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Vortices, Superfluid turbulence & Condensate Formation in Bose Gases
Thomas Gasenzer
Institut für Theoretische Physik Ruprecht-Karls Universität Heidelberg
Philosophenweg 16 • 69120 Heidelberg • Germany
email: [email protected] : www.thphys.uni-heidelberg.de/~gasenzer
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Heidelberg · Thermalization in Nonabelian Plasmas · 14 December 2011 Thomas Gasenzer
Thanks & credits to...
...my work group in Heidelberg:
Sebastian BockSebastian ErneMartin GärttnerRoman HennigMarkus KarlSteven MatheyBoris NowakNikolai PhilippMaximilian SchmidtJan Schole Dénes SextyMartin TrappeJan Zill
...my former students:
Cédric Bodet ( NEC), Alexander Branschädel ( KIT Karlsruhe), Stefan Keßler ( U Erlangen), Matthias Kronenwett ( R. Berger), Christian Scheppach ( Cambridge, UK), Philipp Struck ( Konstanz), Kristan Temme ( Vienna)
LGFG BaWue
€€€...
ExtreMe Matter Institute
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Heidelberg · Thermalization in Nonabelian Plasmas · 14 December 2011 Thomas Gasenzer
ULTRAcool...
... atoms @ nanokelvins –
trapped only a few mm away from
glass cell @ room temperature
(vacuum of 10-12 Torr, i.e. 10-15 bar,or 10-10 Pa,
≈ atmospheric pressure on the moon)
Heavy-Ion collisions (~1015 K)
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Condensate formationin an ultracold Bose gas
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Heidelberg · Thermalization in Nonabelian Plasmas · 14 December 2011 Thomas Gasenzer
Bose-Einstein condensation
Bose-Einstein condensation (BEC)
bimodaldistribution
Experimental picture after free expansion of the trapped cloud:
dens
ity
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Heidelberg · Thermalization in Nonabelian Plasmas · 14 December 2011 Thomas Gasenzer
Equilibration
Transient, metastable statee.g. Turbulence
Non-thermal fixed point
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Heidelberg · Thermalization in Nonabelian Plasmas · 14 December 2011 Thomas Gasenzer
Superfluid hydro of Bose-condensed Gas
The Gross-Pitaevskii Equation, (g = 4πa0/m)
using defs.
can be written as
Euler equation
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Heidelberg · Thermalization in Nonabelian Plasmas · 14 December 2011 Thomas Gasenzer
Movie 1: Phase evolution & Spectrum
n(k) = *(k)(k)∣angle average
Movie by Jan Schole
http://www.thphys.uni-heidelberg.de/~smp/gasenzer/videos/boseqt.html
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Heidelberg · Thermalization in Nonabelian Plasmas · 14 December 2011 Thomas Gasenzer
Movie 2: Vortex “gas” & Spectrum
n(k) = *(k)(k)∣angle average
http://www.thphys.uni-heidelberg.de/~smp/gasenzer/videos/boseqt.html
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Heidelberg · Thermalization in Nonabelian Plasmas · 14 December 2011 Thomas Gasenzer
Spectrum in 2+1 DB. Nowak, D. Sexty, TG, PRB 84: 020506(R), 2011
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Wave turbulencein an ultracold Bose gas
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Heidelberg · Thermalization in Nonabelian Plasmas · 14 December 2011 Thomas Gasenzer
Wave turbulenceStationary scaling n(k) within inertial region:
log n(k)
log k
n(k) ~ k—ζ
pump
dumpcascadeCascade
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Heidelberg · Thermalization in Nonabelian Plasmas · 14 December 2011 Thomas Gasenzer
Dilute ultracold Bose Gas
Gross-Pitaevskii Equation: (g = 4πa0/m)
Momentum spectrum:
n(k) = *(k)(k)
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Heidelberg · Thermalization in Nonabelian Plasmas · 14 December 2011 Thomas Gasenzer
Bose gas in d spatial dimensions
momentum k
ζ = d
C. Scheppach, J. Berges, TG PRA 81 (10) 033611
n ζQ = d −2/3
n ~ k −ζ
J. Berges, A. Rothkopf, J. Schmidt, PRL 101 (08) 041603
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Heidelberg · Thermalization in Nonabelian Plasmas · 14 December 2011 Thomas Gasenzer
Bose gas in d spatial dimensions
momentum k
ζ = 2
thermalequilibrium
(Rayleigh-Jeans)
C. Scheppach, J. Berges, TG PRA 81 (10) 033611
n
n ~ k −ζ
J. Berges, A. Rothkopf, J. Schmidt, PRL 101 (08) 041603
ζ = dζ
Q = d −2/3
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Heidelberg · Thermalization in Nonabelian Plasmas · 14 December 2011 Thomas Gasenzer
Dyn. QFT: Resummed Vertex
p = ( p0, p):
Vertex bubble resummation:(e.g. 2PI to NLO in 1/N)
[Dynamics: J. Berges, (02); G. Aarts et al., (02);[Nonthermal fixed points: J. Berges, A. Rothkopf, J. Schmidt, PRL (08)]
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Heidelberg · Thermalization in Nonabelian Plasmas · 14 December 2011 Thomas Gasenzer
momentum k
ζQ = d + 2
C. Scheppach, J. Berges, TG PRA 81 (10) 033611 J. Berges, A. Rothkopf, J. Schmidt, PRL 101 (08) 041603; J. Berges, G. Hoffmeister, NPB 813, 383 (2009)
nζ = d + 2 + z
Bose gas in d spatial dimensions n ~ k −ζ
New exponentbeyond Quantum Boltzmann!
ζ = dζ
Q = d −2/3
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Heidelberg · Thermalization in Nonabelian Plasmas · 14 December 2011 Thomas Gasenzer
Dyn. QFT: Resummed Vertex
p = ( p0, p):
Vertex bubble resummation:
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Heidelberg · Thermalization in Nonabelian Plasmas · 14 December 2011 Thomas Gasenzer
Cascades in 2+1 D
pump
pump
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Heidelberg · Thermalization in Nonabelian Plasmas · 14 December 2011 Thomas Gasenzer
Decomposition of Energy
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Heidelberg · Thermalization in Nonabelian Plasmas · 14 December 2011 Thomas Gasenzer
Time evolution of Energy Components (3+1 D)
J. Schole, B. Nowak, D. Sexty, TG (unpublished)
interaction energy
incompressible flow
quantum pressure contr.
compressible flow
vortexformation
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Heidelberg · Thermalization in Nonabelian Plasmas · 14 December 2011 Thomas Gasenzer
Evolution of Zero-Mode
J. Schole, B. Nowak, D. Sexty, TG (unpublished)
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Heidelberg · Thermalization in Nonabelian Plasmas · 14 December 2011 Thomas Gasenzer
Mode Occupations
J. Schole, B. Nowak, D. Sexty, TG (unpublished)
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Heidelberg · Thermalization in Nonabelian Plasmas · 14 December 2011 Thomas Gasenzer
1st-order Coherence
J. Schole, B. Nowak, D. Sexty, TG (unpublished)
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Random vortex distributions
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Heidelberg · Thermalization in Nonabelian Plasmas · 14 December 2011 Thomas Gasenzer
Point vortex model
nk ~ k −4 nk ~ k
−2, k < kpair
nk ~ k −4, k > kpair
B. Nowak, J. Schole, D. Sexty, TG, arXiv:1111.61XX [cond-mat.quant-gas]
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Heidelberg · Thermalization in Nonabelian Plasmas · 14 December 2011 Thomas Gasenzer
Point vortex model in 2+1 D
paired : k-2
random distrib.: k-4
vortex core: k-6
B. Nowak, J. Schole, D. Sexty, TG, arXiv:1111.61XX [cond-mat.quant-gas]
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Heidelberg · Thermalization in Nonabelian Plasmas · 14 December 2011 Thomas Gasenzer
Simulations in 2+1 D
paired : k-2 random distrib.: k-4
thermal: k-2
B. Nowak, J. Schole, D. Sexty, TG, arXiv:1111.61XX [cond-mat.quant-gas]
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Heidelberg · Thermalization in Nonabelian Plasmas · 14 December 2011 Thomas Gasenzer
Vortex position correlationsD
ensi
ty –
Den
sity
cor
rela
tion
s vortex-antivortex
vortex-vortex
minus
Distance between vorticesB. Nowak, J. Schole, D. Sexty, TG, arXiv:1111.61XX [cond-mat.quant-gas]
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Heidelberg · Thermalization in Nonabelian Plasmas · 14 December 2011 Thomas Gasenzer
Vortex-Density Decay in 2d
J. S., B. Nowak, D. Sexty, T. Gasenzer (unpublished)
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Vortices in a Na condensate
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Movie 3: Vortex Lines in 3+1 D
n(k) = *(k)(k)∣angle average
http://www.thphys.uni-heidelberg.de/~smp/gasenzer/videos/boseqt.html
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Heidelberg · Thermalization in Nonabelian Plasmas · 14 December 2011 Thomas Gasenzer
3+1 D simulationsB. Nowak, D. Sexty, TG, PRB 84: 020506(R), 2011
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Heidelberg · Thermalization in Nonabelian Plasmas · 14 December 2011 Thomas Gasenzer
Line vortex model in 3+1 Dsmall circular rings: k-2
paired parallel lines: k-3
independent lines: k-5
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Heidelberg · Thermalization in Nonabelian Plasmas · 14 December 2011 Thomas Gasenzer
Simulations in 3+1 D
small elliptical rings: k-3
thermal: k-2
independent lines: k-5
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Decomposition of flow
solenoidal (incompressible) “condensate”
compressible “sound”“quantum pressure”
k-4
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Acoustic turbulence
compressible “sound”“quantum pressure”
solenoidal (incompressible) “condensate”
total
k-4
B. Nowak, J. Schole, D. Sexty, TG, arXiv:1111.61XX [cond-mat.quant-gas]
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Heidelberg · Thermalization in Nonabelian Plasmas · 14 December 2011 Thomas Gasenzer
Non-thermal fixed point
[Fig. courtesy: J. Berges '08]
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Relativistic scalar field
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Heidelberg · Thermalization in Nonabelian Plasmas · 14 December 2011 Thomas Gasenzer
Strong Turbulence
∂t2−∂ x
2x , t 3 x ,t =0
Simulations of the non-linear Klein-Gordon equation, O(2) symmetry
TG, B. Nowak, D. Sexty, arXiv:1108.0541 [hep-ph]
Initial condition: Highly occupied zero mode, Unoccupied modes with k>0
(video)
See also: http://www.thphys.uni-heidelberg.de/~sexty/videos
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Heidelberg · Thermalization in Nonabelian Plasmas · 14 December 2011 Thomas Gasenzer
Strong Turbulence = Charge Separation
TG, B. Nowak, D. Sexty, arXiv:1108.0541 [hep-ph]cf. also Tkachev, Kofman, Starobinsky, Linde (1998)
Modulus of complex field || vs. mean charge distribution
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Heidelberg · Thermalization in Nonabelian Plasmas · 14 December 2011 Thomas Gasenzer
Strong Turbulence = Charge Separation
TG, B. Nowak, D. Sexty, arXiv:1108.0541 [hep-ph]
Charge density distribution vs. power spectrum
(d = 2, N = 2)
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Heidelberg · Thermalization in Nonabelian Plasmas · 14 December 2011 Thomas Gasenzer
Thanks & credits to...
...my work group in Heidelberg:
Boris NowakMaximilian SchmidtJan Schole Dénes SextySebastian ErneSteven MatheyNikolai PhilippSebastian BockMartin GärttnerMartin TrappeJan ZillRoman Hennig
...my former students:
Cédric Bodet ( NEC), Alexander Branschädel ( KIT Karlsruhe), Stefan Keßler ( U Erlangen), Matthias Kronenwett ( R. Berger), Christian Scheppach ( Cambridge, UK), Philipp Struck ( Konstanz), Kristan Temme ( Vienna)
LGFG BaWue
€€€...
ExtreMe Matter Institute
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Supplementary slides
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Heidelberg · Thermalization in Nonabelian Plasmas · 14 December 2011 Thomas Gasenzer
Time-Evolution of Energy-components (2+1 D)
J. S., B. Nowak, D. Sexty, T. Gasenzer (unpublished)
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Heidelberg · Thermalization in Nonabelian Plasmas · 14 December 2011 Thomas Gasenzer
Time evolution of vortex density
J. Schole, B. Nowak, D. Sexty, TG (unpublished)
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Enstrophy in classical turbulence
V. Yakhot, J. Wanderer, PRL 93:154502
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Vortex-Density Decay in 2d
J. S., B. Nowak, D. Sexty, T. Gasenzer (unpublished)
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Vortex-Density Decay in 2d
J. S., B. Nowak, D. Sexty, T. Gasenzer (unpublished)
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Heidelberg · Thermalization in Nonabelian Plasmas · 14 December 2011 Thomas Gasenzer
Kinetic Theory
One of the power laws can be explained by a kinetic theory:
∂t nv t =−ndipann
ann=coll
coll=l
v
l~ 1nv
~d
v=1d
d= 1
nv
ndip~nv
⇒ ∂ tnv t ~−nv2 ⇒ nv t ~t
−1
This result is valid under the assumption that the vortices are moving in pairsand that the pairs are homogeneously distributed.
: average pair distance
: average pair velocity
: mean free path
d
v
l
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Vortex-Distribution
J. S., B. Nowak, D. Sexty, T. Gasenzer (unpublished)
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Vortex-Antivortex-Correlations
J. S., B. Nowak, D. Sexty, T. Gasenzer (unpublished)
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Vortex-Antivortex-Correlations
J. S., B. Nowak, D. Sexty, T. Gasenzer (unpublished)
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Heidelberg · Thermalization in Nonabelian Plasmas · 14 December 2011 Thomas Gasenzer
Simulations in 2+1 D E(k) = ω(k)k d – 1n(k)
Remember:q
solenoidal
rotationless
k0
E(k) ~ k ‒5/3
k ‒1
B. Nowak, D. Sexty, TG (arXiv:1012.4437), PRB tbp.
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Heidelberg · Thermalization in Nonabelian Plasmas · 14 December 2011 Thomas Gasenzer
Simulations in 2+1 D E(k) = ω(k)k d – 1n(k)
Remember:q
solenoidal
rotationless
k0
E(k) ~ k ‒5/3
k ‒1
B. Nowak, D. Sexty, TG (arXiv:1012.4437), PRB tbp.
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Vortex velocity distribution
J. Schole, B. Nowak, D. Sexty, TG (unpublished)s. also C.F. White et al., PRL 104 (10); I.A. Min, Phys. Fluids 8 (96)
Lathrop expt. [PRL 101 (08)]
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Heidelberg · Thermalization in Nonabelian Plasmas · 14 December 2011 Thomas Gasenzer
Velocity distributions
Paoletti et al. PRL 101, 154501 (2008):
Min et al. Phys. Fluids 8, 1169 (1996), White et al. PRL 104, 075301 (2010):
Power law tails distinguish classical turbulence from classical turbulence.
Point vortices: Power law tails
Vorticity patches: Gaussian distributions
White et al. PRL 104, 075301 (2010)
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Velocity distributions (Field)
J. S., B. Nowak, D. Sexty, T. Gasenzer (unpublished)
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Heidelberg · Thermalization in Nonabelian Plasmas · 14 December 2011 Thomas Gasenzer
Velocity distributions (Vortices)
J. S., B. Nowak, D. Sexty, T. Gasenzer (unpublished)
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Condensation
N.G. Berloff, S.V. Svistunov, PRA 66, 013603 (2002)
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Simulations in 3+1 D
B. Nowak, D. Sexty, TG (arXiv:1012.4437)
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Energies
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Heidelberg · Thermalization in Nonabelian Plasmas · 14 December 2011 Thomas Gasenzer
Wave Turbulence – e.g. on water [H. Xia et al., EPL 91 (10) 14002]
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Wave Turbulence – e.g. on water [H. Xia et al., EPL 91 (10) 14002]
[Zakharov & Filonenko (67)]
Theory prediction:
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Heidelberg · Thermalization in Nonabelian Plasmas · 14 December 2011 Thomas Gasenzer
Vortex pairs
[T.W. Neely et al. PRL 104 (10)]
Guadeloupe [NASA]
Tucson [AZ]
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Helmholtz Vortex Law
[T.W. Neely et al. PRL 104 (10)]
Guadeloupe [NASA]
Tucson [AZ]
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density n
phase φ
[W. P. Reinhardt, Seattle]
[Wolfgang Ketterle, MIT]
complex field
velocity
Quantum Vortices
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Nonlinear dynamics: Pattern formation
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Nonlinear dynamics: Pattern formation
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Thermalisation dynamics: Turbulence?
[J.
Berg
es]
• Cosmology: Reheating after Inflation••
••••••
•
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Heidelberg · Thermalization in Nonabelian Plasmas · 14 December 2011 Thomas Gasenzer
Turbulence in reheating after inflation
∂t2−∂ x
2x , t 3 x ,t =0
Simulations of the non-linear Klein-Gordon equation,
Kofmann, Linde, Starobinsky (96)Micha, Tkachev, PRL & PRD (04)
Initial condition: Highly occupied zero modeUnoccupied modes with k>0
Turbulent spectrum emerges
Exponent: weak wave turbulence
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Wave turbulence
Turbulent thermalisation after universe inflation
[Micha & Tkachev, PRL 90 (03) 121301, PRD 70 (04) 043538]
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Heidelberg · Thermalization in Nonabelian Plasmas · 14 December 2011 Thomas Gasenzer
Lewis Fry Richardson, FRS (1881-1953)
Big whirls have little whirls that feed on their velocity,and little whirls have lesser whirls and so on to viscosity.
(L.F. Richardson, The supply of energy from and to Atmospheric Eddies, 1920)
Great fleas have little fleas upon their backs to bite 'em,And little fleas have lesser fleas, and so ad infinitum.And the great fleas themselves, in turn, have greater fleas to go on;While these again have greater still, and greater still, and so on.
(Augustus de Morgan, A Budget of Paradoxes, 1872, p. 370)
So, naturalists observe, a fleaHas smaller fleas that on him prey;And these have smaller still to bite 'em;And so proceed ad infinitum.
(Jonathan Swift: Poetry, a Rhapsody, 1733)