Heat Fluctuation in Phononic Transports

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Heat Fluctuation in Phononic Transports
-Exact Large deviation function in heat transfer
distribution-
Keiji Saito (University of Tokyo)
Collaborators Abhishek Dhar (RRI), Yasuhiro
Utsumi (ISSP)
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Phenomena on heat conduction are similar to
electronic cases. In most cases, experiments
on heat conduction are later than electric ones.
But eventually those are done.
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Electric conduction vs.
Thermal conduction
Rectification M. Terraneo, M. Peyrard, and G.
Casati. PRL (2002) B. Li, L. Wang, and G. Casati
PRL (2004) Banbi Hu, L. Yang, Y, Zhang PRL
(2006) D. Segal and A. Nitzan PRL (2005) .. CW.
Chang, et al, Science (2006)

• Electronics
• First Diode (1873)
• Transistor..

i)
ii)
ii) Conductance quantization B.J.Van.Wees
et al, (1988)
Schwab et al.,(2000)
iii) Quantum Noise
iii)
Schottky (1918)
Not Yet..
Full Counting Statistics (1993)
iv) .
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1. Thermal Rectification

A lot of many candidates to see Rectification of
heat current
Sr12Ca2Cu24O41

• Molecular Wire
• Carbon Nanotube
• Magnetic systems
• ......

Magnetic system e.g., Sr2CuO3?CuGeO3,(Sr,La,Ca
)14Cu24O41 .
Mean free path is the order O(103 ) lattice
size and magnetic contribution is dominant in
heat transports
Solgubenko et al., Physica B 284 (2000)
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gapless
gapped
1.
Degree of themalization
gapless
gapped
2.
Bosonization (Gapless spin phonon Gapped
spin) self-consitent reservoir (M.Bolsterli, M.
Rich, and WM Visscher (1970))
Proposal of using magnetic system to see
rectification effects
KS, JPSJ vol.75, 34603 (2006)
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ii) Themal Conductance Quantization

• Carbon Nano-tube
• Nanoscale SiNx

Extremely low temperature Nonlinear effect is
not significant Universal Quantum of
Thermal conductance
K.Schwab et al, Nature (2000) Chui et al., PRL
(2005)
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iii) Quantum Noise
In case of Electric Conduction
Second order Fluctuation (noise)
Schottky (1918)
e
average
noise
Full distribution (Full Counting Statistics)
Levitov-Lesovik (1993)
Yu.V. Nazarov (1999)
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In Electron case, higher order flucutuation Have
been measured in many systems
Third current cumulants Reulet et al. PRL vol.
91, 196601 (2003) G. Gershon et al..,
arXiv0710.1852
Quantum point contact,
Full distribution Sukhorukov et al., Nature
physics, vol. 3 (2007) Fujisawa et al.,
Science, vol. 312, 1634 (2006),
Coupled-dots,
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In Electronic cases, a lot of experiments on
quantum noise Even higher order fluctuation
have been measured. What is characteristics
of fluctuations in heat conduction ?
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Present study Fluctuation of Quantum heat
transport

• 1. Is fluctuation also universal at the regime
  where
• universal thermal conductance can be seen
  ?
• 2. Steady State Fluctuation is valid at all
  temperature region?
• 3. What is characteristics in Heat transfer
  Distribution ?

Cf.Transient version, Jarzynski, PRL(2004)
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Evans, Cohen, Morris PRL (1993) Gallavotti and
Cohen (1995) JL Lebowitz Spohn J Stat (1999)
One word on Fluctuation Theorem
entropy produced during time
Relation between positive entropy production
and negative entropy production at steady state
Established equality in classical system But in
quantum case, open problem as to the validity
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Model
System plus Bath model
The system has arbitrary masses, spring
constants, networks
System
Heat bath in Normal-mode expression
Spectral Density
Current Operator
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Theoretical Framework
Lets write
can be regarded as characteristic function
counting field
Cumulant Generating Function
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i) Measure energy of Left Reservoir
Collapse of Wave function
Iteration
ii) Free time-evolution during t
iii) After t, again measure energy of Left
Reservoir
Collapse of wave function
Transmits from left to right
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From this Protocol , we get the formula
J. Kurchan (2000) Levitov and Lesovik, (1993)
for electric conduction
Full counting statistics
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Generating function
Path-integral with Keldysh Contour
Results
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First Derivative gives average current
Landauer type formula L.G. Rego and G.
Kircrenow PRL (1998) KS, S. Miyashita, S.
Takesue PRE (2000), A.Dhar and D. Roy, J.
Stat (2006), D. Segal , A. Nitzan, and P.
Hanggi JCP (2003), T.Yamamoto, S. Watanabe,
and K. Watanabe PRL (2004)
Second Derivative gives noise
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1. Experimental situation in the
Low temperature regime
Universal Fluctuation
We consider the situation where quantized
conductance is measured
no scatterings at the contacts
Suitable model for reservoirs Rubin-Greer bath
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Fluctuation is also universal, as expected
Higher order noise becomes universal
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2.
What does the distribution look like?
i) Typical Distribution
2 Particles Ohmic-bath
Fixed Temperature difference
NonGaussian, Exponetial Tail
(This holds even at equilibrium state)
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2. Symmetry and Steady state Fluctuation theorem
(SSFT)
Symmetry
KS, A.dhar, PRL (2007)
Relation of saddle points
SSFT
Steady State Fluctuation Theorem holds even for
random masses, random spring constants.
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General Proof is possible for the system with
Nonlinear potentials
Time reversal Symmetry Infinite serises of
perturbation With respect to Nonliniarity
KS, Y. Utsumi arXiv0709.4128
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More about universal fluctuations Relations among
Nonlinear Transport Coefficient
For arbitrary types of reservoirs, and general
systems
measure
In general, heat transfer is a nonlinear
function of
Definition of transport coefficient
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Symmetry reproduces Linear response results
Kubo Formula
Relations among higer-order coefficients, e.g.,
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One word on Electric conduction
KS, Y. Utsumi arXiv0709.4128
Set up
1) Inside cavity, Coulomb interactions exists
2) Magnetic Field B
B
Measure heat and charge
Counting field for charge
Counting field for heat
Affinities
B and B are connected
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Measure only charge at uniform temparature
B
measure
gt Onsager-Casimir relation
Universal Relation among transport coeff.
Symmetrize the coefficients
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Summary

• Exact Large deviation function is obtained
• Steady State Fluctuation theorem is satisfied in
  the whole
• regime of temperatures in the present protocol
• Exponential form in the Distribution is general
• Universal fluctuation in the extremely low
  temperature regime

Problem
In other protocols , SSFT is satisfied ?
Modified ? This protocol corresponds to realistic
experiments ? How can we measure fluctuation in
experiments ?
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3. Finite t effect Direct simulation in
classical system (N2)
Numerical Simulation of the Langevin equation
with the white Gaussian noise
Clealy small t cases deviate from the asymptotic
one.