Alberto Nicolis
Columbia University
String theory in the bathtub
w/ S. Endlich, R. Penco
B. Horn, S. Gubser
W. Irvine, L. Stanzani
C. Barenghi
(EFT) (Strings)
(Exp)
(Num)
(EFT for hydro: w/ L. Delacretaz, S. Dubovsky, T. Gregoire, S. Endlich,
L. Hui, R. Penco, F. Piazza, R. Porto, R. Rattazzi, R. Rosen, S. Sibiryakov,
D. T. Son, J. Wang.
JHEP 0603, JHEP 1104, JHEP 1206, PRD 85 (2012), PRL 110 (2013),
JCAP 1310, PRD 88 (2013), JHEP 1311, PRD 89 (2014), PRD (2014)
hep-th 1303.3289, 1310.2272, 1311.6491, ... )
1
zero T super-fluid vs. ordinary fluid
compressional (sound) sector
Hydrodynamics
Hydrodynamics
transverse (vortex) sector
Hard (gapped)
Soft (gapless)
⇥
⇤
⇥
⇥
⇥v = 0
2
⇥v ⇥= 0
Vortex dynamics (incompressible limit)
For vortex lines
I
= ~v · d~` $
$
~v
I
~
B
Biot-Savart:
~v (~x) =
4⇡
Z
(~x ~x 0 )
0
⇥
d~
x
|~x ~x 0 |3
3
1st order EOM!
Unlike
m~a = F~ext
No room for “forces”
No free initial condition for v
Instantaneous v
determined by geometry
4
For vortex rings
~v =
4⇡R
log(R/a) n
ˆ
Far away:
~ dipole
~v (~x) = B
with
µ
~ = (⇡R2 ) n
ˆ
5
Excitations: Kelvin waves
Two modes overall (6= 2 + 2 )
!± =
2
2⇡
k log(1/ka)
fewer modes than 2-derivative eom
``non-local’’ dispersion relation
6
William Irvine
U. of Chicago
7
other groups...
8
Leandro Stanzani
Oltremare Park, Riccione, Italy
9
Journal of Comparative Psychology
2000,Vol. l14, No. 1,9S-106
Copyright 2000 by the American PsychologicalAssociation,Inc.
0735-7036/00/$5.00 DOI: 10.1037//0735-7036.114.1.98
Bubble Ring Play of Bottlenose Dolphins (Tursiops truncatus):
Implications for Cognition
Brenda McCowan
Lori Marino
University of California, Davis and Six Flags Marine World
Emory University
Erik Vance and Leah Walke
Diana Reiss
Six Flags Marine World
New York Aquarium
Research on the cognitive capacities of dolphins and other cetaceans (whales and porpoises)
has importance for the study of comparative cognition, particularly with other large-brained
social mammals, such as primates. One of the areas in which cetaceans can be compared with
primates is that of object manipulation and physical causality, for which there is an abundant
body of literature in primates. The authors supplemented qualitative observations with
statistical methods to examine playful bouts of underwater bubble ring production and
manipulation in 4 juvenile male captive bottlenose dolphins (Tursiops truncatus). The results
are consistent with the hypothesis that dolphins monitor the quality of their bubble rings and
anticipate their actions during bubble ring play.
Marten et al. described the physics involved in the formation
Ongoing research into the cognitive capacities of dolphins
of the bubble ring as follows:
and other cetaceans (e.g., whales and porpoises) has captured the interest and imagination of both the scientific
Any spherical bigger than about two centimeters in diameter
community and the public (Herman, 1986; Reiss, Mcwill quickly become a ring because of the difference in water
Cowan, & Marino, 1997). Bottlenose dolphins (Tursiops
pressure above and below the bubble. Water pressure increases with depth, so the bottom of the bubble experiences a
truncatus) are gregarious mammals that show a strong
higher pressure than the top does. The pressure from below
propensity for play behavior with physical objects and with
overcomes the surface tension of the sphere, punching a hole
conspecifics. There have been previous reports that both
in the center to create a doughnut shape. As water rushes
captive and wild dolphins produce their own objects of play, 10
through the hole, a vortex forms around the bubble. Any
vortex ring travels in the same direction as the flow through its
termed bubble rings (Marten, Shariff, Psarakos, & White,
In superfluids
Only allowed vortices = quantized vortex
2
⇠
.1
mm
/s for He)
(
=
2⇡~/m
lines w/ Superfluid turbulence = tangled mess of
vortex lines. Decay?
356
M. Ruderm
In pulsars: strong vortex line-flux tube
interactions. Glitches?
Fig. 14.3 Interacting flux-tubes and vortex-lines during initial spin-down
Observed in unitary Fermi gas
(Bulgac et al., PRL 2014)
11
(Ruderman 2009)
How to make sense of
their dynamics?
12
Effective field theory, quick way
L=
EOM:
⇢
Z
~v (~x) =
d ✏ijk X i @t X j @ X k +
4⇡
Magnetostatics
current J~
~
magnetic field B
~
vector potential A
Z
2
(~x ~x 0 )
0
⇥
d~
x
|~x ~x 0 |3
Z
3
d x @ i Aj
Incompressible Hydro
vorticity !
~
velocity field ~v
~
hydrophoton A
2
Z
d d
Z
0
~ · @ 0X
~0
@ X
~ X
~ 0|
|X
~ ·A
~ X,
~ t
d @ X
Note: no Lorentz force
~ ! ~v
B
13
c dictionary between magnetostatics and our system
of vortices in an incom-
Effective field theory, responsible way
DOF:
~ , t)
X(
~ ⇥A
~
~v (~x, t) = r
Symmetries:
translations, rotations
0
! ( , t)
reparametrizations
spontaneously broken Galilei
~ !X
~ + ~v0 t
X
~v ! ~v + ~v0
1
~
~
A ! A + ~v0 ⇥ ~x
2
Invariants:
{
@ i Aj
✏
ijk
2
i
j
X @t X @ X
14
k
~ ·A
~ X,
~ t
@ X
Running tension
One more term allowed by symmetries:
Why not there?
Needed as counterterm:
Z
0
~
~
0
dE
@
X
·
@
X
z
2
0 z
⇠⇢
=⇢
dz
~ X
~0
dz
X
T (µ) ⇠ ⇢
2
log µ
Many computations now simplified
15
T
2
Z
~
d @ X
log R/a
Kelvin waves
Perturbing
✏X@t X@ X + @ X · A(X) +
2
(rA)bulk
(grad. energy from mixing w/ A)
vs.
✏X@t X@ X + T (µ)
q
(no mixing)
Exact NL waves:
2
1 + @ ⇡?
~⇡? (z, t) = (ˆ
x + iˆ
y) ⇥
v=
4⇡
k log(1/ka) ⇥ p
16
0
e
ik(z vt)
1
1 + k2
2
0
Point-particle limit for vortex loops
L=
!
X⇥
n
X
n
~ ⇥ A)
~
µ
~ n · ~x˙ n + µ
~ n · (r
µ
~ n · ~x˙ n
3/2
µn
log µn
⇤
Z
3
d x @ i Aj
X µ
~n · µ
~m
n6=m
Peculiar conservation laws:
X
P~ =
µ
~n
2
3(~µm · rˆ)(~µn · rˆ)
r3
(µn = ⇡Rn2
n
~ =
L
X
n
E=
X
n
~xn ⇥ µ
~n
3/2
µn
log µn +
X µ
~n · µ
~m
n6=m
17
3(~µm · rˆ)(~µn · rˆ)
r3
n)
Coupling to sound/phonons
18
Subsonic regime:
v << cs
(Endlich, Nicolis 2013)
Nearly incompressible
sound waves difficult to excite
treat vortices
non-linearly
treat sound
perturbatively
integrate it out
19
Deformations of the medium
Dof:
volume elements’ positions
I
(⇧x, t)
I = 1, 2, 3
h
20
I
ieq = x
I
Symmetries:
I
I
(h
I
Action:
+a
I
SO(3)
ieq = xI
I
⇥
I
I
S=
Poincaré + internal
}
preserves diagonal combinations)
I
⇤
det J = 1
⇤⇥
(⇥)
Z
I
recover homogeneity/isotropy
d x F (b)
4
b=
fluid vs solid
q
det @µ
I @µ J
(Dubovsky, Gregoire, Nicolis, Rattazzi 2005)
21
Vortex-sound decomposition
=
I
I
@ 0
@xj
I
x, t)
0 (~
+
I
(~x, t)
}
det
(~x, t) =
+
compression
=1
Expand the action in powers of
22
and v0 /cs
Leading interaction
L=
Z
i ~
~
~
~ ·r
~
d x (r ⇥ A) (r ⇥ A)
3
i
+ ...
Ex: sound emission in vortex ring collisions
21 ⇢(R12 1 )2 (R22 2 )2 v 4
10
5
⇠
E
!
·
(R/r)
·
(v/c
)
P =
kin
s
5
10
2⇡
cs r (t)
23
More promising (experimentally)
oscillating vortex ring
⇢ 4 h1
2
P =
(|A
|
1 + |A
5
480⇡cs 4
2
2
2
|
)
!
+
(|A
|
1
2 + |A
1
⇠ Ekin ! · (A/R)2 · (v/cs )5
i
2
2
2
|
)
!
log
R/a
2
2
(Mitsou, Garcia-Saenz)
24
In fact:
log divergent
localized line-sound running coupling
Z
~ r
~ ·~
c(µ) d @ X
(in progress… )
25
Sound mediated vortex-vortex potential
Leading order
Next to leading
order
26
Long range potential:
⇢ q1 q2
V ⇠ 2 · 3 ⇠ Vdipole ⇥ v/cs
cs r
Z
3
2
q⌘
d xv
2
vortex
Detectable?
potential
NO.
force F =
V !
L!
@r V , right?
(eom)
potentially detectable for Vdipole ! 0
27
Who is this hydrophoton anyway?
For cs ! 1 :
~ = ...
r2 A
instantaneous propagation
For finite cs, propagation at cs ?
Hydrophoton = sound ?
Yet:
~ ·A
~=0
r
vs.
unsuppressed interactions
vs.
28
~ ⇥~=0
r
suppressed by (v/cs )
#
?
29
Relativistic generalization
superfluid
(~x, t) = µt + . . .
Lbulk = P (@ )
to couple to defects:
dA(2) / ?d
P (d )
in some gauge:
~ ~
A,
2
$
A[0i] = Ai
F (dA)
k
A[ij] = ✏ijk (x +
!
A[µ⌫]
30
k
)
Z
S!
0
SN
G
µ
⌫
d d⌧ Aµ⌫ @ X @⌧ X +
Z
4
d x F (dA) +
0
SNG
more general than standard NG:
g↵ = ⌘µ⌫ @↵ X µ @ X ⌫ ! g↵ , h↵ , . . .
µ
h↵ = (⌘µ⌫ + uµ u⌫ ) @↵ X @ X
( u ⇠ d ⇠ ?dA )
0
SNG
=
Z
p
⇣
⌘
p
g
2
d d⌧
h G p , (dA)
h
31
⌫
Within the EFT, perturbative expansion in t-derivatives:
v(r ⇠ a) . cs < c
v(`) ⇠ /` ⇠ cs · (a/`) ⌧ cs
@t X ⌧ cs · @ X
32
Quantum effects
(work in progress)
33
Virtual phonons
Sound production/exchange suppressed by (v/cs )#
For liquid helium:
cs ⇠ 200 m/s
2
⇠ .1 mm /s
v ⇠ cs at r ⇠ a
Measurable perturbative effects at r
34
a ?
Rotons in Helium 4
E
gap
phonons
rotons
1/a
p
@E
v=
@p
>
0
=
<
usually thought of as microscopic vortex rings.
can we check? what does it mean?
35
(Donnelly 1997)
Summary
Vortex lines and rings: very unconventional
mechanical systems
Important degrees in freedom in superfluids
EFT: efficient tool to understand them
Only tool to couple them to sound
Looking forward to experiments
36
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