Charge transport in two dimensions limited by strong short-range scatterers: Going beyond parabolic dispersion and Born approximation

We investigate the conductivity of charge carriers confined to a two-dimensional system with the non-parabolic dispersion $k^N$ with $N$ being an arbitrary natural number. A delta-shaped scattering potential is assumed as the major source of disorder. We employ the exact solution of the Lippmann-Schwinger equation to derive an analytical Boltzmann conductivity formula valid for an arbitrary scattering potential strength. The range of applicability of our analytical results is assessed by a numerical study based on the finite size Kubo formula. We find that for any $N>1$, the conductivity demonstrates a linear dependence on the carrier concentration in the limit of a strong scattering potential strength. This finding agrees with the conductivity measurements performed recently on chirally stacked multilayer graphene where the lowest two bands are non-parabolic and the adsorbed hydrocarbons might act as strong short-range scatterers.Comment: Substantially revised version, as accepted to PRB: 8 pages, 3 figure

Pair-factorized steady states on arbitrary graphs

Stochastic mass transport models are usually described by specifying hopping rates of particles between sites of a given lattice, and the goal is to predict the existence and properties of the steady state. Here we ask the reverse question: given a stationary state that factorizes over links (pairs of sites) of an arbitrary connected graph, what are possible hopping rates that converge to this state? We define a class of hopping functions which lead to the same steady state and guarantee current conservation but may differ by the induced current strength. For the special case of anisotropic hopping in two dimensions we discuss some aspects of the phase structure. We also show how this case can be traced back to an effective zero-range process in one dimension which is solvable for a large class of hopping functions.Comment: IOP style, 9 pages, 1 figur

Exact diffusion coefficient of self-gravitating Brownian particles in two dimensions

We derive the exact expression of the diffusion coefficient of a self-gravitating Brownian gas in two dimensions. Our formula generalizes the usual Einstein relation for a free Brownian motion to the context of two-dimensional gravity. We show the existence of a critical temperature T_{c} at which the diffusion coefficient vanishes. For T<T_{c} the diffusion coefficient is negative and the gas undergoes gravitational collapse. This leads to the formation of a Dirac peak concentrating the whole mass in a finite time. We also stress that the critical temperature T_{c} is different from the collapse temperature T_{*} at which the partition function diverges. These quantities differ by a factor 1-1/N where N is the number of particles in the system. We provide clear evidence of this difference by explicitly solving the case N=2. We also mention the analogy with the chemotactic aggregation of bacteria in biology, the formation of ``atoms'' in a two-dimensional (2D) plasma and the formation of dipoles or supervortices in 2D point vortex dynamics

Mass condensation on networks

We construct classical stochastic mass transport processes for stationary states which are chosen to factorize over pairs of sites of an undirected, connected, but otherwise arbitrary graph. For the special topology of a ring we derive static properties such as the critical point of the transition between the liquid and the condensed phase, the shape of the condensate and its scaling with the system size. It turns out that the shape is not universal, but determined by the interplay of local and ultralocal interactions. In two dimensions the effect of anisotropic interactions of hopping rates can be treated analytically, since the partition function allows a dimensional reduction to an effective one-dimensional zero-range process. Here we predict the onset, shape and scaling of the condensate on a square lattice. We indicate further extensions in the outlook

Distance traveled by random walkers before absorption in a random medium

We consider the penetration length $l$ of random walkers diffusing in a medium of perfect or imperfect absorbers of number density $\rho$. We solve this problem on a lattice and in the continuum in all dimensions $D$, by means of a mean-field renormalization group. For a homogeneous system in $D>2$, we find that $l\sim \max(\xi,\rho^{-1/2})$, where $\xi$ is the absorber density correlation length. The cases of D=1 and D=2 are also treated. In the presence of long-range correlations, we estimate the temporal decay of the density of random walkers not yet absorbed. These results are illustrated by exactly solvable toy models, and extensive numerical simulations on directed percolation, where the absorbers are the active sites. Finally, we discuss the implications of our results for diffusion limited aggregation (DLA), and we propose a more effective method to measure $l$ in DLA clusters.Comment: Final version: also considers the case of imperfect absorber

Nonlinear mean-field Fokker-Planck equations and their applications in physics, astrophysics and biology

We discuss a general class of nonlinear mean-field Fokker-Planck equations [P.H. Chavanis, Phys. Rev. E, 68, 036108 (2003)] and show their applications in different domains of physics, astrophysics and biology. These equations are associated with generalized entropic functionals and non-Boltzmannian distributions (Fermi-Dirac, Bose-Einstein, Tsallis,...). They furthermore involve an arbitrary binary potential of interaction. We emphasize analogies between different topics (two-dimensional turbulence, self-gravitating systems, Debye-H\"uckel theory of electrolytes, porous media, chemotaxis of bacterial populations, Bose-Einstein condensation, BMF model, Cahn-Hilliard equations,...) which were previously disconnected. All these examples (and probably many others) are particular cases of this general class of nonlinear mean-field Fokker-Planck equations

Self-gravitating Brownian systems and bacterial populations with two or more types of particles

We study the thermodynamical properties of a self-gravitating gas with two or more types of particles. Using the method of linear series of equilibria, we determine the structure and stability of statistical equilibrium states in both microcanonical and canonical ensembles. We show how the critical temperature (Jeans instability) and the critical energy (Antonov instability) depend on the relative mass of the particles and on the dimension of space. We then study the dynamical evolution of a multi-components gas of self-gravitating Brownian particles in the canonical ensemble. Self-similar solutions describing the collapse below the critical temperature are obtained analytically. We find particle segregation, with the scaling profile of the slowest collapsing particles decaying with a non universal exponent that we compute perturbatively in different limits. These results are compared with numerical simulations of the two-species Smoluchowski-Poisson system. Our model of self-attracting Brownian particles also describes the chemotactic aggregation of a multi-species system of bacteria in biology

Newtonian gravity in d dimensions

We study the influence of the dimension of space on the thermodynamics of the classical and quantum self-gravitating gas. We consider Hamiltonian systems of self-gravitating particles described by the microcanonical ensemble and self-gravitating Brownian particles described by the canonical ensemble. We present a gallery of caloric curves in different dimensions of space and discuss the nature of phase transitions as a function of the dimension d. We also provide the general form of the Virial theorem in d dimensions and discuss the particularity of the dimension d=4 for Hamiltonian systems and the dimension d=2 for Brownian systems

Tuning the shape of the condensate in spontaneous symmetry breaking

We investigate what determines the shape of a particle condensate in situations when it emerges as a result of spontaneous breaking of translational symmetry. We consider a model with particles hopping between sites of a one-dimensional grid and interacting if they are at the same or at neighboring nodes. We predict the envelope of the condensate and the scaling of its width with the system size for various interaction potentials and show how to tune the shape from a delta-peak to a rectangular or a parabolic-like form.Comment: 4 pages, 2 figures, major revision, the title has been change