] by where [20,21,23] by where infimum is taken over all subdivisions The integral staircase function S α C τ (t) is defined in [20,21] by where d 0 is an arbitrary and fixed real number.The γ-dimension of a set The C τ -limit of a function g : C τ → is given by If l exists, then we have The C τ -continuity of a function g : C τ → is defined by The C τ -derivative of f (t if the limit exists.
In Figure 1 is defined in [20,21,23] and approximately given by For more details, we refer the reader to [20,21].
The characteristic function of the middle-τ Cantor set is defined in [23] by The delta function on middle-τ Cantor set, which is called fractal Gaussian noise, is defined by and ).

Fractal Langevin Equation with Different Coefficients
In this section, we study over-damped and under-damped Langevin equations.

Fractal Over-Damped Langevin Equation
Consider over-damped fractal Langevin equation where The fractal mean square displacement (FMSD) of random walk corresponding to Equation ( 15) is given by where α and γ are fractal space and time dimensions, respectively.Using upper bound of staircase function, namely By substituting Equation (18) into Equation ( 17), we obtain In Figure 2, we plot Equation (19), in which the red, blue, and green lines are to super-, normaland sub-diffusion, respectively.

Fractal Under-Damped Langevin Equation
Let us consider the fractal under-damped Langevin equation as follows where γ 0 (s −γ/α ) and D 0 = T/mγ 0 are called fractal friction coefficient and fractal diffusion constant, respectively [26].Let D γ K,t x(t) = v K (t), then, by Equation ( 20), we obtain Using Equation ( 21), we get which is named FMSD of the fractal under-damped Langevin equation.Utilizing upper bound of S γ K (t) < t γ , we obtain Replacing the short time t 1 γ 0 into Equation ( 22), we obtain

Conclusions
In this work, we have studied fractal scaled Brownian motion, the fractal under-damped Langevin equation, and the fractal over-damped Langevin equation.The stochastic Langevin equations with different diffusion coefficients are considered to give different fractal mean square displacements.The results obtained in this manuscript are generalizations of the known results for the ordinary Langevin equation and scaled Brownian motion.Moreover, we obtain different conditions that are related to the dimensions of space and time.