Moving quantum baths: From black holes to dry friction

Moving quantum baths:
From black holes to dry
friction
Alejandro Jenkins
(Costa Rica)
work with Robert Alicki (Gdańsk)
Seminario de Gravitación y Física-Matemática
Cinvestav
Ciudad de México
12 Mar. 2020
1

Superradiance I
• Predicted by Zel’dovich in 1971 for rotating black
hole (BH)
• Incident electric ﬁeld:
2
0
= ⌦tComoving frame:
m⌦ > !Superradiance for
E0
⇠ exp [im 0
i (! m⌦) t]
E ⇠ exp [i (m !t)]

Superradiance II
• Off-equilibrium, unlike Dicke’s
superradiance (1954)
• Current interest in “BH bombs” (Press &
Teukolsky, 1972) for axion searches
• also, superradiance of gravity waves
• See: Brito, Cardoso & Pani,
Superradiance (Springer, 2015)
[arXiv:1501.06570 [gr-qc]] + refs. therein
• holographic superconductors &
superﬂuids from superradiant instability
of charged BH’s
Hartnoll, Herzog & Horowitz, PRL 101, 031601
(2008) [arXiv:0803.3295 [hep-th]]
3

critical wind speed V = v
“What Zel’dovich knew”
4
Galileo:
@⇠
@t
!
@⇠
@t
+ V
@⇠
@x
= i!⇠ + V · ik⇠
Zel’dovich, JETP Lett. 14, 180 (1971);
Sov. Phys. JETP 35, 1085 (1971)
⇠ ⇠ exp [ik(x vt)] ! = kv
Air
Water
x
*
* Thorne, 2013
= ik⇠(v V )<latexit sha1_base64="ZqUUAp7GO5gJX0BDF2a4SN1498c=">AAAB+3icbVBNS8NAEJ3Ur1q/Yj16WSxCPbQkIuhFKHrxWMF+QBvKZrtpl242YXdTWkL/ihcPinj1j3jz37htc9DWBwOP92aYmefHnCntON9WbmNza3snv1vY2z84PLKPi00VJZLQBol4JNs+VpQzQRuaaU7bsaQ49Dlt+aP7ud8aU6lYJJ70NKZeiAeCBYxgbaSeXbxFFYZGqDthqDxGFdS86Nklp+osgNaJm5ESZKj37K9uPyJJSIUmHCvVcZ1YeymWmhFOZ4VuomiMyQgPaMdQgUOqvHRx+wydG6WPgkiaEhot1N8TKQ6Vmoa+6QyxHqpVby7+53USHdx4KRNxoqkgy0VBwpGO0DwI1GeSEs2nhmAimbkVkSGWmGgTV8GE4K6+vE6al1XXqbqPV6XaXRZHHk7hDMrgwjXU4AHq0AACE3iGV3izZtaL9W59LFtzVjZzAn9gff4AHg6R5A==</latexit><latexit sha1_base64="ZqUUAp7GO5gJX0BDF2a4SN1498c=">AAAB+3icbVBNS8NAEJ3Ur1q/Yj16WSxCPbQkIuhFKHrxWMF+QBvKZrtpl242YXdTWkL/ihcPinj1j3jz37htc9DWBwOP92aYmefHnCntON9WbmNza3snv1vY2z84PLKPi00VJZLQBol4JNs+VpQzQRuaaU7bsaQ49Dlt+aP7ud8aU6lYJJ70NKZeiAeCBYxgbaSeXbxFFYZGqDthqDxGFdS86Nklp+osgNaJm5ESZKj37K9uPyJJSIUmHCvVcZ1YeymWmhFOZ4VuomiMyQgPaMdQgUOqvHRx+wydG6WPgkiaEhot1N8TKQ6Vmoa+6QyxHqpVby7+53USHdx4KRNxoqkgy0VBwpGO0DwI1GeSEs2nhmAimbkVkSGWmGgTV8GE4K6+vE6al1XXqbqPV6XaXRZHHk7hDMrgwjXU4AHq0AACE3iGV3izZtaL9W59LFtzVjZzAn9gff4AHg6R5A==</latexit><latexit sha1_base64="ZqUUAp7GO5gJX0BDF2a4SN1498c=">AAAB+3icbVBNS8NAEJ3Ur1q/Yj16WSxCPbQkIuhFKHrxWMF+QBvKZrtpl242YXdTWkL/ihcPinj1j3jz37htc9DWBwOP92aYmefHnCntON9WbmNza3snv1vY2z84PLKPi00VJZLQBol4JNs+VpQzQRuaaU7bsaQ49Dlt+aP7ud8aU6lYJJ70NKZeiAeCBYxgbaSeXbxFFYZGqDthqDxGFdS86Nklp+osgNaJm5ESZKj37K9uPyJJSIUmHCvVcZ1YeymWmhFOZ4VuomiMyQgPaMdQgUOqvHRx+wydG6WPgkiaEhot1N8TKQ6Vmoa+6QyxHqpVby7+53USHdx4KRNxoqkgy0VBwpGO0DwI1GeSEs2nhmAimbkVkSGWmGgTV8GE4K6+vE6al1XXqbqPV6XaXRZHHk7hDMrgwjXU4AHq0AACE3iGV3izZtaL9W59LFtzVjZzAn9gff4AHg6R5A==</latexit><latexit sha1_base64="ZqUUAp7GO5gJX0BDF2a4SN1498c=">AAAB+3icbVBNS8NAEJ3Ur1q/Yj16WSxCPbQkIuhFKHrxWMF+QBvKZrtpl242YXdTWkL/ihcPinj1j3jz37htc9DWBwOP92aYmefHnCntON9WbmNza3snv1vY2z84PLKPi00VJZLQBol4JNs+VpQzQRuaaU7bsaQ49Dlt+aP7ud8aU6lYJJ70NKZeiAeCBYxgbaSeXbxFFYZGqDthqDxGFdS86Nklp+osgNaJm5ESZKj37K9uPyJJSIUmHCvVcZ1YeymWmhFOZ4VuomiMyQgPaMdQgUOqvHRx+wydG6WPgkiaEhot1N8TKQ6Vmoa+6QyxHqpVby7+53USHdx4KRNxoqkgy0VBwpGO0DwI1GeSEs2nhmAimbkVkSGWmGgTV8GE4K6+vE6al1XXqbqPV6XaXRZHHk7hDMrgwjXU4AHq0AACE3iGV3izZtaL9W59LFtzVjZzAn9gff4AHg6R5A==</latexit>

Irreversibility I
5
˙Ewind
˙Ewave > 0 available for dissipation in the air
Ewind =
p2
2m
;
See: Bekenstein & Schiffer, PRD 58, 064014 (1998)
˙pwind = ˙pwaveMom. conservation:
˙Ewind =
p
m
· ˙pwave = V · (f~k)
˙Ewave = f~! = f~vk ; V > v , ˙Ewind > ˙Ewave

Superradiance III
“Zel’dovich doesn’t understand general
relativity well enough to compute what a
black hole should do, so instead he
computes the behavior of a spinning
metal sphere, he then asserts that a BH
will behave analogously, and he wakes
me up at 6:30 a.m. to test his assertion.”
— Thorne, Black Holes & Time Warps
(1994)
• Motivated Hawking’s 1974 result on
thermal radiation by static BHs
6

Slow Mo Guys™
7
Excerpted from: https://www.youtube.com/watch?v=zs7x1Hu29Wc
Critical speed for shatter: Ω = 23,000 rpm

Shock waves
• “It turns out, very interestingly, that
once the object is moving faster
than the speed of sound, it will make
sound. It is not necessary that it
have a certain tone vibrational
character.”
• Superradiance condition
corresponds to “anomalous Doppler
shift” in Ginzburg-Frank theory for
Vavilov-Cherenkov radiation
• “counterparts in acoustics,
chromodynamics and, strictly
speaking, in any ﬁeld theory” —
Ginzburg, Prog. Optics 32, 267
(1993)
8
— Feynman Lectures, “Waves”
vol. I, no. 51,

Rankine-Hugoniot
“[Shocks were] understood quite early on, more or less in this way, by
Stokes. However, he was persuaded by his former student Rayleigh and
others that such discontinuities were impossible, because they would
violate energy conservation. With a deference that professors traditionally
show their students, Stokes believed Rayleigh." — Thorne & Blandford,
Modern Classical Physics (2017)
9
downstream
front
back
shock
upstream
P1, u1,
V1 = 1/ρ1
P2, u2,
V2 = 1/ρ2
v2v1
FIGURE 17.10 Terminology and notation for a shock front
and the flow into and out of it.
The Rankine-Hugoniot relations (17.29) then take the forms:
ρ2v2 = ρ1v1 = j , (17.30a)
P2 + ρ2v2
2 = P1 + ρ1v2
1, (17.30b)
possible downstream state
shock adiabat
increasing entropy
upstream state
slope–j2
s = const
s = const
P2
P1
V2 V1 V
P
FIGURE 17.11 Shock adiabat. The pressure and specific volume V = 1/ρ in the upstream
flow are P1 and V1, and in the downstream flow they are P2 and V2. The dashed curves
are ordinary adiabats (curves of constant entropy per unit mass s). The thick curve
is the shock adiabat, the curve of allowed downstream states (V2, P2) for a given
upstream state (V1, P1). The actual location of the downstream state on this adiabat
is determined by the mass flux j flowing through the shock: the slope of the dotted
line connecting the upstream and downstream states is −j2.
equations will actually give a curve that extends away from (V1, P1) in both directions,
up-leftward and down-rightward. Only the up-leftward portion is compatible with an
increase of entropy across the shock; the down-rightward portion requires an entropy

Singularities
“Radiation during the uniform motion of various
sources is a universal phenomenon rather than an
eccentricity” — Ginzburg, Phys. Usp. 39, 973 (1996)
“Quantum mechanics helps understand classical
mechanics” — ‘Paradoksov’, Sov. Phys. Uspekhi 9,
618 (1967)
“It seems to be a good principle that the prediction of
a singularity by a physical theory indicates that the
theory has broken down.” — Hawking & Ellis (1973)
10

Quantum ﬁeld
11
Hf =
X
m,↵,!
! a†
m↵(!)am↵(!), Lz
f =
X
m,↵,!
m a†
m↵(!)am↵(!)
h
am↵(!), a†
m0↵0 (!0
)
i
±
= mm0 ↵↵0 !!0
[am↵(!), am0↵0 (!0
)]± = [a†
m↵(!), a†
m0↵0 (!0
)]± = 0
Hint =
X
m,↵,!
am↵(!) ⌦ B†
m↵(!) + a†
m↵(!) ⌦ Bm↵(!)
Linear ﬁeld-bath coupling:
R. Alicki & AJ, Ann. Phys. (NY) 395, 69 (2018)
[arXiv:1702.06231]
Work in units

Rotating bath
12
[Lz
b, Bm↵(!)] = mBm↵(!),
⇥
Lz
b, B†
m↵(!)
⇤
= mB†
m↵(!)
ei Lz
b Bm↵(!)e i Lz
b = eim
Bm↵(!) ,
ei Lz
b B†
m↵(!)e i Lz
b = e im
B†
m↵(!)
B k ⌘ B†
k
Bk with k ⌘ {m↵!}Short-hand:
Effective bath Hamiltonian:
(à la Born-Oppenheimer)
He↵
b = H0
b ⌦Lz
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Cylindrical symmetry:

KMS
13
0
kl( x) = e x 0
l k(x)KMS condition:
Fluctuation-dissipation: 0
kl(x) =
Z 1
1
eixt
D
B†
k(t)Bl
E
b
dt
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Turning on bath rotation: Hb ! He↵
b = H0
b ⌦Lz
b<latexit sha1_base64="wTm6jz48ZGfUNQLNKgcMwO29IHk=">AAACLHicbVBLSwMxGMz6rPVV9eglWApeLLsiqAehqIceBCtYW+jWkk2/bUOT3SXJCnXpH/LiXxHBQxWv/g7TB6W2DoRMZvKRzHgRZ0rbdt9aWFxaXllNraXXNza3tjM7uw8qjCWFMg15KKseUcBZAGXNNIdqJIEIj0PF61wN/MoTSMXC4F53I6gL0gqYzyjRRmpkrouNxJUCez3s6hBPTo/DHXy/hy+mVBsfYfdWQIvgm4n43Mhk7bw9BJ4nzphk0RilRubdbYY0FhBoyolSNceOdD0hUjPKoZd2YwURoR3SgpqhARGg6skwbQ/njNLEfijNCjQeqtMTCRFKdYVJlBNEt9WsNxD/82qx9s/qCQuiWENARw/5Mceml0F1uMkkUM27hhAqmfkrpm0iCdWm4LQpwZmNPE/Kx/nzvHN3ki1cjttIoX10gA6Rg05RARVRCZURRS/oDfXRp/VqfVhf1vfo6oI1ntlDf2D9/AKIXqax</latexit><latexit sha1_base64="wTm6jz48ZGfUNQLNKgcMwO29IHk=">AAACLHicbVBLSwMxGMz6rPVV9eglWApeLLsiqAehqIceBCtYW+jWkk2/bUOT3SXJCnXpH/LiXxHBQxWv/g7TB6W2DoRMZvKRzHgRZ0rbdt9aWFxaXllNraXXNza3tjM7uw8qjCWFMg15KKseUcBZAGXNNIdqJIEIj0PF61wN/MoTSMXC4F53I6gL0gqYzyjRRmpkrouNxJUCez3s6hBPTo/DHXy/hy+mVBsfYfdWQIvgm4n43Mhk7bw9BJ4nzphk0RilRubdbYY0FhBoyolSNceOdD0hUjPKoZd2YwURoR3SgpqhARGg6skwbQ/njNLEfijNCjQeqtMTCRFKdYVJlBNEt9WsNxD/82qx9s/qCQuiWENARw/5Mceml0F1uMkkUM27hhAqmfkrpm0iCdWm4LQpwZmNPE/Kx/nzvHN3ki1cjttIoX10gA6Rg05RARVRCZURRS/oDfXRp/VqfVhf1vfo6oI1ntlDf2D9/AKIXqax</latexit><latexit sha1_base64="wTm6jz48ZGfUNQLNKgcMwO29IHk=">AAACLHicbVBLSwMxGMz6rPVV9eglWApeLLsiqAehqIceBCtYW+jWkk2/bUOT3SXJCnXpH/LiXxHBQxWv/g7TB6W2DoRMZvKRzHgRZ0rbdt9aWFxaXllNraXXNza3tjM7uw8qjCWFMg15KKseUcBZAGXNNIdqJIEIj0PF61wN/MoTSMXC4F53I6gL0gqYzyjRRmpkrouNxJUCez3s6hBPTo/DHXy/hy+mVBsfYfdWQIvgm4n43Mhk7bw9BJ4nzphk0RilRubdbYY0FhBoyolSNceOdD0hUjPKoZd2YwURoR3SgpqhARGg6skwbQ/njNLEfijNCjQeqtMTCRFKdYVJlBNEt9WsNxD/82qx9s/qCQuiWENARw/5Mceml0F1uMkkUM27hhAqmfkrpm0iCdWm4LQpwZmNPE/Kx/nzvHN3ki1cjttIoX10gA6Rg05RARVRCZURRS/oDfXRp/VqfVhf1vfo6oI1ntlDf2D9/AKIXqax</latexit><latexit sha1_base64="wTm6jz48ZGfUNQLNKgcMwO29IHk=">AAACLHicbVBLSwMxGMz6rPVV9eglWApeLLsiqAehqIceBCtYW+jWkk2/bUOT3SXJCnXpH/LiXxHBQxWv/g7TB6W2DoRMZvKRzHgRZ0rbdt9aWFxaXllNraXXNza3tjM7uw8qjCWFMg15KKseUcBZAGXNNIdqJIEIj0PF61wN/MoTSMXC4F53I6gL0gqYzyjRRmpkrouNxJUCez3s6hBPTo/DHXy/hy+mVBsfYfdWQIvgm4n43Mhk7bw9BJ4nzphk0RilRubdbYY0FhBoyolSNceOdD0hUjPKoZd2YwURoR3SgpqhARGg6skwbQ/njNLEfijNCjQeqtMTCRFKdYVJlBNEt9WsNxD/82qx9s/qCQuiWENARw/5Mceml0F1uMkkUM27hhAqmfkrpm0iCdWm4LQpwZmNPE/Kx/nzvHN3ki1cjttIoX10gA6Rg05RARVRCZURRS/oDfXRp/VqfVhf1vfo6oI1ntlDf2D9/AKIXqax</latexit>
Rotational invariance:
⌦
kk(x) = 0
kk(x + m⌦)<latexit sha1_base64="cTX+IqUXMjqVGSuxwM/eKXb3qow=">AAACGXicbVBNSwMxFMzWr1q/qh69BIvQIpRdEdSDUPTizQrWFrpreZumbWiyuyRZaVn6O7z4V7x4UPGoJ/+NabsHbR0IDDPzeHnjR5wpbdvfVmZhcWl5JbuaW1vf2NzKb+/cqTCWhNZIyEPZ8EFRzgJa00xz2ogkBeFzWvf7l2O//kClYmFwq4cR9QR0A9ZhBLSRWnnH7YIQcO9eC9qFVtLvj4qDEj7HqW5PJFwcHIpppNTKF+yyPQGeJ05KCihFtZX/dNshiQUNNOGgVNOxI+0lIDUjnI5ybqxoBKQPXdo0NABBlZdMThvhA6O0cSeU5gUaT9TfEwkIpYbCN0kBuqdmvbH4n9eMdefUS1gQxZoGZLqoE3OsQzzuCbeZpETzoSFAJDN/xaQHEog2beZMCc7syfOkdlQ+Kzs3x4XKRdpGFu2hfVREDjpBFXSFqqiGCHpEz+gVvVlP1ov1bn1MoxkrndlFf2B9/QDve5++</latexit><latexit sha1_base64="cTX+IqUXMjqVGSuxwM/eKXb3qow=">AAACGXicbVBNSwMxFMzWr1q/qh69BIvQIpRdEdSDUPTizQrWFrpreZumbWiyuyRZaVn6O7z4V7x4UPGoJ/+NabsHbR0IDDPzeHnjR5wpbdvfVmZhcWl5JbuaW1vf2NzKb+/cqTCWhNZIyEPZ8EFRzgJa00xz2ogkBeFzWvf7l2O//kClYmFwq4cR9QR0A9ZhBLSRWnnH7YIQcO9eC9qFVtLvj4qDEj7HqW5PJFwcHIpppNTKF+yyPQGeJ05KCihFtZX/dNshiQUNNOGgVNOxI+0lIDUjnI5ybqxoBKQPXdo0NABBlZdMThvhA6O0cSeU5gUaT9TfEwkIpYbCN0kBuqdmvbH4n9eMdefUS1gQxZoGZLqoE3OsQzzuCbeZpETzoSFAJDN/xaQHEog2beZMCc7syfOkdlQ+Kzs3x4XKRdpGFu2hfVREDjpBFXSFqqiGCHpEz+gVvVlP1ov1bn1MoxkrndlFf2B9/QDve5++</latexit><latexit sha1_base64="cTX+IqUXMjqVGSuxwM/eKXb3qow=">AAACGXicbVBNSwMxFMzWr1q/qh69BIvQIpRdEdSDUPTizQrWFrpreZumbWiyuyRZaVn6O7z4V7x4UPGoJ/+NabsHbR0IDDPzeHnjR5wpbdvfVmZhcWl5JbuaW1vf2NzKb+/cqTCWhNZIyEPZ8EFRzgJa00xz2ogkBeFzWvf7l2O//kClYmFwq4cR9QR0A9ZhBLSRWnnH7YIQcO9eC9qFVtLvj4qDEj7HqW5PJFwcHIpppNTKF+yyPQGeJ05KCihFtZX/dNshiQUNNOGgVNOxI+0lIDUjnI5ybqxoBKQPXdo0NABBlZdMThvhA6O0cSeU5gUaT9TfEwkIpYbCN0kBuqdmvbH4n9eMdefUS1gQxZoGZLqoE3OsQzzuCbeZpETzoSFAJDN/xaQHEog2beZMCc7syfOkdlQ+Kzs3x4XKRdpGFu2hfVREDjpBFXSFqqiGCHpEz+gVvVlP1ov1bn1MoxkrndlFf2B9/QDve5++</latexit><latexit sha1_base64="cTX+IqUXMjqVGSuxwM/eKXb3qow=">AAACGXicbVBNSwMxFMzWr1q/qh69BIvQIpRdEdSDUPTizQrWFrpreZumbWiyuyRZaVn6O7z4V7x4UPGoJ/+NabsHbR0IDDPzeHnjR5wpbdvfVmZhcWl5JbuaW1vf2NzKb+/cqTCWhNZIyEPZ8EFRzgJa00xz2ogkBeFzWvf7l2O//kClYmFwq4cR9QR0A9ZhBLSRWnnH7YIQcO9eC9qFVtLvj4qDEj7HqW5PJFwcHIpppNTKF+yyPQGeJ05KCihFtZX/dNshiQUNNOGgVNOxI+0lIDUjnI5ybqxoBKQPXdo0NABBlZdMThvhA6O0cSeU5gUaT9TfEwkIpYbCN0kBuqdmvbH4n9eMdefUS1gQxZoGZLqoE3OsQzzuCbeZpETzoSFAJDN/xaQHEog2beZMCc7syfOkdlQ+Kzs3x4XKRdpGFu2hfVREDjpBFXSFqqiGCHpEz+gVvVlP1ov1bn1MoxkrndlFf2B9/QDve5++</latexit>
(diagonal in m)
⌦
kk( x) = e (x m⌦) ⌦
k k(x)<latexit sha1_base64="bDAZfo3X6RvlZS/nOVPZtaQ9qd0=">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</latexit><latexit sha1_base64="bDAZfo3X6RvlZS/nOVPZtaQ9qd0=">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</latexit><latexit sha1_base64="bDAZfo3X6RvlZS/nOVPZtaQ9qd0=">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</latexit><latexit sha1_base64="bDAZfo3X6RvlZS/nOVPZtaQ9qd0=">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</latexit>
✓
⌘
1
kBT
◆
<latexit sha1_base64="ZVbpETMoo2qQ2k++qDAXbilmkMs=">AAACGXicbVDLSgNBEJz1GeMr6tHLYBDiJeyKoN5CvHiMkJhANoTZSW8yZPbhTG8gLPsdXvwVLx5UPOrJv3HyOGhiQUNR1U13lxdLodG2v62V1bX1jc3cVn57Z3dvv3BweK+jRHFo8EhGquUxDVKE0ECBElqxAhZ4Epre8GbiN0egtIjCOo5j6ASsHwpfcIZG6hYcV4KPJep6gIy68JCIEXV9xXjqZOmwm7oqoNWM1jPqKtEf4Fm3ULTL9hR0mThzUiRz1LqFT7cX8SSAELlkWrcdO8ZOyhQKLiHLu4mGmPEh60Pb0JAFoDvp9LWMnhqlR/1ImQqRTtXfEykLtB4HnukMGA70ojcR//PaCfpXnVSEcYIQ8tkiP5EUIzrJifaEAo5ybAjjSphbKR8wkwuaNPMmBGfx5WXSOC9fl527i2KlOk8jR47JCSkRh1ySCrklNdIgnDySZ/JK3qwn68V6tz5mrSvWfOaI/IH19QOKKaAl</latexit><latexit sha1_base64="ZVbpETMoo2qQ2k++qDAXbilmkMs=">AAACGXicbVDLSgNBEJz1GeMr6tHLYBDiJeyKoN5CvHiMkJhANoTZSW8yZPbhTG8gLPsdXvwVLx5UPOrJv3HyOGhiQUNR1U13lxdLodG2v62V1bX1jc3cVn57Z3dvv3BweK+jRHFo8EhGquUxDVKE0ECBElqxAhZ4Epre8GbiN0egtIjCOo5j6ASsHwpfcIZG6hYcV4KPJep6gIy68JCIEXV9xXjqZOmwm7oqoNWM1jPqKtEf4Fm3ULTL9hR0mThzUiRz1LqFT7cX8SSAELlkWrcdO8ZOyhQKLiHLu4mGmPEh60Pb0JAFoDvp9LWMnhqlR/1ImQqRTtXfEykLtB4HnukMGA70ojcR//PaCfpXnVSEcYIQ8tkiP5EUIzrJifaEAo5ybAjjSphbKR8wkwuaNPMmBGfx5WXSOC9fl527i2KlOk8jR47JCSkRh1ySCrklNdIgnDySZ/JK3qwn68V6tz5mrSvWfOaI/IH19QOKKaAl</latexit><latexit sha1_base64="ZVbpETMoo2qQ2k++qDAXbilmkMs=">AAACGXicbVDLSgNBEJz1GeMr6tHLYBDiJeyKoN5CvHiMkJhANoTZSW8yZPbhTG8gLPsdXvwVLx5UPOrJv3HyOGhiQUNR1U13lxdLodG2v62V1bX1jc3cVn57Z3dvv3BweK+jRHFo8EhGquUxDVKE0ECBElqxAhZ4Epre8GbiN0egtIjCOo5j6ASsHwpfcIZG6hYcV4KPJep6gIy68JCIEXV9xXjqZOmwm7oqoNWM1jPqKtEf4Fm3ULTL9hR0mThzUiRz1LqFT7cX8SSAELlkWrcdO8ZOyhQKLiHLu4mGmPEh60Pb0JAFoDvp9LWMnhqlR/1ImQqRTtXfEykLtB4HnukMGA70ojcR//PaCfpXnVSEcYIQ8tkiP5EUIzrJifaEAo5ybAjjSphbKR8wkwuaNPMmBGfx5WXSOC9fl527i2KlOk8jR47JCSkRh1ySCrklNdIgnDySZ/JK3qwn68V6tz5mrSvWfOaI/IH19QOKKaAl</latexit><latexit sha1_base64="ZVbpETMoo2qQ2k++qDAXbilmkMs=">AAACGXicbVDLSgNBEJz1GeMr6tHLYBDiJeyKoN5CvHiMkJhANoTZSW8yZPbhTG8gLPsdXvwVLx5UPOrJv3HyOGhiQUNR1U13lxdLodG2v62V1bX1jc3cVn57Z3dvv3BweK+jRHFo8EhGquUxDVKE0ECBElqxAhZ4Epre8GbiN0egtIjCOo5j6ASsHwpfcIZG6hYcV4KPJep6gIy68JCIEXV9xXjqZOmwm7oqoNWM1jPqKtEf4Fm3ULTL9hR0mThzUiRz1LqFT7cX8SSAELlkWrcdO8ZOyhQKLiHLu4mGmPEh60Pb0JAFoDvp9LWMnhqlR/1ImQqRTtXfEykLtB4HnukMGA70ojcR//PaCfpXnVSEcYIQ8tkiP5EUIzrJifaEAo5ybAjjSphbKR8wkwuaNPMmBGfx5WXSOC9fl527i2KlOk8jR47JCSkRh1ySCrklNdIgnDySZ/JK3qwn68V6tz5mrSvWfOaI/IH19QOKKaAl</latexit>

Markovian master
equation (MME)
14
˙⇢(t) = i [Hf , ⇢(t)] + L⇢(t) = i [Hf , ⇢(t)]
+
1
2
X
m,↵,!
⌦
m↵(!)
n ⇥
am↵(!)⇢(t), a†
m↵(!)
⇤
+
⇥
am↵(!), ⇢(t)a†
m↵(!)
⇤
]
+e (! m⌦)
⇥
a†
m↵(!)⇢(t), am↵(!)
⇤
+
⇥
a†
m↵(!), ⇢(t)am↵(!)
⇤ o
"(k) = exp [ (! m⌦)] ⌦
m↵(!)Pumping rate:
Decay rate: #(k) ⌘ ⌦
m↵(!) ⌘ ⌦
kk(!)<latexit sha1_base64="ugiXKbaOQE1HbSObBz3CHhJ15+0=">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</latexit><latexit sha1_base64="ugiXKbaOQE1HbSObBz3CHhJ15+0=">AAACSnicbVBNSxxBFOxZTdQ1ias5emlcAutFZoKgR9FDvKmQVWFnM7zpfTvbTH+M3T3KMuzvy8WTN3+ElxwSJBd71gU/CxqKqnq815UWglsXhrdBY27+w8eFxaXm8qfPX1Zaq2unVpeGYZdpoc15ChYFV9h13Ak8LwyCTAWepflB7Z9dorFcq59uXGBfQqb4kDNwXkpaEGcgJSTxQF8pMEZf0U6+SWO8KPkljX/U5q/4SGIGSSVpDKIYwYR2Yl1LT8HsRTDPnyJJqx1uhVPQtySakTaZ4Thp3fhjWClROSbA2l4UFq5fgXGcCZw049JiASyHDHueKpBo+9W0ign95pUBHWrjn3J0qj6fqEBaO5apT0pwI/vaq8X3vF7phrv9iquidKjY46JhKajTtO6VDrhB5sTYE2CG+1spG4EB5nz7TV9C9PrLb8np963I85Pt9t7+rI5Fsk42SIdEZIfskUNyTLqEkd/kjvwl/4Lr4E9wH/x/jDaC2cxX8gKN+QcIL7Le</latexit><latexit sha1_base64="ugiXKbaOQE1HbSObBz3CHhJ15+0=">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</latexit><latexit sha1_base64="ugiXKbaOQE1HbSObBz3CHhJ15+0=">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</latexit>
k = {m, ↵, !}<latexit sha1_base64="X56HfzX/Ja9jjNaWQo/FE4cs0pc=">AAACBHicbZDLSgMxFIYz9VbrbdRlN8EiuJAyI4JuhKIblxXsBTpDOZOmbWgyGZKMUIYu3Pgqblwo4taHcOfbmLaz0NYDIR//fw7J+aOEM20879sprKyurW8UN0tb2zu7e+7+QVPLVBHaIJJL1Y5AU85i2jDMcNpOFAURcdqKRjdTv/VAlWYyvjfjhIYCBjHrMwLGSl23PMJXOMiwOMUB8GQI9paCDgAHk65b8arerPAy+DlUUF71rvsV9CRJBY0N4aB1x/cSE2agDCOcTkpBqmkCZAQD2rEYg6A6zGZLTPCxVXq4L5U9scEz9fdEBkLrsYhspwAz1IveVPzP66SmfxlmLE5SQ2Myf6ifcmwkniaCe0xRYvjYAhDF7F8xGYICYmxuJRuCv7jyMjTPqr7lu/NK7TqPo4jK6AidIB9doBq6RXXUQAQ9omf0it6cJ+fFeXc+5q0FJ585RH/K+fwB7d6WXQ==</latexit><latexit sha1_base64="X56HfzX/Ja9jjNaWQo/FE4cs0pc=">AAACBHicbZDLSgMxFIYz9VbrbdRlN8EiuJAyI4JuhKIblxXsBTpDOZOmbWgyGZKMUIYu3Pgqblwo4taHcOfbmLaz0NYDIR//fw7J+aOEM20879sprKyurW8UN0tb2zu7e+7+QVPLVBHaIJJL1Y5AU85i2jDMcNpOFAURcdqKRjdTv/VAlWYyvjfjhIYCBjHrMwLGSl23PMJXOMiwOMUB8GQI9paCDgAHk65b8arerPAy+DlUUF71rvsV9CRJBY0N4aB1x/cSE2agDCOcTkpBqmkCZAQD2rEYg6A6zGZLTPCxVXq4L5U9scEz9fdEBkLrsYhspwAz1IveVPzP66SmfxlmLE5SQ2Myf6ifcmwkniaCe0xRYvjYAhDF7F8xGYICYmxuJRuCv7jyMjTPqr7lu/NK7TqPo4jK6AidIB9doBq6RXXUQAQ9omf0it6cJ+fFeXc+5q0FJ585RH/K+fwB7d6WXQ==</latexit><latexit sha1_base64="X56HfzX/Ja9jjNaWQo/FE4cs0pc=">AAACBHicbZDLSgMxFIYz9VbrbdRlN8EiuJAyI4JuhKIblxXsBTpDOZOmbWgyGZKMUIYu3Pgqblwo4taHcOfbmLaz0NYDIR//fw7J+aOEM20879sprKyurW8UN0tb2zu7e+7+QVPLVBHaIJJL1Y5AU85i2jDMcNpOFAURcdqKRjdTv/VAlWYyvjfjhIYCBjHrMwLGSl23PMJXOMiwOMUB8GQI9paCDgAHk65b8arerPAy+DlUUF71rvsV9CRJBY0N4aB1x/cSE2agDCOcTkpBqmkCZAQD2rEYg6A6zGZLTPCxVXq4L5U9scEz9fdEBkLrsYhspwAz1IveVPzP66SmfxlmLE5SQ2Myf6ifcmwkniaCe0xRYvjYAhDF7F8xGYICYmxuJRuCv7jyMjTPqr7lu/NK7TqPo4jK6AidIB9doBq6RXXUQAQ9omf0it6cJ+fFeXc+5q0FJ585RH/K+fwB7d6WXQ==</latexit><latexit sha1_base64="X56HfzX/Ja9jjNaWQo/FE4cs0pc=">AAACBHicbZDLSgMxFIYz9VbrbdRlN8EiuJAyI4JuhKIblxXsBTpDOZOmbWgyGZKMUIYu3Pgqblwo4taHcOfbmLaz0NYDIR//fw7J+aOEM20879sprKyurW8UN0tb2zu7e+7+QVPLVBHaIJJL1Y5AU85i2jDMcNpOFAURcdqKRjdTv/VAlWYyvjfjhIYCBjHrMwLGSl23PMJXOMiwOMUB8GQI9paCDgAHk65b8arerPAy+DlUUF71rvsV9CRJBY0N4aB1x/cSE2agDCOcTkpBqmkCZAQD2rEYg6A6zGZLTPCxVXq4L5U9scEz9fdEBkLrsYhspwAz1IveVPzP66SmfxlmLE5SQ2Myf6ifcmwkniaCe0xRYvjYAhDF7F8xGYICYmxuJRuCv7jyMjTPqr7lu/NK7TqPo4jK6AidIB9doBq6RXXUQAQ9omf0it6cJ+fFeXc+5q0FJ585RH/K+fwB7d6WXQ==</latexit>
(multi-index )

Kinetic eq.
15
¯nm↵(!, t) ⌘ Tr ⇢(t)a†
m↵(!)am↵(!)
Average occupation number of a single mode:
˙¯nm↵(!, t) =
⇣
⌦
m↵(!)
h
1 (±)e (! m⌦)
i⌘
¯nm↵(!, t)
+ ⌦
m↵(!)e (! m⌦)
obeys equation:
with (+) for bosons and (-) for fermions
) loc[!] =
✓
1
m⌦
!
◆
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e loc[!]!
⌘
"(k)
#(k)
= e (! m⌦)
<latexit sha1_base64="a4Bk21DTP3IfhhCnI42VDm6dK90=">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</latexit><latexit sha1_base64="a4Bk21DTP3IfhhCnI42VDm6dK90=">AAACYnicbVFNTxsxFPRuodC0hVCO7cEiqhQOoN2qUrlUQuXSW0FqACnert46b4MVf2xtLyha7Z/srScu/SH1JhHi60mWRzPvje1xUUnhfJL8jeIXa+svNzZf9V6/ebu13d95d+5MbTmOuJHGXhbgUAqNIy+8xMvKIqhC4kUxO+n0i2u0Thj9088rzBRMtSgFBx+ovD/HX80BK9BD3jCrqDS8HTOjcAoZXe4tZfi7FteUlRZ4w6agFOSsrsBaczOc7bd33MTc6DuWfqWdOV24D5deB4qyHx3Yb/P+IDlMFkWfgnQFBmRVp3n/T/DntULtuQTnxmlS+awB6wWX2PZY7bACPoMpjgPUoNBlzSKiln4MzISWxoalPV2w9ycaUM7NVRE6Ffgr91jryOe0ce3Lo6wRuqo9ar48qKwl9YZ2edOJsMi9nAcA3IpwV8qvIATpw6/0Qgjp4yc/BeefDtOAzz4Pjr+t4tgk78keGZKUfCHH5Ds5JSPCyW20Hm1F29G/uBfvxLvL1jhazeySBxV/+A82jLdd</latexit><latexit sha1_base64="a4Bk21DTP3IfhhCnI42VDm6dK90=">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</latexit><latexit sha1_base64="a4Bk21DTP3IfhhCnI42VDm6dK90=">AAACYnicbVFNTxsxFPRuodC0hVCO7cEiqhQOoN2qUrlUQuXSW0FqACnert46b4MVf2xtLyha7Z/srScu/SH1JhHi60mWRzPvje1xUUnhfJL8jeIXa+svNzZf9V6/ebu13d95d+5MbTmOuJHGXhbgUAqNIy+8xMvKIqhC4kUxO+n0i2u0Thj9088rzBRMtSgFBx+ovD/HX80BK9BD3jCrqDS8HTOjcAoZXe4tZfi7FteUlRZ4w6agFOSsrsBaczOc7bd33MTc6DuWfqWdOV24D5deB4qyHx3Yb/P+IDlMFkWfgnQFBmRVp3n/T/DntULtuQTnxmlS+awB6wWX2PZY7bACPoMpjgPUoNBlzSKiln4MzISWxoalPV2w9ycaUM7NVRE6Ffgr91jryOe0ce3Lo6wRuqo9ar48qKwl9YZ2edOJsMi9nAcA3IpwV8qvIATpw6/0Qgjp4yc/BeefDtOAzz4Pjr+t4tgk78keGZKUfCHH5Ds5JSPCyW20Hm1F29G/uBfvxLvL1jhazeySBxV/+A82jLdd</latexit>
Local temperature:

Fermions
16
¯nm↵(!, t) = exp
n
⌦
m↵(!)
h
1 + e (! m⌦)
i
t
o
¯nm↵(!, 0)
+
h
1 exp
n
⌦
m↵(!)
h
1 + e (! m⌦)
i
t
oi
¯nm↵(!, 1)
¯nm↵(!, 1) =
h
e (! m⌦)
+ 1
i 1
with asymptotic
Fermi-Dirac distribution with inv. temp. β and
chem. pot. mΩ

Bosons
17
¯nm↵(!, t) = exp
n
⌦
m↵(!)
h
1 e (! m⌦)
i
t
o
¯nm↵(!, 0)
+
h
1 exp
n
⌦
m↵(!)
h
1 e (! m⌦)
i
t
oi
¯nm↵(!, 1)
For ! > m⌦:
¯nm↵(!, 1) =
h
e (! m⌦)
1
i 1
with asymptotic
Bose-Einstein distribution with inv. temp. β and
chem. pot. mΩ

Superradiance
• Bosonic modes with ω < mΩ are unstable
• Occupation numbers grow exponentially with time
• Knowing ρ(t), we may compute entropy
production, heat current, and work
• Ω appears only through shift:
• Effective leads to population
inversion for modes with ω < mΩ
18
⌦
kk(x) = 0
kk(x + m⌦)<latexit sha1_base64="cTX+IqUXMjqVGSuxwM/eKXb3qow=">AAACGXicbVBNSwMxFMzWr1q/qh69BIvQIpRdEdSDUPTizQrWFrpreZumbWiyuyRZaVn6O7z4V7x4UPGoJ/+NabsHbR0IDDPzeHnjR5wpbdvfVmZhcWl5JbuaW1vf2NzKb+/cqTCWhNZIyEPZ8EFRzgJa00xz2ogkBeFzWvf7l2O//kClYmFwq4cR9QR0A9ZhBLSRWnnH7YIQcO9eC9qFVtLvj4qDEj7HqW5PJFwcHIpppNTKF+yyPQGeJ05KCihFtZX/dNshiQUNNOGgVNOxI+0lIDUjnI5ybqxoBKQPXdo0NABBlZdMThvhA6O0cSeU5gUaT9TfEwkIpYbCN0kBuqdmvbH4n9eMdefUS1gQxZoGZLqoE3OsQzzuCbeZpETzoSFAJDN/xaQHEog2beZMCc7syfOkdlQ+Kzs3x4XKRdpGFu2hfVREDjpBFXSFqqiGCHpEz+gVvVlP1ov1bn1MoxkrndlFf2B9/QDve5++</latexit><latexit sha1_base64="cTX+IqUXMjqVGSuxwM/eKXb3qow=">AAACGXicbVBNSwMxFMzWr1q/qh69BIvQIpRdEdSDUPTizQrWFrpreZumbWiyuyRZaVn6O7z4V7x4UPGoJ/+NabsHbR0IDDPzeHnjR5wpbdvfVmZhcWl5JbuaW1vf2NzKb+/cqTCWhNZIyEPZ8EFRzgJa00xz2ogkBeFzWvf7l2O//kClYmFwq4cR9QR0A9ZhBLSRWnnH7YIQcO9eC9qFVtLvj4qDEj7HqW5PJFwcHIpppNTKF+yyPQGeJ05KCihFtZX/dNshiQUNNOGgVNOxI+0lIDUjnI5ybqxoBKQPXdo0NABBlZdMThvhA6O0cSeU5gUaT9TfEwkIpYbCN0kBuqdmvbH4n9eMdefUS1gQxZoGZLqoE3OsQzzuCbeZpETzoSFAJDN/xaQHEog2beZMCc7syfOkdlQ+Kzs3x4XKRdpGFu2hfVREDjpBFXSFqqiGCHpEz+gVvVlP1ov1bn1MoxkrndlFf2B9/QDve5++</latexit><latexit sha1_base64="cTX+IqUXMjqVGSuxwM/eKXb3qow=">AAACGXicbVBNSwMxFMzWr1q/qh69BIvQIpRdEdSDUPTizQrWFrpreZumbWiyuyRZaVn6O7z4V7x4UPGoJ/+NabsHbR0IDDPzeHnjR5wpbdvfVmZhcWl5JbuaW1vf2NzKb+/cqTCWhNZIyEPZ8EFRzgJa00xz2ogkBeFzWvf7l2O//kClYmFwq4cR9QR0A9ZhBLSRWnnH7YIQcO9eC9qFVtLvj4qDEj7HqW5PJFwcHIpppNTKF+yyPQGeJ05KCihFtZX/dNshiQUNNOGgVNOxI+0lIDUjnI5ybqxoBKQPXdo0NABBlZdMThvhA6O0cSeU5gUaT9TfEwkIpYbCN0kBuqdmvbH4n9eMdefUS1gQxZoGZLqoE3OsQzzuCbeZpETzoSFAJDN/xaQHEog2beZMCc7syfOkdlQ+Kzs3x4XKRdpGFu2hfVREDjpBFXSFqqiGCHpEz+gVvVlP1ov1bn1MoxkrndlFf2B9/QDve5++</latexit><latexit sha1_base64="cTX+IqUXMjqVGSuxwM/eKXb3qow=">AAACGXicbVBNSwMxFMzWr1q/qh69BIvQIpRdEdSDUPTizQrWFrpreZumbWiyuyRZaVn6O7z4V7x4UPGoJ/+NabsHbR0IDDPzeHnjR5wpbdvfVmZhcWl5JbuaW1vf2NzKb+/cqTCWhNZIyEPZ8EFRzgJa00xz2ogkBeFzWvf7l2O//kClYmFwq4cR9QR0A9ZhBLSRWnnH7YIQcO9eC9qFVtLvj4qDEj7HqW5PJFwcHIpppNTKF+yyPQGeJ05KCihFtZX/dNshiQUNNOGgVNOxI+0lIDUjnI5ybqxoBKQPXdo0NABBlZdMThvhA6O0cSeU5gUaT9TfEwkIpYbCN0kBuqdmvbH4n9eMdefUS1gQxZoGZLqoE3OsQzzuCbeZpETzoSFAJDN/xaQHEog2beZMCc7syfOkdlQ+Kzs3x4XKRdpGFu2hfVREDjpBFXSFqqiGCHpEz+gVvVlP1ov1bn1MoxkrndlFf2B9/QDve5++</latexit>
He↵
b = H0
b ⌦Lz
b<latexit sha1_base64="ebFkieLLYS1L/KftWLKDk95t5nM=">AAACHnicbZDLSgMxFIYzXmu9VV26CRbBjWVGxMtCKLrpQrCCtYW2Dpn0TBuazAxJRqhD38SNr+LGhYrgSt/GtB28tB4I+fn+c0jO70WcKW3bn9bU9Mzs3HxmIbu4tLyymltbv1ZhLClUaMhDWfOIAs4CqGimOdQiCUR4HKpe92zgV29BKhYGV7oXQVOQdsB8Rok2yM0dlNykIQX2+jfDG3y/j0/wD7XxLm5cCGgTfP4N79xc3i7Yw8KTwklFHqVVdnPvjVZIYwGBppwoVXfsSDcTIjWjHPrZRqwgIrRL2lA3MiACVDMZ7tfH24a0sB9KcwKNh/T3REKEUj3hmU5BdEeNewP4n1ePtX/UTFgQxRoCOnrIjznWIR6EhVtMAtW8ZwShkpm/YtohklBtIs2aEJzxlSdFZa9wXHAu9/PF0zSNDNpEW2gHOegQFVEJlVEFUXSPHtEzerEerCfr1XobtU5Z6cwG+lPWxxf86aFK</latexit><latexit sha1_base64="ebFkieLLYS1L/KftWLKDk95t5nM=">AAACHnicbZDLSgMxFIYzXmu9VV26CRbBjWVGxMtCKLrpQrCCtYW2Dpn0TBuazAxJRqhD38SNr+LGhYrgSt/GtB28tB4I+fn+c0jO70WcKW3bn9bU9Mzs3HxmIbu4tLyymltbv1ZhLClUaMhDWfOIAs4CqGimOdQiCUR4HKpe92zgV29BKhYGV7oXQVOQdsB8Rok2yM0dlNykIQX2+jfDG3y/j0/wD7XxLm5cCGgTfP4N79xc3i7Yw8KTwklFHqVVdnPvjVZIYwGBppwoVXfsSDcTIjWjHPrZRqwgIrRL2lA3MiACVDMZ7tfH24a0sB9KcwKNh/T3REKEUj3hmU5BdEeNewP4n1ePtX/UTFgQxRoCOnrIjznWIR6EhVtMAtW8ZwShkpm/YtohklBtIs2aEJzxlSdFZa9wXHAu9/PF0zSNDNpEW2gHOegQFVEJlVEFUXSPHtEzerEerCfr1XobtU5Z6cwG+lPWxxf86aFK</latexit><latexit sha1_base64="ebFkieLLYS1L/KftWLKDk95t5nM=">AAACHnicbZDLSgMxFIYzXmu9VV26CRbBjWVGxMtCKLrpQrCCtYW2Dpn0TBuazAxJRqhD38SNr+LGhYrgSt/GtB28tB4I+fn+c0jO70WcKW3bn9bU9Mzs3HxmIbu4tLyymltbv1ZhLClUaMhDWfOIAs4CqGimOdQiCUR4HKpe92zgV29BKhYGV7oXQVOQdsB8Rok2yM0dlNykIQX2+jfDG3y/j0/wD7XxLm5cCGgTfP4N79xc3i7Yw8KTwklFHqVVdnPvjVZIYwGBppwoVXfsSDcTIjWjHPrZRqwgIrRL2lA3MiACVDMZ7tfH24a0sB9KcwKNh/T3REKEUj3hmU5BdEeNewP4n1ePtX/UTFgQxRoCOnrIjznWIR6EhVtMAtW8ZwShkpm/YtohklBtIs2aEJzxlSdFZa9wXHAu9/PF0zSNDNpEW2gHOegQFVEJlVEFUXSPHtEzerEerCfr1XobtU5Z6cwG+lPWxxf86aFK</latexit><latexit sha1_base64="ebFkieLLYS1L/KftWLKDk95t5nM=">AAACHnicbZDLSgMxFIYzXmu9VV26CRbBjWVGxMtCKLrpQrCCtYW2Dpn0TBuazAxJRqhD38SNr+LGhYrgSt/GtB28tB4I+fn+c0jO70WcKW3bn9bU9Mzs3HxmIbu4tLyymltbv1ZhLClUaMhDWfOIAs4CqGimOdQiCUR4HKpe92zgV29BKhYGV7oXQVOQdsB8Rok2yM0dlNykIQX2+jfDG3y/j0/wD7XxLm5cCGgTfP4N79xc3i7Yw8KTwklFHqVVdnPvjVZIYwGBppwoVXfsSDcTIjWjHPrZRqwgIrRL2lA3MiACVDMZ7tfH24a0sB9KcwKNh/T3REKEUj3hmU5BdEeNewP4n1ePtX/UTFgQxRoCOnrIjznWIR6EhVtMAtW8ZwShkpm/YtohklBtIs2aEJzxlSdFZa9wXHAu9/PF0zSNDNpEW2gHOegQFVEJlVEFUXSPHtEzerEerCfr1XobtU5Z6cwG+lPWxxf86aFK</latexit>

Superradiance, cont.
• Independent birth-death process for each mode k of
quantum ﬁeld coupled to heat bath:
• Stimulated emission (+) of bosons gives positive
feedback
• Pauli exclusion (-) makes feedback negative
• Decay γ↓(k) and pumping γ↑(k) rates obey Fermi
golden rule
See: Alicki, Int. J. Theor. Phys. 16, 351 (1977);
Alicki & Lendi, Quantum Dynamical Semigroups & Applications, 2nd ed., (2007), I, 1.3
19
˙Pn(k; t) = #(k)(n + 1)Pn+1(k; t) + "(k) [1 ± (n 1)] Pn 1(k; t)
[ #(k)n + "(k)(1 ± n)] Pn(k; t)

Van de Graaf
20
I1
I2 I2
I3
I4
V1 V2
I6
Figure 4.18.
Currents and potentials at the nodes of a
network.
machine, too, there is a downward-directed electric field. Here charge
carriers can be moved against the field if they are stuck to a nonconduct-
ing belt. They are stuck so tightly that they can’t slide backward along the
belt in the generally downward electric field. (They can still be removed
from the belt by a much stronger field localized at the brush in the termi-
nal. We need not consider here the means for putting charge on and off
the belt near the pulleys.) The energy needed to pull the belt is supplied
from elsewhere – usually by an electric motor connected to a power line,
but it could be a gasoline engine, or even a person turning a crank. This
Van de Graaff generator is in effect a battery with an electromotive force,
under these conditions, of V0 volts.
In ordinary batteries it is chemical energy that makes the charge car-
riers move through a region where the electric field opposes their motion.
That is, a positive charge carrier may move to a place of higher electric
potential if by so doing it can engage in a chemical reaction that will
+++++
+
+ + + + +
+
+++++
++
+–
–
–
–
–
–
–
–
–
–
+
+
+
+
+
+
+
+
+
+
V = V0
V = 0
R
I
EEEE
Figure 4.19.
In the Van de Graaff generator, charge carriers
are mechanically transported in a direction
opposite to that in which the electric field would
move them.
yield more energy than it costs to climb the electrical hill.
To see how this works, let us examine one particular voltaic cell.
Voltaic cell is the generic name for a chemical source of electromotive
force. In the experiments of Galvani around 1790 the famous twitching
frogs’ legs had signaled the chemical production of electric current. It
was Volta who proved that the source was not “animal electricity,” as
Galvani maintained, but the contact of dissimilar metals in the circuit.
Volta went on to construct the first battery, a stack of elementary cells,
each of which consisted of a zinc disk and a silver disk separated by
cardboard moistened with brine. The battery that powers your flashlight
comes in a tidier package, but the principle of operation is the same.
Several kinds of voltaic cells are in use, differing in their chemistry but
having common features: two electrodes of different material immersed
in an ionized fluid, or electrolyte.
As an example, we’ll describe the lead–sulfuric acid cell which is
the basic element of the automobile battery. This cell has the important
property that its operation is readily reversible. With a storage battery
made of such cells, which can be charged and discharged repeatedly,
energy can be stored and recovered electrically.
A fully charged lead–sulfuric acid cell has positive plates that hold
“Charge carriers can be moved
against the field if they are stuck
to a nonconducting belt. They are
stuck so tightly that they can’t
slide backward along the belt in
the generally downward electric
field […] We need not consider
here the means for putting charge
on and off the belt.”
– Purcell, Electricity & Magnetism,
3rd ed., (2013 [1963]), sec. 4.9

Minimal tribo-generator
21
E =
I
C
E · ds =
d
dt
Z
B · da = 0
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Faraday-Maxwell:
See: Alicki & AJ, arXiv:1904.11997 [cond-mat.mes-hall]

Open system
22
Hx
0 =
X
,m
!x( , m)c†
x( , m)cx( , m) for x = a, b
<latexit sha1_base64="kQ/BlU63nFn4l1uuEmR5vpeW+58=">AAACVnicbVFNSyNBEO0Zv+PX7HrcS2MQFCTMqKAXQdaLRxeMCpk41HQqk8bu6aG7RxKG+CN3L+tP8SL2xCAaLWh49V4VVfU6LQQ3NgyfPH9ufmFxaXmlsbq2vrEZ/Ph5bVSpGbaZEkrfpmBQ8BzblluBt4VGkKnAm/T+vNZvHlAbrvIrOyqwKyHLeZ8zsI5KAnmRhHdDekpjU8qkig3PJOxTOaaxkphBMqS77+QeZcnwLu5BlqGe5T+mj480HqRqWPWVHtdZPQH20yRohq1wEvQriKagSaZxmQR/455ipcTcMgHGdKKwsN0KtOVM4LgRlwYLYPeQYcfBHCSabjWxZUx3HNOjbgX3cksn7MeOCqQxI5m6Sgl2YGa1mvxO65S2f9KteF6UFnP2NqhfCmoVrT2mPa6RWTFyAJjmblfKBqCBWfcTDWdCNHvyV3B90IoOWwd/jppnv6d2LJNfZJvskogckzNyQS5JmzDyjzx7vjfn/fde/AV/6a3U96Y9W+RT+MErik6wzw==</latexit>
HX
0 =
X
,m
!X(, m)c†
X(, m)cX(, m) for X = A, B
<latexit sha1_base64="q6kNbSwaOwk6Df9i4586EKW4+pk=">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</latexit>
At rest:
Ha
⌦ =
X
,m
[!a( , m) m⌦] c†
a( , m)ca( , m)
<latexit sha1_base64="uZKCFXIOZoL29DeI7AQuChhtFJI=">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</latexit>
HA
⌦ =
X
,m
[!A(, m) m⌦] c†
A(, m)cA(, m)
<latexit sha1_base64="nvfZrInQD0+SVrwNRRseaNFl0T0=">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</latexit>
Motion:

Open system, cont.
23
Surface-bulk tunneling:
Hx
X =
X
, ,m
gx
X(, , m)c†
X(, m)cx( , m) + h.c.
<latexit sha1_base64="CXDR5QPgo0Hsemns6AJBxoz1xvw=">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</latexit>
Htot = Ha
⌦ + Hb
0 + HA
⌦ + HB
0 + Ha
A + Ha
B + Hb
A + Hb
B<latexit sha1_base64="AgcH3XpTbAgCTWGoWQDNsinDLJE=">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</latexit>
Total Hamiltonian:

Kinetic eqs.
24
KMS:
aA
" ( , m) = e (!a( ,m) µA) aA
# ( , m)<latexit sha1_base64="3Va4J5CrTMMhOxaKX5ARlyEyejo=">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</latexit><latexit sha1_base64="3Va4J5CrTMMhOxaKX5ARlyEyejo=">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</latexit><latexit sha1_base64="3Va4J5CrTMMhOxaKX5ARlyEyejo=">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</latexit><latexit sha1_base64="3Va4J5CrTMMhOxaKX5ARlyEyejo=">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</latexit>
aB
" ( , m) = e (!a( ,m) m⌦ µB) aB
# ( , m)<latexit sha1_base64="B09SoTq2+Kp9xeoyuikfYhDoMxo=">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</latexit><latexit sha1_base64="B09SoTq2+Kp9xeoyuikfYhDoMxo=">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</latexit><latexit sha1_base64="B09SoTq2+Kp9xeoyuikfYhDoMxo=">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</latexit><latexit sha1_base64="B09SoTq2+Kp9xeoyuikfYhDoMxo=">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</latexit>
Steady state: na = ¯na ⌘ aA
" + aB
" / a
= const.<latexit sha1_base64="N2qT/SrSRfKhk5J5zjgCon1iodE=">AAACVHicbZFLaxsxFIU1k6RJnTZ102U3IiaQUnBnSiHdFJJ00S4TiJOA5Qx3ZI0trMdUupPGDPMj00Whv6SbLqpxvMjrguDjnHv1OMpLJT0myZ8oXllde7a+8byz+eLl1qvu6+0zbyvHxYBbZd1FDl4oacQAJSpxUToBOlfiPJ99bf3zK+G8tOYU56UYaZgYWUgOGKSsOzMZ0C+U5eBoi0z8qOQVZUoUuEfZBLSGjFUlOGd/XtZw2ND3T8hHDWVOTqb4jn6g7FtrXy72neb2uubWeOw3WbeX9JNF0ceQLqFHlnWcdX+xseWVFga5Au+HaVLiqAaHkivRdFjlRQl8BhMxDGhACz+qF6E0dDcoY1pYF5ZBulDvTtSgvZ/rPHRqwKl/6LXiU96wwuLzqJamrFAYfntQUSmKlrYJ07F0gqOaBwDuZLgr5VNwwDH8QyeEkD588mM4+9hPA5986h0cLePYIG/JDtkjKdknB+Q7OSYDwskN+RuRKIp+R//ilXjttjWOljNvyL2Kt/4DhMayNA==</latexit><latexit sha1_base64="N2qT/SrSRfKhk5J5zjgCon1iodE=">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</latexit><latexit sha1_base64="N2qT/SrSRfKhk5J5zjgCon1iodE=">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</latexit><latexit sha1_base64="N2qT/SrSRfKhk5J5zjgCon1iodE=">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</latexit>
b
⌘ bA
" + bA
# + bB
" + bB
#<latexit sha1_base64="BR0wAlzOTcOS1AauksB8T0HVA1E=">AAACVnicdVFNSwMxFMxu1db6terRS7AIglB2RdBjrQc9VrAqdNvyNk3bYJJdk6xSlv5JvehP8SKm7eJX9UFgMjPvJZlECWfa+P6r4xYWFpeKpeXyyura+oa3uXWt41QR2iQxj9VtBJpyJmnTMMPpbaIoiIjTm+jubKLfPFClWSyvzCihbQEDyfqMgLFU1xPhOQgBnQiH9D5lDzgcTPbdME1Aqfixk0WnY3zwSffiR/mn8OWv/+evj7texa/608LzIMhBBeXV6HpPdgJJBZWGcNC6FfiJaWegDCOcjsthqmkC5A4GtGWhBEF1O5vGMsZ7lunhfqzskgZP2e8dGQitRyKyTgFmqH9rE/IvrZWa/kk7YzJJDZVkdlA/5djEeJIx7jFFieEjC4AoZu+KyRAUEGN/omxDCH4/eR5cH1YDiy+PKrV6HkcJ7aBdtI8CdIxq6AI1UBMR9IzeHNcpOC/Ou7voFmdW18l7ttGPcr0P9pq1LA==</latexit><latexit sha1_base64="BR0wAlzOTcOS1AauksB8T0HVA1E=">AAACVnicdVFNSwMxFMxu1db6terRS7AIglB2RdBjrQc9VrAqdNvyNk3bYJJdk6xSlv5JvehP8SKm7eJX9UFgMjPvJZlECWfa+P6r4xYWFpeKpeXyyura+oa3uXWt41QR2iQxj9VtBJpyJmnTMMPpbaIoiIjTm+jubKLfPFClWSyvzCihbQEDyfqMgLFU1xPhOQgBnQiH9D5lDzgcTPbdME1Aqfixk0WnY3zwSffiR/mn8OWv/+evj7texa/608LzIMhBBeXV6HpPdgJJBZWGcNC6FfiJaWegDCOcjsthqmkC5A4GtGWhBEF1O5vGMsZ7lunhfqzskgZP2e8dGQitRyKyTgFmqH9rE/IvrZWa/kk7YzJJDZVkdlA/5djEeJIx7jFFieEjC4AoZu+KyRAUEGN/omxDCH4/eR5cH1YDiy+PKrV6HkcJ7aBdtI8CdIxq6AI1UBMR9IzeHNcpOC/Ou7voFmdW18l7ttGPcr0P9pq1LA==</latexit><latexit sha1_base64="BR0wAlzOTcOS1AauksB8T0HVA1E=">AAACVnicdVFNSwMxFMxu1db6terRS7AIglB2RdBjrQc9VrAqdNvyNk3bYJJdk6xSlv5JvehP8SKm7eJX9UFgMjPvJZlECWfa+P6r4xYWFpeKpeXyyura+oa3uXWt41QR2iQxj9VtBJpyJmnTMMPpbaIoiIjTm+jubKLfPFClWSyvzCihbQEDyfqMgLFU1xPhOQgBnQiH9D5lDzgcTPbdME1Aqfixk0WnY3zwSffiR/mn8OWv/+evj7texa/608LzIMhBBeXV6HpPdgJJBZWGcNC6FfiJaWegDCOcjsthqmkC5A4GtGWhBEF1O5vGMsZ7lunhfqzskgZP2e8dGQitRyKyTgFmqH9rE/IvrZWa/kk7YzJJDZVkdlA/5djEeJIx7jFFieEjC4AoZu+KyRAUEGN/omxDCH4/eR5cH1YDiy+PKrV6HkcJ7aBdtI8CdIxq6AI1UBMR9IzeHNcpOC/Ou7voFmdW18l7ttGPcr0P9pq1LA==</latexit><latexit sha1_base64="BR0wAlzOTcOS1AauksB8T0HVA1E=">AAACVnicdVFNSwMxFMxu1db6terRS7AIglB2RdBjrQc9VrAqdNvyNk3bYJJdk6xSlv5JvehP8SKm7eJX9UFgMjPvJZlECWfa+P6r4xYWFpeKpeXyyura+oa3uXWt41QR2iQxj9VtBJpyJmnTMMPpbaIoiIjTm+jubKLfPFClWSyvzCihbQEDyfqMgLFU1xPhOQgBnQiH9D5lDzgcTPbdME1Aqfixk0WnY3zwSffiR/mn8OWv/+evj7texa/608LzIMhBBeXV6HpPdgJJBZWGcNC6FfiJaWegDCOcjsthqmkC5A4GtGWhBEF1O5vGMsZ7lunhfqzskgZP2e8dGQitRyKyTgFmqH9rE/IvrZWa/kk7YzJJDZVkdlA/5djEeJIx7jFFieEjC4AoZu+KyRAUEGN/omxDCH4/eR5cH1YDiy+PKrV6HkcJ7aBdtI8CdIxq6AI1UBMR9IzeHNcpOC/Ou7voFmdW18l7ttGPcr0P9pq1LA==</latexit>
a
⌘ aA
" + aA
# + aB
" + aB
#<latexit sha1_base64="g8rqO/jrJRe3RIMGcNKVmAye1LU=">AAACVnicdVFNSwMxFMxu1db6terRS7AIglB2RdBjrQc9VrAqdNvyNk3bYJJdk6xSlv5JvehP8SKm7eJX9UFgMjPvJZlECWfa+P6r4xYWFpeKpeXyyura+oa3uXWt41QR2iQxj9VtBJpyJmnTMMPpbaIoiIjTm+jubKLfPFClWSyvzCihbQEDyfqMgLFU1xPhOQgBHcAhvU/ZAw4Hk303TBNQKn7sZHA6xgefdC9+lH8KX/76f/76uOtV/Ko/LTwPghxUUF6NrvdkJ5BUUGkIB61bgZ+YdgbKMMLpuBymmiZA7mBAWxZKEFS3s2ksY7xnmR7ux8ouafCU/d6RgdB6JCLrFGCG+rc2If/SWqnpn7QzJpPUUElmB/VTjk2MJxnjHlOUGD6yAIhi9q6YDEEBMfYnyjaE4PeT58H1YTWw+PKoUqvncZTQDtpF+yhAx6iGLlADNRFBz+jNcZ2C8+K8u4tucWZ1nbxnG/0o1/sA7ge1Jw==</latexit><latexit sha1_base64="g8rqO/jrJRe3RIMGcNKVmAye1LU=">AAACVnicdVFNSwMxFMxu1db6terRS7AIglB2RdBjrQc9VrAqdNvyNk3bYJJdk6xSlv5JvehP8SKm7eJX9UFgMjPvJZlECWfa+P6r4xYWFpeKpeXyyura+oa3uXWt41QR2iQxj9VtBJpyJmnTMMPpbaIoiIjTm+jubKLfPFClWSyvzCihbQEDyfqMgLFU1xPhOQgBHcAhvU/ZAw4Hk303TBNQKn7sZHA6xgefdC9+lH8KX/76f/76uOtV/Ko/LTwPghxUUF6NrvdkJ5BUUGkIB61bgZ+YdgbKMMLpuBymmiZA7mBAWxZKEFS3s2ksY7xnmR7ux8ouafCU/d6RgdB6JCLrFGCG+rc2If/SWqnpn7QzJpPUUElmB/VTjk2MJxnjHlOUGD6yAIhi9q6YDEEBMfYnyjaE4PeT58H1YTWw+PKoUqvncZTQDtpF+yhAx6iGLlADNRFBz+jNcZ2C8+K8u4tucWZ1nbxnG/0o1/sA7ge1Jw==</latexit><latexit sha1_base64="g8rqO/jrJRe3RIMGcNKVmAye1LU=">AAACVnicdVFNSwMxFMxu1db6terRS7AIglB2RdBjrQc9VrAqdNvyNk3bYJJdk6xSlv5JvehP8SKm7eJX9UFgMjPvJZlECWfa+P6r4xYWFpeKpeXyyura+oa3uXWt41QR2iQxj9VtBJpyJmnTMMPpbaIoiIjTm+jubKLfPFClWSyvzCihbQEDyfqMgLFU1xPhOQgBHcAhvU/ZAw4Hk303TBNQKn7sZHA6xgefdC9+lH8KX/76f/76uOtV/Ko/LTwPghxUUF6NrvdkJ5BUUGkIB61bgZ+YdgbKMMLpuBymmiZA7mBAWxZKEFS3s2ksY7xnmR7ux8ouafCU/d6RgdB6JCLrFGCG+rc2If/SWqnpn7QzJpPUUElmB/VTjk2MJxnjHlOUGD6yAIhi9q6YDEEBMfYnyjaE4PeT58H1YTWw+PKoUqvncZTQDtpF+yhAx6iGLlADNRFBz+jNcZ2C8+K8u4tucWZ1nbxnG/0o1/sA7ge1Jw==</latexit><latexit sha1_base64="g8rqO/jrJRe3RIMGcNKVmAye1LU=">AAACVnicdVFNSwMxFMxu1db6terRS7AIglB2RdBjrQc9VrAqdNvyNk3bYJJdk6xSlv5JvehP8SKm7eJX9UFgMjPvJZlECWfa+P6r4xYWFpeKpeXyyura+oa3uXWt41QR2iQxj9VtBJpyJmnTMMPpbaIoiIjTm+jubKLfPFClWSyvzCihbQEDyfqMgLFU1xPhOQgBHcAhvU/ZAw4Hk303TBNQKn7sZHA6xgefdC9+lH8KX/76f/76uOtV/Ko/LTwPghxUUF6NrvdkJ5BUUGkIB61bgZ+YdgbKMMLpuBymmiZA7mBAWxZKEFS3s2ksY7xnmR7ux8ouafCU/d6RgdB6JCLrFGCG+rc2If/SWqnpn7QzJpPUUElmB/VTjk2MJxnjHlOUGD6yAIhi9q6YDEEBMfYnyjaE4PeT58H1YTWw+PKoUqvncZTQDtpF+yhAx6iGLlADNRFBz+jNcZ2C8+K8u4tucWZ1nbxnG/0o1/sA7ge1Jw==</latexit>
;
nb = ¯nb ⌘ bA
" + bB
" / b
= const.<latexit sha1_base64="Z2dbtewpcIQKWng+tyHxIAnVzlU=">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</latexit><latexit sha1_base64="Z2dbtewpcIQKWng+tyHxIAnVzlU=">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</latexit><latexit sha1_base64="Z2dbtewpcIQKWng+tyHxIAnVzlU=">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</latexit><latexit sha1_base64="Z2dbtewpcIQKWng+tyHxIAnVzlU=">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</latexit>
Kinetic eq.: ˙nx = xA
" + xB
"
xA
# + xB
# + xA
" + xB
" nx<latexit sha1_base64="5Q+KceZNkJNfxtzMfcclaKs9Lfo=">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</latexit>

Tribocurrents I
25
ja = aA
"
aA
# + aA
" ¯na =
aA
"
aB
# 1 e m⌦
a<latexit sha1_base64="O1N75Vni9nF+0Fl14Y7qpBVVieE=">AAACsHicbVFNb9QwEHXCV1m+FjhysVghFaFWmwoJLkilHOBGEWy70jobTbyTrFvbiewJ1SrK7+POjX+D94OqtDuSpaf3ZuaNZ/JaK0/D4Z8ovnX7zt17O/d7Dx4+evyk//TZia8aJ3EkK125cQ4etbI4IkUax7VDMLnG0/z801I//YnOq8r+oEWNqYHSqkJJoEBl/V9nGfAPXJRgDGSiqcG56mLawseO73GhsaDdS3VWXdgr+pvtZcKpck6vucjBcbtuXziQ7bbsba2Pun/GSZgBp63IkYAbLr4aLOHSoWvF52X1NFBZfzDcH66C3wTJBgzYJo6z/u/gKRuDlqQG7yfJsKa0BUdKaux6ovFYgzyHEicBWjDo03a18I6/CsyMF5ULzxJfsVcrWjDeL0weMg3Q3F/XluQ2bdJQ8T5tla0bQivXRkWjOVV8eT0+Uw4l6UUAIJ0Ks3I5h7BcCjfuhSUk1798E5wc7CcBf3s7ODzarGOHvWAv2S5L2Dt2yL6wYzZiMtqLvkciSuODeBxnMaxT42hT85z9F/HZX0TP1FM=</latexit><latexit sha1_base64="O1N75Vni9nF+0Fl14Y7qpBVVieE=">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</latexit><latexit sha1_base64="O1N75Vni9nF+0Fl14Y7qpBVVieE=">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</latexit><latexit sha1_base64="O1N75Vni9nF+0Fl14Y7qpBVVieE=">AAACsHicbVFNb9QwEHXCV1m+FjhysVghFaFWmwoJLkilHOBGEWy70jobTbyTrFvbiewJ1SrK7+POjX+D94OqtDuSpaf3ZuaNZ/JaK0/D4Z8ovnX7zt17O/d7Dx4+evyk//TZia8aJ3EkK125cQ4etbI4IkUax7VDMLnG0/z801I//YnOq8r+oEWNqYHSqkJJoEBl/V9nGfAPXJRgDGSiqcG56mLawseO73GhsaDdS3VWXdgr+pvtZcKpck6vucjBcbtuXziQ7bbsba2Pun/GSZgBp63IkYAbLr4aLOHSoWvF52X1NFBZfzDcH66C3wTJBgzYJo6z/u/gKRuDlqQG7yfJsKa0BUdKaux6ovFYgzyHEicBWjDo03a18I6/CsyMF5ULzxJfsVcrWjDeL0weMg3Q3F/XluQ2bdJQ8T5tla0bQivXRkWjOVV8eT0+Uw4l6UUAIJ0Ks3I5h7BcCjfuhSUk1798E5wc7CcBf3s7ODzarGOHvWAv2S5L2Dt2yL6wYzZiMtqLvkciSuODeBxnMaxT42hT85z9F/HZX0TP1FM=</latexit>
jb = bB
# + bB
" ¯nb
aB
" =
bA
#
bB
" e m⌦
1
b<latexit sha1_base64="fo2Z9r0H1DsXvjq0nQqfMA8FSOU=">AAACsHicdVFNbxMxEPUuHy3hK9AjF4sIqQil2q2QygWptAe4UQRpI8Wb1diZ3bi1vSvbSxWt9vdx58a/wUkDKk0ZydLTmzcffsNrJZ1Pkl9RfOfuvftb2w96Dx89fvK0/+z5qasaK3AkKlXZMQeHShoceekVjmuLoLnCM35xvMyffUfrZGW++UWNmYbSyEIK8IHK+z/Oc07fU6aw8LuUlaA15GxWXRqwtrqctvyoo2/+Jpr6Gs2sLOf+NWUcLDWhz3BTB0EX2hcWRHtb9w/df3qvFsJpO2QcPVBN2WeNJXRhSPpncteyj8vaKe/y/iDZS1ZBN0G6BgOyjpO8/zPsIRqNxgsFzk3SpPZZC9ZLobDrscZhDeICSpwEaECjy9qV4R19FZgZLSobnvF0xV6vaEE7t9A8KDX4ubuZW5K35SaNL95lrTR149GIq0FFo6iv6PJ6dCYtCq8WAYCwMuxKxRyCuT7cuBdMSG9+eROc7u+lAX95Ozg8WtuxTV6Ql2SXpOSAHJJP5ISMiIiG0deIRVm8H4/jPIYraRyta3bIPxGf/wZb6NRq</latexit><latexit sha1_base64="fo2Z9r0H1DsXvjq0nQqfMA8FSOU=">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</latexit><latexit sha1_base64="fo2Z9r0H1DsXvjq0nQqfMA8FSOU=">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</latexit><latexit sha1_base64="fo2Z9r0H1DsXvjq0nQqfMA8FSOU=">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</latexit>
J = e
0
@
X
,m
ja( , m) +
X
0,m
jb( 0
, m)
1
A
<latexit sha1_base64="GuWLWot0gav1GQI+uw4Ju1Zn9n4=">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</latexit><latexit sha1_base64="GuWLWot0gav1GQI+uw4Ju1Zn9n4=">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</latexit><latexit sha1_base64="GuWLWot0gav1GQI+uw4Ju1Zn9n4=">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</latexit><latexit sha1_base64="GuWLWot0gav1GQI+uw4Ju1Zn9n4=">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</latexit>
Net current:
nX(x) ⌘
1
e (x µX ) + 1<latexit sha1_base64="tk9EtFKwTBwk17uXiaFSKFKudI0=">AAACGXicbZDNSsNAFIUn/lv/oi7dDBahRSyJCLosunGpYG2hqWEyvWmHTiZxZlJaQl7Dja/ixoUiLnXl2zitXWjrgYGPc+/lzj1BwpnSjvNlzc0vLC4tr6wW1tY3Nrfs7Z1bFaeSQo3GPJaNgCjgTEBNM82hkUggUcChHvQuRvV6H6RisbjRwwRaEekIFjJKtLF82xF+ozQoYw/uU9bHXigJzdw8g7vMC0CT0gAfYS9K/UY5x4fYzX276FScsfAsuBMooomufPvDa8c0jUBoyolSTddJdCsjUjPKIS94qYKE0B7pQNOgIBGoVja+LMcHxmnjMJbmCY3H7u+JjERKDaPAdEZEd9V0bWT+V2umOjxrZUwkqQZBfxaFKcc6xqOYcJtJoJoPDRAqmfkrpl1iwtEmzIIJwZ0+eRZujyuu4euTYvV8EscK2kP7qIRcdIqq6BJdoRqi6AE9oRf0aj1az9ab9f7TOmdNZnbRH1mf37XWntI=</latexit><latexit sha1_base64="tk9EtFKwTBwk17uXiaFSKFKudI0=">AAACGXicbZDNSsNAFIUn/lv/oi7dDBahRSyJCLosunGpYG2hqWEyvWmHTiZxZlJaQl7Dja/ixoUiLnXl2zitXWjrgYGPc+/lzj1BwpnSjvNlzc0vLC4tr6wW1tY3Nrfs7Z1bFaeSQo3GPJaNgCjgTEBNM82hkUggUcChHvQuRvV6H6RisbjRwwRaEekIFjJKtLF82xF+ozQoYw/uU9bHXigJzdw8g7vMC0CT0gAfYS9K/UY5x4fYzX276FScsfAsuBMooomufPvDa8c0jUBoyolSTddJdCsjUjPKIS94qYKE0B7pQNOgIBGoVja+LMcHxmnjMJbmCY3H7u+JjERKDaPAdEZEd9V0bWT+V2umOjxrZUwkqQZBfxaFKcc6xqOYcJtJoJoPDRAqmfkrpl1iwtEmzIIJwZ0+eRZujyuu4euTYvV8EscK2kP7qIRcdIqq6BJdoRqi6AE9oRf0aj1az9ab9f7TOmdNZnbRH1mf37XWntI=</latexit><latexit sha1_base64="tk9EtFKwTBwk17uXiaFSKFKudI0=">AAACGXicbZDNSsNAFIUn/lv/oi7dDBahRSyJCLosunGpYG2hqWEyvWmHTiZxZlJaQl7Dja/ixoUiLnXl2zitXWjrgYGPc+/lzj1BwpnSjvNlzc0vLC4tr6wW1tY3Nrfs7Z1bFaeSQo3GPJaNgCjgTEBNM82hkUggUcChHvQuRvV6H6RisbjRwwRaEekIFjJKtLF82xF+ozQoYw/uU9bHXigJzdw8g7vMC0CT0gAfYS9K/UY5x4fYzX276FScsfAsuBMooomufPvDa8c0jUBoyolSTddJdCsjUjPKIS94qYKE0B7pQNOgIBGoVja+LMcHxmnjMJbmCY3H7u+JjERKDaPAdEZEd9V0bWT+V2umOjxrZUwkqQZBfxaFKcc6xqOYcJtJoJoPDRAqmfkrpl1iwtEmzIIJwZ0+eRZujyuu4euTYvV8EscK2kP7qIRcdIqq6BJdoRqi6AE9oRf0aj1az9ab9f7TOmdNZnbRH1mf37XWntI=</latexit><latexit sha1_base64="tk9EtFKwTBwk17uXiaFSKFKudI0=">AAACGXicbZDNSsNAFIUn/lv/oi7dDBahRSyJCLosunGpYG2hqWEyvWmHTiZxZlJaQl7Dja/ixoUiLnXl2zitXWjrgYGPc+/lzj1BwpnSjvNlzc0vLC4tr6wW1tY3Nrfs7Z1bFaeSQo3GPJaNgCjgTEBNM82hkUggUcChHvQuRvV6H6RisbjRwwRaEekIFjJKtLF82xF+ozQoYw/uU9bHXigJzdw8g7vMC0CT0gAfYS9K/UY5x4fYzX276FScsfAsuBMooomufPvDa8c0jUBoyolSTddJdCsjUjPKIS94qYKE0B7pQNOgIBGoVja+LMcHxmnjMJbmCY3H7u+JjERKDaPAdEZEd9V0bWT+V2umOjxrZUwkqQZBfxaFKcc6xqOYcJtJoJoPDRAqmfkrpl1iwtEmzIIJwZ0+eRZujyuu4euTYvV8EscK2kP7qIRcdIqq6BJdoRqi6AE9oRf0aj1az9ab9f7TOmdNZnbRH1mf37XWntI=</latexit>
Fermi-Dirac:
ja( , m) ⇠ aA
"
aB
# ⇠ nA(!a( , m))[1 nB(!a( , m) m⌦)]<latexit sha1_base64="9osm1RE23zXZzxnb/pJKEkixpbI=">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</latexit><latexit sha1_base64="9osm1RE23zXZzxnb/pJKEkixpbI=">AAACc3icbZFLaxsxEMe121fqNI2TQC85dIgTsCEOuyXQHBP30ltTqJOAd7PMyvJaiR6LpG0wi79AP15u/Ra99F75BXl0QPDn99doRjN5Kbh1UfQ7CF+8fPX6zdrbxvq7jfebza3tC6srQ1mfaqHNVY6WCa5Y33En2FVpGMpcsMv89svMv/zJjOVa/XCTkqUSC8VHnKLzKGv+usmwnVheSDwE2QEvJUBSoJSYJVWJxui76xrPpis21HdqRXvTRYLKztqJlqzAR691BhBD17u9NqxseFitCzL5NuPQgTRrtqKjaB7wXMRL0SLLOM+a974ZWkmmHBVo7SCOSpfWaByngk0bSWVZifQWCzbwUqFkNq3nM5vCgSdDGGnjj3Iwpw8zapTWTmTub0p0Y/vUm8H/eYPKjU7SmquyckzRRaFRJcBpmC0Ahtww6sTEC6SG+16BjtEgdX5NDT+E+OmXn4uLT0ex19+PW6e95TjWyC7ZI20Sk8/klHwl56RPKPkTfAg+BhD8DXfDvXB/cTUMljk75FGE3X+1lrnF</latexit><latexit sha1_base64="9osm1RE23zXZzxnb/pJKEkixpbI=">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</latexit><latexit sha1_base64="9osm1RE23zXZzxnb/pJKEkixpbI=">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</latexit>
For µ 1 ,<latexit sha1_base64="xOLRELguIb3A9CMK/rmnMVgH1FA=">AAAB/HicbZDNSsNAFIUn9a/Wv2iXbgaL4EJKIoIui25cVrC10IQymd6kQ2eSMDMRQmhfxY0LRdz6IO58G6dtFtp6YODj3Hu5d06Qcqa043xblbX1jc2t6nZtZ3dv/8A+POqqJJMUOjThiewFRAFnMXQ00xx6qQQiAg6Pwfh2Vn98AqlYEj/oPAVfkChmIaNEG2tg170ANMGeyLAXRdjF03M8sBtO05kLr4JbQgOVag/sL2+Y0ExArCknSvVdJ9V+QaRmlMOk5mUKUkLHJIK+wZgIUH4xP36CT40zxGEizYs1nru/JwoilMpFYDoF0SO1XJuZ/9X6mQ6v/YLFaaYhpotFYcaxTvAsCTxkEqjmuQFCJTO3YjoiklBt8qqZENzlL69C96LpGr6/bLRuyjiq6BidoDPkoivUQneojTqIohw9o1f0Zk2tF+vd+li0Vqxypo7+yPr8AV0ik0Q=</latexit><latexit sha1_base64="xOLRELguIb3A9CMK/rmnMVgH1FA=">AAAB/HicbZDNSsNAFIUn9a/Wv2iXbgaL4EJKIoIui25cVrC10IQymd6kQ2eSMDMRQmhfxY0LRdz6IO58G6dtFtp6YODj3Hu5d06Qcqa043xblbX1jc2t6nZtZ3dv/8A+POqqJJMUOjThiewFRAFnMXQ00xx6qQQiAg6Pwfh2Vn98AqlYEj/oPAVfkChmIaNEG2tg170ANMGeyLAXRdjF03M8sBtO05kLr4JbQgOVag/sL2+Y0ExArCknSvVdJ9V+QaRmlMOk5mUKUkLHJIK+wZgIUH4xP36CT40zxGEizYs1nru/JwoilMpFYDoF0SO1XJuZ/9X6mQ6v/YLFaaYhpotFYcaxTvAsCTxkEqjmuQFCJTO3YjoiklBt8qqZENzlL69C96LpGr6/bLRuyjiq6BidoDPkoivUQneojTqIohw9o1f0Zk2tF+vd+li0Vqxypo7+yPr8AV0ik0Q=</latexit><latexit sha1_base64="xOLRELguIb3A9CMK/rmnMVgH1FA=">AAAB/HicbZDNSsNAFIUn9a/Wv2iXbgaL4EJKIoIui25cVrC10IQymd6kQ2eSMDMRQmhfxY0LRdz6IO58G6dtFtp6YODj3Hu5d06Qcqa043xblbX1jc2t6nZtZ3dv/8A+POqqJJMUOjThiewFRAFnMXQ00xx6qQQiAg6Pwfh2Vn98AqlYEj/oPAVfkChmIaNEG2tg170ANMGeyLAXRdjF03M8sBtO05kLr4JbQgOVag/sL2+Y0ExArCknSvVdJ9V+QaRmlMOk5mUKUkLHJIK+wZgIUH4xP36CT40zxGEizYs1nru/JwoilMpFYDoF0SO1XJuZ/9X6mQ6v/YLFaaYhpotFYcaxTvAsCTxkEqjmuQFCJTO3YjoiklBt8qqZENzlL69C96LpGr6/bLRuyjiq6BidoDPkoivUQneojTqIohw9o1f0Zk2tF+vd+li0Vqxypo7+yPr8AV0ik0Q=</latexit><latexit sha1_base64="xOLRELguIb3A9CMK/rmnMVgH1FA=">AAAB/HicbZDNSsNAFIUn9a/Wv2iXbgaL4EJKIoIui25cVrC10IQymd6kQ2eSMDMRQmhfxY0LRdz6IO58G6dtFtp6YODj3Hu5d06Qcqa043xblbX1jc2t6nZtZ3dv/8A+POqqJJMUOjThiewFRAFnMXQ00xx6qQQiAg6Pwfh2Vn98AqlYEj/oPAVfkChmIaNEG2tg170ANMGeyLAXRdjF03M8sBtO05kLr4JbQgOVag/sL2+Y0ExArCknSvVdJ9V+QaRmlMOk5mUKUkLHJIK+wZgIUH4xP36CT40zxGEizYs1nru/JwoilMpFYDoF0SO1XJuZ/9X6mQ6v/YLFaaYhpotFYcaxTvAsCTxkEqjmuQFCJTO3YjoiklBt8qqZENzlL69C96LpGr6/bLRuyjiq6BidoDPkoivUQneojTqIohw9o1f0Zk2tF+vd+li0Vqxypo7+yPr8AV0ik0Q=</latexit>
aA
"
aB
# ⇠ [µB+m⌦, µA](!a( , m))<latexit sha1_base64="0cN/a3Wnj+riJNCueAp3xC2wkGE=">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</latexit>
Only surface modes satisfying m𝛺 < 𝜇A - 𝜇B contribute to ja

Tribocurrents II
26
ja = aA
"
aA
# + aA
" ¯na =
aA
"
aB
# 1 e m⌦
a<latexit sha1_base64="O1N75Vni9nF+0Fl14Y7qpBVVieE=">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</latexit><latexit sha1_base64="O1N75Vni9nF+0Fl14Y7qpBVVieE=">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</latexit><latexit sha1_base64="O1N75Vni9nF+0Fl14Y7qpBVVieE=">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</latexit><latexit sha1_base64="O1N75Vni9nF+0Fl14Y7qpBVVieE=">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</latexit>
jb = bB
# + bB
" ¯nb
aB
" =
bA
#
bB
" e m⌦
1
b<latexit sha1_base64="fo2Z9r0H1DsXvjq0nQqfMA8FSOU=">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</latexit><latexit sha1_base64="fo2Z9r0H1DsXvjq0nQqfMA8FSOU=">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</latexit><latexit sha1_base64="fo2Z9r0H1DsXvjq0nQqfMA8FSOU=">AAACsHicdVFNbxMxEPUuHy3hK9AjF4sIqQil2q2QygWptAe4UQRpI8Wb1diZ3bi1vSvbSxWt9vdx58a/wUkDKk0ZydLTmzcffsNrJZ1Pkl9RfOfuvftb2w96Dx89fvK0/+z5qasaK3AkKlXZMQeHShoceekVjmuLoLnCM35xvMyffUfrZGW++UWNmYbSyEIK8IHK+z/Oc07fU6aw8LuUlaA15GxWXRqwtrqctvyoo2/+Jpr6Gs2sLOf+NWUcLDWhz3BTB0EX2hcWRHtb9w/df3qvFsJpO2QcPVBN2WeNJXRhSPpncteyj8vaKe/y/iDZS1ZBN0G6BgOyjpO8/zPsIRqNxgsFzk3SpPZZC9ZLobDrscZhDeICSpwEaECjy9qV4R19FZgZLSobnvF0xV6vaEE7t9A8KDX4ubuZW5K35SaNL95lrTR149GIq0FFo6iv6PJ6dCYtCq8WAYCwMuxKxRyCuT7cuBdMSG9+eROc7u+lAX95Ozg8WtuxTV6Ql2SXpOSAHJJP5ISMiIiG0deIRVm8H4/jPIYraRyta3bIPxGf/wZb6NRq</latexit><latexit sha1_base64="fo2Z9r0H1DsXvjq0nQqfMA8FSOU=">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</latexit>
• Only surface modes satisfying m𝛺 > 𝜇A - 𝜇B contribute to
jb
• Sign of J depends on relative magnitudes of
and
• As charging increases |𝜇A - 𝜇B|, fewer modes contribute
to corresponding jx and more modes to the opposing jx
aA
"
aB
# / a
<latexit sha1_base64="Vho8AaUxIdcrMeaiKy3cgE0v5b4=">AAACHnicbZDLSgMxFIYzXmu9VV26CRbBVZ2pii5rXeiygr1Ab5xJ0zY0yQxJxlKGPokbX8WNC0UEV/o2phdBWw8Efr7/nCTn90POtHHdL2dhcWl5ZTWxllzf2NzaTu3slnQQKUKLJOCBqvigKWeSFg0znFZCRUH4nJb93tXIL99TpVkg78wgpHUBHcnajICxqJk6q3VACGjWohCUCvqNGC6HP6wV9OUPzQ/xMa5dj4wGNFNpN+OOC88LbyrSaFqFZurDXkYiQaUhHLSuem5o6jEowwinw2Qt0jQE0oMOrVopQVBdj8frDfGhJS3cDpQ90uAx/T0Rg9B6IHzbKcB09aw3gv951ci0L+oxk2FkqCSTh9oRxybAo6xwiylKDB9YAUQx+1dMuqCAGJto0obgza48L0rZjHeSyd6epnP5aRwJtI8O0BHy0DnKoRtUQEVE0AN6Qi/o1Xl0np03533SuuBMZ/bQn3I+vwF5IaNG</latexit>
bA
#
bB
" / b
<latexit sha1_base64="7Yxy4yFNgR/NDTaLHyap8HEqHeg=">AAACHnicbZDLSgMxFIYzXmu9VV26CRbBVZ2pii5rXeiygr1Ab5xJ0zY0yQxJxlKGPokbX8WNC0UEV/o2phdBW38I/HznHE7O74ecaeO6X87C4tLyympiLbm+sbm1ndrZLekgUoQWScADVfFBU84kLRpmOK2EioLwOS37vatRvXxPlWaBvDODkNYFdCRrMwLGombqrNYBIaBZawV9CUoF/UbsXw5/aBT+sPwQH+Pa9Qg3/GYq7WbcsfC88aYmjaYqNFMfdgGJBJWGcNC66rmhqcegDCOcDpO1SNMQSA86tGqtBEF1PR6fN8SHlrRwO1D2SYPH9PdEDELrgfBtpwDT1bO1EfyvVo1M+6IeMxlGhkoyWdSOODYBHmWFW0xRYvjAGiCK2b9i0gUFxNhEkzYEb/bkeVPKZryTTPb2NJ3LT+NIoH10gI6Qh85RDt2gAioigh7QE3pBr86j8+y8Oe+T1gVnOrOH/sj5/AZ+E6NJ</latexit>

Tribocurrent III
• May explain “triboelectric series”
• Also, why current ﬂows in opposite directions at Van
de Graaf generator’s two terminals
• Qualitatively different from any CP-violation in high-
energy theory
• May be compatible with mosaic of charging at the
~ 1 𝜇m roughness scale
27

Mosaic of surface charge
28
uniformly positively and the other, uniformly
negatively (1, 7, 15, 21–24) (Fig. 1A, upper right).
These assumptions, however, make it difficult to
explain numerousexperimentalobservationswhere-
by different particles made of the same bulk mate-
rial (25) or even different macroscopic regions
The starting point of the present study is a recent
observation that contact electrification can occur
between flat pieces of identical materials (32). Ac-
cording to the conventional view of contact elec-
trification, this should not happen because the
chemical potentials of the two surfaces/materialsare
Fig. 1. Possible scenarios of contact elec-
trification and experimental KFM surface
potentialmaps.Thetraditionalview(A,upper
right) assumes that upon contact and sep-
aration, one surface charges uniformly positively and
the other, uniformly negatively. In contrast, the mosaic
picture (A, lower right) assumes that each surface is a union of regions that
have different propensities to accept or donate charge. Upon contact charg-
ing, these regions are then expected to develop different charge polarities.
The mosaic picture of contact electrification is validated by the potential
maps shown in (B to D). The map in (B) corresponds to PDMS before contact
electrification. Such approximately uniform profiles were also observed for
other native materials tested. In contrast, contact-charged surfaces in (C) and
(D) feature a mosaic of (+) and (–) regions. (C) PDMS charged negatively
against another PDMS piece. (D) PC contact-charged positively against
PDMS. The left column has two-dimensional (2D) projections of the maps; on
the right, the corresponding 3D maps are displayed. Color scale is adjusted to
–1 V to 1 V in all images. Net surface charge densities on the contact-
charged pieces (measured by Faraday cup before KFM measurements) were
(B) 0.005, (C) –0.20, and (D) 0.16 nC/cm2
. All images have a scan area of
4.5 mm by 4.5 mm. Scale bars, 500 nm.
Department of Chemistry and Department of Chemical and
Biological Engineering, Northwestern University, 2145 Sheridan
Road, Evanston, IL 60208, USA.
*To whom correspondence should be addressed. E-mail:
grzybor@northwestern.edu
15 JULY 2011 VOL 333 SCIENCE www.sciencemag.org308
www.scienceDownloadedfrom
Source: Baytekin et al., Science 333, 308 (2011)

Basic phenomenology
• In cylindrical coords. k = (km, kz)
• Fermi wave vector
• Open-circuit tribo-voltage:
29
|m⌦| = |kmVs|  kFVs<latexit sha1_base64="wSaogi88czLMB1rWXAhuVQwOl4o=">AAACInicbVDLSgMxFM34rPU16tJNsAiuykwV1IVQFMSdFewDOmXIpLdtaDIzJhmhTPstbvwVNy4UdSX4MaYPUFsPXDg5515u7glizpR2nE9rbn5hcWk5s5JdXVvf2LS3tisqSiSFMo14JGsBUcBZCGXNNIdaLIGIgEM16F4M/eo9SMWi8Fb3YmgI0g5Zi1GijeTbp30ssHctoE1wH5+Z6voCV/zUkwKrgXl7HO6MOBIuBz+Wb+ecvDMCniXuhOTQBCXffveaEU0EhJpyolTddWLdSInUjHIYZL1EQUxol7ShbmhIBKhGOjpxgPeN0sStSJoKNR6pvydSIpTqicB0CqI7atobiv959US3ThopC+NEQ0jHi1oJxzrCw7xwk0mgmvcMIVQy81dMO0QSqk2qWROCO33yLKkU8u5hvnBzlCueT+LIoF20hw6Qi45REV2hEiojih7QE3pBr9aj9Wy9WR/j1jlrMrOD/sD6+gZNwKJW</latexit>
e oc = |µA µB|at zero current . ~kFVs<latexit sha1_base64="Tx5ES3DUePbDm55FeUTPGiZxdus=">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</latexit>
kF
p
k2
m + k2
z<latexit sha1_base64="P7p+saoUi88XIEGjxReG3ITxZSE=">AAACDHicbVBPS8MwHE39O+e/qUcvwSEIwminoMehIB4nuD+w1pJm6RaatF2SCrP0A3jxq3jxoIhXP4A3v43p1oNuPgh5vPd+JL/nxYxKZZrfxsLi0vLKammtvL6xubVd2dltyygRmLRwxCLR9ZAkjIakpahipBsLgrjHSMcLLnO/c0+EpFF4q8YxcTgahNSnGCktuZVq4Ka24PAqg/aAjKAtR0KlwV3d5fAY5vdDplNmzZwAzhOrIFVQoOlWvux+hBNOQoUZkrJnmbFyUiQUxYxkZTuRJEY4QAPS0zREnEgnnSyTwUOt9KEfCX1CBSfq74kUcSnH3NNJjtRQznq5+J/XS5R/7qQ0jBNFQjx9yE8YVBHMm4F9KghWbKwJwoLqv0I8RAJhpfsr6xKs2ZXnSbtes05q9ZvTauOiqKME9sEBOAIWOAMNcA2aoAUweATP4BW8GU/Gi/FufEyjC0Yxswf+wPj8ATACmm0=</latexit>

Pheno., cont.
• For kF ≃1010 m-1 (inter-atomic) and VS ≃1
m/s, we get 𝜙oc ≲ 10-5 V
• Voltage enhanced by mechanical
separation of electriﬁed surfaces
• Increase from inter-atomic to 1 m scale
gives 𝜙oc ≲ 105 V, as in Van de Graaf
generator
30

Triboluminescence
• See: Collins, Camara, Van Cleve &
Putterman, Rev. Sci. Instrum. 89,
013901 (2018)
• visible light emitted at separation
≃10 𝜇m
• 𝜌 ≃1010 e / cm2 seen in tape-
peeling experiments; requires max.
slippage vel. 2 orders of magnitude
above peeling rate ≃1 cm/s
• X-ray production preceded by
large acoustic (stick-slip)
oscillations
See also: Karasaev, Krotova & Deryagin,
Dokl. Akad. Nauk. SSSR 88, 777 (1953);
Deryagin, Krotova & Smilga, Adhesion of Solids (1978)
31
Camara, Escobar, Hird & Putterman,
Nature 455, 1089 (2008)
https://www.youtube.com/watch?
time_continue=1&v=LQBjRF9mX1Y

Feynman
Feynman studied friction on highly polished surfaces
hoping —and mostly failing— to understand how
friction works. He tried to make a theory of how
wind makes ocean waves grow; as he said later, ‘We
put our foot in a swamp and we pulled it up
muddy.’
— J. Gleick, Genius: The Life and Science of Richard
Feynman (1992), p. 9

Dry friction
• Microphysics of viscosity not mysterious
(ﬂuctuation-dissipation)
• Different from dry friction
“Formerly [it] was thought to be that the surfaces were
full of irregularities and friction originated in lifting the
slider over the bumps; but this cannot be, for there is
no loss of energy in that process, whereas power is
in fact consumed. The mechanism of power loss is
that as the slider snaps over the bumps, the bumps
deform and then generate waves and atomic motions
and, after a while, heat, in the two bodies.”
— Feynman Lectures I, 12
33

Poroelasticity
• “Biot slow compressional
wave” has non-linear
dispersion rel., v → 0 as k → 0
• Observations of corresponding
Rayleigh (surface) waves:
Chen et al., Int. J. Solids Struct.
48, 2402 (2011)
• Goldstone theorem + complete
breaking of translation
symmetry by roughness; see
Nielsen & Chadha, Nucl. Phys.
B 105, 445 (1976)
34
k
lim
k!0
v(k) = 0
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v =
!
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phase vel.:

Tribology
• these non-linear Rayleigh phonons can be
superradiated at very low speeds
• Superradiated modes soon thermalize in bulk
• Power P ∝ (V - v), implies F ~ const. for V ≫ v.
Proportional to pressure and area of contact, jointly
(Amontons-Coulomb)
• cf. Frank-Tamm formula for Cherenkov radiation:
35
dP
d!
=
e2
!V
c2
✓
1
c2
n2
rV 2
◆
⇥(nrV c)
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Moving quantum baths: From black holes to dry friction

Moving quantum baths: From black holes to dry friction

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