Nonlinear inhomogeneous Fokker-Planck equations: entropy and free-energy time evolution
NNonlinear inhomogeneous Fokker–Planck equations: Entropy and free-energy timeevolution
Gabriele Sicuro ∗ and Peter Rapčan † Centro Brasileiro de Pesquisas Físicas, Rua Dr. Xavier Sigaud, 150, 22290-180, Rio de Janeiro, Brazil
Constantino Tsallis ‡ Centro Brasileiro de Pesquisas Físicas, and National Institute of Science and Technology for Complex Systems,Rua Dr. Xavier Sigaud, 150, 22290-180, Rio de Janeiro, Brazil andSanta Fe Institute, 1399 Hyde Park Road, Santa Fe, New Mexico, 87501 USA (Dated: December 19, 2016)We extend a recently introduced free-energy formalism for homogeneous Fokker–Planck equationsto a wide, and physically appealing, class of inhomogeneous nonlinear Fokker–Planck equations.In our approach, the free-energy functional is expressed in terms of an entropic functional andan auxiliary potential, both derived from the coefficients of the equation. With reference to theintroduced entropic functional, we discuss the entropy production in a relaxation process towardsequilibrium. The properties of the stationary solutions of the considered Fokker–Planck equationsare also discussed.
I. INTRODUCTION
Since the seminal work of Einstein [1] on the Brow-nian motion, linear Fokker-Planck equations (FPEs) [2]have played a central role in the study of normal diffu-sion processes and in the investigation of nonequilibriumin general. It is well known, however, that many physi-cal phenomena are associated to an anomalous diffusivebehavior, that cannot be properly described by a linearFPE. For this reason, nonlinear FPEs [3, 4], alongsidewith fractional linear FPEs [5, 6], have become naturalcandidates for modeling anomalous diffusion processes.Models based on nonlinear FPEs are indeed able to re-produce the experimentally observed dispersion laws. Inthe last decades, nonlinear FPEs have been put in rela-tion with generalized thermostatistics [7] and successfullydescribe diffusion in porous media [8], stellar dynamicsand turbulence [9], or surface dynamics [10]. Similarly,fluctuations in granular media can be properly treated bymeans of nonlinear FPEs [11], as recently experimentallyverified by Combe et al. [12]. In Ref. [13] a nonlinearFPE was adopted to model the evolution of stock pricereturns, finding a remarkable agreement with the marketdata.The reconstruction of the microscopical dynamics cor-responding to a given nonlinear FPE is, however, a non-trivial task. In this regard, Borland [14] proposed a phe-nomenological model, in which the evolution at the mi-croscopic level can be simulated to successfully reproducethe macroscopic quantities: the equations of motion forthe microscopic components, however, depend on the so-lution of the nonlinear FPE itself. Macroscopic and mi-croscopic evolution are therefore coupled, suggesting that ∗ [email protected] † [email protected] ‡ [email protected] the model can be used as a heuristic description only, asstressed by the author herself.The nontrivial relation between macroscopic and mi-croscopic evolution in nonlinear FPEs may be relevantin the study of thermodynamics. To be more precise,let us recall that a diffusion process in d dimensionscan be studied in the 2 d -dimensional one-particle phasespace, considering a particle distribution density f ( r , v , t )around the space position r and the velocity value v at time t . The evolution of f is described by the so-called Klein–Kramers equation (KKE, or FPE in the one-particle phase space) [15], having the formd f d t = (cid:20) ∂∂t + v · ∇ r + 1 m F ( r ) · ∇ v (cid:21) f = K [ f ] , (1)where F ( r ) is an external force acting on the particle ofmass m , and K [ f ] is a functional of f determined bythe underlying kinetics. The Boltzmann equation is aparticular case of Eq. (1). The possible emergence ofnonlinearity in the KKE is due to the structure of thefunctional K [ f ]. Kaniadakis [16] proposed a very generalkinetic interaction principle that is able to unify manyrelevant particular cases in one single picture. He alsoassociated to f an entropic functional S ( f ), satisfyingthe H theorem such thatd f d t + ∇ v · (cid:20) D ( v ) γ ( f ) ∇ v δ S ( f ) δf (cid:21) = 0 , (2)where D ( v ) is a velocity-dependent diffusion coefficientand γ is a function of f only, related to the kinet-ics K . The relation between a KKE and the corre-sponding equations for ρ r ( r , t ) = R f ( r , v , t ) d d v and ρ v ( v , t ) := R f ( r , v , t ) d d r (usually called Smoluchowskiequation, or SE, and FPE, respectively) is however nottrivial [17]. For example, for a Brownian particle in afluid, we have [2, 16] K [ f ] = λ ∇ v · ( v f + λD ∇ v f ) . (3) a r X i v : . [ c ond - m a t . s t a t - m ec h ] D ec In the expression above, λ is a friction coefficient and D > ∂ρ r ∂t + ∇ r · (cid:18) ρ r F λm + D ∇ r ρ r (cid:19) = 0 , (4)to be associated with a proper initial condition ρ r ( r ,
0) = R f ( r , v ,
0) d d v . This equation, however, is obtained as-suming λ (cid:29)
1. It is expected that, for a generic valueof λ , the evolution of ρ r should depend on f ( r , v ,
0) andnot on ρ r ( r ,
0) only, and that, therefore, Eq. (4) must becorrected. A first study of the corrections to Eq. (4) wasperformed by Wilemski [17], and later by Chaturvedi andShibata [18] and San Miguel and Sancho [19] in the caseof a position dependent external force in the one dimen-sional case. In Refs. [20, 21] the exact SE correspondingto a given KKE was obtained, in which the coefficientsof the SE depend on the coefficients of the KKE and,moreover, on its initial condition f ( r , v , et al. [22] recently introduced a remarkable set of ideas,now called macroscopic fluctuation theory, for a nonequi-librium thermodynamics of systems described by a gen-eral evolution law of the type of a nonlinear SE [23],avoiding reference to the microscopical details.A different approach to the thermodynamics of non-linear FPEs [24] has been independently proposed bySchwämmle et al. [25, 26]. Their formalism is analogousto the one introduced in Ref. [16] for KKEs and it wasinspired by the contribution of Plastino and Plastino [7].They considered a nonlinear FPE for a probability den-sity ρ in (1 + 1) dimensions, in the form ∂ t ρ ( x, t ) + ∂ x j ( x, ρ ) = 0 ,j ( x, ρ ) = χ [ ρ ( x, t )] E ( x ) − Dω [ ρ ( x, t )] ∂ x ρ ( x, t ) , lim x →±∞ j ( x, ρ ) = 0 . (5)Here χ ( ρ ) and ω ( ρ ) are positive quantities depending onthe density ρ only, D is a positive diffusion constant,and E ( x ) is an external field. They showed that it ispossible to construct a free-energy functional F ( ρ ) whichis consistent with thermodynamics and which satisfies ∂ t F ( ρ ) ≤
0. The expression for F ( ρ ) is given explic-itly in terms of the coefficients of the equation. Follow-ing this approach, nonlinear FPEs are typically (but notalways) associated to entropic functionals that are dif-ferent from the Boltzmann–Gibbs entropy, obtained inthe linear case. The stationary distribution of a given nonlinear FPE coincides with the one that maximizesthe corresponding entropy with an appropriate energyconstraint. Due to the fact that nonlinear FPEs ap-pear in the study of vortex diffusion in superconduc-tors, and inspired by the formalism above, Andrade et al. [27] claimed that a nonextensive thermostatistics, differ-ent from the Boltzmann–Gibbs one, is necessary for thestudy of the overdamped motion of interacting vorticesat zero temperature [28, 29].In the present paper we show that the approach ofSchwämmle, Curado and Nobre can be generalized to inhomogeneous nonlinear FPEs, i.e., to nonlinear FPEshaving a diffusion coefficient D depending on x , D = D ( x ). Nonlinear inhomogeneous FPEs describe, for ex-ample, anomalous diffusion processes in which, in ad-dition to the nonlinearity, a local inhomogeneity of themedium is present, inducing a space dependent frictioncoefficient [30]. Position dependent, or velocity depen-dent, diffusion coefficients are not uncommon in the lit-erature. For example, in Ref. [16] a velocity dependentdiffusion coefficient appears in the study of a general-ized kinetics, developed in the proper one-particle phasespace. We will focus, however, on the Fokker–Planck pic-ture. In Sec. II we introduce, in particular, a functional Φthat has (locally) the structure of a free energy rescaledby D ( x ). In the linear homogeneous case, assuming theEinstein–Smoluchowski relation D ∝ T between the dif-fusion coefficient and the temperature of the bath, ourrescaled functional reduces to Φ ∝ T − F , where F is theusual free energy. We also study the time evolution ofthe entropy in a relaxation process towards equilibrium.In Sec. III we discuss the relation between Φ and the sta-tionary solution of the considered nonlinear FPE, and apossible definition of a temperature-like quantity in thiscontext. In Sec. IV, we study, as a particular example,a FPE for diffusion processes in inhomogeneous porousmedia. Finally, in Sec. V, we give our conclusions. II. MODIFIED FREE-ENERGY FUNCTIONAL
A general nonlinear Fokker–Planck equation in (1 + 1)dimensions describes the evolution with time of a proba-bility density function ρ ( x, t ) defined on the open inter-val Ξ := ( x − , x + ) of the real line. We admit Ξ ≡ R as aparticular case. The equation has the general form of aprobability conservation law ∂ρ ( x, t ) ∂t + ∂j ( x, ρ ) ∂x = 0 . (6a)The current of probability j introduced above has thestructure [22, 31, 32] j ( x, ρ ) := E ( x ) χ [ ρ ( x, t )] − D ( x ) ω [ ρ ( x, t )] ∂ x ρ ( x, t ) , (6b)where χ ( ρ ) > mobility and E ( x ) is a drift co-efficient related to the presence of an external potential V ( x ), E ( x ) = − d V ( x )d x . (7)In the present paper, we assume that the diffusion coef-ficient D ( x, ρ ) := D ( x ) ω ( ρ ) in Eq. (6b) is in this specificfactorized form; we will also assume that both the fac-tors are strictly positive, i.e., D ( x ) > ω ( ρ ) >
0, for(almost) all values of their arguments. Equation (6b) istypically obtained through a set of approximations froma microscopical model and assuming a linear response tothe action of the external field. The type of approxima-tions strongly depend on the considered model and on theassumptions about the underlying dynamics. Moreover,in the one-dimensional case the conservation of probabil-ity implies some additional constraints on j , namely, thefact that the current has the same value on the bound-ary for all values of t . We will consider reflecting bound-ary conditions [33] for the probability current j , i.e., ourproblem has the form ∂ t ρ ( x, t ) + ∂ x j ( x, ρ ) = 0 ,j ( x, ρ ) = χ [ ρ ( x, t )] E ( x ) − D ( x ) ω [ ρ ( x, t )] ∂ x ρ ( x, t ) , lim x → x ± j ( x, ρ ) = 0 . (8)Reflecting boundary conditions imply that the stationarystate % ( x ) has j ( x, % ) = 0 on the entire domain, i.e.,the stationary solution is the equilibrium solution, andnonequilibrium stationary solutions are not allowed [34].Schwämmle et al. [25, 26] observed that a FPE in theform in Eq. (6) can be associated to a trace-form free-energy-like functional decreasing in time. In the originalpaper, D was supposed to be a constant. Inspired bytheir result, we will show that a similar functional can beobtained in the inhomogeneous case. In particular, wesearch for a functional in the formΦ( ρ ) := Z Ξ φ [ x, ρ ( x, t )] d x := ¯ U ( ρ ) − S ( ρ ) , (9)such that the following inequality holds,d Φd t ≤ , t ≥ . (10)Observe that Φ has the structure of a free-energy rescaledby the temperature. The first term¯ U ( ρ ) := Z Ξ ¯ V ( x ) ρ ( x, t ) d x (11)corresponds to an “energy contribution” expressed interms of an auxiliar potential ¯ V ( x ), in general differentfrom V ( x ). The second term corresponds to an entropiccontribution S ( ρ ) := Z Ξ s [ ρ ( x, t )] d x, s (0) = s (1) = 0 . (12) Both the form of ¯ V ( x ) and of s ( ρ ) can be determined byimposing the condition in Eq. (10). This result is some-times called the “ H theorem” in the literature [25, 35, 36].In particular, the Boltzmann–Gibbs entropy will be re-covered for linear FPEs, while other commonly used gen-eralized entropies are naturally associated to a wide classof nonlinear FPEs [25, 37, 38]. Differentiating Eq. (9) wehaved Φd t = Z Ξ ∂ρ∂t ∂φ ( x, y ) ∂y (cid:12)(cid:12)(cid:12)(cid:12) y = ρ d x = − Z Ξ (cid:20) − E ( x ) χ ( ρ ) + D ( x ) ω ( ρ ) ∂ρ∂x (cid:21) ∂∂x ∂φ ( x, y ) ∂y (cid:12)(cid:12)(cid:12)(cid:12) y = ρ d x = − Z Ξ d x D ( x ) χ ( ρ ) (cid:20) − E ( x ) D ( x ) + ω ( ρ ) χ ( ρ ) ∂ρ∂x (cid:21) × " d ¯ V ( x )d x − d s ( y )d y (cid:12)(cid:12)(cid:12)(cid:12) y = ρ ∂ρ∂x . (13)We have omitted the explicit dependency on x and t in ρ for simplicity of notation. Comparing the previous resultwith the condition in Eq. (10), we have that the inequal-ity is always satisfied ifd ¯ V ( x )d x = − E ( x ) D ( x ) = 1 D ( x ) d V ( x )d x , (14a) − d s ( ρ )d ρ = ω ( ρ ) χ ( ρ ) , s (0) = s (1) = 0 . (14b)Similar equations have been obtained by Schwämmle et al. [25, 26] in the case of a homogeneous Fokker–Planckequation, i.e., D ( x ) ≡ D = constant. In particular, werecover the same expression for the entropy, while thepotential is replaced by a more general “effective” po-tential that reduces to the usual one in the homogeneouscase. Indeed, in the case of a constant friction coefficient,the natural identification ¯ V ( x ) ≡ βV ( x ) holds, where β = D − . Being D ( x ) >
0, if V ( x ) is bounded frombelow, then ¯ V ( x ) is bounded from below as well. Thisimplies that the functional Φ is bounded from below. Westress again that the expression for Φ and the inequalityin Eq. (10) has been obtained assuming zero current onthe boundary of the domain. A. Properties of the functional Φ Let us now discuss some properties of the function Φ.As in the homogeneous case, s ( ρ ) is strictly concave, andthe rescaled free-energy Φ is strictly convex with respectto ρ , δ Φ δρ = − d s ( y )d y (cid:12)(cid:12)(cid:12)(cid:12) y = ρ ( x,t ) = ω ( ρ ) χ ( ρ ) > . (15)The relation above can be written as ω ( ρ ) = χ ( ρ ) ∂ ρ φ ( x, ρ ). Moreover, Eq. (6) becomes ∂ρ∂t = ∂∂x (cid:18) χ ( ρ ) D ( x ) ∂∂x δ Φ δρ (cid:19) ,j ( x, ρ ) = − D ( x ) χ ( ρ ) ∂∂x δ Φ δρ . (16)In a relaxation process towards the equilibrium distribu-tion % ( x ), using the boundary condition in Eq. (8), thetime derivative of the entropy S ( ρ ) isd S ( ρ )d t = Z Ξ j ( x, ρ ) χ ( ρ ) D ( x ) d x − Z Ξ j ( x, ρ ) E ( x ) D ( x ) d x. (17)Equation (17) is, up to a global positive factor, a general-ization of a corresponding expression obtained by Casas et al. [39] for the case D ( x ) = D = constant. In partic-ular, the first integral can be identified with the entropyproduction contribution, and it is always positive. Bothterms in the equation approach zero as ρ ( x, t ) → % ( x ).Finally, introducing the average energy U ( ρ ) := Z Ξ V ( x ) ρ ( x, t ) d x, (18)the energy dissipation rate is given by˙ U = − Z Ξ E ( x ) j ( x, ρ ) d x. (19)In the case of a homogeneous medium, D ( x ) ≡ D , wecan define the free energy density and the free energy as f ( x, ρ ) := Dφ ( x, ρ ) , F ( ρ ) := Z Ξ f ( x, ρ ) d x, (20)respectively. The free energy density satisfies the relation D ( ρ ) := Dω ( ρ ) = χ ( ρ ) ∂ ρ f ( x, ρ ) , (21)that has the structure of a local Einstein fluctuation–dissipation relation. Moreover, the FPE can be writtenas ∂ρ∂t = ∂∂x (cid:18) χ ( ρ ) ∂∂x δFδρ (cid:19) , j ( x, ρ ) = − χ ( ρ ) ∂∂x δFδρ . (22)In the homogeneous case, Eq. (17) has a more clear inter-pretation. Indeed, in an electromagnetic analogy, we canthink of j as a current of charges in an external electricfield E , flowing in a medium whose resistance is given by χ . Therefore, the first term plays the role of a dissipationpower contribution, whilst the second term is related tothe rate of energy exchange between the external fieldand the charges, and it indeed coincides with Eq. (19) upto a constant multiplicative factor. III. EQUILIBRIUM SOLUTION ON THE REALLINE
Let us assume now that our domain is the real line,i.e., Ξ ≡ R . Adopting reflecting boundary conditions, itis immediately seen that the equation for the stationarysolution of Eq. (6) is the equation for the equilibriumstate % , i.e. j ( x, % ) = 0 ⇔ d % ( x )d x = E ( x ) χ ( % ) D ( x ) ω ( % ) . (23)On the other hand, because of the required integrabilityof % , we ask lim x →±∞ % ( x ) = lim x →±∞ d % ( x )d x = 0 . (24)To prove that the limit distribution, for any initial con-dition, is uniquely identified, and coincides with % , wefirst follow the arguments of Frank and Daffertshofer [35]for the homogeneous case, with proper modifications toadapt them to our case. From Eq. (23), we have E ( x ) D ( x ) = − d ¯ V ( x )d x = ω ( % ) χ ( % ) d % ( x )d x = − dd x " d s ( z )d z (cid:12)(cid:12)(cid:12)(cid:12) z = % . (25)Let us now introduce the functionexp s ( x ) := (cid:20) d s d x (cid:21) − ( − x ) , (26)inverse of the function ∂ p s ( p ) evaluated in − x . The func-tion in Eq. (26) exists, being ∂ ρ s ( ρ ) < ∂ ρ s ( ρ ) is strictly decreasing for ρ >
0. Denoting by x := − lim ρ → ∂ ρ s ( ρ ) < , (27)the function exp s ( x ) is positive and strictly increasingin ( x , + ∞ ) (here x can be also not finite) and weimpose that it is identically zero in ( −∞ , x ), beinglim x → x +0 exp s ( x ) = 0. For future convenience, we definealsolog s ( x ) = exp − s ( x ) , log s ( x ) : R +0 → ( x , + ∞ ) , (28)inverse function of exp s ( x ). For a linear FPE, s ( p ) = − p ln p , then exp s ( x ) = e x − and log s ( x ) = ln x + 1. Bymeans of the introduced function, we can write, for some c to be determined, % ( x ) = exp s (cid:2) c − ¯ V ( x ) (cid:3) . (29)In the previous expression, we have supposed that thearbitrary additive constant in ¯ V is somehow fixed. It isimportant to prove that the normalization constant c inEq. (29) exists, and that it is uniquely identified. Forthis purpose, observe that the function h ( y ) := + ∞ Z −∞ exp s (cid:2) y − ¯ V ( x ) (cid:3) d x (30)is strictly monotonically increasing, being h ( y ) = − + ∞ Z −∞ d xs (cid:8) exp s (cid:2) y − ¯ V ( x ) (cid:3)(cid:9) > . (31)Moreover, we have that lim y →−∞ h ( y ) = 0 and h ( y ) ≥ + ∞ Z −∞ θ (cid:0) y − V ( x ) (cid:1) exp s (cid:2) y − ¯ V ( x ) (cid:3) d x ≥ exp s (1) + ∞ Z −∞ θ (cid:0) y − V ( x ) (cid:1) d x y →∞ −−−→ + ∞ . (32)It follows that the normalization constant exists and it isunique. The constant c , uniquely identified, satisfies theidentity c ≡ ¯ V ( x ) + log s [ % ( x )]. Using the fact that ¯ V isdefined up to an additive constant, we can absorb c inthe auxiliar potential, in such a way that¯ V ( x ) = − log s [ % ( x )] , (33)This gives δ ρ Φ | ρ = % = 0. Moreover, the convexity of Φimplies that % is a minimum for Φ. On the other hand,from Eq. (13) we have thatd Φd t (cid:12)(cid:12)(cid:12)(cid:12) ρ = % = 0 . (34)The uniqueness of the stationary solution in the hypothe-ses specified, together with the result in Eq. (10), guar-antees that the limit distribution is the stationary distri-bution.This result can be proven in a different, and more ex-plicit, way, without invoking Eq. (10). Let us supposethat the entropy density s ( ρ ) is known from the coeffi-cients in Eq. (6). From the functional S ( ρ ), and follow-ing again the approach of Frank and Daffertshofer [35],we can construct the following generalized divergence be-tween two probability distribution densities ρ = ρ ( x, t )and ρ = ρ ( x, t ),∆ s ( ρ k ρ ) := S ( ρ ) − S ( ρ )++ + ∞ Z −∞ " ( ρ ( x, t ) − ρ ( x, t )) ∂s ( z ) ∂z (cid:12)(cid:12)(cid:12)(cid:12) z = ρ d x. (35)The quantity ∆ s is always non-negative, due to thestrict concavity of the function s ( ρ ). In particular,∆ s ( ρ k ρ ) = 0 ⇔ ρ = ρ . In the case of Boltzmann–Gibbs entropy, s ( ρ ) = − ρ ln ρ , ∆ s is the Kullback–Leiblerdivergence [40]∆ BG ( ρ k ρ ) ≡ D KL ( ρ k ρ ) := + ∞ Z −∞ ρ ( x, t ) ln ρ ( x, t ) ρ ( x, t ) d x. (36) If we consider now the case ρ ≡ % ( x ), unique stationarysolution of our equation, and ρ = ρ ( x, t ) a solution atthe time t of the considered nonlinear FPE for a giveninitial condition, we haved ∆ s ( ρ, % )d t = − ∞ Z −∞ D ( x ) χ ( ρ ) ω ( % ) χ ( % ) ∂%∂x (cid:12)(cid:12)(cid:12)(cid:12) % = %% = ρ ! d x ≤ . (37)It follows then thatlim t →∞ ρ ( x, t ) = % ( x ) . (38) A. Effective temperature
It is clear that, in the general case, a definition of atemperature-like quantity is difficult, and the meaningitself of “temperature” might depend on the specific con-sidered model [41]. A possible definition of effective in-verse temperature ¯ β , however, can be given by the usualthermodynamic relation¯ β := ∂S ( % ) ∂U ( % ) , (39)expressing with the variation of entropy respect to theaverage energy with reference to the equilibrium state,which is supposed to be unique. Observing that % is aminimum of the functional Φ( ρ ), we can write the previ-ous relation as ¯ β = ∂ ¯ U ( % ) ∂U ( % ) . (40)In particular, if D ( x ) ≡ D = constant, then we obtainthe well-known result ¯ β = 1 D > . (41) IV. A NONLINEAR FPE FOR DIFFUSION ININHOMOGENEOUS POROUS MEDIA
From the observations above, it is evident that theknowledge of the external potential V ( x ) and of the en-tropic form s ( ρ ) is not sufficient to identify the station-ary distribution. On the other hand, the observation ofa specific limit distribution does not allow us to inferthe entropic form, unless a careful investigation of thestructure of the effective potential is performed. Thissimple fact has been already pointed out in the study ofelementary probabilistic toy models [42–44]. To furtherexemplify it and apply our formalism, let us consider, forexample, the nonlinear inhomogeneous FPE for a fluid ina porous medium. The conservation of mass imposesd ρ ( x, t )d t = ∂ρ ( x, t ) ∂t + ∂∂x [ ˙ xρ ( x, t )] = 0 (42)for the density ρ ( x, t ). On the other hand, in the case ofdiffusion in a porous medium, Darcy’s law holds [8],˙ x = E ( x ) − κ ( x ) ∂P ( x, t ) ∂x , κ ( x ) > , (43)where P ( x, t ) is the pressure of the fluid at the position x and at time t , and E ( x ) = − ∂ x V ( x ) is an externalforce (we are working in the overdamping limit) result-ing from an external potential V ( x ). The dependence ofthe coefficient κ ( x ) on x expresses exactly the lack of ho-mogeneity of the medium (e.g., a porosity depending on x ). Imposing the equation for polytropic gases P ( x, t ) = αρ λ ( x, t ) , λ > , α > , (44)we get the porous medium equation in the presence of anexternal field ∂ρ ( x, t ) ∂t = ∂∂x (cid:20) − E ( x ) ρ ( x, t ) + D ( x ) ρ ν − ( x, t ) ∂ρ ( x, t ) ∂x (cid:21) , (45)where we have defined ν = λ + 1 to uniform our notationwith that adopted in Refs. [7, 11], and D ( x ) := α ( ν − κ ( x ) . (46)Diffusion equations similar to Eq. (45) have been pro-posed, for example, in Refs. [7, 16, 45] with E ( x ) = − γx , γ >
0, for the velocity distribution of one particle in onedimension. In particular, Plastino and Plastino [7] con-sidered the case D ( x ) ≡ D >
0. Kaniadakis and Lapenta[45] assumed a time-dependent diffusion coefficient D ( t )that is homogeneous in space. In Ref. [45] the linear( ν = 1) inhomogeneous case is also considered. Equa-tions (14) take the formd ¯ V ( x )d x = − E ( x ) D ( x ) , (47a)d s ( ρ )d ρ = − ρ ν − , s (0) = s (1) = 0 . (47b)The last equation gives, in particular, s ( ρ ) = ρ − ρ ν ν ( ν − ≡ s ν ( ρ ) ν , (48)where s q ( ρ ) is the nonadditive entropy introduced inRefs. [46, 47] S q ( ρ ) := + ∞ Z −∞ s q [ ρ ( x, t )] d x = 1 − R + ∞−∞ ρ q ( x, t ) d xq − . (49)Equation (23) gives the stationary solution % ( x ) = ρ e − ¯ V ( x ) − ¯ V (0) ρν − − ν , (50)where ρ is fixed by the normalization condition. In theprevious expression we have introduced the so-called q -exponential function with q ∈ R ,e xq := [1 + (1 − q ) x ] − q + , with [ x ] + := x θ ( x ) . (51) The q -exponential is indeed related to the exp s q ( x ) inEq. (26), being exp s q ( x ) = e x − q − q . (52)Even at fixed entropic form, we can therefore obtain awide class of limit distributions by an appropriate choiceof the argument of the q -exponential, and in particularof D ( x ). Namely, to have a limit distribution % ( x ), itsuffices that E ( x ) D ( x ) = [ % ( x )] ν − d % ( x )d x . (53)As an example, let us consider a q -Gaussian limit dis-tribution % q ( x ) = √ aC q e − ax q , a > , q < , (54)where we have introduced the normalization constant C q := √ π Γ ( − q ) (3 − q ) √ − q Γ (cid:0) − q − q ) (cid:1) if q < , √ π if q = 1 , √ π Γ (cid:0) − q q − (cid:1) √ q − ( q − ) if q ∈ (1 , . (55)The Gaussian distribution with parameter a > q →
1. It is immediatelyseen that the following relation between E and D musthold, D ( x ) = − (cid:2) − a (1 − q ) x (cid:3) + ax [ % q ( x )] ν − E ( x ) . (56)If, for example, we consider the case of Boltzmann–Gibbsentropy ( ν = 1), we have D ( x ) = − (cid:2) − a (1 − q ) x (cid:3) + ax E ( x ) . (57)In particular, in the presence of an external harmonicpotential, V ( x ) = γx ⇒ E ( x ) = − γx, γ > , (58)it is sufficient to modulate D ( x ) as D ( x ) = γ (cid:2) − a (1 − q ) x (cid:3) + a , q ≥ , (59)to obtain as a stationary distribution a q -Gaussian, de-spite the fact that the entropy associated to the consid-ered equation is the Boltzmann–Gibbs entropy and theexternal potential is harmonic. Observe also that for q → + we recover the well known result D ( x ) = constant. Aparticular case, in a different formalism, has been ana-lyzed in Refs. [45, 48] for a specific choice of the coef-ficients E ( x ) and D ( x ), that indeed satisfy the relationabove. V. CONCLUSIONS AND PERSPECTIVES
In the present paper we have discussed a generalizationto the inhomogeneous case of the free-energy formalismintroduced by Schwämmle et al. [25] for the study of non-linear FPEs. We have shown that a modified free-energyfunctional Φ, defined on the space of distributions, can beintroduced, in such a way that the stationary solution isa minimum for Φ. This functional is explicitly expressedin terms of the coefficients of the considered FPE, andit involves an auxiliary potential and an entropic den-sity. We have also shown that, in a relaxation processtowards the stationary distribution on the real line, Φdecreases monotonically, reaching the minimum value onthe stationary distribution. Some basic properties of thestationary solutions of nonlinear FPEs have been ana-lyzed. In particular, the solutions can be expressed interms of a generalized exponential function associated tothe entropy, having the auxiliary potential appearing inΦ as argument. We have then applied our formalism toa nonlinear FPE for the macroscopic description of dif-fusion processes in inhomogeneous porous media. The full understanding of the thermodynamical mean-ing of the discussed free-energy functionals is still an ac-tive research topic [49]. As discussed in the Introduction,a FPE is usually obtained from a KKE that properly de-scribes the evolution of the density in the one-particlephase space. In this sense the relation between the ther-modynamical functionals introduced for a general non-linear FPE and the ones for the corresponding KKE de-serves further analysis. Further investigations are alsoneeded in light of the fast-developing field of stochas-tic thermodynamics [50, 51] and in relation to macro-scopic fluctuation theory [22, 23]. The possible connec-tions between the entropy associated to a FPE followingthe recipe presented here, and fluctuation theory mightshed new light on foundational aspects of thermodynam-ics, apart from possible experimental applications. Wehope to address these problems in future publications.
VI. ACKNOWLEDGMENTS
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