Fractionally diffusing passing through the saddle point of metastable potential
The diffusion of a fractional Brownian particle passing over the saddle point is studied in the field of the metastable potential. The barrier escaping probability is found to be greatly related to the fractional exponent . Properties are revealed to move reversely in the opposite direction of diffusion when is relatively large despite of the zero-approximating effective friction of the system. This is very anomalous to the standard Brownian motion.
pacs:05.70.Ce, 05.30.-d, 05.40.Ca
Fractional Brownian motion (fBm) is a quintessential model for the stochastic processes which is characterized by long-memory stationary and Gauss-distributed increments fbm1 (); fbm2 (); fbm3 (). Different from the standard Brownian motion (sBm), fBm is generally believed to originate from the fractional Gaussian noise (fGn) fgn2 (); fgn1 () whose autocorrelation is characterized by the Hurst exponent . For which, corresponds to the standard Brownian motion, and denote the sub- or super-diffusive cases, respectively. Theoretically, no matter fBm or sBm, both of them can be categorized in the Langevin unification of fractional motions fms1 () in the study of anomalous diffusions ad01 (); ad02 (); ad03 ().
Since it offers an alternative model of random processes displaying the Joseph effectjef1 (), fBm and its variants have attracted considerable attention in the past few decades fbm2 (); levy (). Enormous efforts have been made on the establishment and development of relevant theories. However, comprehensively review on the previous studies one can find, little information has been got concerning on the dynamical details of a fBm particle diffusing passing over the saddle point of a matastable potential energy surface (PES). This maybe because of the fractional calculus difficulties immersed in the procedure of solving the kinetic differential equations. But recently we find that this can be easily achieved from another road of mathematical deriving. Therefore in this paper, we report our recent study on this point.
The paper is organized as follows: in Sec.II, kinematic relations of the diffusion particles are obtained by Laplacian solving the fractional Langevin equation. The probability of successfully escaping from the potential well is computed in Sec.III for a characteristic visualization. Sec.IV serves as a short summary of present results where some discussions are also made for a further consideration.
Ii fractional Langevin equation and its solution
where is the fractional Gaussian noise (fGn) which is zero mean with autocorrelation . is the diffusion coefficient identified by the Gamma function .
Although Eq.(1) is the standard presentation of fBm, it is mathematically intractable due to the fractional calculus. But fortunately we find in our recent study that it can be transformed to a more conventional form of fractional GLE by using of the Gamma function and the relation between the external force and the potential energy . Mathematically it can also be derived from a generalized system-plus-reservoir model of harmonic oscillators hotb (). i.e.
and more conveniently
where is the frictional kernel with fractional exponent and the strength constant. is the renewed fGn and the fluctuation-dissipation theorem fdt1 ().
The dominant advantage of Eq. (3) lies in its easy processing. The solution of it, namely also the equation of motion for the diffusing particle, can be easily obtained by a series of Laplacian transformation. For an example, in the particular case of an inverse harmonic potential , we find after some algebra
in which the mean position of the particle along the transport direction is given by
and the variance of reads
where is namely the response function and is the Laplacian transformation of the friction kernel .
Iii diffusion dynamics of fractional damping system
In the studies here and following, we extend the results aforesaid to the more realistic case of a matestable potential to reveal the diffusion dynamics of a fractional damping (FD) system described by Eq. (3) chyx (). This means to set
where denotes the barrier height and indicates the frequency near the potential well as well as the saddle point. Although some new derivations should be made, but noticing that can be approximated to be an inverse harmonic potential in the neighbourhood of the saddle point, this would not be much of a problem.
For the study of diffusion dynamics, one of the basic task is to compute the successful rate of the particle escaping from the potential well. Mathematically, it always emerges to be a complementary error function due to the Gaussian property of noise and linearity of the GLE. i.e.
yielding from a short integration on the reduced distribution function where
is resulted from the joint probability density function (PDF) of the system adelm ()
with the vector and the matrix of second moments
Now everything is ready before tracking the trajectories to find the dynamical details.
In the calculations here and following, we rescale all the quantities so that dimensionless unit such as is used. Firstly in Figs. 1 and 2, we plot the time dependent varying of and (namely also ) at various . From which we can see that increases quickly when is relatively small. But as the increasing of , suddenly decays into a negative diverging function of . The critical point lies in the nearest neighbourhood of in the particular case of what is considered here. This is a very nontrivial result, because from the view point of PDF evolution, indicates the center of the wave packets and the width. Therefore it reveals that the center of the wave packet may move in the opposite direction of diffusion.
However, the value of is always positive increasing. This means, as the forwarding of the center the width the wave packet is always expanding. Therefore, despite of the unusual movement of the center one may always expect a steady barrier escaping probability. As is shown in Fig. 3, the probability for a particle to diffusing passing the saddle point tends to be a steady one (defined as ) in the long time limit no matter what is the value of . The occurrence of such a nontrivial result may probably be caused by the property of long-range correlations (the “Joseph effect”) immersed in the standard process of fBm fbm2 ().
In order to get more details about the fBm, we plot in Fig. 4 the time dependent varying of (the steady value of ) as a function of the fractional exponent . From which we can see that decays monotonously as the increasing of . At first thought, this is considered to be caused by a strong resistance from the friction kernel. However, when we check through the time-dependent varying of , a startling result is found that the effective friction of the system tends to be zero as the increasing of . As is shown in Fig. 5 for more explicitness. But what is the real reason then? Zero damping will lead to low barrier escaping, right?
In order to answer these questions, we make a thorough investigation on the kinematical properties of the system. We find that the mean velocity of the particle can be obtained from a short derivation from , i.e.
And succeeding calculation reveals that tends to be positive when is relatively small, but will turn to be negative at large . As is shown in Fig. 6 for more details. The critical point is identical to what we have found hereinbefore. Therefore we have known the reasons now: for the fBm process with large , the effective friction of the system is very strong at the beginning. Particle inserted in this dissipative environment loses its initial kinetic energy quickly. After a very short period of time, it gets a reversed kinetic energy to move in the opposite direction of diffusion despite of the zero-approximating damping. This results in a low probability of barrier escaping (small ) in the long time.
Iv summary and discussion
In conclusion, we have studied in this paper the diffusion dynamical properties of the fractional damping system which may induce fBm. Several anomalous phenomena are revealed such as the reverse moving of the particles at large , the low at zero-approximating effective friction and so on. All these results have shown us a vivid picture much different from the standard Brownian motion.
Since theoretically fBm may be a model particularly relevant to subcellular transport, unbiased translocation sd26 (); sd27 (), the dispersion of apoferritin proteins in crowded dextran solutions sd11 () and lipid molecules in lipid bilayer sd12 (). Therefore we believe that deep research will be greatly indeed in the near future in some disciplinary field such as sub-diffusing mRNA molecules sd10 (), RNA proteins and chromosomal loci within E. coli cells sd04 (). And the results we have got here will in no doubt motivate the continuous demystification of these seemingly simple subjects.
This work was supported by the Shandong Province Science Foundation for Youths under the Grant No.ZR2011AQ016, the Open Project Program of State Key Laboratory of Theoretical Physics, Institute of Theoretical Physics, Chinese Academy of Sciences, China under the Grant No.Y4KF151CJ1, the National Natural Science Foundation of China under the Grant No.11275259 and 91330113.
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