U. BRISKOT et al. PHYSICAL REVIEW B 92, 115426 (2015) and therefore is not a conserved quantity. However, it is conserved in the collinear scattering processes and hence the corresponding relaxation rate does not contain the logarithmic enhancement. Finally, the imbalance current j I is proportional to the sign of the quasiparticle energy and to the velocity. Similarly to the electric current, it does not experience logarithmically enhanced relaxation. The imbalance current is related to the quasiparticle number or imbalance density [10] n I = n + + n − , where n + and n − are the particle numbers in the upper (conduction) and lower (valence) bands. Neglecting the Auger processes, quasiparticle recombination due to, e.g., electron-phonon interaction, and three-particle collisions due to weak coupling, one finds that n + and n − are conserved independently. In this case, which will be considered in the rest of the paper, not only the total charge density n = n + − n − , but also the quasiparticle density n I is conserved.At times longer than τ g , physical observables can be described within the macroscopic (or hydrodynamic) approach. The existence of the three slow-relaxing modes in graphene implies a peculiar two-step thermalization.Short-time electron-electron scattering (at time scales up to τ g ) establishes the so-called "unidirectional thermalization" [24]: the collinear scattering singularity implies that the electron-electron interaction is more effective along the same direction. Within linear response [18], one can express the nonequilibrium distribution function in terms of the three macroscopic currents j , j E , and j I . The currents can then be found from the macroscopic equations. The currents j and j I are not conserved and can be relaxed by the electron-electron interaction. Close to charge neutrality, the corresponding relaxation rates can be estimated as [6,40] g . These rates enter the macroscopic equations as frictionlike terms. The macroscopic linear-response theory has the same form on time scales shorter or longer than τ ee .Beyond linear response, the scattering processes characterized by the time scale τ ee play an important role in thermalizing quasiparticles moving in different directions and thus lead to establishing the local equilibrium. This is the starting point for derivation of the nonlinear hydrodynamics, which is valid at time scales much longer than τ ee . In view of conservation of the particle number, energy, and momentum, as well as independent conservation of the number of particles in the two bands in graphene, we may write the local equilibrium distribution function as [12,14] where ε λ,k = λv g k denotes the energies of the electronic states with the momentum k in the band λ = ±, μ λ (r) the local chemical potential, the local temperature is encoded in β(r) = 1/T (r), and u(r) is the hydrodynamic velocity field which we define in the following (this field should not be confused with quasiparticle velocities v). The distribution function (1) follows from the sta...

The dynamic conductivity σ(ω) of graphene in the presence of diagonal white noise disorder and quantizing magnetic field B is calculated. We obtain analytic expressions for σ(ω) in various parametric regimes ranging from the quasiclassical Drude limit corresponding to strongly overlapping Landau levels (LLs) to the extreme quantum limit where the conductivity is determined by the optical selection rules of the clean graphene. The nonequidistant LL spectrum of graphene renders its transport characteristics quantitatively different from conventional 2D electron systems with parabolic spectrum. Since the magnetooscillations in the semiclassical density of states are anharmonic and are described by a quasi-continuum of cyclotron frequencies, both the ac Shubnikov-de Haas oscillations and the quantum corrections to σ(ω) that survive to higher temperatures manifest a slow beating on top of fast oscillations with the local energy-dependent cyclotron frequency. Both types of quantum oscillations possess nodes whose index scales as ω 2 . In the quantum regime of separated LLs, we study both the cyclotron resonance transitions, which have a rich spectrum due to the nonequidistant spectrum of LLs, and disorder-induced transitions which violate the clean selection rules of graphene. We identify the strongest disorder-induced transitions in recent magnetotransmission experiments. We also compare the temperature-and chemical potential-dependence of σ(ω) in various frequency ranges from the dc limit allowing intra-LL transition only to the universal high-frequency limit where the Landau quantization provides a small B-dependent correction to the universal value of the interband conductivity σ = e 2 /4 of the clean graphene.

We present a theoretical analysis of the relaxation cascade of a photoexcited electron in graphene in the presence of RPA screened electron-electron interaction. We calculate the relaxation rate of high energy electrons and the jump-size distribution of the random walk constituting the cascade which exhibits fat tails. We find that the statistics of the entire cascade are described by Lévy flights with constant drift instead of standard drift-diffusion in energy space. The Lévy flight manifests nontrivial scaling relations of the fluctuations in the cascade time, which is related to the problem of the first passage time of Lévy processes. Furthermore we determine the transient differential transmission of graphene after an excitation by a laser pulse taking into account the fractional kinetics of the relaxation dynamics.

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