Potential Optimal Gait Performance of Mauch S-N-S Prosthetic Knee Configurations as Predicted by Dynamic Modeling

Patients with prosthetic legs routinely suffer from abnormal gait patterns which can cause health issues and eventually lower the quality of their lives. Despite the half-century advance in the technology of prosthetic knees, from the purely mechanical to microprocessor controlled systems, patient testing suggests that very little progress has been made in the quality of the kinetics and kinematics of amputee gait. Moreover, the cost of microprocessor controlled prosthetic knees may be 10 times more than the purely mechanical knees. While prosthetic knees have become more complex and expensive, it is not proven that the prosthetic knee is a central factor limiting amputee patient gait. The goal of this project is to determine the degree to which the Mauch S-N-S prosthetic knee limits the ability of a subject to achieve a close to normal gait pattern. In this research, we developed dynamic models of the Mauch S-N-S prosthetic knee based on gait-like motion tests of a Mauch knee cylinder and used the dynamic models in computational simulations to determine the best achievable gait, on the basis of obtaining near-to-normal gait kinematics and kinetics. Idealized assumptions were made for patient performance capability and characteristics of the other prosthetic leg components, to obtain the desired focus on knee capabilities and limitations. The results indicate that even with this relatively old technology prosthetic knee, subjects have the potential to walk much more normally than the patient-test data indicates. An extension of the study showed the significant interaction of the prosthetic knee and ankle with respect to achieving optimal gait. The methodology of this study can be applied to evaluation other knees, prosthetic components and prosthetic systems combining these component

Potential Optimal Gait Performance of Mauch S-N-S Prosthetic Knee Configurations as Predicted by Dynamic Modeling

Patients with prosthetic legs routinely suffer from abnormal gait patterns which can cause health issues and eventually lower the quality of their lives. Despite the half-century advance in the technology of prosthetic knees, from the purely mechanical to microprocessor controlled systems, patient testing suggests that very little progress has been made in the quality of the kinetics and kinematics of amputee gait. Moreover, the cost of microprocessor controlled prosthetic knees may be 10 times more than the purely mechanical knees. While prosthetic knees have become more complex and expensive, it is not proven that the prosthetic knee is a central factor limiting amputee patient gait. The goal of this project is to determine the degree to which the Mauch S-N-S prosthetic knee limits the ability of a subject to achieve a close to normal gait pattern. In this research, we developed dynamic models of the Mauch S-N-S prosthetic knee based on gait-like motion tests of a Mauch knee cylinder and used the dynamic models in computational simulations to determine the best achievable gait, on the basis of obtaining near-to-normal gait kinematics and kinetics. Idealized assumptions were made for patient performance capability and characteristics of the other prosthetic leg components, to obtain the desired focus on knee capabilities and limitations. The results indicate that even with this relatively old technology prosthetic knee, subjects have the potential to walk much more normally than the patient-test data indicates. An extension of the study showed the significant interaction of the prosthetic knee and ankle with respect to achieving optimal gait. The methodology of this study can be applied to evaluation other knees, prosthetic components and prosthetic systems combining these component

Theories of Linear Response in BCS Superfluids and How They Meet Fundamental Constraints

We address the importance of symmetry and symmetry breaking on linear response theories of fermionic BCS superfluids. The linear theory of a noninteracting Fermi gas is reviewed and several consistency constraints are verified. The challenge to formulate linear response theories of BCS superfluids consistent with density and spin conservation laws comes from the presence of a broken U(1)$_{\textrm{EM}}$ symmetry associated with electromagnetism (EM) and we discuss two routes for circumventing this. The first route follows Nambu's integral-equation approach for the EM vertex function, but this method is not specific for BCS superfluids. We focus on the second route based on a consistent-fluctuation-of-the order-parameter (CFOP) approach where the gauge transformation and the fluctuations of the order parameter are treated on equal footing. The CFOP approach allows one to explicitly verify several important constraints: The EM vertex satisfies not only a Ward identity which guarantees charge conservation but also a $Q$-limit Ward identity associated with the compressibility sum rule. In contrast, the spin degrees of freedom associated with another U(1)$_z$ symmetry are not affected by the Cooper-pair condensation that breaks only the U(1)$_{\textrm{EM}}$ symmetry. As a consequence the collective modes from the fluctuations of the order parameter only couple to the density response function but decouple from the spin response function, which reflects the different fates of the two U(1) symmetries in the superfluid phase. Our formulation lays the ground work for application to more general theories of BCS-Bose Einstein Condensation crossover both above and below $T_c$.Comment: Review on gauge invariance and charge-spin difference of BCS theory. 27 pages, 1 figure. Some typos have been correcte

Shear Viscosity of Uniform Fermi Gases with Population Imbalance

The shear viscosity plays an important role in studies of transport phenomena in ultracold Fermi gases and serves as a diagnostic of various microscopic theories. Due to the complicated phase structures of population-imbalanced Fermi gases, past works mainly focus on unpolarized Fermi gases. Here we investigate the shear viscosity of homogeneous, population-imbalanced Fermi gases with tunable attractive interactions at finite temperatures by using a pairing fluctuation theory for thermodynamical quantities and a gauge-invariant linear response theory for transport coefficients. In the unitary and BEC regimes, the shear viscosity increases with the polarization because the excess majority fermions cause gapless excitations acting like a normal fluid. In the weak BEC regime the excess fermions also suppress the noncondensed pairs at low polarization, and we found a minimum in the ratio of shear viscosity and relaxation time. To help constrain the relaxation time from linear response theory, we derive an exact relation connecting some thermodynamic quantities and transport coefficients at the mean-field level for unitary Fermi superfluids with population imbalance. An approximate relation beyond mean-field theory is proposed and only exhibits mild deviations from numerical results.Comment: 11 pages, 4 figure

Fundamental Constraints on Linear Response Theories of Fermi Superfluids Above and Below TcT_c

We present fundamental constraints required for a consistent linear response theory of fermionic superfluids and address temperatures both above and below the transition temperature $T_c$. We emphasize two independent constraints, one associated with gauge invariance (and the related Ward identity) and another associated with the compressibility sum rule, both of which are satisfied in strict BCS theory. However, we point out that it is the rare many body theory which satisfies both of these. Indeed, well studied quantum Hall systems and random-phase approximations to the electron gas are found to have difficulties with meeting these constraints. We summarize two distinct theoretical approaches which are, however, demonstrably compatible with gauge invariance and the compressibility sum rule. The first of these involves an extension of BCS theory to a mean field description of the BCS-Bose Einstein condensation crossover. The second is the simplest Nozieres Schmitt- Rink (NSR) treatment of pairing correlations in the normal state. As a point of comparison we focus on the compressibility $\kappa$ of each and contrast the predictions above $T_c$. We note here that despite the compliance with sum rules, this NSR based scheme leads to an unphysical divergence in $\kappa$ at the transition. Because of the delicacy of the various consistency requirements, the results of this paper suggest that avoiding this divergence may repair one problem while at the same time introducing others.Comment: 13 pages, 2 figures. A review on gauge-invariant linear response theories of BCS-BEC crossover and NSR theor

Density and Spin Linear Response of Atomic Fermi Superfluids with Population Imbalance in BCS-BEC Crossover

We present a theoretical study of the density and spin (representing the two components) linear response of Fermi superfluids with tunable attractive interactions and population imbalance. In both linear response theories, we find that the fluctuations of the order parameter must be treated on equal footing with the gauge transformations associated with the symmetries of the Hamiltonian so that important constraints including various sum rules can be satisfied. Both theories can be applied to the whole BCS-Bose-Einstein condensation crossover. The spin linear responses are qualitatively different with and without population imbalance because collective-mode effects from the fluctuations of the order parameter survive in the presence of population imbalance, even though the associated symmetry is not broken by the order parameter. Since a polarized superfluid becomes unstable at low temperatures in the weak and intermediate coupling regimes, we found that the density and spin susceptibilities diverge as the system approaches the unstable regime, but the emergence of phase separation preempts the divergence.Comment: 15 pages, 5 figure