Nonlinear Fokker–Planck equations, H – theorem, and entropies

Accepted Manuscript

Nonlinear Fokker – Planck Equations, H – theorem, and Entropies M.A.F. dos Santos, M.K. Lenzi, E.K. Lenzi PII: DOI: Reference:

S0577-9073(17)30502-6 10.1016/j.cjph.2017.07.003 CJPH 291

To appear in:

Chinese Journal of Physics

Received date: Revised date: Accepted date:

27 April 2017 1 July 2017 2 July 2017

Please cite this article as: M.A.F. dos Santos, M.K. Lenzi, E.K. Lenzi, Nonlinear Fokker – Planck Equations, H – theorem, and Entropies, Chinese Journal of Physics (2017), doi: 10.1016/j.cjph.2017.07.003

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ACCEPTED MANUSCRIPT Highlights

• Nonlinear Fokker Planck equation and connections with the H theorem. • Fluctuation of particles and conditions with the H theorem.

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• Entropies related to the H theorem and conditions.

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Nonlinear Fokker – Planck Equations, H – theorem, and Entropies M. A. F. dos Santos1 , M. K. Lenzi2 , E. K. Lenzi1∗ 1

Departamento de F´ısica, Universidade Estadual de Ponta Grossa,

Av. General Carlos Cavalcanti, 4748 - Ponta Grossa, PR 84030-900, Brazil 2

Departamento de Engenharia Qu´ımica,

Universidade Federal do Paran´ a, Av. Cel. Francisco H. dos Santos,

Abstract

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210 - Jardim das Americas - Curitiba, PR 81531 - 990, Brazil

We analyze the H - theorem like to systems subjected to a process that implies in a nonconservation of the number of particles. Firstly, we consider the system governed by a linear Fokker Planck

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equation with a source (or sink) term. After, we investigate the nonlinear situations including the Tsallis entropy. We also obtain for these cases the entropy production in order to verify that the entropy in these situations system increases.

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Keywords: nonlinear Fokker Planck equation; reaction process; H-theorem

∗

Corresponding Author Electronic Address: [email protected]

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ACCEPTED MANUSCRIPT I.

INTRODUCTION

The comprehension in the microscopic point of view of the dynamics behind of the thermodynamics is usually established in terms of an internal energy and entropy of a system. This comprehension is one of the most important challenge [1, 2] in physics. A relevant contribution for this subject, which is also a central point of in the statistical mechanics, was established by Boltzmann: the H-theorem [3, 4] in which the concept of the entropy is

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present. The entropy represents a modern concept in physics. In particular, it enables a direct connection between the microscopic (described by statistical mechanics) and macroscopic (described by thermodynamics) quantities of a system. Typically, this connection may be performed by the entropy S = kB ln Ω (Boltzmann-Gibbs entropy (BG)) in which Ω represents the microstates accessible to the system. One of the main characteristics present

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in this context is the short - range interactions and the presence of the Markovian processes in which the memory effects are absent. Situations with long - range interactions, memory effects, long range correlations have also investigated with alternative entropic forms among them the R´enyi [5] and Tsallis entropies [6]. The last one has been successfully applied in several physical situations such as including description of chaos in the solar plasma [7],

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high-energy collisions [8], on study of mensure complexity [9], anomalous diffusion [10], and non–equilibrium situations in polymers and biopolymers [11–13]. H–theorem plays an

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important role in these scenarios by imposing a relation between general fluctuation, entropy and internal energy. For the Boltzmann-Gibbs’ entropy, this theorem reveals that

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the dynamics may be related to a linear Fokker – Planck equation [14] which is essentially connected to a Markov processes. For the Tsallis entropy, this theorem shows that nonlinear

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Fokker – Planck equation (NFPE) [15, 16] is the suitable dynamic equation to govern the relaxation processes. It is worth to mention that the NFPE has been used to investigated

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motion of particles in porous media [17–19], dynamics of surface growth [20], dynamics of interacting vortices in disordered superconductors [21], and motion of overdamped particles through narrow channels [22]. In Ref. [15, 16], a framework is developed to connect a general entropic form with its corresponding NPFE to accomplish the H theorem. Here, we investigate how may be established a relation among fluctuation, entropy and internal energy for a system subjected to a process, that implies in a nonconservation of the number of particles. For this, we consider that the particles present in the system are 3

ACCEPTED MANUSCRIPT governed by the following equation, ∂ ∂ P (x, t) = − J (x, t) − α(t)P (x, t), ∂t ∂x

(1)

∂ P (x, t), ∂x

(2)

with α(t) ≥ 0 and J (x, t) = f (x)Ξ[P ] − DΩ[P ]

in which D (D ∝ T , i.e., Einstein – Smoluchowski relation) is a constant with dimensions

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of energy and the external drift, force f (x) is associated with a potential, i.e., f (x) = −dφ(x)/dx. The functionals Ω[P (x, t)] and Ξ[P (x, t)] should satisfy certain mathematical requirements, e.g., positiveness [15, 16]. In particular, for α = 0, Ω[P ] = 1, and Ξ[P ] = P (x, t) the usual form of the Fokker – Planck equation [14] is recovered from Eq. (1). For α(t) 6= 0, particles may be removed (α(t) > 0) from system and, consequently, the survival

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probability is not conserved. By using Eqs. (1) and (2), firstly, we analyze the H–theorem like in the perspective of the linear Fokker – Planck equation with a source (or sink) term. After, we consider nonlinear situations including the Tsallis entropy. We also investigate for these cases the entropy production in order to verify that the entropy of the system

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increases when irreversible processes [23–26] are present, in agreement with the H – theorem

II.

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and, consequently, the second law of thermodynamics.

FOKKER-PLANCK EQUATION, REACTION AND BOLTZMANN THERMO-

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STATISTIC

Let us start our discussion by considering the standard Fokker-Planck equation with a

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linear source (or sink) term. This case implies for Eq. (2) that Ω[P ] = 1 and Ξ[P ] = P (x, t),

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yielding

J (x, t) = f (x)P (x, t) − kB T

∂ P (x, t). ∂x

(3)

Following, we consider in our analysis the grand potential (Φ) which is related to the nonconservation of the number of particles, in contrast to the situations analyzed in Refs. [15, 16] in terms of the free Helmholtz energy (F ) characterized by the conservation of the number of particles. Thus, we have that Φ = F − µN

and 4

F = U − TS ,

(4)

ACCEPTED MANUSCRIPT in which U is internal energy and S is entropy. In the context of the usual statistical mechanics U= with φ(x) = −

Rx

−∞

f (x0 )dx0 , N =

Z +∞

R +∞ −∞

S = −kB

−∞

φ(x)P (x, t)dx

(5)

dxP (x, t), and

Z +∞ −∞

P (x, t) ln P (x, t)dx ,

(6)

the H – theorem may be verified by showing that for α(t) 6= 0 the sign of the

dΦ dt

dF dt

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corresponding to the Boltzmann entropy in which kB is the Boltzmann constant. For α = 0, ≤ 0. In the similar fashion, we analyzed

in order to obtain the conditions necessaries to establish an

equivalent situation as the one found for the H – theorem.

By using previous definitions, it is possible to show that

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∂P dΦ Z +∞ (φ + kB T ln P + kB T − µ) = dx dt −∞ ∂t ! Z +∞ ∂J (φ + kB T ln P + kB T − µ) − = − αP dx . ∂x −∞

(7)

Equation (7) can be rewritten as follows:

(8)

Z +∞ dF ∂J (φ + kB T ln P + kB T − µ) =− dx . dt ∂x −∞

(9)

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dF dΦ = − α(t) (Φ + N kB T ) , dt dt

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in which

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Now, we first analyze the sign of Eq. (9) and after the behavior of Eq. (8) in order to analyze the changes produced by the particle fluctuations. For Eq. (9), after performing

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some calculations with J (±∞, t) = 0 and supposing that φ(x) is confining potential. It is

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possible to show that

Z +∞ dF J (x, t)2 =− dx ≤ 0 . dt −∞ P (x, t)

(10)

By using Eq. (10), the sign of Eq. (9) may be established. For Eq. (8), one can show that e

−

Rt 0

dt0 α(t)





Z +∞ Rt 0 d J (x, t)2 dt α(t) 0 Φ(t)e =− dx − α(t)N kB T dt −∞ P (x, t)

(11)

and, consequently, −

e

Rt 0

dt0 α(t)





Rt 0 d Φ(t)e 0 dt α(t) ≤ 0 . dt

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(12)

ACCEPTED MANUSCRIPT Thus, the situations that is verified the Eq. (12) satisfy a H–theorem like when particles are removed from the system. It is worth to mention that particular case α = 0 recovers the classical form of the H – theorem for closed system when N = entropy production [23–26] leads to obtain that

R +∞ −∞

ρ(x, t)dx = const. The

in which 1 Z +∞ J 2 (x, t) dx T −∞ P (x, t)

Π= and

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Z +∞ 1 ∂P dS = −kB dxJ (x, t) − α(t)(S − N kB ) , dt P ∂x −∞ = Π − Ψ − α(t)(S − N kB ),

(14)

(15)

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1 Z +∞ Ψ= dxf (x)J (x, t) , T −∞

(13)

Eqs (14) and (15) represents the entropy-production contribution and the exchanges of entropy between the system and its neighborhood, respectively. The last term in Eq. (13) is related to the changes of the particle numbers present in the system. In particular, if α(t) → 0 we can verify that Ψ = Π, implying that the entropy-production is equal to the

NONLINEAR FOKKER – PLANCK EQUATION AND TSALLIS STATISTICS

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III.

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exchanges of entropy.

Now, we extend the previous analysis by considering that current density has a nonlinear

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dependence on Ω[P ], i.e., Ω[P ] ∝ P q−1 (x, t), enable us to related the dynamics with porous media equation which is connected to the Tsallis statistics. For this case, Eq. (3) can be

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written as

J (x, t) = f (x)P (x, t) − kT

∂ q P (x, t) . ∂x

(16)

which accomplishes the entropy proposed by Tsallis [6]: Sq = −k

Z +∞ −∞

P (x, t) lnq P (x, t)dx ,

(17)

in which lnq P = (P q−1 − 1)/(q − 1). Equation (17) recovers the Boltzmann – Gibbs entropy for q = 1. Applying the procedure of previous section, we obtain that !

dΦ Z +∞ qP q−1 − 1 ∂P = φ + kT −µ dx dt −∞ q−1 ∂t 6

ACCEPTED MANUSCRIPT Z +∞

=

−∞

qP q−1 − 1 φ + kT −µ q−1

!

!

∂J − − αP dx, ∂x

(18)

which, after some calculations, can be written as dF dΦ = − α(t) (Φ + (1 − q)T Sq + N kT ) dt dt

(19)

with !

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Z +∞ qP q−1 − 1 dF ∂J φ + kT =− −µ dx . dt q−1 ∂x −∞

(20)

Note in Eq. (19) the presence of the parameter q, related to the degree of nonextensivity [2] of the system, which is also related to the dynamic aspect obtained from Eq. (16). After, performing some calculations, one can show for Eq. (20) that

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Z +∞ 2 J (x, t) dF =− dx ≤ 0 dt −∞ P (x, t)

(21)

as occurs for Eq. (10), in which D = kT . Thus, one can show that e−

Rt 0

dt0 α(t)

  Z +∞ Z +∞ Rt 0 d J (x, t)2 P q (x, t)dx dx − α(t)kT Φ(t)e 0 dt α(t) = − dt P (x, t) −∞ −∞

−

e

Rt 0



dt0 α(t)



Rt 0 d Φ(t)e 0 dt α(t) ≤ 0 . dt

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and, consequently,

(22)

(23)

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Equation (23) is formally equal to the one in found the Boltzmann– Gibbs entropy, i.e., Eq. (12). Similar feature is found for the entropy production which in this case can be

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summarized in terms of the following equation (24)

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dSq = Π − Ψ + α(t)(qSq − kN ) , dt

in which Π and Ψ are formally given by Eqs. (14) and (15). Note that Eq. (24) has an explicitly dependence on the parameter q in the term related to the contribution of the

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fluctuation of the particle number. In addition, for q = 1 it recovers Eq. (13) obtained for the Boltzmann entropy.

IV.

ARBITRARY ENTROPY, H-THEOREM, AND ENTROPY PRODUCTION

The previous analysis about the Tsallis entropy leads us to an explicit dependence on q in Eqs. (19) and (24) evidencing the dynamic effect on the thermodynamic. This motives 7

ACCEPTED MANUSCRIPT us to consider a general situation to investigate how above result may depend on the choice of the entropic form. In this case, we consider that S=k

Z +∞

G[P (x, t)]dx

−∞

(25)

with Ω[P ] and Ξ[P ] to be determined by the condition on Eq. (2). By substituting this equation in Eqs. (1) and (2), and perform some calculations, one can obtain that !

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∂G ∂P dΦ Z +∞ φ − kT = −µ dx dt ∂P ∂t −∞ ! ! Z +∞ ∂G ∂J φ − kT −µ − − αP dx.e = ∂P ∂x −∞

(26)

The sign of Eq. (35) may be established by analyzing first the sign of the quantity Υ=−

!

Z +∞

∂G ∂J φ − kT −µ dx . ∂P ∂x

−∞

(27)

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In these sense, one can use J (±∞, t) = 0 as boundary condition and relation D = kT , we obtain Z +∞

Υ=−

−∞

Ω[P ] ∂P dφ Ξ[P ] + kT dx Ξ[P ] ∂x

!2

dx ,

(28)

and also the following relation [15, 16, 27]

d2 G Ω[P ] = 2 dP Ξ[P ]

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(29)

−

e

dt0 α(t)



−∞



Rt 0 d Φ(t)e 0 dt α(t) = Υ − α(t)(T S + kT hG 0 i) dt

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R +∞

0

(30)

dxP ∂P G and with G + P ∂P G ≥ 0, one can verify the condition

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in which hG 0 i =

Rt

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to show that Υ ≤ 0. We can rewrite Eq. (26) as

−

e

Rt 0

dt0 α(t)





Rt 0 d Φ(t)e 0 dt α(t) ≤ 0 . dt

(31)

This result enables us to analyze other entropies such as the Kaniadakis [28, 29] subjected

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to a nonlinear diffusion equation which satisfies the previous conditions. For the entropy production, we have that dS 1 Z +∞ J 2 (x, t) = dx dt T −∞ Ξ(x, t) 1 Z +∞ − f (x)J (x, t)dx + α(t)khG 0 i . T −∞

(32)

The first two terms are equivalent to Π and Ψ present in Eqs.(13) and (24). The last term corresponds to the contribution of the fluctuation of the number of particles. 8

ACCEPTED MANUSCRIPT V.

´ RENYI LIKE ENTROPY

Let us analyze the implications of considering a R´enyi like entropy in the previous scenarios in which the number of particles is not conserved. We consider the following entropic form:

R +∞ −∞

R +∞ −∞

P q (x, t)dx i N

(33)

P (x, t)dx. For q → 1, the Boltzmann – Gibbs entropy is recovered, i.e.

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in which N =

h kN Sq = ln 1−q

Sq → SBG , and if the number of particles is conserved, i.e. N = 1, the standard form of the R´enyi entropy is recovered. We would like to underline that considering the standard form of the R´enyi entropy when

R +∞ −∞

P (x, t)dx 6= const does not recover, in the limit of

q → 1, the Boltzmann – Gibbs entropy. For this reason, we have modified it in order to

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cover both situations (particle conservation or not) having the Boltzmann – Gibbs entropy as limit when q → 1. In addition, a class of nonlinear diffusion equations related to Eq. (33) is obtained from Eq. (2) when the following equation

P q−2 (x, t) Ω[P ] = qN R +∞ q , Ξ[P ] −∞ P (x, t)dx

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is satisfied.

(34)

By using Eq. (4), we can obtain the time derivative of the gran potential and establish

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the condition required to verify an extension of the H – theorem, as in the previous sections. In particular, we have that

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T dΦ Z +∞ N qP q−1 (x, t) φ − Sq − kT = R ++∞ dt N 1 − q −∞ P q (x, t)dx −∞ 1 + kT −µ 1−q

!

!

∂J − − α(t)P dx . ∂x

(35)

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In order to establish the sign of Eq. (35), we may first analyze the sign of the quantity Υ = − = −

Z +∞ −∞

Z +∞ −∞

!

qP q−1 (x, t) 1 T N ∂J φ − Sq − kT + kT −µ dx R +∞ q N 1 − q −∞ P (x, t)dx 1−q ∂x

dφ qP q−2 (x, t) ∂P + kT N R +∞ q Ξ dx −∞ P (x, t)dx ∂x

in which D = kT , that implies in Υ = −

R +∞ −∞

!2

dx

J 2 /P dx ≤ 0 and, consequently, we have that

Z +∞ d J (x, t)2 Φ(t) = − dx − α(t) (Φ + kT N ) , dt −∞ P (x, t)

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(36)

(37)

ACCEPTED MANUSCRIPT which implies in −

e

Rt 0

dt0 α(t)





Rt 0 d Φ(t)e 0 dt α(t) ≤ 0 . dt

(38)

Equation (37) is formally the same result that one previously obtained for the Boltzmann – Gibbs entropy, i.e. Eq. (8). This aspect may be connected to the additivity both found in R´enyi entropy and the Boltzmann – Gibbs entropy. leads to

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Now, we may extend our analysis for the entropy production related to Eq. (33) that ! Z +∞ 1 N qP q−1 (x, t) 1 dS = dx Sq + k −k R +∞ dt N 1 − q −∞ 1−q −∞ P q (x, t)dx

×

!

∂J − α(t)P (x, t) . − ∂x

(39)

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In previous equation, employing the Einstein – Smoluchowski relation (D = kT ), we obtain Z +∞ dS P q−2 (x, t) ∂P = −α(t) (S − kN ) − qkN dxJ (x, t) R +∞ q dt −∞ −∞ P (x, t)dx ∂x = −α(t) (S − kN ) + Π − Ψ .

(40)

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Thus, we also verify that the entropy production for the R´enyi entropy is formally equal to

VI.

CONCLUSION

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the one previously obtained for the Boltzmann – Gibbs entropy.

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We have analyzed an H – theorem like for systems described by linear and/or nonlinear Fokker-Planck equations by taking into account a nonconservation of the particle numbers, R +∞

dxP (x, t) 6= constant. Our analysis is based on the grand canonic potential which

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i.e.,

−∞

admits the fluctuation of the particle numbers, since that we related, −∞

−∞

dxP (x, t) with

dxP (x, t) = N . We first investigate the H–theorem and the production entropy

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N , i.e.,

R +∞

R +∞

in the context of the linear Fokker – Planck equation, which is related to the Boltzmann – Gibbs entropy, when an addition term is considered in this equation in order to lead us to a nonconservation of the particle numbers. In this context, we have shown that an H – theorem like can be verified for general relation (12). For the entropy – production, we obtain an additional contribution related to the variations of system entropy and the particle numbers. Following, we have considered a nonlinear Fokker – Planck equation and 10

ACCEPTED MANUSCRIPT the Tsallis entropy. For this case, we have also obtained a H – theorem like and the entropy production depending on the entropic parameter q when N 6= constant. After, we considered a general situation by taking into account a general entropic form. Situations related to the R´enyi entropy were analyzed in another section due to the mathematical structure of the entropy which cannot be written in the form S = k

R∞

−∞

dxG [P (x, t)] as the case considered

in Sec. IV. We have also shown that the conditions obtained for this entropy are the same as in Boltzmann – Gibbs entropy as well as the entropy production. Such features exist

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because both entropies, R´enyi entropy and Boltzmann-Gibbs entropy, obey the property of additivity. Finally, we hope that the results shown here may be useful to discuss situations characterized by a nonconservation of the particle numbers.

ACKNOWLEDGMENTS

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VII.

We would like to thank CNPq (Brazilian agency) for the partial financial support. We

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also thank the INCT/CNPq – SC.

[1] C. Tsallis, Nonextensive Statistical Mechanics and Thermodynamics: Historical Background

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and Present Status, Nonextensive Statistical Mechanics and Its Applications, pp. 3-80, S. Abe Y. Okamoto (Eds.) (Springer-Verlag, Berlin, 2001). [2] C. Tsallis, Introduction to Nonextensive Statistical Mechanics: Approaching a Complex

PT

World, (Springer-Verlag, Berlin, 2009).

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[3] L. Boltzmann, (1896). Reply to Zermelos Remarks On the Theory of Heat, in Brush, S.G. (1966). Kinetic Theory Volume 2. Irreversible Processes pp. 218-229. Oxford: Pergamon Press. [4] L. Boltzmann, (1896). On Zermelos Paper On the Mechanical Explanation of Irreversible

AC

Processes, in Brush, S.G. (1966). Kinetic Theory Volume 2. Irreversible Processes (pp. 238245). Oxford: Pergamon Press.

[5] A. R´enyi, Probability Theory, (North Holland, Amsterdam, 1970). [6] C. Tsallis, Possible generalization of Boltzmann-Gibbs statistics, J. Stat. Phys. 52, 479-487 (1988). [7] L. P. Karakatsanis, G. P. Pavlos, M. N. Xenakis, Tsallis non-extensive statistics, intermittent

11

ACCEPTED MANUSCRIPT turbulence, SOC and chaos in the solar plasma. Part two: Solar flares dynamics, Physica A 392, 3920-3944 (2013). [8] W. M. Alberico, A. Lavagno, Non-extensive statistical effects in high-energy collisions, Eur. J. Phys. A. 40, 313-323 (2009). [9] C. Tsallis, Entropic nonextensivity: a possible measure of complexity, Chaos, Solitons and Fractals. 13, 371-391 (2002).

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[10] C. Tsallis, E. K. Lenzi, Anomalous diffusion: nonlinear fractional FokkerPlanck equation, Chem. Phys. 284, 341-347 (2002).

[11] W.-J. Ma, and C.-K. Hu, Generalized statistical mechanics and scaling behavior for nonequilibrium polymer chains I: monomers connected by rigid bonds, J. Phys. Soc. Jpn. 79, 024005 (2010).

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[12] W.-J. Ma, and C.-K. Hu, Generalized statistical mechanics and scaling behavior for nonequilibrium polymer chains II: monomers connected by springs, J. Phys. Soc. Jpn. 79, 024006 (2010).

[13] W.-J. Ma, and C.-K. Hu, Physical mechanism for biopolymers to aggregate and maintain in non-equilibrium states, Scientific Reports 7, Article number: 3105 (2017)

(Springer, Berlim, 2010).

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[14] H. Risken, T. Frank, The Fokker-Planck Equation: Methods of Solution and Applications,

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[15] V. Schw¨ammle, F.D. Nobre, E.M.F. Curado, Consequences of the H theorem from nonlinear Fokker-Planck equations, Phys. Rev. E. 76, 041123 (2007).

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[16] V. Schw¨ammle, E.M.F. Curado, F. D. Nobre, A general nonlinear Fokker-Planck equation and its associated entropy, Eur. Phys. J. B. 58, 159-165 (2007).

CE

[17] M. Muskat, Flow of homogeneous fluids through porous media, (McGraw-Hill, New York), (1937).

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[18] B. Berkowitz, H. Scher, Anomalous Transport in Random Fracture Networks, Phys. Rev. Lett. 79, 4038 (1997).

[19] A. Klemm, H. P. Muller, R. Kimmich, NMR microscopy of pore-space backbones in rock, sponge, and sand in comparison with random percolation model objects, Phys. Rev. E. 55, 4413 (1997). [20] H. Spohn, Surface dynamics below the roughening transition, J. Phys. I France 3, 69-81 (1993). [21] S. Zapperi, A. A. Moreira, J. S. Andrade Jr., Flux front penetration in disordered supercon-

12

ACCEPTED MANUSCRIPT ductors, Phys. Rev. Lett. 86, 3622-3625 (2001). [22] P. Barrozo, A. A. Moreira, J. A. Aguiar, J. S. Andrade Jr., Model of overdamped motion of interacting magnetic vortices through narrow superconducting channels, Phys. Rev. B 80, 104513 (2009). [23] I. Prigogine, Introduction to the Thermodynamics of Irreversible Processes (John Wiley and Sons, New York, 1967).

tions (John Wiley and Sons, New York, 1971).

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[24] P. Glansdorff and I. Prigogine, Thermodynamic Theory of Structure, Stability, and Fluctua-

[25] G. Nicolis and I. Prigogine, Self-Organization in Nonequilibrium Systems (John Wiley and Sons, New York), (1977).

[26] S. R. de Groot, P. Mazur, Non-Equilibrium Thermodynamics, (Dover Publications, New York,

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1984).

[27] G. A. Casas, F.D. Nobre, E.M.F. Curado, Entropy production and nonlinear Fokker-Planck equations, Phys. Rev. E 86, 061136 (2012).

[28] G. Kaniadakis, Non-linear kinetics underlying generalized statistics, Physica A 296, 405-425 (2001).

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[29] G. Kaniadakis, Statistical mechanics in the context of special relativity, Phys. Rev. E 66,

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CE

PT

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056125 (2002).

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