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Unravelings provide a probabilistic representation of solutions of master equations and a method of computation of the density operator dynamics. The trajectories generated by unravelings may also be treated as real -- as in the stochastic collapse models. While averages of linear functionals of the unraveling trajectories can be calculated from the master equation, the situation is different for nonlinear functionals, thanks to the corrections with nonzero expected values, coming from the It\^o formula. Two types of nonlinear functionals are considered here: variance, and entropy. The corrections are calculated explicitly for two types of unravelings, based on Poisson and Wiener processes. In the case of entropy, these corrections are shown to be negative, expressing the localization introduced by the Lindblad operators.

The Gorini-Kossakowski-Sudarshan-Lindblad master equation (ME) governs the density matrix of open quantum systems (OQSs). When an OQS is subjected to weak continuous measurement, its state evolves as a stochastic quantum trajectory, whose statistical average solves the ME. The ensemble of such trajectories is termed an unraveling of the ME. We propose a method to operationally distinguish unravelings produced by the same ME in different measurement scenarios, using nonlinear averages of observables over trajectories. We apply the method to the paradigmatic quantum nonlinear system of resonance fluorescence in a two-level atom. We compare the Poisson-type unraveling, induced by direct detection of photons scattered from the two-level emitter, and the Wiener-type unraveling, induced by phase-sensitive detection of the emitted field. We show that a quantum-trajectory-averaged variance is able to distinguish these measurement scenarios. We evaluate the performance of the method, which can be readily extended to more complex OQSs, under a range of realistic experimental conditions., Comment: 5 pages, 3 figures, with supplementary material, revised version following the published version

We study systems of differential equations driven by fast-oscillating functions of a Wiener process. In the limit when the frequency of oscillations goes to infinity, we identify the limiting SDE systems which are driven by several independent Wiener processes. The solutions of the original equations converge to those of the limiting system in law. The limiting equations include additional drift terms which have the form of Stratonovich corrections. Our result can thus be interpreted as a reqularization of SDE systems. The systems we study are motivated by experimental work on the motion of light-sensitive robots and on motility-induced phase separation in bacterial colonies.

We revisit the Haake-Lewenstein-Wilkens (HLW) approach to Edwards-Anderson (EA) model of Ising spin glass [Phys. Rev. Lett. 55, 2606 (1985)]. This approach consists in evaluation and analysis of the probability distribution of configurations of two replicas of the system, averaged over quenched disorder. This probability distribution generates squares of thermal copies of spin variables from the two copies of the systems, averaged over disorder, that is the terms that enter the standard definition of the original EA order parameter, qEA. We use saddle point/steepest descent method to calculate the average of the Gaussian disorder in higher dimensions. This approximate result suggest that qEA >0 at 0

We study the small mass limit of the equation describing planar motion of a charged particle of a small mass $\mu$ in a force field, containing a magnetic component, perturbed by a stochastic term. We regularize the problem by adding a small friction of intensity $\e>0$. We show that for all small but fixed frictions the small mass limit of $q_{\mu, \e}$ gives the solution $q_\e$ to a stochastic first order equation, containing a noise-induced drift term. Then, by using a generalization of the classical averaging theorem for Hamiltonian systems by Freidlin and Wentzell, we take the limit of the slow component of the motion $q_\e$ and we prove that it converges weakly to a Markov process on the graph obtained by identifying all points in the same connected components of the level sets of the magnetic field intensity function.

We study the dynamics of the center of mass of a Brownian particle levitated in a Paul trap. We focus on the overdamped regime in the context of levitodynamics, comparing theory with our numerical simulations and experimental data from a nanoparticle in a Paul trap. We provide an exact analytical solution to the stochastic equation of motion, expressions for the standard deviation of the motion, and thermalization times by using the WKB method under two different limits. Finally, we prove the power spectral density of the motion can be approximated by that of an Ornstein-Uhlenbeck process and use the found expression to calibrate the motion of a trapped particle.

The paper studies a class of quantum stochastic differential equations, modeling an interaction of a system with its environment in the quantum noise approximation. The space representing quantum noise is the symmetric Fock space over L^2(R_+). Using the isomorphism of this space with the space of square-integrable functionals of the Poisson process, the equations can be represented as classical stochastic differential equations, driven by Poisson processes. This leads to a discontinuous dynamical state reduction which we compare to the Ghirardi-Rimini-Weber model. A purely quantum object, the norm process, is found which plays the role of an observer (in the sense of Everett [H. Everett III, Reviews of modern physics, 29.3, 454, (1957)]), encoding all events occurring in the system space. An algorithm introduced by Dalibard et al [J. Dalibard, Y. Castin, and K. M{\o}lmer, Physical review letters, 68.5, 580 (1992)] to numerically solve quantum master equations is interpreted in the context of unravellings and the trajectories of expected values of system observables are calculated., Comment: 34 pages, 2 figures

We study homogenization for a class of generalized Langevin equations (GLEs) with state-dependent coefficients and exhibiting multiple time scales. In addition to the small mass limit, we focus on homogenization limits, which involve taking to zero the inertial time scale and, possibly, some of the memory time scales and noise correlation time scales. The latter are meaningful limits for a class of GLEs modeling anomalous diffusion. We find that, in general, the limiting stochastic differential equations (SDEs) for the slow degrees of freedom contain non-trivial drift correction terms and are driven by non-Markov noise processes. These results follow from a general homogenization theorem stated and proven here. We illustrate them using stochastic models of particle diffusion., Comment: 66 pages

We study the small-mass (overdamped) limit of Langevin equations for a particle in a potential and/or magnetic field with matrix-valued and state-dependent drift and diffusion. We utilize a bootstrapping argument to derive a hierarchy of approximate equations for the position degrees of freedom that are able to achieve accuracy of order $m^{\ell/2}$ over compact time intervals for any $\ell\in\mathbb{Z}^+$. This generalizes prior derivations of the homogenized equation for the position degrees of freedom in the $m\to 0$ limit, which result in order $m^{1/2}$ approximations. Our results cover bounded forces, for which we prove convergence in $L^p$ norms, and unbounded forces, in which case we prove convergence in probability., Comment: 49 pages

The presence of a delay between sensing and reacting to a signal can determine the long-term behavior of autonomous agents whose motion is intrinsically noisy. In a previous work [M. Mijalkov, A. McDaniel, J. Wehr, and G. Volpe, Phys. Rev. X 6, 011008 (2016)], we have shown that sensorial delay can alter the drift and the position probability distribution of an autonomous agent whose speed depends on the illumination intensity it measures. Here, using theory, simulations, and experiments with a phototactic robot, we generalize this effect to an agent for which both speed and rotational diffusion depend on the illumination intensity and are subject to two independent sensorial delays. We show that both the drift and the probability distribution are influenced by the presence of these sensorial delays. In particular, the radial drift may have positive as well as negative sign, and the position probability distribution peaks in different regions depending on the delay. Furthermore, the presence of multiple sensorial delays permits us to explore the role of the interaction between them., Comment: 13 pages, 6 figures

We study the small mass limit (or: the Smoluchowski-Kramers limit) of a class of quantum Brownian motions with inhomogeneous damping and diffusion. For Ohmic bath spectral density with a Lorentz-Drude cutoff, we derive the Heisenberg-Langevin equations for the particle's observables using a quantum stochastic calculus approach. We set the mass of the particle to equal $m = m_{0} \epsilon$, the reduced Planck constant to equal $\hbar = \epsilon$ and the cutoff frequency to equal $\Lambda = E_{\Lambda}/\epsilon$, where $m_0$ and $E_{\Lambda}$ are positive constants, so that the particle's de Broglie wavelength and the largest energy scale of the bath are fixed as $\epsilon \to 0$. We study the limit as $\epsilon \to 0$ of the rescaled model and derive a limiting equation for the (slow) particle's position variable. We find that the limiting equation contains several drift correction terms, the quantum noise-induced drifts, including terms of purely quantum nature, with no classical counterparts., Comment: 29 pages

This paper studies homogenization of stochastic differential systems. The standard example of this phenomenon is the small mass limit of Hamiltonian systems. We consider this case first from the heuristic point of view, stressing the role of detailed balance and presenting the heuristics based on a multiscale expansion. This is used to propose a physical interpretation of recent results by the authors, as well as to motivate a new theorem proven here. Its main content is a sufficient condition, expressed in terms of solvability of an associated partial differential equation ("the cell problem"), under which the homogenization limit of an SDE is calculated explicitly. The general theorem is applied to a class of systems, satisfying a generalized detailed balance condition with a position-dependent temperature., Comment: 32 pages

We study a class of systems whose dynamics are described by generalized Langevin equations with state-dependent coefficients. We find that in the limit, in which all the characteristic time scales vanish at the same rate, the position variable of the system converges to a homogenized process, described by an equation containing additional drift terms induced by the noise. The convergence results are obtained using the main result in \cite{hottovy2015smoluchowski}, whose version is proven here under a weaker spectral assumption on the damping matrix. We apply our results to study thermophoresis of a Brownian particle in a non-equilibrium heat bath., Comment: The contents and results of the paper have been revised and corrected from a previous version

We study the dynamics of a class of Hamiltonian systems with dissipation, coupled to noise, in a singular (small mass) limit. We derive the homogenized equation for the position degrees of freedom in the limit, including the presence of a {\em noise-induced drift} term. We prove convergence to the solution of the homogenized equation in probability and, under stronger assumptions, in an $L^p$-norm. Applications cover the overdamped limit of particle motion in a time-dependent electromagnetic field, on a manifold with time-dependent metric, and the dynamics of nuclear matter., Comment: 51 pages

The theory of quantum Brownian motion describes the properties of a large class of open quantum systems. Nonetheless, its description in terms of a Born-Markov master equation, widely used in the literature, is known to violate the positivity of the density operator at very low temperatures. We study an extension of existing models, leading to an equation in the Lindblad form, which is free of this problem. We study the dynamics of the model, including the detailed properties of its stationary solution, for both constant and position-dependent coupling of the Brownian particle to the bath, focusing in particular on the correlations and the squeezing of the probability distribution induced by the environment, Comment: 15 pages, 10 figures

We study damped geodesic motion of a particle of mass $m$ on a Riemannian manifold, in the presence of an external force and noise. Lifting the resulting stochastic differential equation to the orthogonal frame bundle, we prove that, as $m \to 0$, its solutions converge to solutions of a limiting equation which includes a {\it noise-induced drift} term. A very special case of the main result presents Brownian motion on the manifold as a limit of inertial systems., Comment: 48 pages

Noisy dynamical models are employed to describe a wide range of phenomena. Since exact modeling of these phenomena requires access to their microscopic dynamics, whose time scales are typically much shorter than the observable time scales, there is often need to resort to effective mathematical models such as stochastic differential equations (SDEs). In particular, here we consider effective SDEs describing the behavior of systems in the limits when natural time scales became very small. In the presence of multiplicative noise (i.e., noise whose intensity depends upon the system's state), an additional drift term, called noise-induced drift, appears. The nature of this noise-induced drift has been recently the subject of a growing number of theoretical and experimental studies. Here, we provide an extensive review of the state of the art in this field. After an introduction, we discuss a minimal model of how multiplicative noise affects the evolution of a system. Next, we consider several case studies with a focus on recent experiments: Brownian motion of a microscopic particle in thermal equilibrium with a heat bath in the presence of a diffusion gradient, and the limiting behavior of a system driven by a colored noise modulated by a multiplicative feedback. This allows us to present the experimental results, as well as mathematical methods and numerical techniques that can be employed to study a wide range of systems. At the end we give an application-oriented overview of future projects involving noise-induced drifts, including both theory and experiment., Comment: 19 pages, 11 figures

Collective motions emerging from the interaction of autonomous mobile individuals play a key role in many phenomena, from the growth of bacterial colonies to the coordination of robotic swarms. For these collective behaviours to take hold, the individuals must be able to emit, sense and react to signals. When dealing with simple organisms and robots, these signals are necessarily very elementary, e.g. a cell might signal its presence by releasing chemicals and a robot by shining light. An additional challenge arises because the motion of the individuals is often noisy, e.g. the orientation of cells can be altered by Brownian motion and that of robots by an uneven terrain. Therefore, the emphasis is on achieving complex and tunable behaviors from simple autonomous agents communicating with each other in robust ways. Here, we show that the delay between sensing and reacting to a signal can determine the individual and collective long-term behavior of autonomous agents whose motion is intrinsically noisy. We experimentally demonstrate that the collective behaviour of a group of phototactic robots capable of emitting a radially decaying light field can be tuned from segregation to aggregation and clustering by controlling the delay with which they change their propulsion speed in response to the light intensity they measure. We track this transition to the underlying dynamics of this system, in particular, to the ratio between the robots' sensorial delay time and the characteristic time of the robots' random reorientation. Supported by numerics, we discuss how the same mechanism can be applied to control active agents, e.g. airborne drones, moving in a three-dimensional space., Comment: 8 pages, 5 figures

We consider a general stochastic differential delay equation (SDDE) with state-dependent colored noises and derive its limit as the time delays and the correlation times of the noises go to zero. The work is motivated by an experiment involving an electrical circuit with noisy, delayed feedback. An Ornstein-Uhlenbeck process is used to model the colored noise. The main methods used in the proof are a theorem about convergence of solutions of stochastic differential equations by Kurtz and Protter and a maximal inequality for sums of a stationary sequence of random variables by Peligrad and Utev., Comment: This is a pre-print of an article published in Journal of Statistical Physics. The final authenticated version is available online at: https://doi.org/10.1007/s10955-019-02242-2

We consider a family of percolation models in which geometry and connectivity are defined by two independent random processes. Such models merge characteristics of discrete and continuous percolation. We develop an algorithm allowing effective computation of both universal and modelspecific percolation quantities in the case when both random processes are Poisson processes. The algorithm extends percolation algorithm by Newman and Ziff (M.E.J. Newman and R.M. Ziff, Phys Rev E, 64(1):016706, 2001) to handle inhomogeneous lattices. In particular, we use the proposed method to compute critical exponents and cluster density distribution in two and three dimensions for the model of parallel random tubes connected randomly by bonds, which models the connectivity properties of activated carbon., Comment: 10 pages, 8 figures

We investigate the effect of a unidirectional quenched random field on the anisotropic quantum spin-1/2 $XY$ model, which magnetizes spontaneously in the absence of the random field. We adopt mean-field approach to show that spontaneous magnetization persists even in the presence of this random field but the magnitude of magnetization gets suppressed due to disorder, and the system magnetizes in the directions parallel and transverse to the random field. Our results are obtained by analytical calculations within perturbative framework and by numerical simulations. Interestingly, we show that it is possible to enhance a component of the magnetization in presence of the disorder field provided we apply an additional constant field in the $XY$ plane. Moreover, we derive generalized expressions for the critical temperature and the scalings of the magnetization near the critical point for the $XY$ spin system with arbitrary fixed quantum spin angular momentum., Comment: 11 pages, 6 figures

We analyze the microscopic model of quantum Brownian motion, describing a Brownian particle interacting with a bosonic bath through a coupling which is linear in the creation and annihilation operators of the bath, but may be a nonlinear function of the position of the particle. Physically, this corresponds to a configuration in which damping and diffusion are spatially inhomogeneous. We derive systematically the quantum master equation for the Brownian particle in the Born-Markov approximation and we discuss the appearance of novel terms, for various polynomials forms of the coupling. We discuss the cases of linear and quadratic coupling in great detail and we derive, using Wigner function techniques, the stationary solutions of the master equation for a Brownian particle in a harmonic trapping potential. We predict quite generally Gaussian stationary states, and we compute the aspect ratio and the spread of the distributions. In particular, we find that these solutions may be squeezed (super-localized) with respect to the position of the Brownian particle. We analyze various restrictions to the validity of our theory posed by non-Markovian effects and by the Heisenberg principle. We further study the dynamical stability of the system, by applying a Gaussian approximation to the time dependent Wigner function, and we compute the decoherence rates of coherent quantum superpositions in position space. Finally, we propose a possible experimental realization of the physics discussed here, by considering an impurity particle embedded in a degenerate quantum gas., Comment: 23 pages, 5 figures

We study ground states of Ising models with random ferromagnetic couplings, proving the triviality of all zero-temperature metastates. This unexpected result sheds a new light on the properties of these systems, putting strong restrictions on their possible ground state structure. Open problems related to existence of interface-supporting ground states are stated and an interpretation of the main result in terms of first-passage and random surface models in a random environment is presented.

We consider classical spin models of two- and three-dimensional spins with continuous symmetry and investigate the effect of a symmetry-breaking unidirectional quenched disorder on the magnetization of the system. We work in the mean-field regime. We show, by perturbative calculations and numerical simulations, that although the continuous symmetry of the magnetization is lost due to disorder, the system still magnetizes in specific directions, albeit with a lower value as compared to the case without disorder. The critical temperature, at which the system starts magnetizing, as well as the magnetization at low and high temperature limits, in presence of disorder, are estimated. Moreover, we treat the SO(n) n-component spin model to obtain the generalized expressions for the near-critical scalings, which suggest that the effect of disorder in magnetization increases with increasing dimension. We also study the behavior of magnetization of the classical XY spin model in the presence of a constant magnetic field, in addition to the quenched disorder. We find that in presence of the uniform magnetic field, disorder may enhance the component of magnetization in the direction that is transverse to the disorder field., Comment: 18 pages, 12 figures, REVTeX4-1; significantly extends and improves upon the results in arXiv:0712.0829

We consider a general multidimensional stochastic differential delay equation (SDDE) with state-dependent colored noises. We approximate it by a stochastic differential equation (SDE) system and calculate its limit as the time delays and the correlation times of the noises go to zero. The main result is proven using a theorem about convergence of stochastic integrals by Kurtz and Protter. It formalizes and extends a result that has been obtained in the analysis of a noisy electrical circuit with delayed state-dependent noise, and may be used as a working SDE approximation of an SDDE modeling a real system where noises are correlated in time and whose response to noise sources depends on the system's state at a previous time.

We study a class of systems of stochastic differential equations describing diffusive phenomena. The Smoluchowski-Kramers approximation is used to describe their dynamics in the small mass limit. Our systems have arbitrary state-dependent friction and noise coefficients. We identify the limiting equation and, in particular, the additional drift term that appears in the limit is expressed in terms of the solution to a Lyapunov matrix equation. The proof uses a theory of convergence of stochastic integrals developed by Kurtz and Protter. The result is sufficiently general to include systems driven by both white and Ornstein-Uhlenbeck colored noises. We discuss applications of the main theorem to several physical phenomena, including the experimental study of Brownian motion in a diffusion gradient., Comment: This paper has been corrected from a previous version. Author Austin McDaniel has been added. Lemma 2 has been rewritten, Lemma 3 added, previous version's Lemma 3 moved to Lemma 4. 20 pages, 1 figure

This paper presents an elementary proof of Lifschitz tail behavior for random discrete Schr\"{o}dinger operators with a Bernoulli-distributed potential. The proof approximates the low eigenvalues by eigenvalues of sine waves supported where the potential takes its lower value. This is motivated by the idea that the eigenvectors associated to the low eigenvalues react to the jump in the values of the potential as if the gap were infinite.

This paper explores a system of interacting `soft core' bosons in the Gross-Pitaevskii mean-field approximation in a random Bernoulli potential. First, a condition for delocalization of the ground state wave function is proved which depends on the number of particles and interaction strength. Using this condition, asymptotics for ground state energy per particle are derived in the large system limit for small values of the coupling constant. Our methods directly describe the shape of the ground state in a given realization of the random potential., Comment: 18 pages

Intrinsically noisy mechanisms drive most physical, biological and economic phenomena, from stock pricing to phenotypic variability. Frequently, the system's state influences the driving noise intensity, as, for example, the actual value of a commodity may alter its volatility or the concentration of gene products may regulate their expression. All these phenomena are often modeled using stochastic differential equations (SDEs). However, an SDE is not sufficient to fully describe a noisy system with a multiplicative feedback, because it can be interpreted according to various conventions -- in particular, It\^{o} calculus and Stratonovitch calculus --, each of which leads to a qualitatively different solution. Which convention to adopt must be determined case by case on the basis of the available experimental data; for example, the SDE describing electrical circuits driven by a noise are known to obey Stratonovich calculus. Once such an SDE-convention pair is determined, it c an be employed to predict the system's behavior under new conditions. Here, we experimentally demonstrate that the convention for a given physical system may actually vary under varying operational conditions. We show that, under certain conditions, a noisy electric circuit shifts to obey It\^o calculus, which may dramatically alter the system's long term stability. We track such Stratonovich-to-It\^o transition to the underlying dynamics of the system and, in particular, to the ratio between the driving noise correlation time and the feedback delay time. We briefly discuss ramifications of our conclusions for biology and economics: the possibility of similar transitions and their dramatic consequences should be recognized and accounted for where SDEs are employed to predict the evolution of complex phenomena., Comment: 9 pages, 5 figures

We analyze the disordered Riemannian geometry resulting from random perturbations of the Euclidean metric. We focus on geodesics, the paths traced out by a particle traveling in this quenched random environment. By taking the point of the view of the particle, we show that the law of its observed environment is absolutely continuous with respect to the law of the random metric, and we provide an explicit form for its Radon-Nikodym derivative. We use this result to prove a "local Markov property" along an unbounded geodesic, demonstrating that it eventually encounters any type of geometric phenomenon. We also develop in this paper some general results on conditional Gaussian measures. Our Main Theorem states that a geodesic chosen with random initial conditions (chosen independently of the metric) is almost surely not minimizing. To demonstrate this, we show that a minimizing geodesic is guaranteed to eventually pass over a certain "bump surface," which locally has constant positive curvature. By using Jacobi fields, we show that this is sufficient to destabilize the minimizing property., Comment: 55 pages. Supplementary material at arXiv:1206.4940

This is supplementary material for the main Geodesics article by the authors. In Appendix A, we present some general results on the construction of Gaussian random fields. In Appendix B, we restate our Shape Theorem, specialized to the setting of this article. In Appendix C, we state some straightforward consequences on the geometry of geodesics for a random metric. In Appendix D, we provide a rapid introduction to Riemannian geometry for the unfamiliar reader. In Appendix E, we present some analytic estimates which we use in the article. In Appendix F, we present the construction of the conditional mean operator for Gaussian measures. In Appendix G, we describe Fermi normal coordinates, which we use in our construction of the bump metric., Comment: 12 pages. Main article: arxiv:1206.4939

The Brownian motion of microscopic particles is driven by the collisions with the molecules of the surrounding fluid. The noise associated with these collisions is not white, but coloured due, e.g., to the presence of hydrodynamic memory. The noise characteristic time scale is typically of the same order as the time over which the particle's kinetic energy is lost due to friction (inertial time scale). We demonstrate theoretically that, in the presence of a temperature gradient, the interplay between these two characteristic time scales can have measurable consequences on the particle long-time behaviour. Using homogenization theory, we analyse the infinitesimal generator of the stochastic differential equation describing the system in the limit where the two characteristic times are taken to zero; from this generator, we derive the thermophoretic transport coefficient, which, we find, can vary in both magnitude and sign, as observed in experiments. Furthermore, studying the long-term stationary particle distribution, we show that particles can accumulate towards the colder (positive thermophoresis) or the warmer (negative thermophoresis) regions depending on the dependence of their physical parameters and, in particular, their mobility on the temperature., Comment: 9 pages, 4 figures

We consider the dynamics of systems with arbitrary friction and diffusion. These include, as a special case, systems for which friction and diffusion are connected by Einstein fluctuation-dissipation relation, e.g. Brownian motion. We study the limit where friction effects dominate the inertia, i.e. where the mass goes to zero (Smoluchowski-Kramers limit). {Using the It\^o stochastic integral convention,} we show that the limiting effective Langevin equations has different drift fields depending on the relation between friction and diffusion. {Alternatively, our results can be cast as different interpretations of stochastic integration in the limiting equation}, which can be parametrized by $\alpha \in \mathbb{R}$. Interestingly, in addition to the classical It\^o ($\alpha=0$), Stratonovich ($\alpha=0.5$) and anti-It\^o ($\alpha=1$) integrals, we show that position-dependent $\alpha = \alpha(x)$, and even stochastic integrals with $\alpha \notin [0,1]$ arise. Our findings are supported by numerical simulations., Comment: 11 pages, 5 figures

We investigate the interplay between disorder and interactions in a Bose gas on a lattice in presence of randomly localized impurities. We compare the performance of two theoretical methods, the simple version of multi-orbital Hartree-Fock and the common Gross-Pitaevskii approach, showing how the former gives a very good approximation to the ground state in the limit of weak interactions, where the superfluid fraction is small. We further prove rigorously that for this class of disorder the fractal dimension of the ground state d* tends to the physical dimension in the thermodynamic limit. This allows us to introduce a quantity, the fractional occupation, which gives insightful information on the crossover from a Lifshits to a Bose glass. Finally, we compare temperature and interaction effects, highlighting similarities and intrinsic differences., Comment: Revised and corrected manuscript, new figures

In this paper, we show the that the ground state energy of the one dimensional Discrete Random Schroedinger Operator with Bernoulli Potential is controlled asymptotically as the system size N goes to infinity by the random variable \ell_N, the length the longest consecutive sequence of sites on the lattice with potential equal to zero. Specifically, we will show that for almost every realization of the potential the ground state energy behaves asymptotically as $\frac{\pi^2}{\ell_N+1)^2}$ in the sense that the ratio of the quantities goes to one.

We study a family of quadratic stochastic differential equations in the plane, motivated by applications to turbulent transport of heavy particles. Using Lyapunov functions, we find a critical parameter value $\alpha_{1}=\alpha_{2}$ such that when $\alpha_{2}>\alpha_{1}$ the system is ergodic and when $\alpha_{2}<\alpha_{1}$ solutions are not defined for all times. H\"{o}rmander's hypoellipticity theorem and geometric control theory are also utilized., Comment: 35 pages, 6 figures

Having a priori knowledge of the force acting on a noisy system it is possible to solve the issue relative to the interpretation of multiplicative noise terms (aka Ito-Stratonovich dilemma). We experimentally show that for a Brownian particle the anti-Ito convention is correct. This permits us to reconcile force measurements based on the equilibrium distribution and on the drift velocity of a Brownian particle in a diffusion gradient.

The study of microsystems and the development of nanotechnologies require new techniques to measure piconewton and femtonewton forces at microscopic and nanoscopic scales. Amongst the challenges, there is the need to deal with the ineluctable thermal noise, which, in the typical experimental situation of a spatial diffusion gradient, causes a spurious drift. This leads to a correction term when forces are estimated from drift measurements [Phys. Rev. Lett. 104, 170602 (2010)]. Here, we provide a systematic study of such effect comparing the forces acting on various Brownian particles derived from equilibrium distribution and drift measurements. We discuss the physical origin of the correction term, its dependence on wall distance, particle radius, and its relation to the convention used to solve the respective stochastic integrals. Such correction term becomes more significant for smaller particles and is predicted to be in the order of several piconewtons for particles the size of a biomolecule., Comment: 10 pages, 13 figures

We study a simple stochastic differential equation that models the dispersion of close heavy particles moving in a turbulent flow. In one and two dimensions, the model is closely related to the one-dimensional stationary Schroedinger equation in a random delta-correlated potential. The ergodic properties of the dispersion process are investigated by proving that its generator is hypoelliptic and using control theory.

We observe signatures of disorder-induced order in 1D XY spin chains with an external, site-dependent uni-axial random field within the XY plane. We numerically investigate signatures of a quantum phase transition at T=0, in particular an upsurge of the magnetization in the direction orthogonal to the external magnetic field, and the scaling of the block-entropy with the amplitude of this field. Also, we discuss possible realizations of this effect in ultra-cold atom experiments.

We demonstrate how the ineluctable presence of thermal noise alters the measurement of forces acting on microscopic and nanoscopic objects. We quantify this effect exemplarily for a Brownian particle near a wall subjected to gravitational and electrostatic forces. Our results demonstrate that the force measurement process is prone to artifacts if the noise is not correctly taken into account., Comment: 4 Pages, 4 Figures, Accepted

Riemannian first-passage percolation (FPP) is a continuum model, with a distance function arising from a random Riemannian metric in $\R^d$. Our main result is a shape theorem for this model, which says that large balls under this metric converge to a deterministic shape under rescaling. As a consequence, we show that smooth random Riemannian metrics are geodesically complete with probability one.

We consider a mixture of a superfluid Fermi gas of ultracold atoms and a Bose-Einstein condensate of molecules possessing a continuous U(1) (relative phase) symmetry. We study the effects that a spatially random photo-associative-dissociative symmetry breaking coupling of the systems. Such coupling allows to control the relative phase between a superfluid order parameter of the Fermi system and the condensate wavefunction of molecules for temperatures below the BCS critical temperature. The presented mechanism of phase control belongs to the general class of disorder-induced order phenomena that rely on breaking of continuous symmetry., Comment: 4 pages, 2 figures

We study strategies for establishing long-distance entanglement in quantum networks. Specifically, we consider networks consisting of regular lattices of nodes, in which the nearest neighbors share a pure, but non-maximally entangled pair of qubits. We look for strategies that use local operations and classical communication. We compare the classical entanglement percolation protocol, in which every network connection is converted with a certain probability to a singlet, with protocols in which classical entanglement percolation is preceded by measurements designed to transform the lattice structure in a way that enhances entanglement percolation. We analyze five examples of such comparisons between protocols and point out certain rules and regularities in their performance as a function of degree of entanglement and choice of operations., Comment: 12 pages, 17 figures, revtex4. changes from v3: minor stylistic changes for journal reviewer, minor changes to figures for journal editor

We consider a classical spin model, of two-dimensional spins, with continuous symmetry, and investigate the effect of a symmetry breaking unidirectional quenched disorder on the magnetization of the system. We work in the mean field regime. We show, by numerical simulations and by perturbative calculations in the low as well as in the high temperature limits, that although the continuous symmetry of the magnetization is lost, the system still magnetizes, albeit with a lower value as compared to the case without disorder. The critical temperature at which the system starts magnetizing, also decreases with the introduction of disorder. However, with the introduction of an additional constant magnetic field, the component of magnetization in the direction that is transverse to the disorder field increases with the introduction of the quenched disorder. We discuss the same effects also for three-dimensional spins., Comment: 12 pages, 12 figures, RevTeX4

In order to stabilize the behavior of noisy systems, confining it around a desirable state, an effort is required to suppress the intrinsic noise. This noise suppression task entails a cost. For the important case of thermal noise in an overdamped system, we show that the minimum cost is achieved when the system control parameters are held constant: any additional deterministic or random modulation produces an increase of the cost. We discuss the implications of this phenomenon for those overdamped systems whose control parameters are intrinsically noisy, presenting a case study based on the example of a Brownian particle optically trapped in an oscillating potential., Comment: 6 pages

We propose and analyze a general mechanism of disorder-induced order in two-component Bose-Einstein condensates, analogous to corresponding effects established for XY spin models. We show that a random Raman coupling induces a relative phase of pi/2 between two BECs and that the effect is robust. We demonstrate it in 1D, 2D and 3D at T=0 and present evidence that it persists at small T>0. Applications to phase control in ultracold spinor condensates are discussed., Comment: 4 pages, 3 figures

We propose a general mechanism of random-field-induced order (RFIO), in which long-range order is induced by a random field that breaks the continuous symmetry of the model. We particularly focus on the case of the classical ferromagnetic XY model on a 2D lattice, in a uniaxial random field. We prove rigorously that the system has spontaneous magnetization at temperature T=0, and we present strong evidence that this is also the case for small T>0. We discuss generalizations of this mechanism to various classical and quantum systems. In addition, we propose possible realizations of the RFIO mechanism, using ultracold atoms in an optical lattice. Our results shed new light on controversies in existing literature, and open a way to realize RFIO with ultracold atomic systems., Comment: published version; 8 pages, 3 figures