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A primer on electron transport | |

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Nanoscale systems | |

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Generating currents | |

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Finite versus infinite systems | |

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Electron sources | |

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Intrinsic nature of the transport problem | |

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Measuring currents | |

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Microscopic states | |

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The current operator | |

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The measurement process | |

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Complete measurement and pure states | |

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The statistical operator and macro-states | |

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Pure and mixed states | |

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Quantum correlations | |

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Time evolution of the statistical operator | |

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Random or partially specified Hamiltonians | |

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Open quantum systems | |

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Equilibrium statistical operators | |

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Current measurement and statistical operator truncation | |

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One current, different viewpoints | |

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Summary and open questions | |

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Exercises | |

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Drude model, Kubo formalism and Boltzmann equation | |

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Drude model | |

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Resistance, coherent and incoherent transport | |

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Relaxation vs. dephasing | |

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Mean-free path | |

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The meaning of momentum relaxation time | |

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Kubo formalism | |

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The current-current response function | |

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The use of Density-Functional Theory in the Kubo approach | |

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The fluctuation-dissipation theorem | |

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Ohmic vs. ballistic regimes | |

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Chemical, electrochemical and electrostatic potentials | |

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Drift-diffusion equations | |

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Diffusion coefficient of an ideal electron gas in the non-degenerate limit | |

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Generalization to spin-dependent transport | |

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Distribution functions | |

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Boltzmann equation | |

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Approach to local equilibrium | |

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Entropy, loss of information, and macroscopic irreversibility | |

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The classical statistical entropy | |

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Quantum statistical entropy | |

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Information content of the N- and one-particle statistical operators | |

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Entropy of open quantum systems | |

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Loss of information in the Kubo formalism | |

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Loss of information with stochastic Hamiltonians | |

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Entropy associated with the measurement of currents | |

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Summary and open questions | |

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Exercises | |

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Landauer approach | |

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Formulation of the problem | |

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Local resistivity dipoles and the "field response" | |

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Conduction from transmission | |

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Scattering boundary conditions | |

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Transmission and reflection probabilities | |

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Total current | |

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Two-probe conductance | |

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The Lippmann-Schwinger equation | |

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Time-dependent Lippmann-Schwinger equation | |

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Time-independent Lippmann-Schwinger equation | |

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Green's functions and self-energy | |

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Relation to scattering theory | |

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The S matrix | |

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Relation between the total Green's function and the S matrix | |

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The transfer matrix | |

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Coherent scattering of two resistors in series | |

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Incoherent scattering of two resistors in series | |

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Relation between the conductance and the transfer matrix | |

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Localization, ohmic and ballistic regimes | |

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Four-probe conductance in the non-invasive limit | |

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Single-channel case | |

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Geometrical "dilution" | |

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Multi-channel case | |

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Multi-probe conductance in the invasive limit | |

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Floating probes and dephasing | |

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Generalization to spin-dependent transport | |

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Spin-dependent transmission functions | |

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Multi-probe conductance in the presence of a magnetic field | |

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Local resistivity spin dipoles and dynamical effects | |

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The use of Density-Functional Theory in the Landauer approach | |

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Summary and open questions | |

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Exercises | |

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Non-equilibrium Green's function formalism | |

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Formulation of the problem | |

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Contour ordering | |

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Equilibrium Green's functions | |

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Time-ordered Green's functions | |

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Dyson's equation for interacting particles | |

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More Green's functions | |

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The spectral function | |

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Contour-ordered Green's functions | |

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Equations of motion for non-equilibrium Green's functions | |

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Application to steady-state transport | |

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Coulomb blockade | |

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Quantum kinetic equations | |

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Summary and open questions | |

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Exercises | |

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Noise | |

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The moments of the current | |

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Shot noise | |

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The classical (Poisson) limit | |

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Quantum theory of shot noise | |

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Counting statistics | |

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Thermal noise | |

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Summary and open questions | |

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Exercises | |

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Electron-ion interaction | |

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The many-body electron-ion Hamiltonian | |

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The adiabatic approximation for a current-carrying system | |

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The phonon subsystem | |

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Electron-phonon coupling in the presence of current | |

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Inelastic current | |

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Inelastic current from standard perturbation theory | |

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Inelastic current from the NEGF | |

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Local ionic heating | |

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Lattice heat conduction | |

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Thermopower | |

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Current-induced forces | |

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Elastic vs. inelastic contribution to electro-migration | |

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One force, different definitions | |

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Local resistivity dipoles and the force sign | |

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Forces at equilibrium | |

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Forces out of equilibrium | |

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Are current-induced forces conservative? | |

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Local ionic heating vs. current-induced forces | |

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Summary and open questions | |

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Exercises | |

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The micro-canonical picture of transport | |

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Formulation of the problem | |

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Transport from a finite-system point of view | |

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Initial conditions and dynamics | |

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Electrical current theorems within dynamical DFTs | |

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Closed and finite quantum systems in a pure state | |

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Closed quantum systems in a pure state with arbitrary boundary conditions | |

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Current in open quantum systems | |

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Closure of the BBGKY hierarchy | |

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Functional approximations and loss of information | |

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Transient dynamics | |

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Properties of quasi-steady states | |

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Variational definition of quasi-steady states | |

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Dependence of quasi-steady states on initial conditions | |

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A non-equilibrium entropy principle | |

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Approach to steady state in nanoscale systems | |

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Definition of conductance in the micro-canonical picture | |

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Summary and open questions | |

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Hydrodynamics of the electron liquid | |

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The Madelung equations for a single particle | |

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Hydrodynamic form of the Schrodinger equation | |

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Quantum Navier-Stokes equations | |

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Conductance quantization from hydrodynamics | |

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Viscosity from Time-Dependent Current Density-Functional Theory | |

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Functional approximation, loss of information, and dissipative dynamics | |

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Effect of viscosity on resistance | |

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Turbulent transport | |

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Local electron heating | |

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Electron heat conduction | |

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Hydrodynamics of heat transfer | |

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Effect of local electron heating on ionic heating | |

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Summary and open questions | |

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Exercises | |

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Appendices | |

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A primer on second quantization | |

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The quantum BBGKY hierarchy | |

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The Lindblad equation | |

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The Lindblad theorem | |

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Derivation of the Lindblad equation | |

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Steady-state solutions | |

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Ground-state Density-Functional Theory | |

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The Hohenberg-Kohn theorem | |

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The Kohn-Sham equations | |

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Generalization to grand-canonical equilibrium | |

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The local density approximation and beyond | |

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Time-Dependent DFT | |

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The Runge-Gross theorem | |

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The time-dependent Kohn-Sham equations | |

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The adiabatic local density approximation | |

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Time-Dependent Current DFT | |

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The current density as the main variable | |

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The exchange-correlation electric field | |

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Approximate formulas for the viscosity | |

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Stochastic Time-Dependent Current DFT | |

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The stochastic Schrodinger equation | |

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Derivation of the quantum master equation | |

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The theorem of Stochastic TD-CDFT | |

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Inelastic corrections to current and shot noise | |

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Hydrodynamic form of the Schrodinger equation | |

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Equation of motion for the stress tensor | |

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Cut-off of the viscosity divergence | |

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Bernoulli's equation | |

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References | |

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Index | |