1,440 research outputs found
Equilibrium of a Brownian particle with coordinate dependent diffusivity and damping: Generalized Boltzmann distribution
Fick's law for coordinate dependent diffusivity is derived. Corresponding
diffusion current in the presence of coordinate dependent diffusivity is
consistent with the form as given by Kramers-Moyal expansion. We have obtained
the equilibrium solution of the corresponding Smoluchowski equation. The
equilibrium distribution is a generalization of the Boltzmann distribution.
This generalized Boltzmann distribution involves an effective potential which
is a function of coordinate dependent diffusivity. We discuss various
implications of the existence of this generalized Boltzmann distribution for
equilibrium of systems with coordinate dependent diffusivity and damping.Comment: 11 pages, 1 figur
Quantum theta functions and Gabor frames for modulation spaces
Representations of the celebrated Heisenberg commutation relations in quantum
mechanics and their exponentiated versions form the starting point for a number
of basic constructions, both in mathematics and mathematical physics (geometric
quantization, quantum tori, classical and quantum theta functions) and signal
analysis (Gabor analysis).
In this paper we try to bridge the two communities, represented by the two
co--authors: that of noncommutative geometry and that of signal analysis. After
providing a brief comparative dictionary of the two languages, we will show
e.g. that the Janssen representation of Gabor frames with generalized Gaussians
as Gabor atoms yields in a natural way quantum theta functions, and that the
Rieffel scalar product and associativity relations underlie both the functional
equations for quantum thetas and the Fundamental Identity of Gabor analysis.Comment: 38 pages, typos corrected, MSC class change
Evaluation of a surrogate contact model of TKA
INTRODUCTION: Simultaneous prediction of body-level dynamics and detailed joint mechanics in the frame of musculoskeletal (MS) modeling represents still a highly computationally demanding task. Marra et al. (2014) recently presented and validated a MS model capable of concurrent prediction of muscle forces, knee ligament forces, tibiofemoral (TF) and patellofemoral (PF) contact forces in a MS model of Total Knee Arthroplasty (TKA) [1]. Simulation time for one complete gait cycle was in the order of 3 hours, and the iterative process that solved the equilibrium in the knee joint was thought to be the main source of overhead. Surrogate modeling techniques were suggested [2]. In this study, we develop a surrogate contact model of TKA to decrease the simulation time in the MS simulation. We hypothesize that the algorithm that allows the muscle fibers to wrap around the bones constitutes another source of overhead in the MS model. Therefore, we will also evaluate the performances of the surrogate model with and without muscle wrapping. METHODS: The original tibial component from our TKA model [1] was split in a medial and lateral hemi-part and fixed to the ground, whereas the femoral part was left with 6 degrees of freedom (DOF). The contacting pairs exchanged three forces and three moments, which were assumed functions of the relative pose only. Translations (X, Y, Z) were defined relative to the tibial component frame and rotations of the femoral component (RotX, RotY, RotZ) were described with Cardan angles, using the z-y-x rotation sequence. Similarly to Lin et al (2010) [3] we identified two sensitive directions, Y and RotX and, therefore, we defined a sample point as composed by four pose parameters and the two loads in the sensitive directions: X, Fy, Z, Tx, RotY, RotZ. Reference load-pose data were obtained from four simulations of gait, squat, chair-rise, and right-turn trials using the original contact model. The design space was populated using the Hammersley quasi-random sequence and adopting a multi-domain approach, as proposed by Eskinazi and Fregly (2015) [2]. One domain consisted of 20 data points per each frame of the four dynamic simulations, spanning the boundaries of ± 1 standard deviation from the time-varying reference envelopes. A second domain of 2500 points was generated in the principal component space of the reference load-pose data of each dynamic simulation, with boundaries enlarged by 50%. A third domain of 1000 data points represented one-side-contact situations, in which Tx was bounded to ± 4 Nm. In total, 36000 data points were sampled in the three different domains. Data points were evaluated using the original contact model (Fig. 1) by repeated Force-Dependent Kinematics (FDK) analyses. Data points which did not lead to equilibrium were discarded. The remaining 27620 points were randomly subdivided into a training (70%) and testing (30%) group. Three separate Feed-Forward Artificial Neural Networks (FFANN), consisting of four inner layers of 20 hyperbolic tangent sigmoid neurons each, were configured within the Neural Network Toolbox in MATLAB 8.1 (The MathWorks Inc., Natick, MA, 2013). The first network was trained to learn the relationships between the four âtwo medial and two lateralâ sensitive loads (output) and the six pose parameters (input). Two other networks âone medial and one lateralâ were trained separately to learn the relationships between the remaining loads of each side (output) and all the pose parameters plus the two sensitive loads from each side (input). We used the popular Levenberg-Marquardt training algorithm in conjunction with Bayesian regularization to avoid over-fitting. Stopping criterion was a training time of two hours for each network. The trained networks were translated to custom C++ DLL functions for successive inclusion in our MS model. The surrogate contact model replaced the original contact model and one gait trial was simulated with 4 different combination of the following model settings: original versus surrogate contact model, wrapping enabled versus disabled. RESULTS: The contact sampling model required 238 hours to evaluate the 36000 data points. Predicted tibiofemoral compressive forces under all simulated cases are shown in Fig. 2. A comparison with experimental measurements (eTibia line) is also shown. Surrogate model predictions showed a very good agreement with the original model counterparts. Fig. 3 summarizes the computation times: simulations took the longest when muscle wrapping was enabled and the benefits of using the surrogate model became evident only when the wrapping algorithm was switched off, leading to a 6x speed-up. Simulation time with the original contact model decreased by a factor of 8 by switching off the wrapping algorithm. DISCUSSION: The use of FFANN-based surrogate contact model, in place of the original rigid contact model, could substantially reduce the simulation time of a full gait cycle down to 3 minutes, when the wrapping algorithm was turned off. Such improvement could not be achieved when using the wrapping algorithm. This enlightens another important source of overhead in MS modeling âthe muscle wrapping algorithmâ which unexpectedly was found to dominate the simulation time. At each FDK iteration, the wrapping algorithm needs to be solved as well, introducing overhead. If the wrapping algorithm is slower than the contact algorithm, then the computation time of each step will be dominated by the former, leaving only a small fraction to be gained from the latter. SIGNIFICANCE: We showed that surrogate contact model could reduce the simulation time in a MS model of TKA down to a level which allows parametric studies and/or optimization to be feasible. We also discovered that the muscle wrapping algorithm constituted an unexpectedly large source of overhead during dynamic simulations. These represent new and important findings for the MS modeling community. REFERENCES: [1] M. A. Marra, V. Vanheule, R. Fluit, B. H. F. J. M. Koopman, J. Rasmussen, N. J. J. Verdonschot, and M. S. Andersen, âA Subject-Specific Musculoskeletal Modeling Framework to Predict in Vivo Mechanics of Total Knee Arthroplasty.,â J. Biomech. Eng., Nov. 2014. [2] I. Eskinazi and B. J. Fregly, âSurrogate modeling of deformable joint contact using artificial neural networks.,â Med. Eng. Phys., Jul. 2015. [3] Y.-C. Lin, R. T. Haftka, N. V Queipo, and B. J. Fregly, âSurrogate articular contact models for computationally efficient multibody dynamic simulations.,â Med. Eng. Phys., vol. 32, no. 6, pp. 584â94, Jul. 2010. ACKNOWLEDGEMENTS: This study was conducted within the ERC âBioMechToolsâ project, funded by the European Research Council
Epidemic spreading with immunization and mutations
The spreading of infectious diseases with and without immunization of
individuals can be modeled by stochastic processes that exhibit a transition
between an active phase of epidemic spreading and an absorbing phase, where the
disease dies out. In nature, however, the transmitted pathogen may also mutate,
weakening the effect of immunization. In order to study the influence of
mutations, we introduce a model that mimics epidemic spreading with
immunization and mutations. The model exhibits a line of continuous phase
transitions and includes the general epidemic process (GEP) and directed
percolation (DP) as special cases. Restricting to perfect immunization in two
spatial dimensions we analyze the phase diagram and study the scaling behavior
along the phase transition line as well as in the vicinity of the GEP point. We
show that mutations lead generically to a crossover from the GEP to DP. Using
standard scaling arguments we also predict the form of the phase transition
line close to the GEP point. It turns out that the protection gained by
immunization is vitally decreased by the occurrence of mutations.Comment: 9 pages, 13 figure
Changing industrial metabolism: methods for analysis
Research in the field of "industrial metabolism" traditionally has been focused on measuring and describing physical flows of economic systems. The "metabolism" of economic systems, however, changes over time, and measuring material flows is insufficient to understand this process. Understanding the relation between economic activities and material flows can help to unravel the socio-economic causes of these physical flows. Three issues are addressed: The importance of spatial scales and trade flows, empirical analysis of relations between economic development and material flows, and treatment of behaviour of and interactions between stake-holders. For each of these issues, methods for analysis are suggested
Patchiness and Demographic Noise in Three Ecological Examples
Understanding the causes and effects of spatial aggregation is one of the
most fundamental problems in ecology. Aggregation is an emergent phenomenon
arising from the interactions between the individuals of the population, able
to sense only -at most- local densities of their cohorts. Thus, taking into
account the individual-level interactions and fluctuations is essential to
reach a correct description of the population. Classic deterministic equations
are suitable to describe some aspects of the population, but leave out features
related to the stochasticity inherent to the discreteness of the individuals.
Stochastic equations for the population do account for these
fluctuation-generated effects by means of demographic noise terms but, owing to
their complexity, they can be difficult (or, at times, impossible) to deal
with. Even when they can be written in a simple form, they are still difficult
to numerically integrate due to the presence of the "square-root" intrinsic
noise. In this paper, we discuss a simple way to add the effect of demographic
stochasticity to three classic, deterministic ecological examples where
aggregation plays an important role. We study the resulting equations using a
recently-introduced integration scheme especially devised to integrate
numerically stochastic equations with demographic noise. Aimed at scrutinizing
the ability of these stochastic examples to show aggregation, we find that the
three systems not only show patchy configurations, but also undergo a phase
transition belonging to the directed percolation universality class.Comment: 20 pages, 5 figures. To appear in J. Stat. Phy
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