We investigate random walks on complex networks and derive an exact expression for the mean first-passage time (MFPT) between two nodes. We introduce for each node the random walk centrality C, which is the ratio between its coordination number and a characteristic relaxation time, and show that it determines essentially the MFPT. The centrality of a node determines the relative speed by which a node can receive and spread information over the network in a random process. Numerical simulations of an ensemble of random walkers moving on paradigmatic network models confirm this analytical prediction.

PrefaceThis book is an interdisciplinary book: it tries to teach physicists the basic knowledge of combinatorial and stochastic optimization and describes to the computer scientists physical problems and theoretical models in which their optimization algorithms are needed. It is a unique book since it describes theoretical models and practical situation in physics in which optimization problems occur, and it explains from a physicists point of view the sophisticated and highly efficient algorithmic techniques that otherwise can only be found specialized computer science textbooks or even just in research journals. Traditionally, there has always been a strong scientific interaction between physicists and mathematicians in developing physics theories. However, even though numerical computations are now commonplace in physics, no comparable interaction between physicists and computer scientists has been developed. Over the last three decades the design and the analysis of algorithms for decision and optimization problems have evolved rapidly. Most of the active transfer of the results was to economics and engineering and many algorithmic developments were motivated by applications in these areas. The few interactions between physicists and computer scientists were often successful and provided new insights in both fields. For example, in one direction, the algorithmic community has profited from the introduction of general purpose optimization tools like the simulated annealing technique that originated in the physics community. In the opposite direction, algorithms in linear, nonlinear, and discrete optimization sometimes have the potential to be useful tools in physics, in particular in the study of strongly disordered, amorphous and glassy materials. These systems have in common a highly non-trivial minimal energy configuration, whose characteristic features dominate the physics a t low temperatures. For a theoretical understanding the knowledge of the so called "ground states" of model Hamiltonians, or optimal solutions of appropriate cost functions, is mandatory. To this end an efficient algorithm, applicable to reasonably sized instances, is a necessary condition. The list of interesting physical problems in this context is long, it ranges from disordered magnets, structural glasses and superconductors through polymers, membranes, and proteins to neural networks. The predominant method used by physicists to study these questions numerically are Monte Carlo simulations and/or simulated annealing. These methods are doomed to fail in the most interesting situations. But, as pointed out above, many useful results in optimization algorithms research never reach the physics community, and interesting computational problems in physics do not come to the attention of algorithm designers. We therefore think that there is a definite need VI Preface to intensify the interaction between the computer science and physics communities. We hope that this book will help to extend the bridge between these two groups. Since one...

We study numerically the critical region and the disordered phase of the random transverse-field Ising chain. By using a mapping of Lieb, Schultz and Mattis to non-interacting fermions, we can obtain a numerically exact solution for rather large system sizes, L ≤ 128. Our results confirm the striking predictions of earlier analytical work and, in addition, give new results for some probability distributions and scaling functions.

The low temperature dynamics of the two-and three-dimensional Ising spin glass model with Gaussian couplings is investigated via extensive Monte Carlo simulations. We find an algebraic decay of the remanent magnetization. For the autocorrelation function C(t, t w ) = [ S i (t + t w )S i (t w ) ] av a typical aging scenario with a t/t w scaling is established. Investigating spatial correlations we find an algebraic growth law ξ(t w ) ∼ t α(T ) w of the average domain size.The spatial correlation function G(r, t w ) = [ S i (t w )S i+r (t w ) 2 ] av scales with r/ξ(t w ). The sensitivity of the correlations in the spin glass phase with respect to temperature changes is examined by calculating a time dependent overlap length. In the two dimensional model we examine domain growth with a new method: First we determine the exact ground states of the various samples (of system sizes up to 100 × 100) and then we calculate the correlations between this state and the states generated during a Monte Carlo simulation. 75.10Nr, 75.50Lh, 75.40Mg Typeset using REVT E X

A continuous time cluster algorithm for two-level systems coupled to a dissipative bosonic bath is presented and applied to the sub-Ohmic spin-boson model. When the power s of the spectral function Jomega proportional, variant omegas is smaller than 1/2, the critical exponents are found to be classical, mean-field like. Potential sources for the discrepancy with recent renormalization group predictions are traced back to the effect of a dangerously irrelevant variable.

Cell polarization enables restriction of signalling into microdomains. Polarization of lymphocytes following formation of a mature immunological synapse (IS) is essential for calcium-dependent T-cell activation. Here, we analyse calcium microdomains at the IS with total internal reflection fluorescence microscopy. We find that the subplasmalemmal calcium signal following IS formation is sufficiently low to prevent calcium-dependent inactivation of ORAI channels. This is achieved by localizing mitochondria close to ORAI channels. Furthermore, we find that plasma membrane calcium ATPases (PMCAs) are re-distributed into areas beneath mitochondria, which prevented PMCA up-modulation and decreased calcium export locally. This nano-scale distribution-only induced following IS formation-maximizes the efficiency of calcium influx through ORAI channels while it decreases calcium clearance by PMCA, resulting in a more sustained NFAT activity and subsequent activation of T cells.

A variety of computational models have been developed to describe active matter at different length and time scales. The diversity of the methods and the challenges in modeling active matterranging from molecular motors and cytoskeletal filaments over artificial and biological swimmers on microscopic to groups of animals on macroscopic scales-mainly originate from their out-ofequilibrium character, multiscale nature, nonlinearity, and multibody interactions. In the present review, various modeling approaches and numerical techniques are addressed, compared, and differentiated to illuminate the innovations and current challenges in understanding active matter. The complexity increases from minimal microscopic models of dry active matter toward microscopic models of active matter in fluids. Complementary, coarse-grained descriptions and continuum models are elucidated. Microscopic details are often relevant and strongly affect collective behaviors, which implies that the selection of a proper level of modeling is a delicate choice, with simple models emphasizing universal properties and detailed models capturing specific features. Finally, current approaches to further advance the existing models and techniques to cope with real-world applications, such as complex media and biological environments, are discussed.2

We consider the non-stationary quantum relaxation of the Ising spin chain in a transverse field of strength h. Starting from a homogeneously magnetized initial state the system approaches a stationary state by a process possessing quasi long range correlations in time and space, independent of the value of h. In particular the system exhibits aging (or lack of time translational invariance on intermediate time scales) although no indications of coarsening are present.Non-equilibrium dynamical properties of quantum systems have been of interest recently, experimentally and theoretically. Measurements on magnetic relaxation at low-temperatures show deviations from the classical exponential decay [1], which was explained by the effect of quantum tunneling. On the theoretical side, among others, integrable [2] and non-integrable models [3] were studied in the presence of energy or magnetic currents, as well as the phenomena of quantum aging in systems with long-range [4] and short-range interactions [5].Here we pose a different question: Consider a quantum mechanical interacting many body system described by a Hamilton operatorĤ without any coupling to an external bath, which means that the system is closed. Suppose the system is prepared in a specific state |ψ 0 at time t = 0, which is not an eigenstate of the HamiltonianĤ. Then we are interested in the natural quantum dynamical evolution of this state which is described by the Schrödinger equation and is formally given byObviously the energy E = ψ 0 |Ĥ|ψ 0 is conserved. In particular we want to study the time evolution of the expectation value A(t) of an observable or the twotime correlation function C AB (t 1 , t 2 ) of two observableŝ A andB, defined bywhere H (t) = exp(+iĤt) exp(−iĤt) is the operator A in the Heisenberg picture (withh set to unity) and {ÂB} S = 1/2(ÂB +BÂ) the symmetric product of two operators.One should emphasize that in such a situation one does not expect time translational invariance to hold, which would manifest itself in, for instance, A(t) = A 0 = const. and C AB (t 1 , t 2 ) = C AB (t 1 −t 2 ). There will be a transient regime in which these relations are violated and depending on the complexity of the system this non-equilibrium regime will extend over the whole time axis, in which we would denote it as quantum aging, as it can be observed for instance for the universe, which is (most probably) a closed system.To be concrete we consider the prototype of an interacting quantum systems, the Ising model in a transverse field (TIM) in one dimension defined by the Hamiltonian: