Cumulants of the current and large deviations in the...
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Cumulants of the current and large deviations in the Symmetric Simple Exclusion Process (SSEP)
on graphs
Eric AkkermansPhysics-Technion
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Show selected photi http://www.admin.technion.ac.il/pard/mediaarc/showphoto.asp?photol...
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Benefitted from discussions and collaborations with:
Ohad Sphielberg, Technion,PhysicsBernard Derrida, ENS, Physics, ParisThierry Bodineau, ENS, Maths, Paris
Alex Leibenzon, Technion,Physics+CS
Conference on quantum spectra and transport Yosi Avron birthday,
Hebrew University, Jerusalem,June 30, 2013
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PHYSICAL MOTIVATION OF THIS WORK
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Charge fluctuations in quantum mesoscopic conductors
Current that flows in an electric conductor fluctuates due to the stochastic nature of electron emission and
transport
Study of Transport, Noise and Full Counting Statistics allow to characterize basic physical mechanisms at
work.
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Quantum conductance and shot noiseTwo-terminal conductors
Electric conductance (Landauer)
(a)
ab a
ab
(b)
a' b'
(c) (d)
Tab = tab2
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Noise power is given by the current-current correlation function
where are electronic current operators
(Nyquist fluctuation-dissipation)
Equilibrium noise (V=0)
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Non-equilibrium noise at
Excess noise measures the second cumulant of charge fluctuations :
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Fano factor
is the transmission coefficient along the channel
Tabab
(a)
ab a
ab
(b)
a' b'
(c) (d)
F = S(0,V )! S(0,0)e I
=Tab (1!
ab" Tab )
Tabab"
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Fano factor
F has a universal value 1/3 for weakly disordered “one-dimensional” metals
F = S(0,V )! S(0,0)e I
=Tab (1!
ab" Tab )
Tabab"
Is this result universal ?nature of disorder, geometry, space
dimensionality, extends to higher order cumulants,...
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Classical version of the quantum conductor
Same physical content : particles cannot pile up on the same site (Pauli
principle or quantum crossings in quantum mesoscopic physics)
Defines the classical Symmetric Simple exclusion process (SSEP)
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The SSEP model
For large enough time, the system is in a steady state.
Define the probability of observing particles flowing through the system during a time interval and for 2 reservoirs at densities and
For ,
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All the cumulants are known for arbitrary densities and
The generating function
depends on a single scaling variable
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And the Fano factor is
The Fano factor and all other cumulants are identical to those calculated in the quantum
mesoscopic case.
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How these results generalize to higher space dimensions ?
Numerical results on a Sierpinski gasket fractal network suggests a Fano factor
(Groth et al. PRL 2008)
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How these results generalize to higher space dimensions ?
Numerical results on a Sierpinski gasket fractal network suggests a Fano factor
(Groth et al. PRL 2008)
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Our Result: (T. Bodineau, B. Derrida, O. Shpielberg, E.A, 2013)
1. Large class of graphs (including fractals) can be characterized by an effective length
ρa
ρb
A
B
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2. For large values of , the generating function of the cumulants of the current of the SSEP is the same as for a linear chain, up to a multiplicative function
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ρa
ρb
A
B
2. For large values of , the generating function of the cumulants of the current of the SSEP is the same as for a linear chain, up to a multiplicative function
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2. For large values of , the generating function of the cumulants of the current of the SSEP is the same as for a linear chain, up to a multiplicative function
Thus, the ratio between any pair of cumulants of is the same as for the linear chain. Then,
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ELEMENTS OF THE PROOF
• Use the macroscopic fluctuation theory of Bertini et al. and the additivity principle.
• Alternative description based on Energy/Dirichlet forms: allows to characterize the SSEP and to provide a derivation of the additivity principle.
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The macroscopic fluctuation theory
Basic definitions and results
(Bertini, De Sole, Gabrielli, Jona-Lasinio, and Landim)
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= Number of particles flowing through the system during t
for large t
The large deviation function is defined from the probability
It is the Legendre transform of
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Scaling - Electrical conductance
The large deviation function is a scaling function :
so that scales like an electrical conductance.
(Bodineau,Derrida,Lebowitz - Thouless - Montambaux, E.A.)
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General diffusive system (e.g. SSEP) s.t.,
Weak current through the system : use Fick’s law
+ fluctuations :
For SSEP,
has its maximum for . Close to equilibrium :
Gaussian distribution for the probability,
Additivity principle and large deviation function
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Additivity principle + scaling
L + L’
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Additivity principle + scaling
L + L’
so that,
and
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Macroscopic fluctuation theory for SSEP on a -dimensional domain
ρa
ρb
A
B
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ρa
ρb
A
B
Define the number of particles flowing between the 2 reservoirs :
where is the number of particles transferred
from to during and is an arbitrary function
on site except for
Nothing depends on the choice of the ‘s. We take it a
solution of the Laplace eq.
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Continuous version:
where
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The minimization in the generating function
leads to
where is a Lagrange multiplier field associated to current conservation.
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If one knows the solution of
This results from
The link between and higher dimensions:
in (chain of length L), then we know the solution in any dimension and for any domain !
so that
solve
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where is such that
the generating function in dimensions is
We have the following remarkable identity:
so that
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Since
Then,
with
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The generating function for an arbitrary
domain in -dimensions is the same as the
generating function for the effective
length up to a multiplicative function
independent of
Therefore, for any -dimensional domain, the ratio of any pair of cumulants is the same as
in .
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Analogies between SSEP on a graph and resistor networks
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(Derrida, Bodineau PRL 2004)
Kirchhoff’s rules - Addition in series and in parallel
More generally, using the Δ-Y transformN
N N N/3
N/3 N/3
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Star – mesh transform
• A Two-terminal resistor network always has an equivalent resistor (Helmholtz, Thevenin).
• The equivalent resistor can be obtained through repeated use of the star-mesh transform.
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Star – mesh transform
• A Two-terminal resistor network always has an equivalent resistor (Helmholtz, Thevenin).
• The equivalent resistor can be obtained through repeated use of the star-mesh transform.
The same applies to any SSEP graph
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The SSEP resistor theorem
• For any graph G,
• can be obtained by Kirchhoff's resistor rules
• The theorem applies for any non-eq. process given that
1.The additivity principle applies
2.The scaling assumption applies
3.There is a steady state
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Energy/Dirichlet forms
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Energy forms
A graph with sites and bonds
Each bonds carries a weight - rxy
We define the energy function
Connect the network of resistors to a battery
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Well known exact mapping between electric networks of resistances and random walk on
a lattice
(Doyle & Snell)
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Useful theorem by Beurling and Deny which extends these results to the equivalence between energy forms and symmetric
Markov processes.
This theorem allows to describe the SSEP as an effective conductance network whose
electric energy is the large deviation function.
Moreover, it guarantees the additivity principle (through the concavity property of
the minimum energy)
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Energy forms and SSEP
Consider the energy form EL u,u( ) = u x( )! u y( )"# $%2
rx,y&
u x( ) =! x( )r + D ! x( )( )! x( )2! " x( )( )!" #$
1 2with
EL h j,!a ,!b( )( ) =max!i{ }
j n + D !i( )!!i"# $%2
2n" !i( )i&
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Energy forms and SSEP
The Large Deviation Function is the minimum of an energy form - it is a
conductance
EL h j,!a ,!b( )( ) =max!i{ }
j n + D !i( )!!i"# $%2
2n" !i( )i&
EL h j,!a ,!b( )( ) = FL j,!a ,!b( )
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Summary - further issues
Full counting statistics of quantum mesoscopic conductors is well described by means of the classical 1D SSEP model:
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SSEP - resistor theorem : analogy between electric networks and non-equilibrium stochastic processes.
Energy forms provide a useful framework to derive the large deviation function of symmetric Markov processes.
The additivity principle results from the energy form description.
Extension to more complicated stochastic processes (ASEP) - with phase transitions.
More than 2 reservoirs ?
Random graphs
Back to the quantum case : semi-classical description (A. Pilgram, Sukhorukov).
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