Diffusion under the Effect of Lorentz Force

66 Pages·2022·1.514 MB·English

by  Erik Kalz

BestMasters
Erik Kalz
Diffusion under
the Effect
of Lorentz Force
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Erik Kalz
Diffusion
under the Effect
of Lorentz Force
ErikKalz
Finsterwalde,Germany
ISSN2625-3577 ISSN2625-3615 (electronic)
BestMasters
ISBN978-3-658-39517-9 ISBN978-3-658-39518-6 (eBook)
https://doi.org/10.1007/978-3-658-39518-6
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Abstract
English1
It is generally believed that collisions of particles reduce the self-diffusion coef-
ficient. In this thesis, we show that in systems under the effect of Lorentz force,
which are characterized by diffusion tensors with antisymmetric elements, colli-
sionssurprisinglycanenhanceself-diffusion.Inthesesystems,duetoaninherent
curving effect, the motion of particles is facilitated, instead of hindered by colli-
sions.Consistentwiththiswefindthatthecollectivediffusionremainsunaffected.
Using a geometric model, we theoretically predict a magnetic field governed
crossover from a reduced to an enhanced self-diffusion. The physical interpre-
tation is quantitatively supported by the force autocorrelation function, which
turns negative with increasing the magnetic field. Using Brownian-dynamics
simulations, we validate the predictions.
Deutsch
Man nimmt allgemein an, dass Teilchenkollisionen den Selbstdiffusionskoef-
fizienten verringern. In dieser Arbeit wird gezeigt, dass in Systemen unter
dem Einfluss einer Lorentzkraft, welche durch antisymmetrische Nebendiago-
nalelemente im Diffusionstensor gekennzeichnet sind, Teilchenkollisionen die
Selbstdiffusion erhöhen können. Die charakterischen Nebendiagonalemente sor-
gen direkt dafür, dass Kollisionen zwischen Teilchen die Selbstdiffusion nicht
1Reprintedandeditedabstractwithpermissionfrom[1].Copyright(2022)bytheAmerican
PhysicalSociety.
v
vi Abstract
mehr notwendigerweise verringern. Konsistent mit dem hier vorgeschlagen
physikalischenMechanismusbleibtdiekollektiveDiffusionvonderLorentzkraft
unbeinflusst. Mittels eines geometrischen Modells wird in dieser Arbeit ein
Magnetfeld-getriebener Übergang von Reduktion der Sebstdiffusion durch
TeilchenkollisionenhinzuVerstärkungdurchKollisionenhergeleitet.Dasphysik-
laische Bild zur Erklärung des Effektes wird quantitaiv durch die Kraft-
Autokorrelationsfunktion unterstützt, welche mit steigender Magnetfledstärke
negativ wird. Die Vorhersagen der Theorie werden durch Brownsche-Dynamik
Simulationen bestätigt.
Contents
1 Introduction ................................................... 1
2 Theory ........................................................ 5
2.1 Model Description ......................................... 5
2.1.1 Diffusive Motion under Lorentz Force ................. 5
2.1.2 Probabilistic Description ............................. 7
2.2 Diffusion with Finite Size Effects ............................ 9
2.3 Matched Asymptotic Expansion ............................. 11
2.4 Collision Integral .......................................... 15
2.5 Single Species Model ...................................... 17
2.6 Final Equation ............................................. 19
2.7 Summary ................................................. 21
3 NumericalResults .............................................. 23
3.1 Numerical Method ......................................... 23
3.2 Central Derivative .......................................... 25
3.3 Charged versus Uncharged Particles in 2 Dimensions ........... 28
3.4 Summary ................................................. 31
4 Self-Diffusion .................................................. 33
4.1 Diffusion Coefficients ...................................... 33
4.1.1 Self- and Collective Diffusion ......................... 34
4.1.2 Two-Species Diffusion ............................... 37
4.2 Mechanism of Enhanced Self-Diffusion ....................... 40
4.3 Summary ................................................. 45
vii
viii Contents
5 First-PrinciplesApproachtoSelf-Diffusion ....................... 47
6 ConclusionsandOutlook ....................................... 53
Bibliography ...................................................... 57
List of Figures
Figure 1.1 Probability fluxes with and without Lorentz force .......... 2
Figure 1.2 Collective and self-diffusion ............................ 4
Figure 2.1 Excluded volume ..................................... 8
Figure 2.2 Coordinate change .................................... 12
Figure 3.1 Charged and uncharged particles sensing a boundary ....... 24
Figure 3.2 Central versus forward derivative ....................... 26
Figure 3.3 Numerically updating in the bulk and at a boundary ....... 28
Figure 3.4 Charged and uncharged particles diffusing with hard-core
interactions .......................................... 30
Figure 4.1 Collective and self-diffusion ............................ 34
Figure 4.2 Theoretical predictions and simulation results
for self-diffusion ..................................... 40
Figure 4.3 Ying-Yang figure: collisions of hard-disks ................ 42
Figure 4.4 (Self-) Diffusion for different scenarios .................. 44
Figure 5.1 Modified force autocorrelation function for hard-disks ...... 52
ix