Brownian motion in AdS/CFT

We study Brownian motion and the associated Langevin equation in AdS/CFT. The Brownian particle is realized in the bulk spacetime as a probe fundamental string in an asymptotically AdS black hole background, stretching between the AdS boundary and the horizon. The modes on the string are excited by the thermal black hole environment and consequently the string endpoint at the boundary undergoes an erratic motion, which is identified with an external quark in the boundary CFT exhibiting Brownian motion. Semiclassically, the modes on the string are thermally excited due to Hawking radiation, which translates into the random force appearing in the boundary Langevin equation, while the friction in the Langevin equation corresponds to the excitation on the string being absorbed by the black hole. We give a bulk proof of the fluctuation-dissipation theorem relating the random force and friction. This work can be regarded as a step toward understanding the quantum microphysics underlying the fluid-gravity correspondence. We also initiate a study of the properties of the effective membrane or stretched horizon picture of black holes using our bulk description of Brownian motion.Comment: 54 pages (38 pages + 5 appendices), 5 figures. v2: references added, clarifications in 6.2. v3: clarifications, version submitted to JHE

Active Brownian Particles. From Individual to Collective Stochastic Dynamics

We review theoretical models of individual motility as well as collective dynamics and pattern formation of active particles. We focus on simple models of active dynamics with a particular emphasis on nonlinear and stochastic dynamics of such self-propelled entities in the framework of statistical mechanics. Examples of such active units in complex physico-chemical and biological systems are chemically powered nano-rods, localized patterns in reaction-diffusion system, motile cells or macroscopic animals. Based on the description of individual motion of point-like active particles by stochastic differential equations, we discuss different velocity-dependent friction functions, the impact of various types of fluctuations and calculate characteristic observables such as stationary velocity distributions or diffusion coefficients. Finally, we consider not only the free and confined individual active dynamics but also different types of interaction between active particles. The resulting collective dynamical behavior of large assemblies and aggregates of active units is discussed and an overview over some recent results on spatiotemporal pattern formation in such systems is given.Comment: 161 pages, Review, Eur Phys J Special-Topics, accepte

Brownian motion near a liquid-like membrane

The dynamics of a tracer molecule near a fluid membrane is investigated, with particular emphasis given to the interplay between the instantaneous position of the particle and membrane fluctuations. It is found that hydrodynamic interactions creates memory effects in the diffusion process. The random motion of the particle is then shown to cross over from a ``bulk'' to a ``surface'' diffusive mode, in a way that crucially depends on the elastic properties of the interface.Comment: 7 pages, 1 figur

Mode-Coupling Theory for Active Brownian Particles

We present a mode-coupling theory (MCT) for the high-density dynamics of two-dimensional spherical active Brownian particles (ABP). The theory is based on the integration-through-transients (ITT) formalism and hence provides a starting point for the calculation of non-equilibrium averages in active-Brownian particle systems. The ABP are characterized by a self-propulsion velocity $v_0$, and by their translational and rotational diffusion coefficients, $D_t$ and $D_r$. The theory treats both the translational and the orientational degrees of freedom of ABP explicitly. This allows to study the effect of self-propulsion of both weak and strong persistence of the swimming direction, also at high densities where the persistence length $\ell_p=v_0/D_r$ is large compared to the typical interaction length scale. While the low-density dynamics of ABP is characterized by a single P\'eclet number, $Pe=v_0^2/D_rD_t$, close to the glass transition the dynamics is found to depend on $Pe$ and $\ell_p$ separately. At fixed density, increasing the self-propulsion velocity causes structural relaxatino to speed up, while decreasing the persistence length slows down the relaxation. The theory predicts a non-trivial idealized-glass-transition diagram in the three-dimensional parameter space of density, self-propulsion velocity and rotational diffusivity. The active-MCT glass is a nonergodic state where correlations of initial density fluctuations never fully decay, but also an infinite memory of initial orientational fluctuations is retained in the positions

Chemical Reaction Dynamics within Anisotropic Solvents in Time-Dependent Fields

The dynamics of low-dimensional Brownian particles coupled to time-dependent driven anisotropic heavy particles (mesogens) in a uniform bath (solvent) have been described through the use of a variant of the stochastic Langevin equation. The rotational motion of the mesogens is assumed to follow the motion of an external driving field in the linear response limit. Reaction dynamics have also been probed using a two-state model for the Brownian particles. Analytical expressions for diffusion and reaction rates have been developed and are found to be in good agreement with numerical calculations. When the external field driving the mesogens is held at constant rotational frequency, the model for reaction dynamics predicts that the applied field frequency can be used to control the product composition.Comment: 13 pages, 5 figure

Holograghic Brownian motion in three dimensional G\"{o}del black hole

By using the AdS/CFT correspondence and G\"{o}del black hole background, we study the dynamics of heavy quark under a rotating plasma. In that case we follow Atmaja (JHEP 1304, 021, (2013)) about Brownian motion in BTZ black hole. In this paper we receive some new results for the case of $\alpha^{2}l^{2}\neq1$. This case, we must redefine the angular velocity of string fluctuation. We obtain the time evolution of displacement square that angular velocity and show that it behaves as a Brownian particle in non-relativistic limit. In this plasma, it seems that relating the Brownian motion with physical observables is rather a difficult work. But our results match with Atmaja work in the limit $\alpha^{2}l^{2}\rightarrow1$.Comment: 16 page

Fluctuation and dissipation in de Sitter space

In this paper we study some thermal properties of quantum field theories in de Sitter space by means of holographic techniques. We focus on the static patch of de Sitter and assume that the quantum fields are in the standard Bunch-Davies vacuum. More specifically, we follow the stochastic motion of a massive charged particle due to its interaction with Hawking radiation. The process is described in terms of the theory of Brownian motion in inhomogeneous media and its associated Langevin dynamics. At late times, we find that the particle undergoes a regime of slow diffusion and never reaches the horizon, in stark contrast to the usual random walk behavior at finite temperature. Nevertheless, the fluctuation-dissipation theorem is found to hold at all times.Comment: 1+45 pages, 5 figures. v4: matches published versio

Anomalous transport in the crowded world of biological cells

A ubiquitous observation in cell biology is that diffusion of macromolecules and organelles is anomalous, and a description simply based on the conventional diffusion equation with diffusion constants measured in dilute solution fails. This is commonly attributed to macromolecular crowding in the interior of cells and in cellular membranes, summarising their densely packed and heterogeneous structures. The most familiar phenomenon is a power-law increase of the MSD, but there are other manifestations like strongly reduced and time-dependent diffusion coefficients, persistent correlations, non-gaussian distributions of the displacements, heterogeneous diffusion, and immobile particles. After a general introduction to the statistical description of slow, anomalous transport, we summarise some widely used theoretical models: gaussian models like FBM and Langevin equations for visco-elastic media, the CTRW model, and the Lorentz model describing obstructed transport in a heterogeneous environment. Emphasis is put on the spatio-temporal properties of the transport in terms of 2-point correlation functions, dynamic scaling behaviour, and how the models are distinguished by their propagators even for identical MSDs. Then, we review the theory underlying common experimental techniques in the presence of anomalous transport: single-particle tracking, FCS, and FRAP. We report on the large body of recent experimental evidence for anomalous transport in crowded biological media: in cyto- and nucleoplasm as well as in cellular membranes, complemented by in vitro experiments where model systems mimic physiological crowding conditions. Finally, computer simulations play an important role in testing the theoretical models and corroborating the experimental findings. The review is completed by a synthesis of the theoretical and experimental progress identifying open questions for future investigation.Comment: review article, to appear in Rep. Prog. Phy