Microscopic Theory for Negative Differential Mobility in Crowded Environments

Abstract

We study the behavior of the stationary velocity of a driven particle in an environment of mobile hard-core obstacles. Based on a lattice gas model, we demonstrate analytically that the drift velocity can exhibit a nonmonotonic dependence on the applied force, and show quantitatively that such negative differential mobility (NDM), observed in various physical contexts, is controlled by both the density and diffusion time scale of obstacles. Our study unifies recent numerical and analytical results obtained in specific regimes, and makes it possible to determine analytically the region of the full parameter space where NDM occurs. These results suggest that NDM could be a generic feature of biased (or active) transport in crowded environments.PACS numbers: 05.40.Fb,83.10.Pp Introduction.-Quantifying the response of a complex system to an external force is one of the cornerstone problems of statistical mechanics. In the linear response regime, a fundamental result is the fluctuationdissipation theorem, which relates system response and spontaneous fluctuations. Within the last years a great effort has been devoted to generalizations of this theorem to nonequilibrium situations [1][2][3][4], when the time reversal symmetry is broken, and also to elucidating the effects of the higher order contributions in the external perturbation [5][6][7][8][9][10][11]. From experimental perspective, theoretical understanding of the latter issues is of an utmost importance in several fields, such as active microrheology [12][13][14] and dynamics of nonequilibrium fluids [15,16].A striking example of anomalous behavior beyond the linear regime is the negative response of a particle's velocity to an applied force, observed in diverse situations in which a particle subject to an external force F travels through a medium. The terminal drift velocity V (F ) attained by the driven particle is then a nonmonotonic function of the force: upon a gradual increase of F , the terminal drift velocity first grows as expected from linear response, reaches a peak value and eventually decreases. This means that the differential mobility of the driven particle becomes negative for F exceeding a certain threshold value. Such a counter-intuitive "getting more from pushing less" [17] behavior of the differential mobility (or of the differential conductivity) has been observed for a variety of physical systems and processes, e.g. for electron transfer in semiconductors at low temperatures [18][19][20][21], hopping processes in disordered media [22], transport of electrons in mixtures of atomic gases with reactive collisions [23], far-from-equilibrium quantum spin chains [24], some models of Brownian motors [25,26], soft matter colloidal particles [27], different nonequilibrium systems [17], and also for the kinetically constrained models of glass formers [28][29][30].Apart of these examples, negative differential mobility (NDM) has been observed in the minimal model of a driven lattice gas, which captures many essential features of the behavior in r...

Nonlinear response and emerging nonequilibrium microstructures for biased diffusion in confined crowded environments

“…(14) and (15), and using the definition (10), one can explicitly obtain the TP velocity in the dilute limit, for arbitrary choice of jump probabilities and time scales τ, τ * . Notice that the comparison between the expression for V (F ) obtained in the dilute limit for unconfined geometries [44], following a computation analogous to that reported here, and the analytical result of [41], revealed that the decoupling approximation is indeed exact at lowest order in ρ, for arbitrary values of the time scales τ, τ * . We claim that this statement also holds for confined geometries.…”

“…It is expected to hold in the dilute regime, ρ ≪ 1, and for values of τ * not too large with respect to τ , namely when the dynamics of bath particles is sufficiently fast. More specifically, it has been shown that this approximation provides exact results for V (F ) in the limits of very low and very high densities in unconfined geometries [44]. Notice that the regime of validity of the approximation can be dependent on the specific choice of transition rates.…”

“…Recently, the behavior of V (F ) beyond the linear regime in lattice gas models has received great attention, triggered by the observation of the striking phenomenon of negative differential mobility (NDM), namely a nonmonotonic dependence of the TP velocity on the applied force [39][40][41][42][43][44][45]. Upon a gradual increase of the force F , the velocity grows linearly, as prescribed by the linear response, then approaches a maximal value and further decreases with an increase of F .…”

“…Later, Basu and Maes [43] observed via numerical simulations the same phenomenon in a dynamical environment, where obstacles can diffuse with excluded-volume interactions, and related NDM with the "frenetic" contributions appearing in a nonequilibrium fluctuation-dissipation relation [51][52][53][54][55]. A general analytical approach, accounting for NDM in dynamical environments, and valid for arbitrary density and arbitrary choice of transition rates, has been then proposed and discussed in [44], where the trapping effect induced on the TP by the coupling between the density and the characteristic time scale of bath particles, was indicated as the main physical mechanism responsible for the phenomenon. The study of the dependence of NDM on the microscopic dynamical rules has been further investigated by Baiesi and coworkers [45], who stressed that the particular shape of the obstacles and their coupling with the TP can play a central role in determining the effective trapping mechanism.…”

“…In this regime, remarkably, the differential mobility becomes negative. Moreover, we study in detail the behaviors arising from two specific choices of the TP jump probabilities, considered recently in [41,43,44], both satisfying the LDB, but differing in the explicit dependence on F in the directions orthogonal to the force. We derive the phase chart in the parameter space of the model where NDM is expected to occur for both kinds of jump probabilities.…”