A General Nonlinear Fokker-Planck Equation and its Associated Entropy
A General Nonlinear Fokker-Planck Equation and its Associated
Entropy
Veit Schwämmle, Evaldo M. F. Curado, Fernando D. Nobre
Centro Brasileiro de Pesquisas F́ısicas,
Rua Xavier Sigaud 150, Rio de Janeiro, RJ 22290-180, Brazil
(Dated: October 27, 2018)
Abstract
A recently introduced nonlinear Fokker-Planck equation, derived directly from a master equa-
tion, comes out as a very general tool to describe phenomenologically systems presenting complex
behavior, like anomalous diffusion, in the presence of external forces. Such an equation is char-
acterized by a nonlinear diffusion term that may present, in general, two distinct powers of the
probability distribution. Herein, we calculate the stationary-state distributions of this equation
in some special cases, and introduce associated classes of generalized entropies in order to satisfy
the H-theorem. Within this approach, the parameters associated with the transition rates of the
original master-equation are related to such generalized entropies, and are shown to obey some
restrictions. Some particular cases are discussed.
Keywords: Nonlinear Fokker-Planck Equation, Generalized Entropies, H-Theorem, Nonextensive
Thermostatistics.
PACS numbers: 05.40.Fb, 05.20.-y, 05.40.Jc, 66.10.Cb
http://arxiv.org/abs/0704.0465v1
I. INTRODUCTION
The standard statistical-mechanics formalism, as proposed originally by Boltzmann and
Gibbs (BG), is considered as one of the most successful theories of physics, and it has en-
abled physicists to propose theoretical models in order to derive thermodynamical properties
for real systems, by approaching the problem from the microscopic scale. Such a prescrip-
tion has led to an adequate description of a large diversity of physical systems, essentially
those represented by linear equations and characterized by short-range interactions and/or
short-time memories. Although BG statistical mechanics is well formulated (under certain
restrictions) for systems at equilibrium, the same is not true for out-of-equilibrium systems,
in such a way that most of this theory applies only near equilibrium [1, 2, 3]. One of the
most important phenomenological equations of nonequilibrium statistical mechanics is the
linear Fokker-Planck equation (FPE), that rules the time evolution of the probability distri-
bution associated with a given physical system, in the presence of an external force field [4],
provided that the states of the system can be expressed by a continuum. This equation
deals satisfactorily with many physical situations, e.g., those associated with normal diffu-
sion, and is essentially associated with the BG formalism, in the sense that the Boltzmann
distribution, which is usually obtained through the maximization of the BG entropy under
certain constraints (the so-called MaxEnt principle), also appears as the stationary solution
of the linear FPE [4, 5].
Nevertheless, restrictions to the applicability of the BG statistical-mechanics formalism
have been found in many systems, including for instance, those characterized by nonlinear-
ities, long-range interactions and/or long-time memories, which may present several types
of anomalous behavior, e.g., stationary states far from equilibrium [6, 7, 8]. These anoma-
lous behaviors suggest that a more general theory is required; as a consequence of that,
many attempts have been made, essentially by proposing generalizations of the BG entropy
[9, 10, 11, 12, 13, 14, 15, 16, 17, 18]. Among these, the most successful proposal, so far,
appears to be the one suggested by Tsallis [9], through the introduction of a generalized
entropy, characterized by an index q, in such a way that the BG entropy is recovered in the
limit q → 1. The usual extensivity property of some thermodynamic quantities holds only
for q = 1, and if q 6= 1 such quantities do not increase linearly with the size of the system;
this has led to the so-called nonextensive statistical-mechanics formalism [6, 7, 8].
Among many systems that present unusual behavior, one should mention those char-
acterized by anomalous diffusion, e.g., particle transport in disordered media. A possible
alternative for describing anomalous-transport processes consists in introducing modifica-
tions in the standard FPE. Within the most common procedure, one considers nonlinear
FPEs [19], that in most of the cases come out as simple phenomenological generalizations
of the usual linear FPE [20, 21, 22, 23, 24, 25, 26, 27, 28, 29]. In these nonlinear systems,
interesting new aspects appear, leading to a wide range of open problems, in such a way that
their investigation has led to a new research area in physics, with a lot of applications in
natural systems. It is very common to find the power-law-like probability distribution that
maximizes Tsallis’ entropy as solutions of some nonlinear FPEs [19, 20, 21, 23, 24, 30]. It
seems that the nonextensive statistical mechanics formalism appears to be intimately related
to nonlinear FPEs, motivating an investigation for a better understanding of possible con-
nections between generalized entropies and nonlinear FPEs [19, 20, 21, 25, 29, 31, 32, 33, 34].
Recently, a general nonlinear FPE has been derived directly from a standard master
equation, by introducing nonlinear contributions in the associated transition probabilities,
leading to [35, 36]
∂P (x, t)
∂(A(x)P (x, t))
Ω[P (x, t)]
∂P (x, t)
; Ω[P ] = aµP µ−1+b(2−ν)P ν−1 ,
where a and b are constants, whereas µ and ν are real exponents [38]. The system is in
the presence of an external potential φ(x), associated with a dimensionless force A(x) =
−dφ(x)/dx [φ(x) = −
A(x′)dx′]; herein, we assume analyticity of the potential φ(x) and
integrability of the force A(x) in all space.
In what concerns the functional Ω[P (x, t)], we are assuming its differentiability and in-
tegrability with respect to the probability distribution P (x, t), in such a way that at least
its first derivative exists, i.e., that it should be at least Ω[P ] ∈ C1. Furthermore, this func-
tional should be a positive finite quantity, as expected for a proper diffusion-like term; this
property will be verified later on, as a direct consequence of the H-theorem.
As usual, we assume that the probability distribution, together with its first derivative,
as well as the product A(x)P (x, t), should all be zero at infinity,
P (x, t)|x→±∞ = 0 ;
∂P (x, t)
= 0 ; A(x)P (x, t)|x→±∞ = 0 , (∀t) . (2)
The conditions above guarantee the preservation of the normalization for the probability
distribution, i.e., if for a given time t0 one has that
dx P (x, t0) = 1, then a simple
integration of Eq. (1) with respect to the variable x yields,
dx P (x, t) = − [A(x)P (x, t)]
Ω[P (x, t)]
∂P (x, t)
= 0 , (3)
and so,
dx P (x, t) =
dx P (x, t0) = 1 (∀t) . (4)
In the present work we investigate further properties of the nonlinear FPE of Eq. (1), finding
stationary solutions in several particular cases, and discussing its associated entropies, that
were introduced in order to satisfy the H-theorem. In the next section we present stationary
solutions of this equation; in section III we prove the H-theorem by using Eq. (1), and show
that the validity of this theorem can be directly related to the definition of a general entropic
form associated with this nonlinear FPE. In section IV we discuss particular cases of this
general entropic form and their associated nonlinear FPEs. Finally, in section V we present
our conclusions.
II. STATIONARY STATE
The nonlinear FPE of Eq. (1) is very general and covers several particular cases, e.g., the
one related to Tsallis’ thermostatistics [20, 21, 30]. In this section we will restrict ourselves
to a stationary state, and will derive the corresponding solutions for particular values of the
parameters associated with this equation.
Let us then rewrite Eq. (1) in the form of a continuity equation,
∂P (x, t)
∂j(x, t)
= 0 ; j(x, t) = A(x)P (x, t)− Ω[P (x, t)]
∂P (x, t)
, (5)
in such a way that a stationary solution of Eq. (1), Pst(x), is associated with a stationary
probability flux, jst(x) = constant, which becomes jst(x) = 0, when one uses Eq. (2).
Therefore, using the functional Ω[P ] of Eq. (1), the stationary-state solution satisfies,
A(x) =
st (x) + b(2 − ν)P
st (x)
] ∂Pst(x)
, (6)
which, after integration, becomes
φ0 − φ(x) = a
st (x) + b
2 − ν
ν − 1
P ν−1st (x) , (7)
where φ0 represents a constant. The equation above may be solved easily in some particular
cases, e.g., ν = µ, ν = 2, and µ = 0,
Pst(x) =
; Z(1) =
, (8)
where α = µ, in the cases ν = µ and ν = 2, whereas α = ν, if µ = 0. In the equation above,
[y]+ = y, for y > 0, and zero otherwise. Another type of solution applies for µ = 2ν − 1, or
ν = 2µ− 1,
Pst(x) =
1 +K (φ(x)− φ0)
; Z(2) =
1 +K (φ(x)− φ0)
where
(2ν − 1)(ν − 1)
(2− ν)2
, if µ = 2ν − 1 (α = ν) ,
(3− 2µ)(µ− 1)
, if ν = 2µ− 1 (α = µ) ,
and we are assuming that [1 + K(φ(x) − φ0)] ≥ 0. Some well-known particular cases of
the stationary solutions presented above come out easily, e.g., from Eq. (8) one obtains, in
all three situations that yielded this equation, the exponential solution associated with the
linear FPE in the limit α → 1 [with φ0 ∝ (α− 1)
−1], as well as the generalized exponential
solution related to Tsallis thermostatistics, for α = 2− q, where q denotes Tsallis’ entropic
index.
III. H–THEOREM AND THE ASSOCIATED ENTROPY
In this section we will demonstrate the H-theorem by making use of Eq. (1), and for
that purpose, an entropic form related to this equation will be introduced. Let us therefore
suppose a general entropic form satisfying the following conditions,
dx g[P (x)] ; g[0] = 0 ; g[1] = 0 ;
≤ 0 , (10)
where one should have g[P (x, t)] at least as g[P (x, t)] ∈ C2; in addition to that, let us also
define the free-energy functional,
F = U −
S ; U =
dx φ(x)P (x, t) , (11)
where β represents a Lagrange multiplier, restricted to β ≥ 0. Furthermore, we will show
that this free-energy functional is bounded from below; this condition, together with the
H-theorem [(∂F/∂t) ≤ 0), leads, after a long time, the system towards a stationary state.
A. H–Theorem
The H-theorem for a system that exchanges energy with its surrounding, herein repre-
sented by the potential φ(x), corresponds to a well-defined sign for the time derivative of
the free-energy functional defined in Eq. (11). Using the definitions above,
dx φ(x)P (x, t)−
dx g[P ]
φ(x)−
∂g[P ]
. (12)
Now, one may use the FPE of Eq. (1) for the time derivative of the probability distribution;
carrying out an integration by parts, and using the conditions of Eq. (2), one obtains,
dφ(x)
P (x, t) + Ω[P ]
dφ(x)
∂2g[P ]
. (13)
Usually, one is interested in verifying the H-theorem from a well-defined FPE, together
with a particular entropic form, in such a way that the quantities Ω[P ] and ∂2g[P ]/∂P 2
are previously defined (see, e.g., Refs. [19, 37]). Herein, we follow a different approach,
by assuming that the general Eqs. (1) and (10) should be satisfied; then, we impose the
condition,
∂2g[P ]
Ω[P ]
P (x, t)
, (14)
in such way that,
dx P (x, t)
dφ(x)
Ω[P ]
P (x, t)
≤ 0 . (15)
It should be noticed that Eq. (14), introduced in such a way to provide a well-defined sign
for the time derivative of the free-energy functional, yields two important conditions, as
described next.
(i) Ω[P ] ≥ 0 [cf. Eq. (10)], which is expected for an appropriate diffusion-like term.
(ii) It expresses a relation involving the FPE of Eq. (1) and an associated entropic form,
allowing for the calculation of such an entropic form, given the FPE, and vice-versa. Since
the FPE is a phenomenological equation that specifies the dynamical evolution associated
with a given physical system, Eq. (14) may be useful in the identification of the entropic
form associated with such a system. In particular, the present approach makes it possible
to identify entropic forms associated with some anomalous systems, exhibiting unusual be-
havior, that are appropriately described by nonlinear FPEs, like the one of Eq. (1). As an
illustration of this point, let us consider the simple case of a linear FPE, that describes the
dynamical evolution of many physical systems, essentially those characterized by normal
diffusion. This equation may be recovered from Eq. (1) by choosing µ = ν = 1, in such a
way that Ω[P ] = a+b = D, where D represents a positive constant diffusion coefficient with
units (time)−1. One may now set the Lagrange multiplier β = kB/D, where kB represents
the Boltzmann constant; integrating Eq. (14), and using the conditions of Eq. (10), one gets
the well-known BG entropic form,
g[P ] = −kBP lnP . (16)
In the next section we will explore further the relation of Eq. (14), by analyzing other
particular cases.
The simplest situation for which condition (i) above is satisfied may be obtained by
imposing both terms of the functional Ω[P ] to be positive, which leads to
a ≥ 0 ; µ ≥ 0 , or a < 0 ; µ < 0 , (17)
b ≥ 0 ; ν ≤ 2 , or b < 0 ; ν > 2 . (18)
It should be stressed that it is possible to have Ω[P ] ≥ 0 with less restrictive ranges for
the parameters above. However, an additional property for the free-energy functional of
Eq. (11) to be discussed next, namely, the boundness from below, requires the conditions of
Eqs. (17) and (18), with the additional restriction ν > 0.
Now, integrating Eq. (14) for the general functional Ω[P ] of Eq. (1), and using the
standard conditions for g[P ] defined in Eq. (10), one gets that
g[P ] = −β
P µ + b
ν(ν − 1)
P ν +
aν(1 − ν) + b(2− ν)(1− µ)
(1− µ)(1− ν)ν
. (19)
This entropic form recovers, as particular cases, the BG entropy (e.g., when µ, ν → 1) and
several generalized entropies defined previously in the literature, like those introduced by
Tsallis [9], Abe [10], Borges-Roditi [12], and Kaniadakis [16, 17]. Such particular cases, as
well as their associated FPEs, will be discussed in the next section.
For the simpler situation of an isolated system, i.e., φ(x) = constant, the H-theorem
should be expressed in terms of the time derivative of the entropy, in such a way that
Eq. (13) should be replaced by
∂S[P ]
Ω[P ]
∂2g[P ]
dx Ω[P ]
∂2g[P ]
≥ 0 .
In this case all that one needs is the standard condition associated with the FPE [same
condition (i) above], i.e., Ω[P ] ≥ 0, and the general restrictions of Eq. (10) for the entropy.
A similar result may also be obtained by proving the H-theorem using the master equation
from which Eq. (1) was derived, with the transition probabilities introduced in Ref. [35, 36]
(see the Appendix).
B. Boundness from Below
Above, we have proven that the free-energy functional decreases in time, and so, for the
existence of a stationary state at long times of an evolution process, characterized by a
probability distribution Pst(x), one should have that
F (P (x, t)) ≥ F (Pst(x)) (∀t). (21)
In what follows, we will show this inequality and find the conditions for its validity. There-
fore, using Eqs. (7), (10), and (11), we can write,
F (P (x, t)) =
P (x, t)
φ0 − a
st (x)− b
ν − 1
P ν−1st (x)
g[P ]dx , (22)
and so,
F (Pst)− F (P ) =
(P − Pst)
st + b
ν − 1
P ν−1st
(g[P ]− g[Pst]) dx ,
where we have used the normalization condition for the probabilities. Now, we insert the
entropic form of Eq. (19) in the equation above to obtain,
F (Pst)− F (P ) =
st Γµ[P/Pst] +
b(2− ν)
ν(ν − 1)
P νst Γν [P/Pst]
dx , (24)
where,
Γα[z] = 1− α + αz − z
α (α = µ, ν) . (25)
By analyzing the extrema of the functional Γα[P/Pst], one can see that Γα[P/Pst] ≥ 0 for
0 < α < 1 and Γα[P/Pst] ≤ 0 for α > 1 and α < 0. Therefore, the inequality of Eq. (21)
is satisfied for the range of parameters specified by Eqs. (17) and (18), if one considers
additionally, ν > 0.
For the case of an isolated system, the stationary solution turns out to be the equilibrium
state, which is the one that maximizes the entropy. Therefore, one may use the concavity
property of the entropy [cf. Eq. (10)] in order to get,
S[Peq(x)]− S[P (x, t)] =
(g[Peq]− g[P ])dx ≥ 0 (∀t). (26)
IV. SOME PARTICULAR CASES
In this section we analyze some particular cases of the entropic form of Eq. (19) and using
Eq. (14), we find for each of them, the corresponding functional Ω[P ] of the associated FPE.
In the examples that follow, we will set the Lagrange multiplier β = k/D, with k and D
representing, respectively, a constant with dimensions of entropy and a constant diffusion
coefficient.
(a) Tsallis entropy [9]: This represents the most well-known generalization of the BG entropy,
which has led to the development of the area of nonextensive statistical mechanics [6, 7, 8].
One may find easily that Eq. (19) recovers Tsallis entropy in several particular cases, e.g.,
{b = 0, a = D, µ = q}, {a = 0, b = Dν/(2−ν), ν = q}, and {a = D/2, b = Dν/[2(2−ν)], µ =
ν = q}. For all these cases one may use Eq. (14), in order to get the corresponding functional
Ω[P ],
g[P ] = k
P q − P
, Ω[P ] = qDP q−1 . (27)
With the functional Ω[P ] above, one identifies the nonlinear FPE that presents the well-
known q-exponential, or Tsallis distribution (replacing q → 2 − q), as a time-dependent
solution [20, 21, 25].
(b) Abe entropy [10]: This proposal was inspired in the area of quantum groups, where
certain quantities, usually called q-deformed quantities, are submitted to deformations and
are often required to possess the invariance q ↔ q−1. The Abe entropy may be obtained from
Eq. (19) in the particular case {a = D(q−1)/(q−q−1), b = −Dq(q+1)/[(q−q−1)(q+2)], µ =
−ν = q}, for which
g[P ] = −k
P q − P−q
q − q−1
, Ω[P ] = D
q(q − 1)
q − q−1
P q−1 −
q(q + 1)
q − q−1
P−q−1
. (28)
(c) Borges-Roditi entropy [12]: This consists in another generalization of Tsallis entropy,
where now one has two distinct entropic indices, q and q′, with a more general invariance
q ↔ q′; this case may be obtained from Eq. (19) by choosing {a = D(q − 1)/(q′ − q), b =
Dq′(q′ − 1)/[(q − q′)(2− q′)], µ = q, ν = q′}. One gets,
g[P ] = −k
P q − P q
q − q′
, Ω[P ] = D
q − q′
q(q − 1)P q−1 − q′(q′ − 1)P q
. (29)
(d) Kaniadakis entropy [16, 17]: This is also a two-exponent entropic form, but slightly
different from those presented in examples (b) and (c) above; it may be reproduced from
Eq. (19) by choosing {a = b = D/[2(1 + q)], µ = 1 + q, ν = 1− q}, in such a way that
g[P ] = −
1 + q
P 1+q −
P 1−q
, Ω[P ] =
(P q + P−q) . (30)
Except for the well-known example (a), i.e., Tsallis entropy and its corresponding FPE,
the other three particular cases presented herein were much less explored in the literature.
Their associated FPEs, defined in terms of their respective functionals Ω[P ] above are,
to our knowledge, presented herein for the first time. These equations, whose nonlinear
terms depend essentially in two different powers of the probability distribution, may be
appropriated for anomalous-diffusion phenomena where a crossover between two different
diffusion regimes occurs [24].
V. CONCLUSION
We have analyzed important aspects associated with a recently introduced nonlinear
Fokker-Planck equation, that was derived directly from a master equation by setting non-
linear effects on its transition rates. Such equation is characterized by a nonlinear diffusion
term that may present two distinct powers of the probability distribution; for this reason,
it may reproduce, as particular cases, a large range of nonlinear FPEs of the literature.
We have obtained stationary solutions for this equation in several cases, and some of them
recover the well-known Tsallis distribution. We have proven the H-theorem, and for that,
an important relation involving the parameters of the FPE and an entropic form was intro-
duced. Since the FPE is a phenomenological equation that specifies the dynamical evolution
associated with a given physical system, such a relation may be useful for identifying en-
tropic forms associated with real systems and, in particular, anomalous systems that exhibit
unusual behavior and are appropriately described by nonlinear FPEs. It is shown that, in
the simple case of a linear FPE, the Boltzmann-Gibbs entropy comes out straightforwardly
from this relation. Considering the above-mentioned nonlinear diffusion term, this relation
yields a very general entropic form, which similarly to its corresponding FPE, depends on
two distinct powers of the probability distribution. Apart from Tsallis’ entropy, other en-
tropic forms introduced in the literature are recovered as particular cases of the present one,
essentially those characterized by two entropic indices. Nonlinear FPEs (as well as their
associated entropic forms) whose nonlinear terms depend essentially on two different powers
of the probability distribution, like the ones discussed in the present paper, are good candi-
dates for describing anomalous-diffusion phenomena where a crossover between two different
diffusion regimes may take place. As a typical example, one could have a particle trans-
port in a system composed by two types of disordered media, characterized respectively, by
significantly different grains (e.g., different average sizes), arranged in such a way that the
diffusion process is dominated by one medium, at high densities, and by the other one, at
low densities.
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[38] Even though one could use a simpler notation for the functional Ω[P ], e.g., Ω[P ] = a′Pµ
ν′ , herein we will keep the notation of Eq. (1), as appeared naturally in the derivation of
the above FPE, for a consistency with previous publications [35, 36].
Appendix
In this appendix we will prove the H-theorem directly from the master equation, for an
isolated system (i.e., no external forces). Let us consider a system described in terms of W
discrete stochastic variables; then, Pn(t) represents the probability of finding this system in
a state characterized by the variable n at time t. This probability evolves in time according
to a master equation,
(Pmwm,n − Pnwn,m) , (31)
where wk,l(t) represents the transition probability rate from state k to state l. The nonlinear
effects were introduced through the transition rates [35],
wk,l(∆) =
(δk,l+1 + δk,l−1)[aP
(t) + bP ν−1
(t)] , (32)
where ∆ represents the size of the step of the random walk. Herein, we shall consider a
random walk characterized by ∆ = 1; substituting such a transition rate in the master
equation one gets,
n = 1 :
2 − aP
1 + bP2P
1 − bP1P
2 , (33)
n = W :
W−1 − aP
W + bPW−1P
W − bPWP
W−1 , (34)
n = 2, ..., (W − 1) :
n+1 + P
− 2aP µn + bP
n (Pn+1 + Pn−1)
− bPn
P ν−1n+1 + P
, (35)
where we have treated the borders of the spectrum (n = 1 and n = W ) separately from the
rest. Let us now consider the entropy, S =
g[Pn], satisfying the same conditions specified
in the text [see Eq. (10)]; the H-theorem, to be proven below, is expressed in terms of its
time derivative,
dg[Pn]
≥ 0 . (36)
Then, using Eqs. (33)–(35) one can write this time derivative as,
dg[Pn]
n+1 − aP
n − bPnP
n+1 + bPn+1P
dg[Pn]
n−1 − aP
n + bPn−1P
n − bPnP
The sum indices can be rearranged to yield all summations in the range
, and thus the
time derivative of the entropy becomes,
dg[Pn]
dg[Pn+1]
dPn+1
n+1 − P
− bPnPn+1
P ν−2n+1 − P
. (38)
The negative curvature of the entropic function [cf. Eq. (10)] implies that its first derivative
decays monotonically with Pn. Hence, for Pn+1 > Pn, the condition of Eq. (36) is satisfied
n+1 − P
Pn+1 − Pn
− bPnPn+1
P ν−2n+1 − P
Pn+1 − Pn
≥ 0 , (39)
where we divided the term inside the brackets in Eq.(38) by the difference ∆P = Pn+1−Pn.
Now we consider the limit ∆P → 0 and obtain,
aµP µ−1 + b(2− ν)P ν−1 ≥ 0 , (40)
which corresponds to the condition Ω[P ] ≥ 0 found in the text, when proving the H-theorem
by making use of the FPE of Eq. (1). It should be mentioned that the procedure above works
also for Pn > Pn+1 ; therefore, in this appendix we have proven that the H-theorem holds
for an arbitrary state n and all times t of an isolated system.
	Introduction
	Stationary State
	H–theorem and the Associated Entropy
	H–Theorem
	Boundness from Below
	Some Particular Cases
	Conclusion
	References
	Appendix
