Consequences of coarse-grained Vlasov equations

Morawetz, Klaus; Walke, Rainer

doi:10.1016/s0378-4371(03)00507-7

Cited by 6 publications

(7 citation statements)

References 46 publications

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“…For notational simplicity we will use a shorthand operator representation of the gaussian smoothing defined in the second line of (16). The corresponding coarse-grained Vlasov equation can be obtained by applying the smoothing operator on (4), see [23,40]. 8 Due to the assumed initial conditions (13), the map X(t, q) belongs to the homotopy class of the identity, whose degree is one.…”

Section: General Casementioning

confidence: 99%

“…The corresponding coarse-grained Vlasov equation can be obtained by applying the smoothing operator on (4), see [23,40].…”

Section: General Casementioning

confidence: 99%

“…The coarse-grained phase space density f has many desirable features that f does not have. For instance f is no longer exactly conserved along phase space trajectories which allows us to properly define phase mixing, virialization and entropy production [40,41]. This also makes f better behaved from a numerical point of view since f cannot develop structures below the scales σ x and σ u , in contrast to f which continues to develop ever smaller structures over time.…”

Section: General Casementioning

confidence: 99%

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Solving the Vlasov equation in two spatial dimensions with the Schrödinger method

2017

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We demonstrate that the Vlasov equation describing collisionless self-gravitating matter may be solved with the so-called Schrödinger method (ScM). With the ScM, one solves the Schrödinger-Poisson system of equations for a complex wave function in d dimensions, rather than the Vlasov equation for a 2d-dimensional phase space density. The ScM also allows calculating the d-dimensional cumulants directly through quasi-local manipulations of the wave function, avoiding the complexity of 2d-dimensional phase space.We perform for the first time a quantitative comparison of the ScM and a conventional Vlasov solver in d = 2 dimensions. Our numerical tests were carried out using two types of cold cosmological initial conditions: the classic collapse of a sine wave and those of a gaussian random field as commonly used in cosmological cold dark matter N-body simulations. We compare the first three cumulants, that is, the density, velocity and velocity dispersion, to those obtained by solving the Vlasov equation using the publicly available code ColDICE. We find excellent qualitative and quantitative agreement between these codes, demonstrating the feasibility and advantages of the ScM as an alternative to N-body simulations. We discuss, the emergence of effective vorticity in the ScM through the winding number around the points where the wave function vanishes. As an application we evaluate the background pressure induced by the non-linearity of large scale structure formation, thereby estimating the magnitude of cosmological backreaction. We find that it is negligibly small and has time dependence and magnitude compatible with expectations from the effective field theory of large scale structure. * kopp@fzu.cz † kyriakos vattis@brown.edu ‡ skordis@fzu.cz

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Section: General Casementioning

confidence: 99%

“…The corresponding coarse-grained Vlasov equation can be obtained by applying the smoothing operator on (4), see [23,40].…”

Section: General Casementioning

confidence: 99%

Section: General Casementioning

confidence: 99%

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Solving the Vlasov equation in two spatial dimensions with the Schrödinger method

2017

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“…The corresponding coarse-grained Vlasov equation as given in [44] is easily obtained from the usual Vlasov equation ( 6) by applying the smoothing operator. We employ the following identity for the smoothing operator…”

Section: Coarse-grained Vlasov Equationmentioning

confidence: 99%

Schrödinger method asN-body double and UV completion of dust

Uhlemann¹,

Kopp

Haugg³

2014

Phys. Rev. D

132

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We investigate large-scale structure formation of collisionless dark matter in the phase space description based on the Vlasov (or collisionless Boltzmann) equation whose nonlinearity is induced solely by gravitational interaction according to the Poisson equation. Determining the time evolution of density and peculiar velocity demands solving the full Vlasov hierarchy for the moments of the phase space distribution function. In the presence of long-range interaction no consistent truncation of the hierarchy is known apart from the pressureless fluid (dust) model, which is incapable of describing virialization due to the occurrence of shell-crossing singularities and the inability to generate vorticity and higher cumulants like velocity dispersion. Our goal is to find a simple ansatz for the phase space distribution function that approximates the full Vlasov distribution function without pathologies in a controlled way and therefore can serve as theoretical N-body double and as a replacement for the dust model. We argue that the coarsegrained Wigner probability distribution obtained from a wave function fulfilling the Schrödinger-Poisson equation (SPE) is the sought-after function. We show that its evolution equation approximates the Vlasov equation and therefore also the dust fluid equations before shell crossing, but cures the shell-crossing singularities and is able to describe regions of multistreaming and virialization. This feature was already employed in cosmological simulations of large-scale structure formation by Widrow and Kaiser (1993). The coarse-grained Wigner ansatz allows us to calculate all higher moments from density and velocity analytically, thereby incorporating nonzero higher cumulants in a self-consistent manner. On this basis we are able to show that the Schrödinger method automatically closes the corresponding hierarchy such that it suffices to solve the SPE in order to directly determine density and velocity and all higher cumulants.

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“…Further, due to the finite number of particles used, PIC methods inevitably introduce some amount of random noise in the Vlasov dynamics, which drives the system towards classical Maxwell-Boltzmann thermalization. Therefore, the fermionic character of the electrons is not preserved during time evolution, 12 which constitutes a major drawback for any PIC method. The accuracy of PIC simulations can be somewhat improved by using finite-size particles 13 or by introducing ad hoc collision operators.…”

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confidence: 99%

Vlasov simulations of ultrafast electron dynamics and transport in thin metal films

Manfredi

Hervieux

2004

Phys. Rev. B

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The ultrafast dynamics of electrons and ions in thin metal films has been investigated using a semiclassical model based on self-consistent Vlasov simulations. The Vlasov equation is solved using a very accurate Eulerian scheme that preserves the fermionic character of the electron distribution for all times. With this technique, the electronic transport and thermalization are studied on a time scale of over 150 plasmon cycles. Our results demonstrate that heat transport occurs at a velocity close to the Fermi velocity, in agreement with experimental measurements in thin gold films. We also show that (i) internal electron thermalization can be achieved without including any binary electron-electron collisions and (ii) nonequilibrium electrons begin to interact with the lattice well before the internal electron thermalization is completed. These effects are considerably enhanced by the interaction of nonequilibrium electrons with the film surfaces.

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Consequences of coarse-grained Vlasov equations

Cited by 6 publications

References 46 publications

Solving the Vlasov equation in two spatial dimensions with the Schrödinger method

Solving the Vlasov equation in two spatial dimensions with the Schrödinger method

Schrödinger method asN-body double and UV completion of dust

Vlasov simulations of ultrafast electron dynamics and transport in thin metal films

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