N-dimensional nonlinear Fokker-Planck equation with time-dependent coefficients

Abstract

An N -dimensional nonlinear Fokker-Planck equation is investigated here by considering the time dependence of the coefficients, where drift-controlled and source terms are present. We exhibit the exact solution based on the generalized gaussian function related to the Tsallis statistics. Furthermore, we show that a rich class of diffusive processes, including normal and anomalous ones, can be obtained by changing the time dependence of the coefficients. The existence of the anomalous diffusion and its ubiquity has motivated, in particular, the analytical study by considering nonlinear [13,14,15,16,17,18,19] and fractional [20,21] Fokker-Planck equations, spatial dependence of the diffusion coefficient [17,19,22,23], and temporal dependence of the drift term [18,24].In this brief report, we consider an N -dimensional nonlinear Fokker-Planck equation incorporating the time dependence in every coefficient, where drift-controlled (external force) and source terms are present. In our exact solution, a rich class of anomalous behaviors can arise by choosing appropriate time dependence of the coefficients. These results indicate that the possible anomalies in diffusive processes can appear as a consequence of different causes. This paper contains, essentially, the main results obtained in Refs. [13,14,18] and [24] as special cases.In order to investigate, in an exact way, a large class of anomalous diffusions, let us consider the N -dimensional nonlinear Fokker-Planck equationwhere we incorporate the time dependence in the external force,in the diffusion coefficient D(t), and in the source term α(t). In particular, we consider the case where the space constant term in F(r, t) can be taken with different coefficients. This situation is useful, for example, to study diffusion in the gravitational field [k 1 = (0, 0, k 1z )]. The harmonic potential is considered isotropic. The source term in Eq.(1) can be removed by an appropriate change in the solution:This way, ρ(r, t) obeys the equationwithThus, Eq. (4) has the same structure of Eq. (1) without the source term, but with an additional time dependence of the diffusion coefficient. Observe that this additional time dependence of D(t) is induced by the nonlinear term ρ ν , disappearing when ν = 1.In order to obtain an exact solution for Eq. (4) with the external force term given by Eq. (2), we are going to employ the ansatz ρ(r, t) = 1 Z(t) 1 − (1 − q)β(t) (r − r 0 (t)) 2 1/(1−q)

Elephants can always remember: Exact long-range memory effects in a non-Markovian random walk

“…The last relation is a special case of more general FPE with time-dependent coefficients that has been investigated in several papers [18,19,20]. Whereas the approach in those papers is phenomenologically we demonstrate in the frame of a microscopic model the origin of such FPE.…”

“…As the result we get a FPE with a time-dependent drift term. Recently stochastic processes leading to FPE with time-dependent coefficients, are discussed by several authors [18,19,20]. In view of those more heuristic approaches our model yields a more microscopic foundation for a special realization of a FPE with a time-dependent term.…”

q-entropies and the entanglement dynamics of two-qubits interacting with an environment

“…Research activity on applications of these entropic measures to physics (as contrasted to their restricted, sole application to information theory) started in earnest after Tsallis proposal in 1988 of a thermostatistical formalism based on the S (T ) q entropy [12]. After Tsallis' 1988 pioneering work, the S (T ) q measure has been successfully utilized in connection with several problems both in the classical [14][15][16][17][18][19][20][21][22][23] and the quantum regimes [24][25][26][27][28]. Tsallis entropy is nowadays thought to be of relevance for the study (among others) of systems governed by non linear Fokker-Planck equations [15,16]; systems exhibiting a scaleinvariant occupancy of phase space [17,18]; systems with anomalous thermostatting dynamics [19]; non equilibrium scenarios characterized by temperature fluctuations [20]; systems exhibiting weak chaos [21]; many body systems with interactions of long range relative to the system's size [22]; and biological ecosystems [23].…”

“…After Tsallis' 1988 pioneering work, the S (T ) q measure has been successfully utilized in connection with several problems both in the classical [14][15][16][17][18][19][20][21][22][23] and the quantum regimes [24][25][26][27][28]. Tsallis entropy is nowadays thought to be of relevance for the study (among others) of systems governed by non linear Fokker-Planck equations [15,16]; systems exhibiting a scaleinvariant occupancy of phase space [17,18]; systems with anomalous thermostatting dynamics [19]; non equilibrium scenarios characterized by temperature fluctuations [20]; systems exhibiting weak chaos [21]; many body systems with interactions of long range relative to the system's size [22]; and biological ecosystems [23]. Last, but certainly not least, several authors have explored the relationships between the q-entropies and the phenomenon of quantum entanglement [25][26][27][28][29][30][31][32].…”

Nonlinear Wave Equations Related to Nonextensive Thermostatistics

“…This work is motivated by a recent line of enquiry concerning nonlinear extensions of the Schroedinger, Dirac, and Klein-Gordon equations also linked to this formalism [4][5][6][7][8][9][10][11][12][13]. These nonlinear equations, in turn, exhibit close formal similarities with a family of nonlinear Fokker-Planck equations that govern the evolution of a variegated class of systems and processes in physics, biology and other fields, and have been the focus of intense research efforts in recent years [14][15][16][17][18][19][20][21].…”