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 –two medial and two lateral– sensitive loads (output) and the six pose parameters (input). Two other networks –one medial and one lateral– 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 –the muscle wrapping algorithm– 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, “A Subject-Specific Musculoskeletal Modeling Framework to Predict in Vivo Mechanics of Total Knee Arthroplasty.,” J. Biomech. Eng., Nov. 2014. [2] I. Eskinazi and B. J. Fregly, “Surrogate modeling of deformable joint contact using artificial neural networks.,” Med. Eng. Phys., Jul. 2015. [3] Y.-C. Lin, R. T. Haftka, N. V Queipo, and B. J. Fregly, “Surrogate articular contact models for computationally efficient multibody dynamic simulations.,” Med. Eng. Phys., vol. 32, no. 6, pp. 584–94, Jul. 2010. ACKNOWLEDGEMENTS: This study was conducted within the ERC ‘BioMechTools’ 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