1. Introduction
The Boltzmann-Gibbs (BG) theory of statistical mechanics represents one of the most successful theoretical frameworks of physics [
1,
2]. The proposal of different types of statistical ensembles, and their equivalence in the thermodynamic limit, makes it appropriate for a description of a wide variety of many-particle physical situations. These statistical ensembles can be defined in a very elegant manner, by the introduction of a probability density
, associated with the occurrence of a given physical quantity
at a time
t. Particularly, in the present paper,
will denote the position of a particle in a
N-dimensional space; then, one can define the BG entropy,
where
is the Boltzmann constant, and the integral
will represent herein an integration over all possible positions of the particle [
3]. By maximizing
with respect to
under certain constraints, one obtains the equilibrium distribution
associated to the different statistical ensembles (see, e.g., [
1]). The BG theory is based on a fundamental assumption, namely, the ergodic hypothesis; only if ergodicity holds is that one can replace a given time average by the corresponding average over a statistical ensemble. Additionally, the fact that all thermodynamic quantities are either intensive or extensive is directly related to the short-range character of the interactions among the microscopic constituents of the system. Considering the constraint
the maximization of
leads to the equilibrium microcanonical distribution. The same procedure carried by imposing Equation (
3), as well as an additional constraint for the total energy,
yields the usual Boltzmann weight of the equilibrium canonical distribution.
However, the applicability of the BG theory becomes questionable for systems that violate the ergodic hypothesis, which may occur in systems characterized by long-range and/or competing interactions. The nonextensive statistical mechanics [
4,
5] appears nowadays as a possible alternative for describing physical situations where the BG theory fails. The former emerged from the introduction of the entropic form [
6],
where
k represents a constant with units of entropy, and the entropic index
q is responsible for deviations from BG; notice that
, when
.
The maximization of
under the constraints in Equations (
3) and (
4) yields the stationary distribution [
4],
where
, for
and zero otherwise. As usual,
represents a normalization factor, whereas
is the Lagrange multiplier associated with the energy constraint [Equation (
4)]. However, if one uses Equation (
3) and instead of Equation (
4), one considers a generalized definition for the internal energy [
4,
7],
one obtains the Tsallis distribution,
One should notice that the distributions in Equations (
6) and (
8) present a “duality” property,
, which appears frequently in nonextensive statistical mechanics. A third proposal was introduced in terms of the escort distribution [
8],
yielding a stationary distribution similar to the one in Equation (
8), with both
q and the corresponding Lagrange multiplier
redefined in terms of the quantities appearing in Equation (
8) [
4,
7,
8,
9]. Therefore, the three proposals above for the extremization of
are equivalent to one another, by a proper redefinition of its parameters.
The linear differential equations in physics are, in many cases, valid for media characterized by specific conditions, like homogeneity, isotropy, and translational invariance, with particles interacting through short-range forces and with a dynamical behavior characterized by short-time memories. One of the most important equations of non-equilibrium statistical mechanics is the linear Fokker-Planck equation (FPE); its time-dependent solution, for an external harmonic potential, is given by a Gaussian distribution [
2]. However, many real systems—specially the ones within the realm of complex systems—do not fulfill these requirements, e.g., those characterized by spatial disorder and/or long-range interactions. In some of these cases, the associated equations have to be modified, and very frequently, nonlinear terms are considered in order to take into account such effects. Nowadays the study of nonlinear differential equations has gained a lot of interest and a considerable advance in this area has been attained essentially due to latest advances in computer technology.
In the recent years the linear FPE has been modified in a way to introduce nonlinear terms, so that the study of nonlinear Fokker-Planck equations (NLFPEs) [
10] became an area of a wide interest, particularly due to their applications in many physical phenomena, like those related to anomalous diffusion [
11]. These type of phenomena may be found in the motion of particles in porous media [
12,
13,
14,
15], the dynamics of surface growth [
15], the diffusion of polymer-like breakable micelles [
16], the dynamics of interacting vortices in disordered superconductors [
17,
18], and the motion of overdamped particles through narrow channels [
19], among others. An interesting aspect about the Tsallis distribution is that it appears also as a a stable solution of a NLFPE; in its simplest, one-dimensional form, this equation is given by [
20,
21,
22]
where the external force
is associated with a potential
[
] and
μ is a real number [notice that Equation (
10) recovers the linear FPE in the limit
]. The stationary solution of Equation (
10), for an arbitrary confining potential
, is given by Equation (
6) for
, or by Equation (
8) for
[considering in both cases,
] [
20]. However, for an external force
(
and
constants,
), and the initial condition
, the time-dependent solution of Equation (
10) is given by a
q-Gaussian distribution [
20,
21],
in the case
. In the equation above,
is related to the average value of
, following the same behavior as for
[
21], whereas for the preservation of the norm at all times, the time dependent quantities
and
should obey,
For
, considering times much smaller than those necessary for an approach to the stationary state, one gets a time evolution completely dominated by the diffusion term, which gives
, leading to an anomalous diffusion for any
[
23]. In the limit
one approaches a stable stationary state characterized by
, in which the time-dependent parameters of Equation (
12) take their corresponding stationary values,
and
.
A typical general NLFPE can be written in the following form [
10,
23,
24,
25,
26,
27,
28,
29,
30,
31],
which recovers Equation (
10) if
and
. In Equation (
13) the functionals
and
should satisfy certain mathematical requirements, like positiveness, integrability, and differentiability (at least once) with respect to the probability distribution
[
29,
30]. Moreover, in order to preserve the probability normalization for all times, one should impose the probability distribution, together with its first derivative, as well as the product
, to be zero in the limit
,
The H-theorem represents one of the most important results of nonequilibrium statistical mechanics, providing a well-defined sign for the time-derivative of the free-energy (or the entropy), allowing for an approach to an equilibrium state. In BG statistical mechanics one may prove the H-theorem by making use of the linear FPE [
2]; generalizations of this procedure for NLFPEs have called the attention of many workers in the recent years [
10,
23,
25,
26,
27,
28,
29,
30,
32,
33]. The H-theorem in the case of a system subject to an external potential
corresponds to a well-defined sign for the time derivative of the free-energy functional,
where
γ represents a positive Lagrange multiplier. The entropy may be considered in the general form,
with the condition that
should be at least twice differentiable. One should remind that the entropy may be further formulated in more general forms than the definition above, e.g., by means of a functional of
[
29], or even taking into account the possibility of a nonlocal diffusion coefficient, as done in [
33]; however, since herein our main purpose is to discuss properties related to NLFPEs associated with Tsallis entropy, we will restrict ourselves to Equation (
16).
Imposing a well-defined sign for the time derivative of the free energy (which was taken as
in [
23,
29,
30]), and making use of the NLFPE of Equation (
13), one gets the relation
involving the functionals
and
of the FPE and the entropy defined in Equation (
16). Examples of some entropic forms known in the literature, and their associated FPEs were worked out in [
10,
24,
29,
30,
31].
As a simple illustration, one can see easily that the NLFPE of Equation (
10) in the case
, for which
and
, is related to Tsallis entropy. Substituting these quantities in Equation (
17), integrating and using the conditions of Equation (
16), one obtains [
23,
29],
where
k is defined through
in the present formalism. Then, Tsallis entropy is recovered by
.
The one-dimensional NLFPE of Equation (
10) has been generalized to
N-dimensions [
34,
35],
with the solution of Equation (
11) turning into a
N-dimensional
q-Gaussian distribution,
for
and the external force given by
(
). One should mention that for the
N-dimensional
q-Gaussian distribution above there are two typical values of
q which depend on
N, namely, the value of
q below which the distribution is normalizable,
and the one at which the second moment diverges,
[
36].
In this paper we extend some well known results of one-dimensional NLFPEs to
N dimensions, as described below. In Section II we derive
N-dimensional NLFPEs from a master equation, following the same lines of [
37,
38]. In Section III we prove the H-theorem by making use of
N-dimensional NLFPEs, deriving relations involving terms of the corresponding NLFPE and the entropic form, using both standard [
cf. Equation (
4)], and generalized definitions for the internal energy. In this later case, it is shown that the corresponding NLFPEs have to be modified accordingly. In Section IV we give an special emphasis the class of NLFPEs associated with Tsallis entropy and in particular, to those modified due to generalized definitions of the internal energy. Finally, in Section V we present our conclusions.
2. From Master Equation to Fokker-Planck Equation
The one-dimensional NLFPEs in Equation (
10) and Equation (
13) may be derived directly from a master equation, by introducing nonlinear terms in the corresponding transition probabilities, as done in [
29,
32,
37,
38,
39]. Herein we follow closely the procedure used in [
29,
37,
38] in order to derive the
N-dimensional FPE of Equation (
19).
Let us then consider a system described in terms of discrete
N-dimensional stochastic variables, for which
represents the probability for finding it in a state characterized by the variable
at time
t. The corresponding master equation is given by
where
and
represents the transition probability rate from state
to state
(
i.e.,
is the probability for a transition from state
to state
to occur during the time interval
). For a random walk in an isotropic space,
i.e., the same step size Δ for all directions, one can write the master equation above as
In [
37] the following ansatz for the transition rate was introduced,
where
a and
b are constants that may depend, in principle, on the system under consideration, and
represents an external force. The nonlinear contributions,
and
, correspond to dependences on the probabilities associated to the outgoing and ingoing states, respectively; the motivations for these terms were already presented in [
37]. One should remind that this transition rate recovers the one in the usual derivation of the linear FPE [
2], either for (
), or (
), where
D represents the diffusion constant.
In principle, one may modify the ansatz above to a very general form, as done in [
29]; however, herein we are mostly interested in
N-dimensional NLFPEs associated with Tsallis entropy [like the one of Equation (
19)], and so, we shall introduce a slight modification in the force term of Equation (
23),
where
represents a functional of the probability associated to the outgoing state. The two-dimensional extension of Equation (
24) is given by,
whereas its
N-dimensional form may be written as,
Substituting this transition rate in Equation (
22) one gets, after carrying out the summations over the states
,
where
. Defining
,
, and considering the limit
, one obtains,
or yet,
One should notice that the equation above corresponds to the
N-dimensional generalization of Equation (2.4) in [
37]. Now, following a procedure similar to the one of [
38], this equation may be rewritten as,
As in the one-dimensional case, the two functionals
and
should be both positive finite quantities, integrable, as well as differentiable (at least once) with respect to
; additionally,
should also be monotonically increasing with respect to
. Considering
,
i.e.,
, the equations above correspond to the
N-dimensional generalization of Equations (1.5) in [
38]; also, they recover Equation (
19) of the present paper in the particular cases, (
), (
), (
), for which the
N-dimensional
q-Gaussian distribution of Equation (
20) is a solution.
4. The Family of NLFPEs Associated to Tsallis Entropy
In this section we shall restrict ourselves to those NLFPEs associated with Tsallis entropy. As shown above, these connections are provided by the H-theorem, through relations involving quantities of the NLFPEs and entropic forms. In the case of Equation (
30), the relevant relation is given in Equation (
40), whereas for Equation (
44), the relations are Equation (
40) and Equation (
48). Hence, considering the functional
of Equation (
18) in Equation (
40), one gets,
which, by choosing
leads to the form of NLFPE in Equation (
19), with
,
and a diffusion constant
. This corresponds to the simplest
N-dimensional NLFPE associated with Tsallis entropy, whose time-dependent solution is given by the
q-Gaussian distribution of Equation (
20) [
34]. However, it is possible to define a whole class of FPEs satisfying Equation (
65), as discussed in [
23,
29], essentially by introducing functionals
and
such that
and
. This class of FPEs satisfy the H-theorem for the standard definition of the internal energy in Equation (
33).
A similar analysis follows for those NLFPEs associated with generalized energy definitions. In particular, considering the choice
, for which
, the NLFPE of Equation (
44) turns into
which becomes similar to Equation (
30) if one uses the time-dependent “potential” of Equation (
43). By defining conveniently the functionals
and
, one introduces the following family of NLFPEs,
which, by imposing the functional
of Equation (
18), becomes directly associated with Tsallis entropy,
for an arbitrary functional
restricted to the conditions already imposed for the functionals
and
.
Although finding a time-dependent solution of Equation (
67) may be a hard task, its equilibrium solution can be found easily by using
and
in Equation (
57),
Now, solving for
,
where
is a normalization factor. Notice that the equilibrium solution above applies for any confining potential
; particularly, for an external force
(
) one gets the same
N-dimensional
q-Gaussian solution of Equation (
19), calculated in [
34], which is valid for all times. Therefore, we have shown that Equation (
67), which defines a class of NLFPEs directly related to Tsallis entropy and a generalized definition for the energy, may present complicated time-dependent solutions, although in the long-time limit these solutions should approach the above equilibrium solution, given in terms of a
q-exponential for any confining potential.
Let us now proceed inversely,
i.e., given the solution of Equation (
70) we shall obtain relation Equation (
68) at equilibrium. One has that,
which shows that
, for
. In addition to this,
Substituting Equation (
71) into Equation (
72), one gets,
and using the equilibrium solution of Equation (
70) one sees that the potential-dependent term inside braces on the r.h.s. disappears. Now substituting this result in Equation (
53), one recovers Equation (
68) at equilibrium,
showing the association of the equilibrium solution of Equation (
67) with Tsallis entropy.