Nonlinear mean field Fokker-Planck equations. Application to the chemotaxis of biological populations

Recently, several researchers have tried to extend the usual concepts of thermodynamics and kinetic theory in order to describe complex systems that are characterized by non-Boltzmannian distributions. In that respect, some generalized forms of entropic functionals1 have been introduced. One of the most popular “generalized entropy” is the Tsallis (1988) [9] entropy, but other entropies have been presented by Abe (1997) [10], Borges & Roditi (1998) [11], Kaniadakis (2001) [12], Naudts (2004) [13], and Kaniadakis et al. (2005) [14]. It was later realized that these entropic functionals are special cases of the one-parameter family of entropies introduced earlier by Harvda & Charvat (1967) [15] or of the two-parameters family of entropies introduced by Mittal (1975) [16] and Sharma & Taneja (1975) [17]. Other famous forms of entropies have been presented by Reyni (1970) [18] and Sharma & Mittal (1975) [19]. We refer to Kaniadakis & Lissia [20] for a very interesting discussion of these historical aspects, starting from the early works of Euler in 1779.

Following these developments, some researchers have tried to develop out-of-equilibrium theories associated to a generalized thermodynamical framework. In particular, it has been first shown by Plastino & Plastino (1995) [21] that the Tsallis q-distributions are the steady states of a nonlinear Fokker-Planck equation taking into account anomalous diffusion. This type of equations had been previously considered by mathematicians to describe porous media [22]. The seminal work of Plastino & Plastino [21] has been further developed by Tsallis & Bukman (1996) [23], Stariolo (1997) [24], Borland (1998) [25] and Nobre et al. (2004) [26] among others. On the other hand, Kaniadakis & Quarati (1994) [27] have introduced nonlinear Fokker-Planck equations whose steady states are the Fermi-Dirac 2 and Bose-Einstein statistics. These kinetic equations take into account an exclusion (fermions) or inclusion (bosons) principle leading to quantum-like statistics at equilibrium. The case of intermediate statistics, interpolating between fermions and bosons, has also been considered in [27]. Recently, the bosonic Kramers equation has been studied in [30] and was shown to reproduce the phenomenology of the Bose-Einstein condensation in the canonical ensemble.

The above-mentioned nonlinear Fokker-Planck (NFP) equations are associated with special forms of entropic functionals (Tsallis, Fermi-Dirac, Bose-Einstein). More recently, Martinez et al. (1998) [31], Kaniadakis (2001) [12], Frank (2002) [32] and Chavanis (2003) [33] have studied generalized forms of NFP equations associated with an almost arbitrary entropic functional. They can be viewed as generalized Kramers and Smoluchowski equations where the coefficients of diffusion, friction and drift explicitly depend on the local density of particles. Physically, this can take into account microscopic constraints (exclusion volume effects, steric hindrance, non-extensive effects…) that modify the dynamics of the particles at small scales and lead to non-standard equilibrium distributions 3 . Martinez et al. [31] determined the NFP equation in order to recover, as a steady state, the equilibrium state produced by minimizing a generalized form of free energy at fixed mass. Kaniadakis [12] obtained the NFP equation 2 A generalized Fokker-Planck equation leading to the Fermi-Dirac statistics has also been introduced by Chavanis et al. (1996) [28] in the context of the violent relaxation of collisionless stellar systems described by the Vlasov equation. This is based on the Lynden-Bell’s form of entropy (1968) [29] which becomes similar to the Fermi-Dirac entropy in the two-levels approximation of the theory.

3 Generalized Kramers and Smoluchowski equations describe dissipative system

…(Full text truncated)…