Curl Forces and the Nonlinear Fokker-Planck Equation
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] S e p Curl Forces and the Nonlinear Fokker-Planck Equation
R.S. Wedemann , A.R. Plastino and C. Tsallis , Instituto de Matem´atica e Estat´ıstica,Universidade do Estado do Rio de JaneiroRua S˜ao Francisco Xavier, 524, 20550-900, Rio de Janeiro, RJ, Brazil CeBio y Secretar´ıa de Investigaci´on,Universidad Nacional Buenos Aires - Noroeste,UNNOBA-Conicet, Roque Saenz Pe˜na 456, Junin, Argentina Centro Brasileiro de Pesquisas F´ısicas andNational Institute of Science and Technology for Complex Systems,Rua Xavier Sigaud 150, 22290-180, Rio de Janeiro - RJ, Brazil Santa Fe Institute, 1399 Hyde Park Road, Santa Fe, New Mexico 87501, USA
Nonlinear Fokker-Planck equations endowed with curl drift forces are investigated.The conditions under which these evolution equations admit stationary solutions,which are q -exponentials of an appropriate potential function, are determined. Itis proved that when these stationary solutions exist, the nonlinear Fokker-Planckequations satisfy an H -theorem in terms of a free-energy like quantity involvingthe S q entropy. A particular two dimensional model admitting analytical, time-dependent, q -Gaussian solutions is discussed in detail. This model describes a systemof particles with short-range interactions, performing overdamped motion under drageffects due to a rotating resisting medium. It is related to models that have beenrecently applied to the study of type-II superconductors. The relevance of the presentdevelopments to the study of complex systems in physics, astronomy, and biology, isdiscussed.Keywords: Nonlinear Fokker-Planck Equation, Curl Forces, Nonadditive Entropies, q -Gaussian Exact Solutions PACS numbers: 05.20.-y, 05.70.Ln, 05.90.+m a E-mail address: [email protected]
I. INTRODUCTION
The nonlinear Fokker-Planck equation [1] constitutes a powerful tool for the study ofdiverse phenomena in complex systems [2–8], with applications including (among many oth-ers) type-II superconductors [9], granular media [10], and self-gravitating systems [11, 12].It governs the behavior of a time-dependent density F ( x , t ), where x ∈ ℜ N designates alocation in an N -dimensional configuration space. The evolution of F is determined by twoterms: a nonlinear diffusion [13, 14] term and a linear drift term (more general equationswith nonlinear drift terms have also been proposed, but we are not going to consider themin the present work). In several of the above mentioned applications, the density F is areal physical density (as opposed to a statistical ensemble probability density) describingthe evolving distribution of a set of interacting particles executing overdamped motion, inthe relevant configuration space [8, 15]. In these kind of scenarios, the nonlinear diffusionterm constitutes an effective description of the interaction between the particles, while thedrift term describes the effects of other external forces acting upon them. The nonlin-ear Fokker-Planck equations recently addressed in the literature exhibit several interestingand physically relevant properties. They obey an H -theorem in terms of a free-energy-likequantity [16]. In some important cases, the nonlinear Fokker-Planck equations admit exactanalytical solutions of the q -Gaussian form, that can be interpreted as maximum entropy( q -maxent) densities obtainable from the optimization under appropriate constraints, of thenonadditive power-law entropic functionals, S q [17, 18]. Indeed, there is a deep connectionbetween the nonlinear Fokker-Planck dynamics and the generalized thermostatistics basedon the S q entropies. Although this connection was first pointed out more than twenty yearsago [19], its full physical implications are being systematically explored only in recent years(see, for instance, [2, 15, 20–22] and references therein). A remarkable example of thistrend is given by experimental work on granular media published in 2015 [10] that verifies,within a 2% error and for a wide experimental range, a scale relation predicted in 1996 onthe basis of the theoretical analysis of q -Gaussian solutions of the nonlinear Fokker-Planckequation [23]. The particular case of the nonlinear Fokker-Planck equation with vanishingdrift corresponds to the porous media equation.Virtually all the literature on the nonlinear Fokker-Planck equation and its applicationsdeals with Fokker-Planck equations in which the drift forces K can be derived from apotential function V ( x ), leading to stationary densities which are q -exponentials of thepotential V . In the present contribution, we consider more general scenarios where thedrift force K has, besides a component given by minus the gradient of a potential V ,a term ˜ K that does not come from a potential. In two or three space dimensions, thissituation corresponds to having forces exhibiting a non vanishing rotational or curl , whichare usually referred to as curl forces [24]. The incorporation of curl forces enriches thedynamical features of the nonlinear Fokker-Planck equations, enabling it to describe a widerset of phenomena. Curl forces, although not dynamically fundamental [25], are neverthelessrelevant as useful effective descriptions of diverse physical problems, as for example thenonconservative force fields generated by optical tweezers [26]. Dynamical systems withcurl forces have interesting properties that are not yet fully understood and are the subjectof current research [25, 27]. In the present work, we investigate the behavior of nonlinearFokker-Planck equations under the presence of curl forces. We determine the conditionsunder which these evolution equations admit stationary solutions of the q -maxent formand satisfy an H -theorem. We also discuss in detail a two dimensional example admittinganalytical time-dependent solutions, that describes a set of interacting particles undergoingoverdamped motion, under the drag effect arising from a uniformly rotating medium. II. THE NONLINEAR FOKKER-PLANCK EQUATION
In the present work, we shall consider nonlinear Fokker-Planck equations (NLFP) of theform ∂F∂t = D ∇ [ F − q ] − ∇ · [ F K ] , (1)where F ( x , t ) is a time-dependent density, D is a diffusion constant, K ( x ) is a drift force, and q is a real parameter characterizing the (power-law) nonlinearity appearing in the diffusionterm. The density F is a dimensionless quantity of the form F = ρ ( x , t ) /ρ , where ρ hasdimensions of inverse volume and ρ is a constant with the same dimensions as ρ . Therefore,the dimensional density ρ ( x , t ) obeys the evolution equation ∂ [ ρ/ρ ] /∂t = D ∇ [( ρ/ρ ) − q ] − ∇ · [( ρ/ρ ) K ]. As already mentioned, in the most frequently studied case of Eq.(1), thedrift force K is assumed to arise from a potential function V ( x ), K = − ∇ V . (2)The stationary solutions of the NLFP then satisfy ∇ h D ∇ (cid:0) F − q (cid:1) + F ( ∇ V ) i = 0 . (3)Let us consider the q -statistical ansatz [18] F q = A exp q [ − βV ( x )]= A [1 − (1 − q ) βV ( x )] − q + , (4)where A and β are constants to be determined, and the function exp q ( z ) = [1 + (1 − q ) z ] − q + ,usually referred to as the q -exponential function, vanishes whenever 1 + (1 − q ) z ≤
0. Onefinds that the ansatz given by Eq.(4) complies with the equation D ∇ (cid:0) F − q (cid:1) + F ( ∇ V ) = 0 , (5)if (2 − q ) βD = A q − . (6)It therefore satisfies also equation (3) and constitutes a stationary solution of the NLFPequation. In summary, the q -exponential ansatz (4) is a stationary solution of the NLFPequation, if the drift force K is derived from a potential and A and β satisfy the relation (6).We shall assume that the stationary distribution F q has a finite norm, that is, R F q d N x = I < ∞ . The specific conditions required for F q to have a finite norm (such as the allowedrange of q -values) cannot be stated in general, because they depend on the particular form ofthe potential function V ( x ). Since in many applications the solution of the NLFP equationis interpreted as a physical density (as opposed to a probability density) we assume finitenorm, but not necessarily normalization to unity. The stationary density F q can be regardedas a q -maxent distribution, because it maximizes the nonextensive q -entropic functional S q under the constraints corresponding to the norm and the mean value of the potential V [18, 19].In the limit q →
1, the standard linear Fokker-Planck equation, ∂F∂t = D ∇ F − ∇ · [ F K ] , (7)is recovered. In this limit, the q -maxent stationary density (4) reduces to the exponential,Boltzmann-Gibbs-like density, F BG = 1 Z exp[ − D V ( x )] , (8)with the condition (6) becoming βD = 1, independent of the normalization constant A .The density F BG is normalized to one provided that Z = R exp[ − D V ( x )] d x . The density F BG optimizes the Boltzmann-Gibbs entropy S BG = − R F ln F d x , under the constraints ofnormalization and the mean value h V i of the potential V .Note that a dynamical system with a phase space flux of the form (2) (that is, of a gradientform) evolves always down-hill on the potential energy landscape, so as to minimize thepotential energy function V ( x ). The components { K i , i = 1 , . . . , N } of such a field satisfy ∂K i ∂x j = ∂K j ∂x i = ∂ V∂x i ∂x j , (9)which in two or three dimensions leads to K = − ∇ V ⇐⇒ ∇ × K = 0 . III. NONLINEAR FOKKER-PLANCK EQUATION WITH CURL DRIFTFORCES: STATIONARY SOLUTIONS
Now we consider NLFP equations endowed with drift forces having two terms, one ex-hibiting the gradient form and the other one not arising from the gradient of a potential.That is, we consider drift fields of the form K = G + ˜ K , (10)where the force G is equal to minus the gradient of some potential function V ( x ), whilethe component ˜ K does not come from a potential (that is, ∂ ˜ K i /∂x j = ∂ ˜ K j /∂x i ). Our aimis to determine under which conditions a density proportional to the q -exponential of thepotential V still provides a stationary solution of the NLFP equation, preserving thus thelink between this equation and the generalized nonextensive thermostatistics. Substitutingthe above dirft force K and q -exponential density F q (4) into the stationary NLFP equation(3), one obtains D ∇ [ F − qq ] + ∇ · [ F q ( ∇ V )] − ∇ [ F q ˜ K ] = 0 . (11)It can be verified that, if A and β satisfy (6), the sum of the first two terms in the aboveequation vanish, since F q is a stationary solution of the NLFP equation (3), when the driftfield K consists solely of the gradient field G . In order for F q to comply also with the fullNLFP equation (11), including the drift contribution associated with the non-gradient field˜ K , it is then necessary that ∇ [ F q ˜ K ] = 0 . (12)If the above relation is satisfied, the density F q constitutes a stationary solution of the fullNLFP equation, corresponding to the complete drift force K = − ( ∇ V ) + ˜ K . To have the q -maxent stationary solution, one therefore requires ∇ (cid:16) ˜ K A [1 − (1 − q ) βV ] − q (cid:17) = 0 , (13)which in turn leads to the following relation between the non-gradient drift component ˜ K and the potential function V ( x )[1 − (1 − q ) βV ]( ∇ · ˜ K ) − β ( ˜ K · ∇ V ) = 0 . (14)This is a consistency relation that the potential function V , the non-gradient force field ˜ K , the Lagrange multiplier β , and the entropic parameter q have to satisfy, in order thatthe nonlinear Fokker-Planck equation admits the q -maxent stationary solution (4). Thegeneral β -dependent equation (14) constitutes a rather complicated relation between thenon-gradient field ˜ K and the potential function V , which is difficult to characterize. More-over, this relation depends explicitly on the value of β . This means that for given formsof ˜ K ( x ) and V ( x ), one may have stationary solutions of the q -maxent form (4), only forparticular values of β .It follows from the relation (14) that, in order for the NLFP equation to admit the β -parameterized family of stationary solutions (4), with a continuous allowed range of β -values,two conditions have to be fulfilled. On the one hand, the non-gradient component of thedrift, ˜ K , has to be a divergenceless vector field, ∇ · ˜ K = 0 . (15)On the other hand, ˜ K has to be everywhere orthogonal to the gradient of the potential, ˜ K · ( ∇ V ) = 0 . (16)Notice that conditions (15) and (16) are not only sufficient, but also necessary conditionsfor the ansatz (4) to be a stationary solution of the NLFP equation (1), for a continuousrange of β -values. Indeed, if (4) solves (1) for such a set of β -values, the left hand sideof (14), which is an inhomogeneous linear function of β , has to vanish for an interval ofvalues of β . This clearly implies that both the independent term, and the coefficient of the β -linear term, have to vanish individually, leading in turn to conditions (15) and (16). Itis intersting that these conditions do not explicitly depend on the value of the q -parameter,constituting a q -invariant structure. The stationary solution guaranteed by these conditionsis a physical solution when it is normalizable (otherwise, it is not physical, although stillformally a solution of the NLFP equation). The normalizability of the stationary solutiondepends on the particular shape of the potential V and on the value of q and, as alreadymentioned, can only be studied in a case by case way.In two or three space dimensions, the decomposition K = G + ˜ K , with G = − ∇ V and ∇ · ˜ K = 0, resembles the decomposition of a vector field into a curlless (irrotational) com-ponent and a solenoidal (divergenceless) component arising from the celebrated Helmholtztheorem [28]. We are not, however, imposing the boundary conditions on the fields K , G , and ˜ K , that are usually considered in connection with the Hemlholtz decomposition.Furthermore, we require the point to point orthogonality of the irrotational and the diver-genceless components of K , which is not a condition usually considered in connection withthe Helmholtz decomposition.It is interesting that the Helmholtz-like decomposition (10), with orthogonal irrotationaland divergenceless parts, G · ˜ K = 0, arises naturally in some circumstances. For instance,the most general rotationally invariant vector field in two dimensions has precisely this form.Indeed, such vector fields are of the form G = − g ( r ) e r , ˜ K = l ( r ) e θ , (17)where g ( r ) and l ( r ) are functions of the radial coordinate r = ( x + y ) / and e r and e θ respectively denote the radial and tangential unit vectors. It is clear that the field G in (17)is of the form − ∇ V ( r ) with V ( r ) = R r ′ g ( r ′ ) dr ′ , and that the field ˜ K satisfies ∇ · ˜ K = 0and G · ˜ K = 0.Summing up, we have thus determined that the NLFP equation (1) having a non-potentialdrift force of the form (10) admits, for a continuous range of values of the parameter β , thefamily of q -maxent stationary solutions (4) if and only if the relations (15) and (16) aresatisfied. IV. H -THEOREM We are now going to explore the possibility of formulating an H -theorem for the nonlinearFokker-Planck equations, endowed with a drift term involving a non-vanishing-curl force ˜ K ,not derivable from the potential function V . Let us first consider the time derivative of thepower law entropic functional S q ∗ , with q ∗ = 2 − q . This is a reasonable choice, because q ∗ is precisely the exponent that appears inside the Laplacian term in the NLFP equation (1).The duality q → − q appears frequently in the q -generalized thermostatistical formalism[18]. We have, dS q ∗ dt = q ∗ − q ∗ Z F q ∗ − ∂F∂t d N x = Dq ∗ Z F q ∗ − | ∇ F | d N x + q Z F q ∗ − ( ∇ F ) · ( ∇ V ) d N x + Z F q ∗ (cid:16) ∇ · ˜ K (cid:17) d N x . (18)It is clear that the first term in the above expression is definite positive. However, the secondterm does not have a definite sign. Consequently, the time derivative of S q ∗ does not havea definite sign and the entropic form S q ∗ does not itself verify an H -theorem. The last twoterms in the expression for dS q ∗ dt , describing the contribution of the drift term to the change inthe entropy, suggest that a linear combination of S q ∗ and of the mean value of the potentialfunction V may comply with an H -theorem. The time derivative of h V i = R F V d N x is d h V i dt = Z V ∂F∂t d N x = − q ∗ D Z F q ∗ − ( ∇ F ) · ( ∇ V ) d N x − Z F | ∇ V | d N x + Z F ( ∇ V ) · ˜ K d N x . (19)Combining now equations (18) and (19) one obtains, after some algebra, ddt ( DS q ∗ − h V i ) = Z F (cid:12)(cid:12)(cid:12) q ∗ DF q ∗ − ( ∇ F ) + ∇ V (cid:12)(cid:12)(cid:12) d N x + Z F q ∗ (cid:16) ∇ · ˜ K (cid:17) d N x + Z F ( ∇ V ) · ˜ K d N x . (20)If the curl component ˜ K of the drift force complies with the requirements given by equations(15) and (16), which are necessary and sufficient for the nonlinear Fokker-Planck equation tohave the family of q -maxent stationary solutions (4), it follows from (20) that the nonlinearFokker-Planck equations satisfies the H -theorem, ddt ( DS q ∗ − h V i ) = Z F (cid:12)(cid:12)(cid:12) q ∗ DF q ∗ − ( ∇ F ) + ∇ V (cid:12)(cid:12)(cid:12) d N x = D(cid:12)(cid:12)(cid:12) q ∗ DF q ∗ − ( ∇ F ) + ∇ V (cid:12)(cid:12)(cid:12) E ≥ . (21)It is worth stressing that the conditions (15) and (16) for having stationary q -maxent solu-tions are essentially the same as those for having an H -theorem.There is an interesting consequence of the H theorem, in relation with the uniquenessof the decomposition (10) of the total drift force K into a gradient component G = − ∇ V and an (orthogonal) divergenceless component ˜ K . Let us assume that that total drift forcecan be decomposed in this fashion in two different ways, K = − ∇ V + ˜ K = − ∇ V + ˜ K .If the nonlinear Fokker-Planck equation admits a stationary solution (of finite norm) F st , itfollows from the H -theorem (21) that ∇ V = ∇ V = − q ∗ DF q ∗ − ( ∇ F st ) , (22)which, in turn, implies also that ˜ K = ˜ K . Consequently, if the nonlinear Fokker-Planckequation admits a stationary solution, the decomposition of the total drift force into thesum of a gradient term and a divergenceless term is unique. V. QUADRATIC POTENTIAL AND LINEAR DRIFT
We now consider in detail the case of a quadratic potential V and a linear drift ˜ K .We shall se that in this case the conditions (15) and (16) are required even for having astationary solution of the q -exponential form (4) for one, single value of the parameter β .We assume and potential and a drift field respectively of the forms, V ( x ) = X ij ( a ij x i x j ) + X i ( b i x i ) , (23) ˜ K ( x ) = X j ( c ij x j ) + d i , (24)with the a ij , c ij , b i and d i constant coefficients. We can assume a ij = a ji , although the c ij arenot necessarily symmetric. Equation (14) leads to a set of constraints on these coefficients,thus defining V ( x ) and ˜ K ( x ). If we substitute equations (23) and (24) in Eq.(14), we obtain ( − (1 − q ) β "X ij ( a ij x i x j ) + X i ( b i x i ) k c kk ! − β X k "X i ( c ki x i ) + d k j (( a kj + a jk ) x j ) + b k = 0 . (25)0Equation (25) is a second degree polynomial in the x i ’s that is equal to zero. Since thisequality should hold for any value of x , the coefficients of the different powers of the x i should each be equal to zero. Therefore, by separately equating to zero the independentzero-th, first and second order terms in the left-hand side of Eq.(25), one obtains X k ( c kk − βd k b k ) = 0 , (26a) X k [(1 − q ) c kk b i + c ki b k + ( a ki + a ik ) d k ] = 0 , ∀ i , (26b) X k [(1 − q ) c kk ( a ij + a ji ) + c ki ( a kj + a jk ) + c kj ( a ki + a ik )] = 0 , ∀ i, j . (26c)With symmetric a ij , we shall now assumedet | a ij | 6 = 0 . (27)This assumption is also necessary if V ( x ) should represent a confining potential, leading toa normalizable stationary state of the nonlinear Fokker-Planck equation.If we introduce an appropriate shift in the x i coordinates, it is possible to work using apotential V ( x ) (Eq.(23)) with no linear terms. We thus define x i = x i − r i , (28)so that the r i are constants that can be derived from constraints, as we will show. We canthen express Eq.(23) in terms of the x i as V ( x ) = X ij a ij [ x i x j + ( x i r j + x j r i ) + r i r j ] + X i b i ( x i + r i ) . (29)The linear terms in Eq.(29) are now X i ("X j ( a ij r j + a ji r j ) + b i ) x i , (30)and they will vanish if the r j ’s satisfy, b i + X j ( a ij + a ji ) r j = 0 , or b i + 2 X j a ij r j = 0 , i = 1 , . . . , N . (31)1The N equations (31) can be solved for the r j ’s because the condition in Eq.(27) holds. Theconstant term ( P i b i r i ) + (cid:16)P ij a ij r i r j (cid:17) in the potential V can be ignored and eliminated:since the potential enters the NLFP equation only through its gradient, this constant termhas no physical significance. Therefore, in terms of the shifted coordinates x i , we have V ( x ) = X ij a ij x i x j , (32a)˜ K i ( x ) = X j ( c ij x j ) + d i , (32b)where d i = P j ( c ij r j ) + d i . We thus see that, after an appropriate shift in the phase spacevariables, the problem reduces to that of a homogeneous, quadratic potential.If the associated nonlinear Fokker-Planck equation admits a q -maxent stationary solution,even for one single value of β , it follows from Eq.(26a) that we must have X j c jj = 0 = ⇒ ∇ · ˜ K = 0 , (33)from which it follows that the condition ˜ K · ∇ V = 0 also follows. In other words, for aquadratic potential V and a linear drift ˜ K , if one has a q -maxent stationary solution evenfor one single value of β , it is possible after a coordinates shift to recast the system in termsof a drift field complying with conditions (15) and (16). VI. TWO DIMENSIONAL SYSTEM WITH EXACT TIME-DEPENDENT q -GAUSSIAN SOLUTIONS. We now consider, as an example of a time-dependent solution of a nonlinear Fokker-Planckequation with a ˜ K not arising from a potential, that admits a q -maxent stationary solution,a bi-dimensional system submitted to the following quadratic potential and nongradientlinear drift term. For simplicity of notation, we will name the phase space state variables as x ≡ x and y ≡ x , so that the potential and drift term can be expressed as V ( x ) = a ( x + y ) , (34) ˜ K ( x ) = ( − by, + bx ) . (35)2It can be verified that (34) and (35) satisfy conditions given by equations (15) and (16).The NLFP equation then has the form ∂F∂t = D ∇ [ F − q ] + ∂ [(2 ax + by ) F ] ∂x + ∂ [(2 ay − bx ) F ] ∂y . (36)We propose the ansatz F ( x, y, t ) = η ( t ) (cid:2) − (1 − q )( α ( t ) x + δ ( t ) xy + γ ( t ) y ) (cid:3) − q ) , (37)where η ( t ), α ( t ), δ ( t ) and γ ( t ) are time-dependent parameters. This ansatz has a time-dependent Tsallis q maximum entropy ( q -maxent) form, with the time dependence repre-sented in the parameters η , α , δ and γ . We then define ϕ = 1 − (1 − q )( αx + δxy + γy ) , (38)calculate the terms of the nonlinear Fokker-Planck equation (1) and obtain the followingexpressions ∂F∂t = ˙ ηϕ − q ) − η ( ˙ αx + ˙ δxy + ˙ γy ) ϕ q (1 − q ) , (39a) ∂ F − q ∂x = (2 − q ) η − q (cid:16) − αϕ − q ) + (2 αx + δy ) ϕ q (1 − q ) (cid:17) , (39b) ∂ F − q ∂y = (2 − q ) η − q (cid:16) − γϕ − q ) + (2 γy + δx ) ϕ q (1 − q ) (cid:17) , (39c) ∂ [(2 ax + by ) F ] ∂x = η h aϕ − q ) − (2 ax + by )(2 αx + δy ) ϕ q (1 − q ) i , (39d) ∂ [(2 ay − bx ) F ] ∂y = η h aϕ − q ) − (2 ay − bx )(2 γy + δx ) ϕ q (1 − q ) i . (39e)Next we substitute the right-hand side of the above equations (39) into the NLFP equation(36) and, with some algebra, obtain the following set of ordinary diferential equations forthe time evolution of the parameters η , α , δ and γdηdt = 4 ηa − − q ) Dη − q ( α + γ ) , (40a) dαdt = − (2 − q ) Dη − q (cid:0) α + δ (cid:1) + 4 aα − bδ , (40b) dγdt = − (2 − q ) Dη − q (cid:0) γ + δ (cid:1) + 4 aγ + bδ , (40c) dδdt = − − q ) Dη − q δ ( α + γ ) + 4 aδ + 2 b ( α − γ ) . (40d)3Therefore the q-maxent ansatz (37) will be a solution of the NLFP equation (36), providedthat the functions η ( t ), α ( t ), δ ( t ) and γ ( t ) satisfy the set of four coupled ordinary differentialequations (40).When we interpret the function F q ( x , · · · , x N , t ) (4) as a probability density in phasespace, or as a physical density of particles or other entities, we should require that the norm I of F q is finite, so that I = Z F q dx dx · · · dx N ≤ ∞ . (41)For the density function (37) to have a finite norm, in that expression, we should have αx + δxy + γy = const . >
0, determining the isodensity curves which should correspondto ellipses. Therefore the quadratic form αx + δxy + γy has to be definite positive. Con-sequently, the discriminant ς = αγ − δ dςdt = [ δ − αγ ][(2 − q ) Dη − q ( α + γ ) − a ]= 4 ς [2 a − (2 − q ) Dη − q ( α + γ )] . (43)We see that the value of the discriminant is not constant in time. However, equation (43)implies that the positive character of ς is preserved under the time evolution of the system.For the proposed q -statistical ansatz (37), we find after some algebra that, for q <
1, thenorm (equation 41) is I = πη (2 − q ) q αγ − δ . (44)After some more calculation, it is also possible to verify using the equations of motion (40)that dIdt = ∂I∂η dηdt + ∂I∂α dαdt + ∂I∂γ dγdt + ∂I∂δ dδdt = 0 , (45)so that I is a conserved quantity during the time evolution of the system, as is to be expected.A density F ( x , t ) governed by the partial differential equation (36) can be interpretedas describing the distribution of a set of particles interacting via short range interactions,performing overdamped motion under the drag effects due to a uniformly rotating medium,and confined by an external harmonic potential. To see this, let us consider the equation ofmotion of one individual test particle of this system m ¨ r = − ∇ W int − ∇ W ext − Γ( ˙ r − ˙ r R ) , (46)4where m is the mass of the test particle, W int is the potential function associated with theforces acting on the test particle due to the other particles of the system, W ext is the externalconfining potential, and Γ is a drag coefficient describing the drag forces due to a resistingmedium that rotates uniformly with an angular velocity Ω. Notice that the equation of mo-tion (46) is expressed with respect to an inertial reference frame (with cartesian coordinates( x, y )) and not with respect to the rotating frame where the resisting medium is at rest.With respect to the inertial frame, the local velocity ˙ r R of the medium has components( − Ω y, +Ω x ).Since the interactions between the particles are short-range, we assume that the potentialfunction W int is a function of the local density F , that is W int = D ( F ). In the regime ofoverdamped motion, equation (46) becomes˙ r = − ∇ W int − ∇ W ext + ˙ r R , (47)implying that the velocity ˙ r of a particle in the system, at a given time, is completelydetermined by its location r . It can then be verified, after some calculations, that thecontinuity equation in configuration space, ∂F/∂t = − ∇ ( ˙ r F ), describing the evolution ofthe space density F of a set of articles moving according to the equation of motion (47), isprecisely the NLFP equation (36), after the identifications D Γ = − q − q DF − q , W ext Γ = a r , and b = Ω.An illustrative example of the time evolution of the q -Gaussian solution (37) is providedin Figures 1-4. In these Figures, the parameters α , γ , δ , and η , determining the evolving sizeand shape of the two-dimensional q -Gaussian (37), are depicted as a function of time. Thedifferent curves shown in each Figure correspond to the NLFP equation (36), with q = 0 . D = 0 . a = 1, b = 4, and different initial conditions. The curves were therefore obtainedfrom the numerical integration of the set of coupled ordinary differential equations (40). Allsolutions exhibited correspond to evolving densities normalized to unity (that is, I = 1. Seeequation (44)). The initial conditions are α = 1, δ = 0, with different initial values of theparameter γ , as indicated in the Figures. The initial value of η is calculated from the initialvalues of the other three parameters, using the normalization condition I = 1.It can be appreciated from Figures 1-4 that the different initial densities considered (allhaving the same norm I = 1) relax to the same final stationary distribution (characterizedby the same value of the norm). This stationary distribution is rotationally symmetric.5 α ( t ) t γ = 1.0 γ = 1.2 γ = 1.4 γ = 1.5 γ = 1.6 γ = 1.8 γ = 2.0 γ = 2.5 FIG. 1. Evolution of the parameter α appearing in the time-dependent solution (37) of the NLFPequation (36), for q = 0 .
5. The units employed are defined in terms of the constants D and b appearing in the NLFP equation. The parameter α has dimensions of inverse squared length andis measured in units of b D . The time t is measured in units of b . γ ( t ) t γ = 1.0 γ = 1.2 γ = 1.4 γ = 1.5 γ = 1.6 γ = 1.8 γ = 2.0 γ = 2.5 FIG. 2. Evolution of the parameter γ appearing in the time-dependent solution (37) of the NLFPequation, for q = 0 .
5. The parameter γ has dimensions of inverse squared length. The unitsemployed are the same as in Figure 1. Consequently, the initial asymmetry of the density tends to decrease as the evolution takesplace (the two axis of the isodensity curves tend to become equal to each other). Theoscillatory behavior of the parameter δ , which takes alternating signs as time advances,6 -0.8-0.6-0.4-0.2 0 0.2 0.4 0 0.2 0.4 0.6 0.8 1 1.2 1.4 δ ( t ) t γ = 1.0 γ = 1.2 γ = 1.4 γ = 1.5 γ = 1.6 γ = 1.8 γ = 2.0 γ = 2.5 FIG. 3. Evolution of the parameter δ appearing in the time-dependent solution (37) of the NLFPequation, for q = 0 .
5. The parameter δ has dimensions of inverse squared length. The unitsemployed are the same as in Figure 1. η ( t ) t γ = 1.0 γ = 1.2 γ = 1.4 γ = 1.5 γ = 1.6 γ = 1.8 γ = 2.0 γ = 2.5 FIG. 4. Evolution of the parameter η appearing in the time-dependent solution (37) of the NLFPequation, for q = 0 .
5. The parameter η is dimensionless. The time t is measured in the same unitsas in Figure 1. indicates that the asymmetric density rotates as the evolution proceeds. Note that at thetimes when δ = 0 the axis of the isodensity curves are parallel to the coordinate axis. Thishappens at approximately regular time intervals, indicating that the elliptical isodensitycurves rotate at an approximately constant mean angular velocity. The oscillatory behaviorassociated with the rotation affects the other variables (besides δ ) as well, which also exhibit7oscillations whose amplitudes tend to decrease as the density function F relaxes towards thestationary one. VII. CONCLUSIONS
We investigated the main properties of multi-dimensional NLFP equations involving curldrift forces. We considered drift force fields comprising both an irrotational term G derivedfrom a potential function V ( x ) and a curl, non-gradient term ˜ K . We determined the re-quirements that the two parts G and ˜ K of the drift field have to satisfy, in order for thecorresponding NLFP equation to admit a stationary solution of the q -maxent form (that is,a q -exponential of the potential function V ( x ) associated with the gradient component ofthe drift force). We found that this kind of stationary solution exists for a continuous rangeof values of the parameter β if and only if, the curl part ˜ K is divergenceless and the curlpart is orthogonal to the gradient part G . We also proved that NLFP equations admitting astationary solution also verify an H -theorem, in terms of an appropriate linear combinationof the S q entropic functional and the mean value of the potential V . Finally, we studiedexact analytical time-dependent solutions of a two dimensional NLFP equation, describinga system of interacting particles in an overdamped motion regime, under the drag effectsoriginating on a uniformly rotating medium. The connection between rotation and NLFPequations with curl forces, combined with the connection between q -thermostatistics andself-gravitating systems, indicates that those evolution equations may have applications ingeophysical and astrophysical problems. Previous successful physical applications of NLFPequations also suggest that experimental implementations involving rotating granular ma-terials may also be worth exploring.Another potential field of application of the NLFP dynamics, investigated in the presentwork, is the space-time behavior of some biological systems [29]. Diffusion processes areuseful to model the spread of biological populations [30, 31]. Nonlinear diffusion equationshave been proposed, as effective descriptions of the interaction between the members ofa diffusing biological population [32–34]. On the other hand, drift terms can be used todescribe other effects on the motion of the individuals. In this biological context, since the“forces” are not fundamental but rather the effective result of a set of complex circumstances,it is to be expected that non-gradient forces can be relevant. NLFP equations with non-8gradient drift fields may also be useful in connection with the generalized Boltzmann machineapproach (based on a q -generalization of simulated annealing [35]) to neural network modelsof memory [36], when considering asymmetric neural interactions. Any further developmentsalong these or related lines will be very welcome. Acknowledgments
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