Jan Wehr

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• Piñol, E., Mavrogordatos, T. K., Keys, D., Veyron, R., Sierant, P., García-March, M. A., Grandi, S., Mitchell, M. W., Wehr, J., & Lewenstein, M. (2023). Telling different unravelings apart via nonlinear quantum-trajectory averages. Physical Review Letters.
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  We propose a way to operationally infer different unravelings of the Gorini-Kossakowski-Sudarshan-Lindblad master equation appealing to stochastic conditional dynamics via quantum trajectories. We focus on the paradigmatic quantum nonlinear system of resonance fluorescence for the two most popular unravelings: the Poisson-type, corresponding to direct detection of the photons scattered from the two-level emitter, and the Wiener-type, revealing complementary attributes of the signal to be measured, such as the wave amplitude and the spectrum. We show that a quantum-trajectory-averaged variance, made of single trajectories beyond the standard description offered by the density-matrix formalism, is able to make a distinction between the different environments encountered by the field scattered from the two-level emitter. Our proposal is tested against commonly encountered experimental limitations, and can be readily extended to account for open quantum systems with several degrees of freedom. [Journal_ref: ]
• Wehr, J., Tercas, H., Mendonca, J. T., Lewenstein, M., Khan, M. M., Garcia-march, M. A., & Charalambous, C. (2021). Quantum dynamics of a Bose polaron in a d -dimensional Bose-Einstein condensate. Physical Review A, 103(2). doi:10.1103/physreva.103.023303
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  We study the quantum motion of an impurity atom immersed in a Bose-Einstein condensate in arbitrary dimensions. It was shown, for all dimensions, that the Bogoliubov excitations of the Bose-Einstein condensate act as a bosonic bath for the impurity, where linear coupling is possible for a certain regime of validity, which was assessed only in one dimension. Here we present the detailed derivation of the $d$-dimensional Langevin equations that describe the quantum dynamics of the system, and of the associated generalized tensor that describes the spectral density in the full generality, and assesses the linear assumption in all dimensions. As results, we obtain, when the impurity is not trapped, the mean square displacement in all dimensions, showing that the motion is superdiffusive. We obtain also explicit expressions for the superdiffusive coefficient in the small and large temperature limits. We find that, in the latter case, the maximal value of this coefficient is the same in all dimensions, but is only reachable in one dimension, within the validity of the assumptions. We study also the behavior of the average energy and compare the results for various dimensions. In the trapped case, we study squeezing and find that the stronger position squeezing can be obtained in lower dimensions. We quantify the non-Markovianity of the particle's motion and find that it increases with dimensionality.
• Wehr, J., & Birrell, J. (2020). Langevin equations in the small-mass limit: Higher order approximations. Annales Henri Poincare, 21, 1765-1811.
• Wehr, J., & Keys, D. (2020). Poisson stochastic master equation unravellings and the measurement problem: a quantum stochastic calculus perspective. Journal of Mathematical Physics, 61. doi:https://doi.org/10.1063/1.5133974
• Wehr, J., Cerrai, S., & Zhu, Y. (2020). An Averaging Approach to the Smoluchowski–Kramers Approximation in the Presence of a Varying Magnetic Field. Journal of Statistical Physics, 181, 132-148.
• Wehr, J., Conangla, G., Nwaigwe, D., & Rica, R. (2020). Overdamped dynamics of a Brownian particle levitated in a Paul trap. Physical Review A, 101(5). doi:https://doi.org/10.1103/PhysRevA.101.053823
• Wehr, J., Lim, S. H., & Lewenstein, M. (2020). Homogenization for Generalized Langevin Equations with Applications to Anomalous Diffusion. Annales Henri Poincare, 21, 1813-1871.
• Hottovy, S., & McDaniel, A. (2019). A small delay and correlation time limit of stochastic dierential delay equations with state-dependent colored noise. Journal of Statistical Physics, 175, 19-46.
• Lim, S. H., & Wehr, J. (2019). Homogenization for a Class of Generalized Langevin Equations with an Application to Thermophoresis. JOURNAL OF STATISTICAL PHYSICS, 174(3), 656-691.
• Wehr, J., & Birrell, J. (2019). A homogenization theorem for Langevin systems with an application to Hamiltonian dynamics. Contribution to "Sojourns in probability theory and statistical physics", V. Sidoravicius (ed.), 1, 89-122.
• Wehr, J., & Lim, S. H. (2019). Homogenization for a Class of Generalized Langevin Equations with an Application to Thermophoresis. Journal of Statistical Physics.
• Lewenstein, M., Lim, S. H., Garcia-March, M. A., & Lampo, A. (2018). On the Small Mass Limit of Quantum Brownian Motion with Inhomogeneous Damping and Diusion. Journal of Statistical Physics, 170(2), 351-377.
• Wehr, J., & Birrell, J. (2018). Homogenization of Dissipative, Noisy, Hamiltonian Dynamics. Submitted to Stochastic Analysis (2017). Stochastic Analysis and Its Applications, 128(7). doi:https://doi.org/10.1016/j.spa.2017.09.005
• Wehr, J., & Lim, S. H. (2018). Homogenization for a Class of Generalized Langevin Equations with an Application to Thermophoresis. Journal of Statistical Physics.
• Wehr, J., Lewenstein, M., García-March, M. Á., Lampo, A., & Lim, S. H. (2018). On the Small Mass Limit of Quantum Brownian Motion with Inhomogeneous Damping and Diffusion. Journal of Statistical Physics. doi:10.1007/s10955-017-1907-7
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  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.
• Wehr, J., Volpe, G., Leyman, M., & Ogemark, F. (2018). Tuning phototactic robots with sensorial delays. Physical Review E.
• Birrell, J., Wehr, J., & Birrell, J. (2017). Homogenization of dissipative, noisy, Hamiltonian dynamics. Stochastic Processes and their Applications, 128(7), 2367-2403. doi:10.1016/j.spa.2017.09.005
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  Abstract 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 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.
• Wehr, J., Mijalkov, M., McDaniel, A., & Volpe, G. (2017). Small Mass Limit of a Langevin Equation on a Manifold. Ann. Henri Poincare 18 no. 2, 707-755 (2017), 18(2), 707-755.
• Arguin, L., Newman, C., Stein, D., & Wehr, J. (2016). Zero-temperature fluctuations in short-range spin glasses. Journal of Statistical Physics, 1-10. doi:10.1007/s10955-016-1516-x
• Bera, A., Rakshit, D., Lewenstein, M., Sen, A., Sen, U., & Wehr, J. (2016). Order-from-disorder and critical scalings of spontaneous magnetization in random-field quantum spin systems. Physical Review B, 94. doi:https://doi.org/10.1103/PhysRevB.94.014421
• Birrell, J., Hottovy, S., Volpe, G., & Wehr, J. (2016). Small mass limit of a Langevin equation on a manifold. Annals Henri Poincare. doi:10.1007/s00023-016-0508-3
• Lampo, A., Lim, S. H., Wehr, J., Massignan, P., & Lewenstein, M. (2016). Lindblad model of quantum Brownian motion. Physical Review A, 94(4). doi:10.1103/PhysRevA.94.042123
• Mijalkov, M., McDaniel, A., Volpe, G., & Wehr, J. (2016). Engineering Sensorial Delay to Control Phototaxis and Emergent Collective Behaviors. Physical Review X, 6(1). doi:011008
• Szczygiel, B., Dudynski, M., Kwiatkowski, K., Lewenstein, M., Lapeyre, G. J., & Wehr, J. (2016). Percolation thresholds for discrete-continuous models with nonuniform probabilities of bond formation. Physical Review E. doi:022127
• Volpe, G., & Wehr, J. (2016). Effective drifts in dynamical systems with multiplicative noise: A review of recent progress. Reports on Progress in Physics. doi:https://doi.org/10.1088/0034-4885/79/5/053901
• Wehr, J., & Wasielak, A. (2016). Uniqueness of translation-covariant zero-temperature metastate in disordered Ising ferromagnets. Journal of Statistical Physics. Journal of Statistical Physics, 162(2), 487-494.
• Wehr, J., Duman, O., McDaniel, A., & Volpe, G. (2016). McDaniel, A., Duman, O., Volpe, G., & Wehr, J. An SDE approximation for stochastic differential delay equations with colored state-dependent noise.. Markov Processes and Related Fields, 22(3), 595-628.
• Hottovy, S., McDaniel, A., Volpe, G., & Wehr, J. (2015). The Smoluchowski-Kramers limit of stochastic dierential equations with arbitrary state-dependent friction. Communications in Mathematical Physics, 336, 1259-1283. doi:http://link.springer.com/article/10.1007/s00220-014-2233-4?sa_campaign=email/event/articleAuthor/onlineFirst
• Massignan, P., Lampo, A., Wehr, J., & Lewenstein, M. (2015). Quantum Brownian motion with inhomogeneous damping and diusion. Physical Review A, 91. doi:033627
• Wehr, J., Bishop, M., & Borovyk, V. (2015). Lifschitz Tails for Random Schrdinger Operator in Bernoulli Distributed Potentials. Journal of Statistical Physics, 160(1), 151-162. doi:http://link.springer.com/article/10.1007%2Fs10955-015-1242-9
• Wehr, J., Bishop, M., & Borovyk, V. (2015). Lifschitz Tails for Random Schrödinger Operator in Bernoulli Distributed Potentials.. Journal of Statistical Physics, 160, 151.
• Wehr, J., Sen, U., Sen, A., Rakshit, D., Lewenstein, M., De, A. S., & Bera, A. (2015). Erratum: Classical spin models with broken symmetry: Random-field-induced order and persistence of spontaneous magnetization in the presence of a random field (Physical Review B - Condensed Matter and Materials Physics (2014) 90 (174408)). Physical Review B, 91(9). doi:10.1103/physrevb.91.099905
• LaGatta, T., & Wehr, J. (2014). Geodesics of Random Riemannian Metrics. Communications in Mathematical Physics, 327(1), 181-241.
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  Abstract: 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. © 2014 Springer-Verlag Berlin Heidelberg.
• Wehr, J., Arguin, L., Newman, C., & Stein, D. (2014). Fluctuation Bounds For Interface Free Energies in Spin Glasses. Journal of Statistical Physics.
• Wehr, J., Bera, A., Rakshit, D., Lewenstein, M., Sen(De), A., & Sen, U. (2014). Classical spin models with broken symmetry: Random-field-induced order and persistence of spontaneous magnetization in the presence of a random field A. Bera, D. Rakshit, M. Lewenstein, A. Sen(De), U. Sen, J. Wehr Phys. Rev. B 90, 174408 (2014). Phys. Rev. B, Phys. Rev. B 90, 174408 (2014).
• Wehr, J., Mcdaniel, A., Hottovy, S., & Volpe, G. (2014). Hottovy, S., McDaniel, A., Volpe, G., & Wehr, J. (2014). The Smoluchowski-Kramers limit of stochastic differential equations with arbitrary state-dependent friction. Commun. Math. Phys. DOI: 10.1007/s00220-014-2233-4, DOI: 10.1007/s00220-014-2233-4.
• Bishop, M., & Wehr, J. (2013). Ground state energy of mean field model of interacting bosons in Bernoulli potential. Journal of Mathematical Physics, 54(8).
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  Abstract: This paper explores the ground state energy of a system of interacting "soft core" bosons in a random Bernoulli potential in the Gross-Pitaevskii mean-field approximation. First, we prove a condition for a state to delocalize due to interaction. 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. © 2013 AIP Publishing LLC.
• Pesce, G., McDaniel, A., Hottovy, S., Wehr, J., & Volpe, G. (2013). Stratonovich-to-Itô transition in noisy systems with multiplicative feedback. Nature Communications, 4.
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  PMID: 24217466;Abstract: Intrinsically noisy mechanisms drive most physical, biological and economic phenomena. Frequently, the system's state influences the driving noise intensity (multiplicative feedback). These phenomena are often modelled using stochastic differential equations, which can be interpreted according to various conventions (for example, Itô calculus and Stratonovich calculus), leading to qualitatively different solutions. Thus, a stochastic differential equation-convention pair must be determined from the available experimental data before being able to predict the system's behaviour under new conditions. Here we experimentally demonstrate that the convention for a given system may vary with the operational conditions: we show that a noisy electric circuit shifts from obeying Stratonovich calculus to obeying Itô calculus. We track such a 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 discuss possible implications of our conclusions, supported by numerics, for biology and economics. © 2013 Macmillan Publishers Limited. All rights reserved.
• Birrell, J., Herzog, D. P., & Wehr, J. (2012). The transition from ergodic to explosive behavior in a family of stochastic differential equations. Stochastic Processes and their Applications, 122(4), 1519-1539.
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  Abstract: We study a family of quadratic, possibly degenerate, stochastic differential equations in the plane, motivated by applications to turbulent transport of heavy particles. Using Lyapunov functions, Hrmander's hypoellipticity theorem, and geometric control theory, we find a critical parameter value α 1=α 2 such that when α 2>α 1 the system is ergodic and when α 2
• Bishop, M., & Wehr, J. (2012). Ground State Energy of the One-Dimensional Discrete Random Schrödinger Operator with Bernoulli Potential. Journal of Statistical Physics, 147(3), 529-541.
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  Abstract: In this paper we show the that the ground state energy of the one-dimensional discrete random Schrödinger operator with Bernoulli potential is controlled asymptotically as the system size N goes to infinity by the random variable ℓ 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 π 2/(ℓ N+1) 2 in the sense that the ratio of the quantities goes to one. © 2012 Springer Science+Business Media, LLC.
• Hottovy, S., Volpe, G., & Wehr, J. (2012). Noise-Induced Drift in Stochastic Differential Equations with Arbitrary Friction and Diffusion in the Smoluchowski-Kramers Limit. Journal of Statistical Physics, 146(4), 762-773.
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  Abstract: 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 Itô 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 α∈ℝ. Interestingly, in addition to the classical Itô (α=0), Stratonovich (α=0. 5) and anti-Itô (α=1) integrals, we show that position-dependent α=α(x), and even stochastic integrals with α∉[0,1] arise. Our findings are supported by numerical simulations. © 2012 Springer Science+Business Media, LLC.
• Hottovy, S., Volpe, G., & Wehr, J. (2012). Thermophoresis of Brownian particles driven by coloured noise. EPL, 99(6).
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  Abstract: Brownian motion of microscopic particles is driven by collisions with surrounding fluid molecules. The resulting noise 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 of magnitude as the inertial time-scale over which the particle's kinetic energy is lost due to friction. 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's 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 time-scales are taken to zero keeping their ratio constant and derive the thermophoretic transport coefficient, which, we find, can vary in both magnitude and sign, as observed in experiments. Studying the long-term stationary particle distribution, we show that particles accumulate towards the colder (positive thermophoresis) or the hotter (negative thermophoresis) regions depending on their physical parameters. © Copyright EPLA, 2012.
• Stasińska, J., Massignan, P., Bishop, M., Wehr, J., Sanpera, A., & Lewenstein, M. (2012). The glass to superfluid transition in dirty bosons on a lattice. New Journal of Physics, 14.
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  Abstract: We investigate the interplay between disorder and interactions in a Bose gas on a lattice in the presence of randomly localized impurities. We compare the performance of two theoretical methods, namely the simple version of multi-orbital Hartree-Fock and the common Gross-Pitaevskii approach, and show 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 dtends to the physical dimension in the thermodynamic limit. This allows us to introduce a quantity called the fractional occupation, which gives useful information on the crossover from a Lifshits to a Bose glass. Finally, we compare the temperature and interaction effects and highlight their similarities and intrinsic differences. © IOP Publishing Ltd and Deutsche Physikalische Gesellschaft.
• Brettschneider, T., Volpe, G., Helden, L., Wehr, J., & Bechinger, C. (2011). Force measurement in the presence of Brownian noise: Equilibrium- distribution method versus drift method. Physical Review E - Statistical, Nonlinear, and Soft Matter Physics, 83(4).
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  Abstract: The study of microsystems and the development of nanotechnologies require alternative techniques to measure piconewton and femtonewton forces at microscopic and nanoscopic scales. Among the challenges 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. Here we provide a systematic study of such an effect by 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 and particle radius, and its relation to the convention used to solve the respective stochastic integrals. Such a correction term becomes more significant for smaller particles and is predicted to be on the order of several piconewtons for particles the size of a biomolecule. © 2011 American Physical Society.
• Gawȩdzki, K., Herzog, D. P., & Wehr, J. (2011). Ergodic Properties of a Model for Turbulent Dispersion of Inertial Particles. Communications in Mathematical Physics, 308(1), 49-80.
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  Abstract: 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 Schrödinger equation in a random δ-correlated potential. The ergodic properties of the dispersion process are investigated by proving that its generator is hypoelliptic and using control theory. © 2011 Springer-Verlag.
• Volpe, G., Helden, L., Brettschneider, T., Wehr, J., & Bechinger, C. (2011). Volpe etal. Reply. Physical Review Letters, 107(7).
• Wehr, J., Volpe, G., Helden, L., Brettschneider, T., & Bechinger, C. (2011). Comment on "Influence of Noise and Force Measurements" Reply. Physical Review Letters, 107(7). doi:10.1103/physrevlett.107.078902
• LaGatta, T., & Wehr, J. (2010). A shape theorem for Riemannian first-passage percolation. Journal of Mathematical Physics, 51(5).
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  Abstract: Riemannian first-passage percolation is a continuum model, with a distance function arising from a random Riemannian metric in Rd. 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 of 1. © 2010 American Institute of Physics.
• Niederberger, A., Rams, M. M., Dziarmaga, J., Cucchietti, F. M., Wehr, J., & Lewenstein, M. (2010). Disorder-induced order in quantum XY chains. Physical Review A - Atomic, Molecular, and Optical Physics, 82(1).
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  Abstract: We observe signatures of disorder-induced order in one-dimensional XY spin chains with an external site-dependent uniaxial 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 ultracold atomic experiments. © 2010 The American Physical Society.
• Volpe, G., Helden, L., Brettschneider, T., Wehr, J., & Bechinger, C. (2010). Influence of noise on force measurements. Physical Review Letters, 104(17).
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  Abstract: 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. © 2010 The American Physical Society.
• Korbicz, J. K., Wehr, J., & Lewenstein, M. (2009). Entanglement and quantum groups. Journal of Mathematical Physics, 50(6).
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  Abstract: We describe quantum mechanical entanglement in terms of compact quantum groups. We prove an analog of positivity of partial transpose criterion and formulate a Horodecki-type theorem. © 2009 American Institute of Physics.
• Lapeyre, G. J., Wehr, J., & Lewenstein, M. (2009). Enhancement of entanglement percolation in quantum networks via lattice transformations. Physical Review A - Atomic, Molecular, and Optical Physics, 79(4).
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  Abstract: 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 nonmaximally 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. © 2009 The American Physical Society.
• Niederberger, A., Wehr, J., Lewenstein, M., & Sacha, K. (2009). Disorder-induced phase control in superfluid Fermi-Bose mixtures. EPL, 86(2).
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  Abstract: 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 of a spatially random photo-associative- dissociative symmetry-breaking coupling of the systems. Such coupling allows one to control the relative phase between a superfluid order parameter of the Fermi system and the condensate wave function of molecules for temperatures below the Bardeen-Cooper-Schriefer critical temperature. The presented mechanism of phase control belongs to the general class of phenomena in which disorder interacts with continuous symmetry. Our results show the robustness and wide range of applicability of disorder-induced order and are valid for both disordered and regular couplings. Here, the effect is studied in the case of interacting fermionic and bosonic gases in the superfluid phase. © EPLA, 2009.
• Volpe, G., Wehr, J., Petrov, D., & Rubi, J. M. (2009). Thermal noise suppression: How much does it cost?. Journal of Physics A: Mathematical and Theoretical, 42(9).
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  Abstract: 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. © 2009 IOP Publishing Ltd.
• Korbicz, J. K., Wehr, J., & Lewenstein, M. (2008). Entanglement of positive definite functions on compact groups. Communications in Mathematical Physics, 281(3), 753-774.
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  Abstract: We define and study entanglement of continuous positive definite functions on products of compact groups. We formulate and prove an infinite-dimensional analog of the Horodecki Theorem, giving a necessary and sufficient criterion for separability of such functions. The resulting characterisation is given in terms of mappings of the space of continuous functions, preserving positive definiteness. A relation between the developed group-theoretical formalism and the conventional one, given in terms of density matrices, is established through the non-commutative Fourier analysis. It shows that the presented method plays the role of a "generating function" formalism for the theory of entanglement. © 2008 Springer-Verlag.
• Niederberger, A., Schulte, T., Wehr, J., Lewenstein, M., Sanchez-Palencia, L., & Sacha, K. (2008). Disorder-induced order in two-component Bose-Einstein condensates. Physical Review Letters, 100(3).
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  Abstract: 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 π/2 between the two BECs and that the effect is robust. We demonstrate it in one, two, and three dimensions at T=0 and present evidence that it persists at small T>0. Applications to phase control in ultracold spinor condensates are discussed. © 2008 The American Physical Society.
• Perseguers, S., Cirac, J. I., Acín, A., Lewenstein, M., & Wehr, J. (2008). Entanglement distribution in pure-state quantum networks. Physical Review A - Atomic, Molecular, and Optical Physics, 77(2).
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  Abstract: We investigate entanglement distribution in pure-state quantum networks. We consider the case when nonmaximally entangled two-qubit pure states are shared by neighboring nodes of the network. For a given pair of nodes, we investigate how to generate the maximal entanglement between them by performing local measurements, assisted by classical communication, on the other nodes. We find optimal measurement protocols for both small and large one-dimensional networks. Quite surprisingly, we prove that Bell measurements are not always the optimal ones to perform in such networks. We generalize then the results to simple small two-dimensional (2D) networks, finding again counterintuitive optimal measurement strategies. Finally, we consider large networks with hierarchical lattice geometries and 2D networks. We prove that perfect entanglement can be established on large distances with probability one in a finite number of steps, provided the initial entanglement shared by neighboring nodes is large enough. We discuss also various protocols of entanglement distribution in 2D networks employing classical and quantum percolation strategies. © 2008 The American Physical Society.
• Weinan, E., Wehr, J., & Xin, J. (2008). Breakdown of homogenization for the random Hamilton-Jacobi equations. Communications in Mathematical Sciences, 6(1), 189-197.
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  Abstract: We study the homogenization of Lagrangian functionals of Hamilton-Jacobi equations (HJ) with quadratic nonlinearity and unbounded stationary ergodic random potential in Rd, d ≥ 1. We show that homogenization holds if and only if the potential is bounded from above. When the potential is unbounded from above, homogenization breaks down, due to the almost sure growth of the running maxima of the random potential. If the unbounded randomness appears in the advection term, homogenization may or may not hold depending on the structure of the flow field. In (compressible) unbounded gradient flows, homogenization holds in spite of the unboundedness. In (incompressible) unbounded shear flows, homogenization breaks down again due to unbounded running maxima of the flows. Results for random advection follow from a transformation of the problem to that of HJ with random potential. Analogous effective behavior is present for front speeds in reaction-diffusion-advection equations with unbounded random advection, and may have broader implications for wave propagation in random media. © 2008 International Press.
• Horvai, P., Komorowski, T., & Wehr, J. (2007). Finite time approach to equilibrium in a fractional Brownian velocity field. Journal of Statistical Physics, 127(3), 553-565.
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  Abstract: We consider the solution of the equation r(t) = W(r(t)), r(0) = r 0 > 0 where W(·) is a fractional Brownian motion (f.B.m.) with the Hurst exponent α ∈ (0,1). We show that for almost all realizations of W(·) the trajectory reaches in finite time the nearest equilibrium point (i.e. zero of the f.B.m.) either to the right or to the left of r0, depending on whether W(r0) is positive or not. After reaching the equilibrium the trajectory stays in it forever. The problem is motivated by studying the separation between two particles in a Gaussian velocity field which satisfies a local self-similarity hypothesis. In contrast to the case when the forcing term is a Brownian motion (then an analogous statement is a consequence of the Markov property of the process) we show our result using as the principal tools the properties of time reversibility of the law of the f.B.m., see Lemma 2.4 below, and the small ball estimate of Molchan, Commun. Math. Phys. 205 (1999) 97-111. © 2007 Springer Science+Business Media, LLC.
• Wehr, J., & Xin, J. (2006). Scaling limits of waves in convex scalar conservation laws under random initial perturbations. Journal of Statistical Physics, 122(2), 361-370.
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  Abstract: We study waves in convex scalar conservation laws under noisy initial perturbations. It is known that the building blocks of these waves are shock and rarefaction waves, both are invariant under hyperbolic scaling. Noisy perturbations can generate complicated wave patterns, such as diffusion process of shock locations. However we show that under the hyperbolic scaling, the solutions converge in the sense of distribution to the unperturbed waves. In particular, randomly perturbed shock waves move at the unperturbed velocity in the scaling limit. Analysis makes use of the Hopf formula of the related Hamilton-Jacobi equation and regularity estimates of noisy processes. © 2006 Springer Science+Business Media, Inc.
• Wehr, J., Niederberger, A., Sanchez-Palencia, L., & Lewenstein, M. (2006). Disorder versus the Mermin-Wagner-Hohenberg effect: From classical spin systems to ultracold atomic gases. Physical Review B - Condensed Matter and Materials Physics, 74(22).
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  Abstract: 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 two-dimensional 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. © 2006 The American Physical Society.
• Korbicz, J. K., Cirac, J. I., Wehr, J., & Lewenstein, M. (2005). Hilbert's 17th problem and the quantumness of states. Physical Review Letters, 94(15).
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  Abstract: A state of a quantum system can be regarded as classical (quantum) with respect to measurements of a set of canonical observables if and only if there exists (does not exist) a well defined, positive phase-space distribution, the so called Glauber-Sudarshan P representation. We derive a family of classicality criteria that requires that the averages of positive functions calculated using P representation must be positive. For polynomial functions, these criteria are related to Hilbert's 17th problem, and have physical meaning of generalized squeezing conditions; alternatively, they may be interpreted as nonclassicality witnesses. We show that every generic nonclassical state can be detected by a polynomial that is a sum-of-squares of other polynomials. We introduce a very natural hierarchy of states regarding their degree of quantumness, which we relate to the minimal degree of a sum-of-squares polynomial that detects them. © 2005 The American Physical Society.
• Wehr, J., & Woo, J. M. (2001). Central limit theorems for nonlinear hierarchical sequences of random variables. Journal of Statistical Physics, 104(3-4), 777-797.
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  Abstract: Let a random variable x0 and a function f: [a, b]k → [a, b] be given. A hierarchical sequence {xn: n = 0, 1, 2,...} of random variables is defined inductively by the relation xn = f(xn-1, 1, xn-1, 2...., xn-1 k), where {xn-1, i: i = 1, 2,..., k} is a family of independent random variables with the same distribution as xn-1. We prove a central limit theorem for this hierarchical sequence of random variables when a function f satisfies a certain averaging condition. As a corollary under a natural assumption we prove a central limit theorem for a suitably normalized sequence of conductivities of a random resistor network on a hierarchical lattice.
• Schenkel, A., Wehr, J., & Wittwer, P. (2000). Computer-assisted proofs for fixed point problems in Sobolev spaces. Mathematical Physics Electronic Journal, 6, 1-67.
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  Abstract: In this paper we extend the technique of computer-assisted proofs to fixed point problems in Sobolev spaces. Up to now the method was limited to the case of spaces of analytic functions. The possibility to work with Sobolev spaces is an important progress and opens up many new domains of applications. Our discussion is centered around a concrete problem that arises in the theory of critical phenomena and describes the phase transition in a hierarchical system of random resistors. For this problem we have implemented in particular the convolution product based on the Fast Fourier Transform (FFT) algorithm with rigorous error estimates.
• Wehr, J., & Woo, J. (1998). Absence of geodesics in first-passage percolation on a half-plane. Annals of Probability, 26(1), 358-367.
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  Abstract: An H-geodesic is a doubly infinite path which locally minimizes the passage time in the i.i.d. first passage percolation model on a half-plane H. Under the assumption that the bond passage times are continuously distributed with a finite mean, we prove that, with probability 1, H-geodesics do not exist. As a corollary we show that, with probability 1, any geodesic in the analogous model on the whole plane Z 2 has to intersect all straight lines with rational slopes.
• Berlyand, L., & Wehr, J. (1997). Non-Gaussian limiting behavior of the percolation threshold in a large system. Communications in Mathematical Physics, 185(1), 73-92.
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  Abstract: We study short-range percolation models. In a finite box we define the percolation threshold as a random variable obtained from a stochastic procedure used in actual numerical calculations, and study the asymptotic behavior of these random variables as the size of the box goes to infinity. We formulate very general conditions under which in two dimensions rescaled threshold variables cannot converge to a Gaussian and determine the asymptotic behavior of their second moments in terms of a widely used definition of correlation length. We also prove that in all dimensions the finite-volume percolation thresholds converge in probability to the percolation threshold of the infinite system. The convergence result is obtained by estimating the rate of decay of the limiting distribution function's tail in terms of the correlation length exponent v. The proofs use exponential estimates of crossing probabilities. Substantial parts of the proofs apply in all dimensions.
• Wehr, J. (1997). A lower bound on the variance of conductance in random resistor networks. Journal of Statistical Physics, 86(5-6), 1359-1365.
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  Abstract: We study the conductance of random resistor networks in d≥2 dimensions. It is shown (under some technical assumptions) that if a network exhibits a non-zero conductivity in the infinite-volume limit, then the variance of a finite-volume conductance grows at least like the volume.
• Wehr, J. (1997). A strong law of large numbers for iterated functions of independent random variables. Journal of Statistical Physics, 86(5-6), 1373-1384.
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  Abstract: We study sequences of random variables obtained by iterative procedures, which can be thought of as nonlinear generalizations of the arithmetic mean. We prove a strong law of large numbers for a class of such iterations. This gives rise to the concept of generalized expected value of a random variable, for which we prove an analog of the classical Jensen inequality. We give several applications to models arising in mathematical physics and other areas.
• Wehr, J. (1997). On the number of infinite geodesics and ground states in disordered systems. Journal of Statistical Physics, 87(1-2), 439-447.
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  Abstract: We study first-passage percolation models and their higher dimensional analogs - models of surfaces with random weights. We prove that under very general conditions the number of lines or, in the second case, hypersurfaces which locally minimize the sum of the random weights is with probability one equal to 0 or with probability one equal to + ∞. As corollaries we show that in any dimension d ≥ 2 the number of ground states of an Ising ferromagnet with random coupling constants equals (with probability one) 2 or + ∞. Proofs employ simple large-deviation estimates and ergodic arguments.
• Wehr, J. (1997). On the number of infinite geodesics and ground states in disordered systems. Journal of Statistical Physics. doi:10.1007/bf02181495
• Wehr, J., & Berlyand, L. (1997). Non-Gaussian Limiting Behavior of the Percolation Threshold in a Large System. Communications in Mathematical Physics. doi:10.1007/s002200050082
• Wehr, J., & Xin, J. (1997). Front speed in the Burgers equation with a random flux. Journal of Statistical Physics, 88(3-4), 843-871.
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  Abstract: We study the large-time asymptotic shock-front speed in an inviscid Burgers equation with a spatially random flux function. This equation is a prototype for a class of scalar conservation laws with spatial random coefficients such as the well-known Buckley-Leverett equation for two-phase flows, and the contaminant transport equation in groundwater flows. The initial condition is a shock located at the origin (the indicator function of the negative real line). We first regularize the equation by a special random viscous term so that the resulting equation can be solved explicitly by a Cole-Hopf formula. Using the invariance principle of the underlying random processes and the Laplace method, we prove that for large times the solutions behave like fronts moving at averaged constant speeds in the sense of distribution. However, the front locations are random, and we show explicitly the probability of observing the head or tail of the fronts. Finally, we pass to the inviscid limit, and establish the same results for the inviscid shock fronts.
• Wehr, J., & Xin, J. (1996). White noise perturbation of the viscous shock fronts of the burgers equation. Communications in Mathematical Physics, 181(1), 183-203.
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  Abstract: We study the front dynamics of solutions of the initial value problem of the Burgers equation with initial data being the viscous shock front plus the white noise perturbation. In the sense of distribution, the solutions propagate with the same speed as the unperturbed front, however, the front location is random and satisfies a central limit theorem with the variance proportional to the time t, as t goes to infinity. With probability arbitrarily close to one, the front width is O(1) for large time.
• Berlyand, L., & Wehr, J. (1995). The probability distribution of the percolation threshold in a large system. Journal of Physics A: Mathematical and General, 28(24), 7127-7133.
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  Abstract: We show that the distribution of the percolation threshold in a large finite system does not converge to a Gaussian when the size of the system goes to infinity, provided that the two widely accepted definitions of correlation length are equivalent. The shape of the distribution is thus directly related to the presence or absence of logarithmic corrections in the power law for the correlation length. The result is obtained by estimating the rate of decay of tail of the limiting distribution in terms of the correlation length exponent v. All results are rigorously proven in the 2D case. Generalizations for three dimensions are also discussed.
• Aizenman, M., & Wehr, J. (1990). Erratum: Rounding of first-order phase transitions in systems with quenched disorder (Physical Review Letters). Physical Review Letters, 64(11), 1311-.
• Aizenman, M., & Wehr, J. (1990). Rounding effects of quenched randomness on first-order phase transitions. Communications in Mathematical Physics, 130(3), 489-528.
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  Abstract: Frozen-in disorder in an otherwise homogeneous system, is modeled by interaction terms with random coefficients, given by independent random variables with a translation-invariant distribution. For such systems, it is proven that in d=2 dimensions there can be no first-order phase transition associated with discontinuities in the thermal average of a quantity coupled to the randomized parameter. Discontinuities which would amount to a continuous symmetry breaking, in systems which are (stochastically) invariant under the action of a continuous subgroup of O(N), are suppressed by the randomness in dimensions d≦4. Specific implications are found in the Random-Field Ising Model, for which we conclude that in d=2 dimensions at all (β, h) the Gibbs state is unique for almost all field configurations, and in the Random-Bond Potts Model where the general phenomenon is manifested in the vanishing of the latent heat at the transition point. The results are explained by the argument of Imry and Ma [1]. The proofs involve the analysis of fluctuations of free energy differences, which are shown (using martingale techniques) to be Gaussian on the suitable scale. © 1990 Springer-Verlag.
• Wehr, J., & Aizenman, M. (1990). Fluctuations of extensive functions of quenched random couplings. Journal of Statistical Physics, 60(3-4), 287-306.
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  Abstract: An extensive quantity is a family of functions Ψv of random parameters, indexed by the finite regions V (subsets of ℤd) over which Ψv are additive up to corrections satisfying the boundary estimate stated below. It is shown that unless the randomness is nonessential, in the sense that lim Ψv/|V| has a unique value in the absolute (i.e., not just probabilistic) sense, the variance of such a quantity grows as the volume of V. Of particular interest is the free energy of a system with random couplings; for such Ψv bounds are derived also for the generating function E(etΨ). In a separate application, variance bounds are used for an inequality concerning the characteristic exponents of directed polymers in a random environment. © 1990 Plenum Publishing Corporation.
• Aizenman, M., & Wehr, J. (1989). Rounding of first-order phase transitions in systems with quenched disorder. Physical Review Letters, 62(21), 2503-2506.
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  Abstract: It is shown, by a general argument, that in 2D quenched randomness results in the elimination of discontinuities in the density of the thermodynamic variable conjugate to the fluctuating parameter. Analogous results for continuous symmetry breaking extend to d4. In particular, for random-field models we rigorously prove uniqueness of the Gibbs state 2D Ising systems, and absence of continuous symmetry breaking in the Heisenberg model in d4, as predicted by Imry and Ma. Another manifestation of the general statement is found in 2D random-bond Potts models where a phase transition persists, but ceases to be first order. © 1989 The American Physical Society.

Presentations

• Wehr, J. (2015, December). Equation of motion of a Brownian particle. Mathematical Physics and Probability Seminar. Davis, CA: University of California, Davis.
• Wehr, J. (2015, July). Low lying eigenvalues of the one-dimensional Schr ̈ odinger operator with a Bernoulli random potential. Quantum Many-body Systems, Random Matrices and Disorder. Vienna, Austria: Erwin Schroedinger Institute.
• Wehr, J. (2015, May). Low-energy behavior of one-dimensional disorder boson systems. Randomness in Quantum Physics and Beyond. Barcelona, Spain: Institute for Photonic Sciences.