Asymptotic solutions of a nonlinear diffusive equation in the framework of -generalized statistical mechanics
In the framework of non-equilibrium thermostatistics, irreversible processes can be often described by means of Fokker-Planck equations (FPE’s)  whose time evolutions are characterized by monotonically non-increasing Lyapunov functionals. In a previous work , we derived a non-linear FPE in the picture of a generalized statistical mechanics based on the -entropy , and discussed its relation with the associate Lyapunov functional or Bregman type divergence . Based on the monotonic behavior of the Lyapunov functional, one can show that any initial localized state, which evolves according to the -generalized FPE, converges to a stationary solution which minimizes the Lyapunov functional. In addition, in the linear drift case, the corresponding stationary solution is nothing but the -Gaussian, which is a generalization of Gaussian by replacing the standard exponential with its -generalized version .
As well known, the diffusive equations play a fundamental rôle in the description of several physical phenomena. In particular, the linear diffusive equation arising in the Boltzmann-Gibbs (BG) statistical mechanics has been widely studied jointly with its Gaussian self-similar solution . Similarly, the -generalized diffusive equation arises, in a natural way, in the framework of the Tsallis statistical physics. This is equivalent to the porous medium equation (PME), which is a transport equation of gases or fluids in a porous medium. The main properties of PME are now well known  and its solutions have been widely investigated in literature , in particular its self-similar solution is given by the so-called Barenblatt solution .
In this work we study a different nonlinear diffusive equation obtained in the picture of the -generalized statistical mechanics. As known, a useful method to obtain the solutions of a partial differential equation is based on the study of its Lie symmetries and the related group invariant solutions . We accomplished such analysis for the -diffusive equation considering only the classical Lie symmetries whose generators are functions of the independent and the dependent variables. We will leave the study of the generalized Lie symmetries, whose generators are functions also of the derivatives of the depending variables, to a future work.
In particular, we show that the only group-invariant solutions for the -diffusive equation are the kink-like solutions (a non-normalizable family of self-similar solutions) and the traveling wave solutions (physically irrelevant because they are divergent). Next, we explore numerically the evolutions of a localized initial state, whose spreading is governed by the -diffusive equation. It is shown that, independently from the initial states, the numerical solutions approach to a shape which is asymptotically well fitted by the -Gaussian distribution, although this one is not a group invariant solution of the equation under investigation.
The paper is organized as follows. In the next section 2 we briefly review the -generalized thermostatistics and its associated non-linear FPE. In section 3, after presenting an alternative derivation of the -diffusive equation, we classify its classical Lie symmetries and their related group invariant solutions. Section 4 deals with the numerical analysis and the final section 5 is devoted to summary.
We briefly review the generalized thermostatistics based on the -entropy  given by
where is the -logarithm defined us
The -entropy is an extension of the BG entropy by means of a real parameter .
The inverse function of , namely , given by
is called -exponential.
The -logarithm and -exponential functions are the building blocks of the statistical mechanics based on the -entropy. In particular, the -exponential is a monotonic increasing and convex function, fulfilling the relationship . Moreover, , and , as the ordinary exponential function does. For , the -exponential is well approximated by the standard exponential, whereas for , it asymptotically approaches to a power-law: . In the limit, both and reduce to the standard logarithm and exponential functions, respectively. Accordingly, the -entropy reduces to the BG entropy.
Maximizing under the constraints of the linear kinetic energy average and the normalization
leads to the -Gaussian function
Here is the constant fixing the normalization of the distribution and depends on the Lagrange multiplier which controls the dispersion of the distribution. Finally, the parameters and are -dependent constants given by
respectively, and are related each to the other through the relation
The -entropy and the corresponding statistical mechanics preserve several properties of the BG theory . In particular, the Legendre structure of the thermo-statistics theory has been investigated in  through the introduction of several thermodynamic potentials.
For instance, it is found that the generalized partition function, as a function of , given by
satisfies the Legendre relation
and that the -entropy and the generalized partition function are related through the relationship
The function in equation (Equation 4) is defined by
where the quantity , defined as
fulfills the following proprieties and . In the limit, it reduces to the unity: .
Both functions and play an important rôle in the development of a theory based on the -entropy.
Starting from the definition of the partition function , we can introduce a -generalization of the free-energy, according to
In this way, it was shown that the -free-energy satisfies the Legendre transformation structures  summarized by the following relationships
2.1-generalized Fokker-Planck equation
In a previous work , we have studied the nonlinear FPE associated with the -entropy. It can be introduced as a continuity equation for the density field
with the current density
As shown in  the quantity is a -generalized Lyapunov functional, which is monotonically non-increasing in time.
The -generalized FPE reads
where is a constant diffusion coefficient.
We remark that the current density is the sum of a linear drift current , which describe the standard Uhlenbeck-Ornstein process and a nonlinear diffusive current, given by , which reduces to the standard Fick’s current in the limit. Consequently, in the same limit, equation (Equation 5) reduces to the standard linear FPE
Since , the functional is minimized for the stationary solutions of equation (Equation 5) according to
It is easy to verify that the stationary solution of the -generalized FPE is equal to the optimal maximal entropy distribution , i.e.
with the position .
Without the external potential the drift therm disappear and equation (Equation 5) reduces to a purely -diffusive equation whose explicit form is given by
Equation (Equation 7) is the main subject of our investigation.
3-generalized diffusive equation
3.1A simple physical derivation
First, let us describe a physical derivation of the -diffusive equation (Equation 7). In this respect, we recall that the linear diffusive equation
can be obtained from the following physical relations:
where denotes the average velocity of the fluid and denotes the pressure of the fluid, which is related to the density of the fluid through a suitable equation of state , with .
The continuity equation assures the conservation of the zero-th moment
which corresponds to the total mass of the fluid.
The PME  can be derived from the same physical relations by posing, in the equation of state, :
Here, is a positive real parameter characterizing the degree of porosity of the medium and characterizing also the power-law dependence of polytropic fluid. In other words, is a polytropic index.
The corresponding PME is given by
Interestingly, the same settings can be also applied to the -diffusive equation (Equation 7), merely replacing, in the equation of state, the logarithm with its -generalization: .
In this way, the -diffusive equation (Equation 7) can be derived according to
which describe a special type of barotropic fluid . “A special type” means as follows. From the definition of -logarithm , the last relation can be written as
i.e., a pressure of this type of barotropic fluid depends on the difference between one term with a polytropic index and the other term with .
3.2Classical Lie symmetry analysis
In the following we classify the Lie symmetries of the -diffusive equation.
For the sake of comparison, we first consider the PME . In this case, the symmetry generators are given by (please refer to  for more details)
This means that if is a solution of PME, then
are also solutions.
Generators (Equation 9) and ( ?) describe translations invariance in - and -coordinates, respectively and their group invariant solutions (GIS) are merely the constant functions.
Generators ( ?) and ( ?) describe scaling invariance of independent and dependent variables, respectively. In particular, the GIS for the symmetry has a kink-like shape and in the limit, it reduces to the Erf function.
Further, GIS can be obtained by means of linear combination of the vectors . For instance, from the generator we obtain the traveling wave solutions which are physically irrelevant because they diverge. Differently, the most general self-similar generator is provided by vector . Although the corresponding GIS can not be given in terms of elementary functions, explicit solutions can be obtained for special values of the constant . In particular, for we derive the Barenblatt scale invariant solution 
where is the normalization constant. Note that this solution is nothing but a -Gaussian with .
Next, we study the symmetries for the more complicated -diffusive equation. It is worth noting that the PME is characterized by the single-diffusive term , which leads to the GIS , while the -diffusive equation contains two different powers and . This double nonlinearity in the diffusive term destroys the scaling invariance for the dependent variable .
Thus, the only Lie symmetry generators are
so that, if is a solution of the -generalized diffusive equation then
are also solutions.
Again, the GIS corresponding to the traveling wave generator are physically irrelevant because merely constants or divergent. Differently, the only self-similar generator is and the corresponding GIS has a kink-like shape as shown in figure ?. These solutions, although not normalizable, can describe the evolution of an order parameter belonging to the system. However, as compared to the case of PME, there is no scaling generator like equation and consequently, the -Gaussian is not an exact solution for the -diffusive equation.
To further study the long time behavior of the -diffusion, we performed numerical simulations against various initial probability distribution functions with different shapes.
We used a numerical method developed in  in order to study the approach to the equilibrium of a given Cauchy problem for the one-dimensional generalized diffusive equation
where is an initial localized state and . For computational simplicity, the diffusion coefficient is re-scaled in the independent variables according to the symmetry ( ?), which still holds whatever the function is.
In general, a solution of any diffusive equation spreads out as time increases. Then, simple conventional methods of discretizations in the -coordinate, which are often employed in the numerical simulations, leads to much computational costs, especially for a long time numerical simulation.
In order to overcome the difficulty due especially to the expanding support of , Gosse and Toscani  developed an explicit numerical scheme applicable to one-dimensional differential equations of the type given in .
We here explain the basic points of this numerical scheme. First, we introduce the cumulative distribution function
associated with the probability distribution and its inverse relation
By using equation and integrating by part we obtain
Next, since is non-decreasing in the variable, one can define its pseudo-inverse function for a given value as
Then, we have
Note that does not depend on time . In other words, at each time there exists only one point of whose cumulative distribution function equals to . Therefore, differentiating both sides of equation (Equation 13) w.r.t. we obtain
In the case of we derive the time-evolution equation for as
In short, we compute a time-evolution of the pseudo-inverse function for a given initial condition . Since the computational domain is now restricted we can go around the above expanding support issue for any large time.
For the -diffusion equation, we set
The top plot in figure ? shows a typical time-evolution of with an initial state concentrated around .
The bottom plot in the same figure ? shows the time-evolution for an initial state with double peaks. Note that the double peaks immediately disappear as time evolves and become a single peaked shape, which seems to be asymptotically approaching to the -Gaussian.
We have run several simulations with different initial shapes to confirm this asymptotic behavior.
A further indication on this asymptotic behavior can be obtained by studying the time evolution of the function
which is the logarithmic ratio between the numerical solution of equation (Equation 7) and the trial fitting function
This is a -Gaussian where and are the best fitted parameters for the numerical solution at each time .
In figure ?, we plotted the time evolution of for an initial with a triangle shape. It is clear from this picture that as , i.e. the function (Equation 15) gradually decreases to zero as time evolves, which gives a strong evidence that the numerical solution is asymptotically approaching to the -Gaussian function.
4.2Time dependent transformation
In order to further confirm these asymptotic behaviors let us consider a time-dependent transformation for the -generalized FPE , defined by
that is the -diffusive equation (Equation 7) with the source term .
In the large time limit, the source term becomes negligible and equation (Equation 18) can be well approximated by equation (Equation 7). On the other hand, we recall that the solution of -generalized FPE approaches asymptotically to the stationary state which is the -Gaussian . Therefore, this solution is transformed, by means of equation (Equation 17), into another one that well approximates the solution of the -diffusive equation. In other words, this solution asymptotically approaches to a -Gaussian function.
We have studied the asymptotic behaviors of the -diffusive equation, which is naturally related with the -generalized statistical physics and the -FPE . The analysis based on the standard Lie symmetry showed that the -Gaussian function is not a scale invariant solution of the -diffusion equation. Notwithstanding, the -Gaussian plays a relevant rôle in the study of the time evolution of localized perturbations. In fact, by performing several numerical simulations, with different initial shapes, we have found strong evidence that these solutions, for large time, are well approximated by the -Gaussian. Arguments based on a time-dependent transformation performed on the -generalized FPE also support this result.
Further confirmations of this behavior could be obtained by studying whether a -Gaussian is invariant under asymptotic symmetries  of the -diffusive equation.
This research was partially supported by the Ministry of Education, Science, Sports and Culture (MEXT), Japan, Grant-in-Aid for Scientific Research (C), 19540391, 2008.
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