Stochastic Night Club Current Transport

8/9/2019 Stochastic Night Club Current Transport

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[ c o n d - m a t . m e s - h a l l ] 4 A u g 2

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Fluctuation Statistics in Networks: a Stochastic Path Integral Approach

Andrew N. Jordan,∗ Eugene V. Sukhorukov, and Sebastian PilgramDépartement de Physique Théorique, Université de Genève, CH-1211 Genève 4, Switzerland

We investigate the statistics of fluctuations in a classical stochastic network of nodes joined byconnectors. The nodes carry generalized charge that may be randomly transferred from one nodeto another. Our goal is to find the time evolution of the probability distribution of charges in thenetwork. The building blocks of our theoretical approach are (1) known probability distributions for

the connector currents, (2) physical constraints such as local charge conservation, and (3) a time-scale separation between the slow charge dynamics of the nodes and the fast current fluctuationsof the connectors. We integrate out fast current fluctuations and derive a stochastic path integralrepresentation of the evolution operator for the slow charges. The statistics of charge fluctuationsmay be found from the saddle-point approximation of the action. Once the probability distributionson the discrete network have been studied, the continuum limit is taken to obtain a statistical fieldtheory. We find a correspondence between the diffusive field theory and a Langevin equation withGaussian noise sources, leading nevertheless to non-trivial fluctuation statistics. To complete ourtheory, we demonstrate that the cascade diagrammatics, recently introduced by Nagaev, naturallyfollows from the stochastic path integral. By generalizing the principle of minimal correlations,we extend the diagrammatics to calculate current correlation functions for an arbitrary network.One primary application of this formalism is that of full counting statistics (FCS), the motivationfor why it was developed in the first place. We stress however, that the formalism is suitable forgeneral classical stochastic problems as an alternative approach to the traditional master equation

or Doi-Peliti technique. The formalism is illustrated with several examples: both instantaneous andtime averaged charge fluctuation statistics in a mesoscopic chaotic cavity, as well as the FCS andnew results for a generalized diffusive wire.

PACS numbers: 73.23.b, 02.50.r, 05.40.a, 72.70.+m

I. INTRODUCTION

Consider an exclusive night-club with a long line at theentrance. A bouncer is at the front of the line to keepout the rif-raf. At every time step, a person is acceptedinside the club with probability p, or rejected with prob-ability 1 − p. Inside the club, people stay for a while and

eventually leave. At every time step, the probability aperson leaves is q . We want to answer a question suchas “what is the probability that Q people leave the clubafter t time steps?”.

Assuming that p and q remain constant, the situationis simple and we can easily solve the relevant probabilis-tic problem. However, in realistic situations this rarelyhappens: the management wants to make money. If theclub is almost empty, they instruct the bouncer to beless discriminating, while if the club is almost full, thebouncer is to be more discriminating. Thus, p becomes afunction of the number of people in the club. People willbe more likely to leave if the club is very crowded, so q is

also a function of the number of people inside the club.The problem posed now is much more difficult becauseof the presence of feed-back: the elementary processeschange in response to the cumulative effect of what theyhave accomplished in the past.

This simple example captures all the basic features of the problems we wish to consider. Although the exam-ple was given with people, the actors in the probabilitygame may be any quantity such as charge, energy, heator particles, which we will refer to simply as generalized

charge. Similarly, the night club can be a mesoscopicchaotic cavity,1 a birth-death process,2 a biological mem-brane channel,3 etc.

Historically, general stochastic problems are solvedwith the master equation. The time rate of change of the probability to be in a particular state is given interms of transition rates to other states. This approach

has had great success and leads naturally to the Fokker-Planck and Langevin equations.4 However, once the mas-ter equation is given, the solution is often quite difficultto obtain.

This paper takes a different approach. Rather thanbeginning with a master equation describing the prob-ability of all processes happening in a unit of time, wemake several assumptions from which we can reformulatethe problem. Although these assumptions limit the ap-plicability of the theory, when they apply, the problemsare much easier to solve. The assumptions are:

• The system we are interested in is a composite sys-tem made out of constituent parts. In the nightclub example, the system is made up of three phys-ical regions: outside the front door, the interior of the club, and outside the back door. The decom-position of a larger system into smaller interactingparts is only meaningful for us if there is a separa-tion of time scales. This means that the charge in-side the constituent parts changes on a slower timescale than the fluctuations at the boundaries. Inthe night club example, this simply means that theaverage time a person spends in the club will be

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much longer than the typical time needed to enterthe door.

• Taken alone, the parts of the composite systemhave a finite number of simple properties or pa-rameters. The only property of the night club thatwas relevant for the problem was the total numberof people in it at any given time. The important

element of the line out in front is that it never runsout. All other details are irrelevant.

• In the limit where all parts of the network are verylarge (so that the elementary transport processes donot affect themselves in the short-run), the trans-port probability distributions between elements areknown. In the night club example, the probabilityof getting Q people through the front door after ttime steps (given a constant, large number of peo-ple inside) is easy to find, because we have assumedthat the elementary probability p does not changefrom trial to trial. The transport probability distri-bution is simply the binomial distribution4 where

the probability p is a function of the (approximatelyunchanging) number of people inside. The backdoor distribution is obtained in the same way.

• There are conservation laws that govern the proba-bilistic processes. No matter what probability dis-tributions we have, there are certain rules that mustbe obeyed. The net number of people that enter,stay, and leave the club must be a constant. Thismeans that the time rate of change of the club’s oc-cupancy is given by the people-current in minus thepeople-current out. The people in the line outsideare a special case. There is in principle always a

replacement, so moving one person inside the clubdoesn’t affect the properties of the line.

Now, the strategy is to use this information as thestarting point to find transport statistics for the com-bined interacting system. The main result derived isa path integral expression for the conditional probabil-ity (taking conservation laws into account) for startingand ending with a given amount of charge at each loca-tion after some time has passed. From this conditionalprobability, specific quantities such as transport statis-tics through the system, fluctuation statistics of chargeat a particular location and the like may be found.

One primary application of this formalism is that of full counting statistics (FCS),5,6 the motivation for whyit was developed in the first place.7 FCS describes thefluctuations of currents in electrical conductors. It givesthe distribution of the probability that a certain num-ber of electrons pass a conductor in certain amount of time. Mean current flow and and shot noise1 corre-spond to the first and second cumulant of this distri-bution. The full distribution (defined by all cumulants)provides a full characterization of the transport proper-ties of a electrical conductor in the long time limit. In the

past, FCS was mainly addressed with quantum mechan-ical tools such as the scattering theory5,6,8,9 of coherentconductors, the circuit theory based on Keldysh Greenfunctions,10,11,12,13 or the nonlinear σ model.14 However,a number of works realized that for semi-classical systemswith a large number of conductance channels, shot noisemay be calculated without accounting for the phase co-herence of the electron.15,16,17,18 These works treat the

basic sources of noise quantum mechanically, but calcu-late the spread of the noise throughout the conductorclassically. For specific conductors like diffusive wiresand chaotic cavities, this idea has been extended to thecalculation of third and fourth cumulants via the cas-cade principle19,20 and to the full generating functionof FCS.7,21,22,23 In the present work, we consider anabstract model instead of any particular example anddevelop the mathematical foundations of the proposedsemi-classical procedure to obtain FCS. We introduceand investigate networks of elements with known trans-port statistics and show how the FCS of the entire net-work can be constructed systematically.

The formalism we present is related to a different ap-proach in non-equilibrium statistical physics called theDoi-Peliti technique.24 The idea is that once the basicmaster equation governing the time evolution of proba-bility distributions is given, it may be interpreted as aSchrödinger equation which may be cast into a second-quantized language. This quantum problem is then con-verted into a quantum mechanical path integral (oftenobeying bosonic or fermionic statistics) from which onemay take the continuum limit and use a field theoryrenormalization group approach with diagrammatic per-turbation expansion.25 This approach is useful in manysituations far from equilibrium and has several parallelsto our approach. It has been pointed out that this tech-nique is in some sense the classical limit of the quantummechanical Keldysh formalism,26 the same tool used inthe past to calculate FCS, so this gives another connec-tion with the subject matter we are concerned with.

There are several advantages of our approach. First,we skip the master equation step. If the probability dis-tributions of the connector fluctuations are given, we mayimmediately construct network distributions. Second,from a computational view, our formulation of the prob-lem is much simpler than starting from first principlesfor situations where the ingredients we need are available,and results are much easier to obtain than beginning withthe master equation alone. Thirdly, our formulation also

applies to situations where temporal transition proba-bilities may be large. Finally, the formalism’s physicalorigin is clear, so the needed mathematical objects arewell motivated.

The rest of the paper is organized as follows. In Sec.II, we introduce and develop the general theory. Afterreviewing elements of probability theory, we derive thestochastic path integral for a network of nodes as wellas explore the relationship to the master equation andDoi-Peliti formalism. In Sec. III, the continuum limit


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{Qa2, Λa2

}

{Qc2, Λc2

} {Qc3, Λc3}

{Qc1, Λc

1}

{Qa1, Λa1

} I 13

FIG. 1: An arbitrary network. Each node has charge andcounting variables {Qα,Λα}. The nodes transfer charge viacurrents I αβ through the connectors. The absorbed countingfields (Λaα) are constants by definition of the absorbed chargesQaα (see text). Each node may have an arbitrary number of different charge species, Qα = {Q

1

α,Q2

α, . . . ,Qjα}.

is taken to derive a stochastic field theory and link ourformalism with the Langevin equation point of view. InSec. IV, we develop diagrammatics rules to calculate cu-mulants of the current distribution as well as current cor-

relation functions for an arbitrary network. Sec. V givesseveral applications of the theory to different physical sit-uations. We solve the field theory for the mesoscopic wireand demonstrate universality in multiple dimensions aswell as present new results for the conditional occupationfunction and probability distribution. We also considerthe problem of charge fluctuation statistics (both instan-taneous and time-averaged) in a mesoscopic chaotic cav-ity. Sec. VI contains our conclusions.

II. GENERAL FORMALISM

Once we have the basic elements of our theory (the gen-eralized charges), we must specify some spatial structurethat they move around on. As we noted in the introduc-tion, the essential structure needed to state the problemare simply points we refer to as nodes, joined by con-nectors. This defines a network (see Fig. 1). The stateof each node α is described by one (effectively continu-ous) charge Qα,27 and Q is the charge vector describingthe charge state of the network. The node’s state maybe changed by transport: flow of charges between nodestakes place via the connectors carrying currents I αβ fromnode α to node β . The variation of these charges Qα isgiven by

Qα(t + ∆t) − Qα(t) =β

Qαβ , (1)

where the transmitted charges Qαβ(t) = ∆t0

dt′I αβ(t+t′)are distributed according to P αβ(Qαβ(t)). The fact thatthe probabilities P αβ(t) also depend on the charges Q(t)is one source of the difficulty of the problem.

Assuming that the probability distributions P αβ(which depend parametrically on the state of nodes α andβ ) of the transmitted charges Qαβ are known, we seek the

time evolved probability distribution Γ(Q, t) of the setof charges Q for a given initial distribution Γ(Q, 0). Inother words, one has to find the conditional probability(which we refer to as the evolution operator) U (Q,Q′, t)such that

Γ(Q, t) =

dQ′ U (Q,Q′, t)Γ(Q′, 0) . (2)

We assume that there is a separation of time scales, τ 0 ≪τ C , between the correlation time of current fluctuations,τ 0, and the slow relaxation time of charges in the nodes,τ C . As we will show in the next section, this separation of time scales allows us to derive a stochastic path integralrepresentation for the evolution operator,

U (Qf ,Qi, t) =

DQDΛ exp{S (Q,Λ)}, (3a)

S (Q,Λ) =

t0

dt′[−iΛ · Q̇

+ (1/2)αβ

H αβ(Q, λα − λβ)], (3b)

where the vector Λ has components λα: node variablesconjugated to the Qα that impose charge conservation inthe network.

In the following, we define the functions H αβ as thegenerating functions of the fast currents between nodes αand β . On the time scale ∆t ≫ τ 0, the currents throughisolated connectors are Markovian, so that all cumu-lants (irreducible correlators which are denoted by dou-ble angle brackets) of the transmitted charge (Qαβ)nare linear in ∆t. Following the standard notation inmesoscopic physics,28 we define the current cumulants(Ĩ αβ)n as the coefficients in

(Qαβ)n = ∆t(Ĩ αβ)

n, (4)

where the tilde symbol has been introduced to distinguishthe bare currents of each connector (the sources of noise)from the physical currents I αβ flowing through that sameconnector when it is placed into the network. Then thegenerators H αβ are defined via the equation

(Ĩ αβ)n =

∂ nH αβ(Q, λαβ)

(i∂λαβ)n

λαβ=0

, (5)

and thus contain complete information about the statis-

tics of the noise sources. The λαβ [eventually to be re-placed with with λα − λβ in Eq. (3)] is the generating

variable for the current Ĩ αβ. The notion of current cu-mulants is useful because they are the time independentobjects, and thus have a time independent generators,Eq. (5). The generators H αβ(Q, λαβ) depend in gen-eral on the full vector Q and not just on the generalizedcharges of the neighboring nodes Qα and Qβ. This mayserve to incorporate long range interactions between dis-tant nodes.


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The charge Qαβ transfered through the connectors[characterized by Eqs. (4,5)] may be discrete. However,the charge in the nodes Qα is treated as an effectivelycontinuous variable in Eqs. (1-3). This is justified if manycharges in the node participate in transport. Formally,this limit allows a saddle-point evaluation of the propa-gator (3a).

A. Derivation of the Path Integral.

To derive the path integral Eq. (3), we follow the usualprocedure29 and first discretize time, t = n∆t to derivean expression for U that is valid for propagation over onetime step ∆t. Because of the separation of time scalesτ 0 ≪ τ C , we can consider ∆t as an intermediate timescale,

τ 0 ≪ ∆t ≪ τ C . (6)

The left inequality, τ 0 ≪ ∆t, implies that the transmittedcharges Qαβ are Markovian.

4 This means that charges

transmitted in separate time intervals are uncorrelatedwith each other. While it is not necessary to specify thesource of the current correlation in the general formula-tion, it is worth noting two examples. In a mesoscopicpoint contact, the correlation time τ 0 has the interpreta-tion of the time taken by an electron wavepacket to passthe point contact. In chemical dynamics, it could be thetime taken for a long molecule in solution to traverse afilter.

In a time ∆t, the probability that charge Qαβ is trans-mitted between nodes α and β can be written as theFourier transform of the exponential of a generating func-tion S αβ:

P αβ(Qαβ, ∆t) =

dλαβ

2π exp{−iλαβQαβ + S αβ(λαβ)} .

(7)The definition of the cumulant of transmitted charge is

(Qαβ)n =

∂ nS αβ(λαβ)

(i∂λαβ)n

λαβ=0

. (8)

The Markovian assumption implies that the probabilityof transmitting charge Qαβ in time ∆t followed by chargeQ′αβ in time ∆t

′ through any connector is given by theproduct of independent probability distributions. Thisimplies that the probability of transmitting charge Qαβin time ∆t + ∆t′ may be calculated by finding all waysof independently transferring charge Q′αβ in the first step

and Qαβ − Q′αβ in the second step,

P (Qαβ, ∆t + ∆t′) =

dQ′αβP (Qαβ − Q

′αβ, ∆t

′)

×P (Q′αβ, ∆t) , (9)

which takes the form of a convolution of probabili-ties. Applying a Fourier transform to both sides of

Eq. (9) with argument λαβ decouples the convolution intoproduct of the two Fourier transformed distributions.Eq. (7) implies S αβ(∆t + ∆t′, λαβ) = S αβ(∆t, λαβ) +S αβ(∆t

′, λαβ). It then immediately follows that the gen-erating function must be linear in time. Therefore, a timeindependent H αβ may be introduced: S αβ = ∆t H αβ.The linear dependence of S αβ on time implies that allcharge cumulants (8) will be proportional to time. There-

fore, we define the time independent current cumulants,Eq. (5).

Different connectors are clearly uncorrelated for ∆t ≪τ C , which indicates that the total probability distributionof transmitted charges is a product of the independentprobabilities in each connector:30

P [{Qαβ}] =α>β

P αβ[Qαβ, ∆t]. (10)

Thus far, the analysis is only valid for times much smallerthan τ C . For this case, the charges in the nodes will onlyslightly change. Since we wish to consider longer times,

we need to take into account the fact that charge trans-fer between different nodes will be correlated as chargepiles up inside the nodes. This may be accounted forby imposing charge conservation Eq. (1) during the timeinterval with a delta function,

δ (Qα − Q′α −

β

Qαβ)

=

dλα

2π exp{−iλα[Qα − Q

′α −

β

Qαβ]} . (11)

Here, Q′α is the charge in the node before the time interval

while Qα is the charge accumulated in the node after thetime interval is over. In Eq. (11), λα (referred to as acounting variable ) plays the role of a Lagrange multiplier.The propagator is obtained by multiplying the constraint(11) and the independent probability distribution (10).Representing the probabilities in their Fourier form (7)then yields

Ũ (Q,Q′, Qαβ, ∆t) =α

dλα

2π

α>β

dλαβ

2π exp(S ) ,

S = −iα

λα(Qα − Q′α −

β

Qαβ)

+α>β

[−iλαβQαβ + ∆tH αβ(Q′, λαβ)] . (12)

The full propagator Ũ (Q,Q′, Qαβ, ∆t) still keeps trackof each individual connector contribution Qαβ. We nowintegrate out the fast fluctuations to obtain the dynamicsof the slow variables. This may be done by using theidentity

α λα

β Qαβ =

α>β(λαQαβ + λβQβα) and

Qαβ = −Qβα. The integration over Qαβ gives a deltafunction of argument λαβ − (λα − λβ), so that the λαβ


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integrals may be trivially done. We obtain

U (Q,Q′, ∆t) =α

dλα

2π exp

− i

α

λα(Qα − Q′α)

+∆tα>β

H αβ(Q′, λα − λβ)

. (13)

This is the general result for the one step propagator. If

any two nodes are unconnected, H αβ is zero.An important comment is in order: because H αβ

changes slightly over the time period, which in turn af-fects the probability of transmitting charge through thecontacts, it is not clear at what part of the time stepH αβ should be evaluated. This ambiguity exists becauseour theory is not microscopic. Rather, it takes the mi-croscopic noise generators as an input. This ambiguitygives the freedom of stochastic quantization.31 The sameproblem also occurs in quantum mechanical path inte-grals, and its source there is an ambiguity in operatorordering.32 As we are interested in the large transport-

ing charge limit, γ ≫ 1, and evaluate the integrals inleading order saddle-point approximation, this ambigu-ity will not affect the results.7 For calculations beyondthe large transporting charge limit, the canonical vari-ables Q and Λ need to be properly ordered, which canonly be done with a microscopic theory. For example,the master equation discretized in time as discussed inSec. II D requires the placement of Λ operators in front

of Q operators, since the generating functions H αβ of thetransition probabilities depend on the state of the systemat the beginning of the time period.

To extend the propagator (13) to longer times t = n∆t,we use the composition property of the evolution opera-tor (also known as the Chapman-Kolmogorov equation4).This requires separate {Qα} integrals at each time step,so that for n time steps there will be n − 1 integrals overQ, while each of the n one-step propagators comes withits own Λ integral, Λ = {λα}. Inserting our expressionfor the ∆t step propagator Eq. (13), we find

U (Qf ,Qi, t) =

dΛ0

n−1k=1

dQk dΛk exp

n−1k=0

−iΛk · (Qk+1 − Qk) + ∆tH (Qk,Λk)

, (14a)

with

H (Qk,Λk) =α>β

H αβ [Qα,k; λα,k − λβ,k ] , (14b)

where we have introduced the notations dQk =

α dQα,kand dΛk =

α(dλα,k/2π). We are now in a position totake the continuous time limit. Writing Qk+1 − Qk =∆t Q̇, which is valid because the charge in any nodechanges only slightly over the time scale ∆t, the action of this discrete path integral has the form S = ∆t

nk=1 S k,

which goes over into a time integral in the continu-ous limit. Using the standard path integral notation

DQDΛ =

dΛ0n−1

k=1

dQkdΛk, and invoking the

symmetry H αβ(λα − λβ) = H βα(λβ − λα) we recoverEq. (3). The only explicit constraint on the path inte-gral comes with the charge configurations at the startand finish, Qi and Qf . We also note that H αβ dependson any external parameters such as voltages or chemicalpotentials driving the charge Q.

In the simplest case of one charge and counting vari-able, the form of the path integral is the same as the(Euclidian time) path integral representation of a quan-tum mechanical propagator in phase space with positioncoordinate Q and momentum coordinate λ.32 The differ-ences with the quantum version are that the propagatorevolves probability distributions, not amplitudes (simi-larly to Ref. 25), as well as the fact that the “Hamilto-nian” H = (1/2)

αβ H αβ(Q, λα − λβ) is not really a

Hamiltonian, but rather a current cumulant generatingfunction and therefore is not Hermitian in general. Evenso, because of the similarity we shall refer to H as theHamiltonian from now on.

B. Absorbed Charges, Boundary Conditions and

Correlation Functions.

A useful special case occurs when one has absorbedcharges. These are charges that vanish into (or are in-

jected from) absorbing nodes without altering the sys-tem dynamics. In mesoscopics for example, the absorb-ing nodes are metallic reservoirs. Formally, we dividethe charges into those that are conserved and those thatare absorbed: Q = {Qc,Qa}, where the subset of ab-sorbed charges Qa = {Qaα} does not appear in H αβ. Wedo the same for the corresponding counting variables:

Λ = {Λc

,Λa

}. Because H αβ does not depend on Qa

,these charges may be integrated out by integrating theaction by parts,

i

t0

dt′ Λa · Q̇a = − i

t0

dt′ Qa · Λ̇a

+ i (Λaf · Qaf − Λ

ai · Q

ai ) , (15)

and then functionally integrating over Qa to obtainδ ( Λ̇a), where δ is a functional delta function. This im-mediately constrains the Λa to be constants of motion


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so the functional integration over Λa becomes a normalintegration, DΛa → dΛa. The absorbed kinetic terms inthe action may then be integrated to obtain

U (Qf ,Qi, t) =

dΛa

DQcDΛc exp {S (Q,Λ)} ,

(16a)

S (Q,Λ) = t

0

dt′[−iΛc · Q̇c + (1/2)αβ

H αβ(λα − λβ)]

−iΛa · (Qaf − Qai ). (16b)

Often one is interested in the probability to transmitsome amount of charge through each of the absorbingnodes. By applying a Fourier transform to Eq. (16a)with respect to Qa(t) − Qa(0) we remove the last termin Eq. (16b) and obtain the path integral representationfor the characteristic function Z which generates currentmoments at every absorbing node

Z (Λa) =

DQcDΛc exp{S (Q,Λ)}, (17a)

S (Q,Λ) = t0

dt′[−iΛc · Q̇c

+(1/2)αβ

H αβ(λα − λβ)] . (17b)

Note that the counting variables Λa enter the action(17b) only as a set of constant parameters. The initialcondition in the path integral (17) is given by the initialcharge states Qc(0). There is a choice of the final condi-tion: by fixing the final Qc(t) one obtains the distributionof the conserved charge subject to this constraint, whileby fixing Λc(t) the corresponding characteristic functionis obtained. The choice of Λc(t) = 0 in Eq. (17) gives thecharacteristic function of the absorbed charge under the

condition that the conserved charge is not being moni-tored, i.e. the final charge state is integrated over. There-fore ln Z becomes the generator of the FCS, defining thecharge cumulants at the absorbing node,

[Qaα(t) − Qaα(0)]

n = ∂ n ln Z

∂ (iλaα)n

Λa=0

. (18)

In the long time limit, this quantity is proportional totime, independent of the details of the boundary condi-tions.

Alternatively, in the short time limit one may calculateirreducible correlation functions of absorbed and con-served current fluctuations, I = Q̇. These correlation

functions can be obtained by extending the time integralin (3b) to infinity, introducing sources32 in the action,S → S + i

dt χ(t) · I(t), and applying functional deriva-

tives with respect to χ. Repeating the steps leading toEqs. (17), we find that variables λα in the Hamiltonianin Eq. (17b) have to be shifted λα → λα + χα. Then, theirreducible current correlation function is given by

I α1(t1) · · · I αn(tn) = δ n ln Z [χ]

δiχα1(t1) · · · δiχαn(tn)

χ=0

.

(19)

With these correlation functions, one may calcu-late for example the frequency dependence of currentcumulants.34

C. The Saddle Point Approximation.

If the Hamiltonian has some dimensionless large pref-

actor, then the path integral (3) may be evaluated usingthe saddle point approximation, which is justified below.At the saddle point, (where the first variation of the ac-tion vanishes), we can write equations of motion analo-gous to the Hamiltonian equations of classical mechanics:

i Q̇c = ∂

∂ ΛcH (Qc,Λ), i Λ̇c = −

∂

∂ QcH (Qc,Λ), (20)

where H (Qc,Λ) = (1/2)

αβ H αβ(Qc; λα − λβ). There

may be many saddle point solutions in general, and onehas to sum over all of them. Eqs. (20) are solved sub-

ject to the temporal boundary conditions and generally

describe the relaxation of the conserved charges from theinitial state to a stationary state {Q̄c, Λ̄c} on a time scalegiven by τ C , the dynamical time scale of the nodes. Thesestationary coordinates are functions of any external pa-rameters as well as the (constant) absorbed counting vari-ables Λa. In the saddle point approximation, the actiontakes the form S = S sp + S fluc.7 The term S sp is thecontribution to the action from the solution of the equa-tions (20), which describes the evolution of the systemfrom the initial to the final state. The term S fluc de-scribes fluctuations around the saddle point and is sup-pressed compared to the saddle-point contribution, if theHamiltonian has a large prefactor (in analogy to the -

expansion of quantum mechanics). Physically, the valid-ity condition for the saddle point approximation is thatthere should be many (transporting) charge carriers inthe nodes. For times longer than the charge relaxationtime of the node, the dominant contribution is from thestationary state only, where the saddle-point part of theaction is simply linear in time:

S sp(Q̄, Λ̄) = tH (Q̄, Λ̄), t ≫ τ C . (21)

The linear time dependence of Eq. (21) indicates that thedynamics are Markovian on a long time scale. It is thefact that the contribution S sp emerges in a dominant waywhich makes the approach given here a powerful tool to

analyze the counting statistics of transmitted charge.We now discuss the large parameter that justifies the

saddle point approximation. The boundary conditionson the charge in the absorbing nodes fix a (dimension-less) charge scale of the system, γ . All charges in thenetwork are scaled accordingly, Q → γ Q. We make theassumption that there is a one parameter scaling of theHamiltonian, H → γH . The time is also scaled by τ C ,the time scale of charge relaxation in the nodes. The di-

mensionless action is now S = γ t/τ C0

dt′(−i Q̇λ + τ C H ).


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7

The saddle point action is proportional to γt/τ C , whilethe fluctuation contribution will be of order t/τ C . Wenote that the parameter γ is related to (though not nec-essarily the same as) the separation of time scales, τ C /τ 0,needed to derive the path integral. For the mesoscopicconductors considered in the example section V B of thispaper, the charge scale is set by the maximum number of semiclassical states on the cavity involved in transport,

γ = ∆µN F ≫ 1, the bias times the density of statesat the Fermi level. On the other hand, for the chaoticcavity, τ C /τ 0 = γ /(GL + GR), where GL,R ≫ 1 are thedimensionless conductances of the left and right pointcontact.

D. Relation to the master Equation and Doi-Peliti

Technique.

The evolution operator U (Q,Q′, t) may be interpretedas a Green function of a differential equation which de-termines the propagation in time of an initial probability

distribution Γ(Q). In the theory of stochastic processes,such a differential equation is called a master equation. Anatural question that arises is the relationship of the for-malism presented here to other approaches to stochasticproblems.

The most general type of Markovian master equationfor discrete states and discrete time is of the form

Γn(tk+1) =m

P nm(tk+1, tk)Γm(tk) , (22)

where Γm(tk) is the probability to be in state m at timetk and P nm is the transition probability from state mto state n. The state is described by a vector n =

(n1, . . . , nN ) whose components are the charges nα of each node α. The Markovian assumption implies thattk+1 − tk = ∆t is greater than the correlation time, τ 0. If we further assume that the probability to make a tran-sition to another state is small, P nm ≪ 1 for n = m,so that the transition probability is only linear in ∆t, atransition rate W nm = P nm/∆t may be defined. It thenfollows that we may write a differential master equation,

Γ̇n(t) =m

[W nmΓm(t) − W mnΓn(t)] . (23)

Eq. (23) is the starting point for the Doi-Peliti

technique,24 where one formally maps the space of physical states to the Fock space of states |n =

(a†1)n1 . . . (a†N )

nN |0, where n is the number of charges.The entire state of the system is expressed by a vector|Ψ =

n Γn|n which weights the states |n with their

probabilities Γn. Thus, the master equation (23) may beinterpreted as a many-body Schrödinger equation wherethe rates W mn are incorporated into a Hamiltonian in asecond-quantized form. One may then write a coherent-state path integral over the variables a, and a† for this

many-body quantum system and perform perturbationexpansions along with the renormalization group.25 Thisprocedure eventually involves taking the continuum limitso the discrete charge states become continuous.

Let us now consider how our formalism is related tothe master equation or the Doi-Peliti technique. Accord-ing to the results of Sec. II, our stochastic path integral,Eq.(3) solves the continuum variable version of Eq. (22)

with the transition probabilities given by the one steppropagator U (Q,Q′, ∆t). In general, the transition prob-abilities are neither small nor linear in time for ∆t > τ 0.It is instructive nevertheless to consider the special caseof processes where Hτ 0 ≪ 1, when we can expand theone-step propagator (13) to first order in ∆t,

U (Q,Q′, ∆t) ≈ δ (Q − Q′)

+ ∆t

dΛe−iΛ·(Q−Q

′)H (Q′,Λ). (24)

Defining the Fourier transform of the generating func-tion as H̃ (Q,Q′), the differential equation governing theevolution of a probability distribution of charges Γ(Q) isthen

Γ̇(Q, t) =

dQ′ H̃ (Q,Q′)Γ(Q′, t). (25)

Comparison with the continuous version of the masterequation (23),

Γ̇(Q, t) =

dQ′ [W (Q,Q′)Γ(Q′, t) − W (Q′,Q)Γ(Q, t)],

(26)

indicates that H̃ is related to W . The Hamiltonian maybe expressed in terms of the transition kernel33 as,

H (Q′,Λ) =

dQ

ei(Q−Q′)·Λ − 1

W (Q,Q′), (27)

where the normalization of probability is expressed byH (Q′, 0) = 0. Eq. (27) is an important result, becauseit allows the conversion of the master equation (26) intothe stochastic path integral (3).

We would like to stress that our formalism is not simplyequivalent to the differential master equation (26) (andtherefore the Doi-Peliti technique), but that it allows thetreatment of a complementary class of problems. Our for-malism assumes effectively continuous charge, and thuscannot resolve effects due to the discreteness of charge onthe nodes. Such effects are present in the master equa-tion (23). In contrast, the differential master equationassumption, Hτ 0 ≪ 1 (which simply states that transi-tion probabilities are small in the time interval τ 0) is notrequired. Our formalism is especially important whenthis is not the case, i.e. Hτ 0 ∼ 1.

This is illustrated by the simple example from meso-scopics of two metallic reservoirs connected by a sin-gle electron barrier with hopping probability p and bias∆µ at zero temperature. For a time interval ∆t largerthan the correlation time τ 0 = /∆µ (the time scale for


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8

FIG. 2: A one dimensional lattice of nodes connected on bothends to absorbing reservoirs. This situation could representa series of mesoscopic chaotic cavities connected by quantumpoint contacts.

an electron wavepacket to transverse the barrier), ∆t/τ 0electrons approach the barrier and either are transmittedor reflected. Mathematically, this is a classical binomialprocess with the generator

S = (∆t/τ 0) ln[1 + p(eieλ − 1)]. (28)

As this action is the starting point of many mesoscopicimplementations of the formalism, it is an important ex-ample. Since the action is proportional to the large pa-rameter ∆t/τ 0 > 1, for p ∼ 1 the expansion of exp(S )to first order in ∆t is strictly forbidden, effectively notallowing a first order differential master equation. Onlyin the limit p ≪ 1, (i.e. when Eq. (28) describes a Pois-sonian process) may the logarithm be expanded to firstorder. This suggests that Eq. (26) describes the slowdynamics of systems whose fast transitions are Poisso-nian in nature. A more general type of dynamics suchas the binomial distribution may only be found using thecontinuous charge state master equation in discrete time

(22).

III. THE FIELD THEORY

From the stochastic network, Fig. 1, it is straightfor-ward to go to spatially continuous systems as the spacingbetween the nodes is taken to zero. The goal is to intro-duce a Hamiltonian functional h(ρ, λ) whose argumentsare the charge density ρ and the counting field functionsλ, that are themselves functions of space and time. Wemay then replace (1/2)α,β H α,β → dz h(ρ, λ). Ourdescription is local, so in the model each node is onlyconnected to its nearest neighbors. We first derive theone dimensional field theory with one charge species indetail, and then generalize to multiple dimensions andcharge species.

Consider a series of identical, equidistant nodes sepa-rated by a distance ∆z. This nodal chain could repre-sent a chain of chaotic cavities, Fig. 2, in a mesoscopiccontext.35,36 The sum over α and β becomes a sum overeach node in space connected to its neighbors. The action

for this arrangement is

S =

t0

dt′α

{−λα Q̇α + H (Qα, Qα−1; λα − λα−1)} ,

(29)where for simplicity we have chosen real counting vari-ables, iλα → λα. The imaginary counting variableswill be restored at the end of the section. The only

constraint made on H is that probability is conserved,H (λα − λα−1) = 0 for λα = λα−1. We now derive a lat-tice field theory by formally expanding H in λα − λα−1and Qα − Qα−1. Only differences of the counting vari-ables will appear in the series expansion, while we mustkeep the full Q dependence of the Hamiltonian. If thereare N ≫ 1 nodes in the lattice, for fixed boundary condi-tions the difference between adjacent variables, λα−λα−1and Qα − Qα−1 will be of order 1/N , and therefore pro-vides a good expansion parameter. The expansion of theHamiltonian (29) to second order in the difference vari-ables gives

H = ∂H ∂λα(λα − λα−1) + 12 ∂

2

H ∂λ2α(λα − λα−1)2

+ ∂ 2H

∂Qα∂λα(Qα − Qα−1)(λα − λα−1) , (30)

where the expansion coefficients are evaluated at λα =λα−1 and Qα = Qα−1 and are functions of Qα−1. Termsinvolving only differences of Qα − Qα−1 are zero be-cause H (λα − λα−1) = 0 for λα = λα−1. All terms inEq. (30) need explanation. First, the expression ∂H/∂λαis the local current at zero bias (because the chargesin adjacent nodes are equal) which will usually be zero.There may be circumstances where this term should be

kept,37

but we do not consider them here. The term∂ 2H/∂Qα∂λα = −G(Qα−1) is the linear response of thecurrent to a charge difference. Hence, G is the gener-alized conductance38 of the connector between nodes αand α − 1. ∂ 2H/∂λ2α = C (Qα−1) is the current noisethrough the same connector because H is the generatorof current cumulants.

We are now in a position to take the continuum limitby replacing the node index α with a coordinate z, intro-ducing the fields Q(z), λ(z), and making the expansions

λα − λα−1 → λ′∆z + (1/2)λ′′(∆z)2 + O(∆z)3, (31a)

Qα

− Qα−1

→Q′∆z + (1/2)Q′′(∆z)2 + O(∆z)3. (31b)

The action may now be written in terms of intensive fieldsby scaling away ∆z,

H → h(ρ, λ)∆z, Qα → ρ(z)∆z,

Gα(∆z)2 → D(ρ), C α∆z → F (ρ) , (32)

and taking the limit

α H →

dzh(ρ, λ). One maycheck that expanding the Hamiltonian to higher than sec-ond order in ∆z will result in terms suppressed by powers


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9

of ∆z/L and consequently vanish as ∆z → 0. This scal-ing argument for the field theory is analogous to VanKampen’s size expansion.39 Though the lattice spacing∆z does not appear in the continuum limit, it providesa physical cut-off for any ultra-violet divergences thatmight appear in a loop expansion.

These considerations leave the one dimensional actionas

S = −

t0

dt′ L0

dz

λρ̇ + D ρ′λ′ −

1

2F (λ′)2

. (33)

Here D is the local diffusion constant and F is the lo-cal noise density which are discussed in detail below. Itis very important that these two functionals D, F areall that is needed to calculate current statistics. Classi-cal field equations may be obtained by taking functionalderivatives of the action with respect to the charge andcounting fields: δS/δρ(z) = δS/δλ(z) = 0 to obtain theequations of motion,

˙λ = −

1

2

δF

δρ (λ

′

)

2

− Dλ

′′

, ρ̇ = [−F λ

′

+ Dρ

′

]

′

. (34)

From the charge equation, one can see immediately thatthe term inside the derivative may be interpreted as acurrent density so that local charge conservation is guar-anteed. We have to solve these coupled differential equa-tions subject to the boundary conditions

ρ(t, 0) = ρL(t), ρ(t, L) = ρR(t),

λ(t, 0) = λL(t), λ(t, L) = λR(t), (35)

where ρL(t), ρR(t), λL(t), and λR(t) are arbitrary timedependent functions. Functions ρL(t) and ρR(t) are the

charge densities at the far left and right end of the systemwhich may be externally controlled. Functions λL(t) andλR(t) are the counting variables of the absorbed chargesat the far left and right end which count the current thatpasses them.

Once Eqs. (34) are solved subject to the bound-ary conditions (35), the solutions ρ(z, t) and λ(z, t)should be substituted back into the action (33) andintegrated over time and space. The resulting func-tion, S sp[ρL(t), ρR(t), λL(t), λR(t), t , L] is the generatingfunction for time-dependent cumulants of the currentdistribution. Often, the relevant experimental quan-tities are the stationary cumulants. These are givenby neglecting the time dependence, finding static solu-tions, ρ̇ = λ̇ = 0, and imposing static boundary condi-tions. Similarly to section IID, we can also introducesources

dtdz χ(z, t)ρ(z, t) and calculate density correla-

tion functions.To estimate the contribution of the fluctuations to the

action, it is useful to define dimensionless variables. Theboundary conditions ρL, and ρR provide the charge den-sity scale ρ0 in the problem, so we define ρ(z) = ρ0f (z),where f ∼ 1 is an occupation. We furthermore rescalez → Lz, and t → τ Dt, where τ D = L

2/D is the diffusion

time, thus obtaining

S = −Lρ0

t0

dt′ 10

dz′

λ ḟ + f ′λ′ − F

2Dρ0(λ′)2

.

(36)We assume that the combination F/Dρ0 is of order 1.From Eq. (36), the dimensionless large parameter is γ =ρ0L ≫ 1, i.e. the number of transporting charge carriers.

As in Sec. IIC, the saddle point contribution is of orderγt/τ D, while the fluctuation contribution is of order t/τ D.

Repeating this derivation in multiple dimensions withN charge species ρ = {ρi(r)} and counting fields Λ ={λi(r)}, i = 1, . . . , N yields the action

S = −

t0

dt′ Ω

dr [ Λρ̇ + ∇Λ D̂ ∇ρ − (1/2)∇Λ F̂ ∇Λ ],

(37)where tensor notation is used and we have introducedF̂ ij = ∂ λi∂ λjh and D̂ij = −∂ ρi∂ λjh as general matrixfunctionals of the field vector ρ and coordinate r whichshould be interpreted as noise and diffusion matrices. If

the medium is isotropic, then the vector gradients simplyform a dot product. It should be emphasized that thevectors appearing are vectors of different species of chargefields, as all node delimitation has been accounted forin the spatial integration. The functional integral nowruns over all field configurations that obey the imposedboundary conditions at the surface ∂ Ω. Classical fieldequations may be formally obtained by taking functionalderivatives of the action with respect to the charge andcounting fields as in the 1D case.

As in any field theory, symmetries of the action playan important role because they lead to conserved quan-tities. We first note that the Hamiltonian h(ρ, ∇ρ, ∇Λ)is a functional of ∇Λ alone with no Λ dependence. Thissymmetry is analogous to gauge invariance, and leads tothe equation of motion

ρ̇ + ∇ · j = 0 , j = −D̂∇ρ + F̂ ∇Λ , (38)

which can be interpreted as conservation of the condi-tional current j. The next symmetry is related to theinvariance under a shift in the space and time coordi-nates {δ r, δt}. This symmetry leads to equations analo-gous to the conservation of the local energy/momentumtensor.40 We do not explicitly give this quantity becauseit is rather cumbersome in the general case. However, forthe stationary limit (where ρ̇ and λ̇ vanish) and for sym-

metric diffusion and noise tensors, the one charge speciesconservation law is relatively simple and is given by

m

∇mT mn = 0 , (39a)

T mn = jm(∇nλ) − (∇nρ) ( D̂∇λ)m − h δ mn . (39b)

For the special case of a one dimensional geometry,the Hamiltonian itself is the conserved quantity (seeSec. V A).


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10

In the continuum limit, all terms of higher order in Λare suppressed so that the action is quadratic in the Λvariables. This fact may be viewed as a consequence of the central limit theorem and confirms the observationmade by Nagaev that local noise in the mesoscopic dif-fusive wire (see Sec. V A) is Gaussian.19 To further clar-ify the physical meaning of D and F , and also to makeconnection with previous work,32 we restore the complex

variables, Λ → iΛ, and make a Hubbard-Stratronovichtransformation by introducing an auxiliary vector fieldν ,

exp{−(1/2)∇Λ F̂ ∇Λ}

= (det F̂ )−1

2

Dν exp{−(1/2) ν F̂ −1 ν + iν ∇Λ} . (40)

We may then integrate out the Λ variables, taking ac-count of the boundary terms to obtain,

U = exp

t0

dt′

∂ Ω

ds · (iΛa J)

Dρ Dνδ ( ρ̇ + ∇ · J)

× (det F̂ )−1

2 exp

−1

2 t0

dt′ Ω

dr′ ν F̂ −1 ν

, (41)

where the δ above is a functional delta function, imposingthe Langevin equation

ρ̇ + ∇ · J = 0 , (42a)

J = −D̂∇ρ + ν , (42b)

with a current noise source ν , whose correlator41 is givenby

ν (r, t)ν (r′, t′) = δ (t − t′)δ (r − r′) F̂ (ρ). (42c)

J may be interpreted as the physical current density [not

to be confused with the conditional current density (38)]so that local current conservation is guaranteed, and the(det F̂ )−1/2 serves to normalize the ν probability distri-bution. The role of the boundary term is to count thecurrent J flowing out of the boundary with the count-ing variable Λa, which serves as a Lagrange multiplier.This formula gives an immediate translation between theLangevin approach and full counting statistics, a connec-tion not previously known. The algorithm is as follows:

1. Given a Langevin equation of the form (42), writethe average of the boundary term with source Λa asa path integral (41) over noise and density fields.42

2. Introduce an auxiliary field Λ that takes on thevalue Λa at the boundaries and represents the deltafunction in Eq. (41) imposing current conservation(42a) in Fourier form.

3. Integrate out the Gaussian noise to obtain an actionof the form of Eq. (37).

4. Find where the first variation of the action is zeroand solve the equations of motion subject to theboundary conditions.

5. Insert the solutions back into the action, and do thespace and time integrals. The answer is the currentcumulant generating function.

IV. PERTURBATION THEORY

We have shown in Sec. II C that a large number of par-

ticipating elementary charges justifies the saddle pointapproximation for the generator of counting statistics.While the generator may sometimes be found in closedform,7 in general, it has no compact expression and thecumulants should be found separately at every order.This may be done by expanding S sp(Q,λ,χ) as a seriesin χ and solving the saddle point equations to a given or-der in χ directly. However, there is another approach forevaluating the higher cumulants, the cascade diagram-matics representing higher-order cumulants in terms of the lower ones. It has been introduced by Nagaev inthe context of mesoscopic charge statistics in the diffu-sive wire19 and later extended to the chaotic cavity,20

but without proof. The basic idea is that lower ordercumulants mix in to yield corrections to the bare fluctu-ations of higher order cumulants. This method was usedsuccessfully in Ref. 43 to explain the recent experimentof Ref. 44. In this section, we demonstrate that theserules follow naturally from the stochastic path integralin the same way as Feynman diagrams follow from thequantum mechanical functional integral. In Sec. IV C wepresent another (simpler) method for computing cumu-lants based completely on differential operators obtainedfrom the Hamiltonian equations of motion. In Sec. IV Dwe generalize the cascade diagrammatics to an arbitrarynetwork, and to the case of time-dependent correlators.

A. The Principle of Minimal Correlations.

To motivate the cascade diagrammatics, we refer to aspecific physical system (see the inset of Fig. 8), the meso-scopic chaotic cavity.1 For the purposes of this section,the cavity is a conserving node carrying charge Q, theelectronic reservoirs correspond to the left and right areabsorbing nodes, and the two point contacts are the con-nectors described by Hamiltonians H L, H R (see Fig. 3).Although a detailed description of this system is given inSec. V B, we would like to mention that the mesoscopiccavity is described by an electron distribution functionf , which is fluctuating around its mean value, f 0. Theactual electrical charge in the cavity Q and the occu-pation f are related via the large parameter γ throughQ = γ (f − f 0), where γ = ∆µN F ≫ 1 (the densityof states at the Fermi energy N F times the bias ∆µ) isthe maximum possible number of electrons on the cavitywhich contribute to the transport (see Sec. II C).

The cascade approach builds on the principle of mini-mal correlations developed in Ref. 18: The point contactscreate bare noise Ĩ 2L = ∂

2H L/(∂iλL)2, and Ĩ 2R =


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11

H L

I L I R

H R

{Q,λ}

FIG. 3: Network representing a chaotic cavity. The state of the internal node is described by the variable Q, the charge onthe cavity. The statistics of the connectors are characterizedby the two generating functions H L,R.

∂ 2H R/(∂iλR)2 with no correlation, Ĩ LĨ R = 0 [see Eq.

(5)]. However, for times longer than the average dwelltime of electrons in the cavity, the current conservationrequirement imposes “minimal correlations” on the fluc-tuations of the physical currents I L and I R, which canbe expressed in the form of the Langevin equations,

I L = Ĩ L − GLQ, I R = Ĩ R + GRQ, (43)

where Ĩ L,R are now the sources of bare noise, GL,R arethe generalized conductances of the left and right pointcontact, and Q is the fluctuating charge in the cavity.Current conservation of the physical currents, I L = I R =I , can now be used to obtain

I = GRĨ L + GLĨ R

GL + GR, Q =

Ĩ L − Ĩ RGL + GR

. (44)

Combining powers of I and Q and averaging over thebare noise, we obtain the minimal correlation resultfor arbitrary cumulants QkI lm. In particular, using

Ĩ LĨ R = 0, we find the second cumulant of currentis17,18

I 2 = I 2m = G2RĨ

2L + G

2LĨ

2R

(GL + GR)2 , (45)

where the subscript m denotes the minimal correlationresult. We stress that the bare correlators Ĩ 2L,R arefully determined by the average occupation function f 0of the cavity.

This example demonstrates that a simple redefinitionof the current fluctuations makes it straightforward tofind the noise. Therefore, it came as a surprise45 that the

minimal correlation approach is not sufficient to correctlyobtain higher-order cumulants of current. The reasonfor the failure of the minimal correlation approach hasbeen found recently by Nagaev,19 who showed that fromthe third order cumulant on, there are “cascade correc-tions” to the minimal correlation result, which may beinterpreted as “noise of noise”. For example, the thirdcumulant of current through the mesoscopic cavity,20

I 3 = I 3m + 3 IQm∂

∂QI 2m, (46)

contains a contribution from fluctuations of the chargein the cavity that couples back into the current fluctua-tions. The factor of 3 comes from the fact that there are3 independent currents that the charge fluctuation maybe correlated with. For higher cumulants, there will bemore cascade corrections that may be represented in adiagrammatic form.19,20

B. Derivation of Diagrammatic Rules.

We now present a derivation of these diagrammaticrules for a single node attached between two absorbingnodes. Generalizations to an arbitrary network will sub-sequently be given in Sec. IV D. As we have shown inSec. IIC, the charge scale imposed by the boundary con-ditions, γ , gives a dimensionless large parameter which

justifies the saddle point approximation of the path inte-gral, so that fluctuations around the saddle point are sup-pressed by 1/γ . In the diagrammatic language, we willshow that loop diagrams are suppressed by the same fac-

tor 1/γ . The diagrammatic approach given here is basedon perturbation theory originally developed in quantummechanics.29

Consider the path integral expression of the generatingfunction for the charge absorbed in the left (L) and right(R) node:

Z (χL, χR) =

DQDλ exp

t0

dt′[−i Q̇λ

+H (Q,λ,χL, χR)]

, (47)

where H = H L(Q, λ − χL) + H R(Q, χR − λ). The per-turbation theory is formulated as follows. First, the ex-

ternal counting variables are set to zero, χL = χR = 0.The Hamiltonian H → H L(Q, λ) + H R(Q, −λ) has a sta-tionary saddle point located at {Q0, λ0} that we wishto define as the origin of coordinates. The probabilitydistributions of transferred charge are normalized, so

∂ nQH L,R(Q, λ)|λ=0 = 0, ∀n. (48)

In particular, ∂ QH L(λ)|λ=0 = ∂ QH R(λ)|λ=0 = 0, andtherefore λ0 = 0. Next, ∂ iλH (λ)|λ=0 = I L(Q) −I R(Q) = 0, since H L and H R are the generators of the left and right current respectively. Therefore, Q0 isfixed as the charge in the node such that left and rightconnector currents are equal on average. The stability of the saddle point is guaranteed by the fact that the barenoise correlators, Ĩ 2L,R, are positive. The derivatives∂ iλ∂ QH L = −GL, ∂ iλ∂ QH R = −GR define the gener-alized conductance of each connector, where the currentflows from left to right in both connectors.

The principle of minimal correlation plays an impor-tant role in the cascade diagrammatics. We will showthat this principle is equivalent to exploiting certain free-doms in the path integral in order to postpone the cas-cade corrections to third and higher order cumulants. In


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12

the long-time limit, t ≫ τ C (where 1/τ C = GL+GR is therelaxation rate of the charge in the node), the absorbedcurrent is conserved, I R = I L. Therefore, the currentthrough the node can be defined as weighted average of the left and right connector currents I = (1 − v)I L + vI R,where v is arbitrary constant. The corresponding count-ing variable χ is introduced by substituting χR = vχ andχL = (v − 1)χ. Consider now the second derivative

∂ 2H

∂iχ∂Q

χ=0

= (v − 1)GL + vGR. (49)

We may set it to zero by fixing v = GL/(GL + GR).This is equivalent to imposing conservation of currentfluctuations as in Eq. (44). If we consider further thederivative

∂ 2H

∂iλ∂Q

χ=0

= −(GL + GR), (50)

we have the freedom to scale λ to make the right handside of Eq. (50) equal to −1 [this scaling only alters theχ independent prefactor of Eq. (47)]. The Hamiltoniantakes the new form

H = H L

Q,

GRχ + λ

GL + GR

+ H R

Q,

GLχ − λ

GL + GR

. (51)

We refer to these new variables as minimal correlationcoordinates and will see that they simplify the diagram-matic expansion.

Define δQ(t) = Q(t) − Q0 and δλ(t) = λ(t) − λ0. If we expand the Hamiltonian in a power series in χ, δQ,and δλ, the terms linear in δQ and δλ vanish at the sad-dle point, as well as the (δQ)2 coefficient by Eq. (48)with n = 2. As argued above, in the minimal correlation

coordinates, ∂ iλ∂ QH (Q0, λ0) = −1. With these transfor-mations, we may split the action S as

S = S 0 +

t0

dt′V (t′), S 0 = −i

t0

dt′ δλ(τ C δ Q̇ + δQ),

(52)where V represents the rest of the H power series and willbe treated perturbatively. It should be emphasized thatV is a general nonlinear function of δλ, so unlike mostquantum examples, the full momentum dependence mustbe kept.

In order to formulate the perturbation theory, we addtwo sources, J and K to the action, S → S + dt

′[JδQ +

iKδλ], so that any average of a function of the variablesδQ,δλ may be evaluated by taking functional derivativeswith respect to the sources J , and K , and then setting thesources to zero. In particular, for the generating functionwe can write

Z (χ) =

DQDλ exp

t0

dt′ V (δQ,δλ,χ)

exp

S 0 +

t0

dt′[JδQ + iKδλ]

J,K =0

= exp t

0

dt′ V δ

δJ ,

δ

δiK , χ DQDλ expS 0 +

t

0

dt′[JδQ + iKδλ]J,K =0 (53)

Using S 0 from Eq. (52) we evaluate the integral over Qand λ and obtain:

Z (χ) = exp

t0

dt′V

δ

δJ ,

δ

δiK , χ

W (J, K )

J,K =0

,

(54)where the functional W (J, K ) is

W (J, K ) = exp

t0

dt′dt′′J (t′)D(t′, t′′)K (t′′)

.

(55)The operator D = (τ C ∂ t + 1)−1 is the retarded propaga-tor, and may be found explicitly by inverting the kernelin frequency space,

D(t, t′) =

∞−∞

dω

2π

e−iω(t−t′)

−iτ C ω + 1

= τ −1C Θ(t − t′)exp[−(t − t′)/τ C ]. (56)

It describes the relaxation of the charge Q(t) to the sta-tionary state Q0 with the rate 1/τ C = GL + GR.

Expanding the exponential in Eq. (54) and taking thet ≫ τ C limit, we arrive at the following expression forthe nth current cumulant

I n = t−1 δ n

δ (iχ)n

∞m=1

1

m!

t0

dt′V

δ

δJ ,

δ

δiK , χ

m

×W (J, K )

χ=J =K=0

connected

. (57)

According to the linked cluster expansion,32

by consid-ering ln Z (χ) rather than Z (χ), we have eliminated alldisconnected terms. In order to compare with the resultsof Ref. 20, we introduce a new notation by defining

∂ jQQkI lm ≡ ∂

jQ∂

kiλ∂

liχV (Q0, λ0, χ = 0). (58)

Here QkI lm is the irreducible correlator expressed interms of the noise sources, i.e. the minimal correlationcumulant. In this notation, the expansion of V in a Tay-


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13

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

1 1 1 1

1 1 1 1

1 1 1 1

1 1 1 1

1 1 1 1

1 1 1 1

1 1 1 1

1 1 1 1

= 0.

(c)

= 1,

(a) (b)

(d)

FIG. 4: (a) An n-point current cumulant. (b) The vertexconnecting l external lines with j internal Q lines and k in-ternal λ lines. (c) The propagator connecting λ to Q, equalto 1 in the stationary limit. (d) The vanishing vertex ∂ QI in minimal correlation coordinates.

lor series of all variables takes the form:

V (δQ,δλ,χ) =j,k,l

1

j!k!l! ∂ jQQ

k

I l

m

×[δQ(t)]j[iδλ(t)]k[iχ]l. (59)

Inserting the expansion Eq. (59) into the formula for thecurrent cumulants Eq. (57) gives the formal solution tothe problem. From the form of W (J, K ) and V , we canimmediately read off the diagrammatic rules with the in-ternal lines given by the propagators (56), and the expan-

sion coefficients ∂ jQQkI lm playing the role of vertices.

The following simplifications can be done before therules are finally formulated. First, it is straightforwardto see that loop diagrams are suppressed by powers of

γ −1

. Indeed, according to our single-parameter scalingassumption, the action (52) has a large prefactor γ , whichcan be explicitly displayed, S → γ S , by rescaling thecharge, Q → γ Q. Then it becomes clear that each prop-agator D , represented by an internal line, comes with afactor of γ −1. Each vertex comes from V and thereforehas a factor of γ . If a diagram has I internal lines, E external legs, V vertices and L loops, it will come witha total γ power of V − I . Furthermore, Euler’s formulatells us that V + L − I = 1. Therefore, diagrams with noloops (“tree” diagrams) come with a power of γ , whileloop diagrams are suppressed by the number of loops,γ 1−L. From now on we will concentrate on tree-level di-agrams, since they represent current cumulants at thelevel of the saddle-point approximation.

Second, in the long time limit, t ≫ τ C , each propagator(56) integrated over time gives 1. As a result, since everyvertex is connected to at least one other vertex, all thetime integrals together simply give a factor of t, and thetime dependence cancels on the right hand side of theEq. (57). There are no time integrals in the vertices andthe propagators just give a factor of 1 as in Ref. 20.We are now able to formulate the diagrammatic rules forhigh-order current cumulants:

1. The nth order cumulant I n is a connected n-point function of n external legs I represented bysolid arrows (see Fig. 4a).

2. The external legs must be connected by using ver-tices (see Fig. 4b) and linking internal dashed linesto internal dashed arrows.

3. The vertices ∂ j

Q

I lQkm are represented by a cir-cle with l external legs, k internal outgoing dashedlines, and j internal incoming dashed arrows (seeFig. 4b).

4. Multiply each diagram by the number of inequiva-lent permutations (NIP).

Formally, the vertices ∂ jQI lQkm are the expansion co-

efficients in (59). However, it is important to note thatthey can also be easily evaluated by solving the Langevinequations (43) and expressing the minimal correlationcumulants I lQkm in terms of cumulants of the noise

sources, Ĩ l+kL and Ĩ l+kR . Some vertices are zero,

∂ pH/∂Q p(Q0, λ0)|χ=0 = 0 because of probability conser-vation, but other may or may not be zero depending onthe physical system. Here, the advantage of the min-imal correlation coordinates is made clear: the vertex∂ QI m = 0, and therefore any diagram that containsthis vertex is zero (see Fig. 4d).

To obtain the overall prefactor of a diagram, one canwrite out all the numerical constants and count the num-ber of different ways of producing the same diagram.32

For example, there is the n! from the χ derivatives, the1/m! from the Taylor series of eV , a binomial coefficientfrom expanding V m, and the 1/( j!k!l!) from every vertexwith j + k + l attachments for the different lines. To com-

pensate these factors, we have to do the combinatorics of the number of equivalent terms: interchange the vertices,find the number of different placements of lines on a ver-tex, etc. Often, the number of permutations of the nexternal legs will cancel the m!, and the j !k!l! number of permutations of the internal legs attaching to the vertexwill cancel that factor arising from the Taylor expansion.

Rather than making this expansion, there is a sim-pler method which exploits these cancellations given bycounting the number of inequivalent permutations of thediagram (NIP). The NIP of the diagram is defined by howmany ways the external legs of the diagram may be rela-beled, such that the diagram is not topologically equiv-alent under deformation of the external legs. In otherwords, a diagram with n external legs has n! ways of la-beling them. If this diagram with a given labeling of thelegs may be topologically deformed to give the diagramback with a different labeling, these two sets of labelingsare equivalent permutations. If we write out all the dif-ferent labelings the external legs can have, and cross outevery labeling that is an equivalent permutation of an-other, then the number of labelings that remain is theNIP. This number is most easily found by dividing n! bythe number of equivalent permutations of the diagram.


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14

0 0 0

0 0 0

0 0 0

0 0 0

0 0 0

0 0 0

1 1 1

1 1 1

1 1 1

1 1 1

1 1 1

1 1 1 = + +

(a) (b) (c)

FIG. 5: Tree level contributions to the third cumulant of

transmitted current.

The number of equivalent permutations of the diagramis also called the symmetry factor of the diagram.

We illustrate these two approaches with the third cu-mulant, see Fig. 5. With the simplifications discussedabove, these diagrams may be written as

I 3 = I 3m + 3I Qm∂

∂QI 2m

+ 3I Q2m∂ 2

∂Q2I m . (60)

Note that diagram (c) does not appear in Ref. 20, be-cause it happens to vanish for the chaotic cavity [see alsoEq. (46)]. Referring to the formula (57), the contribu-tions in Eq. (60) are from m = 1, 2, 3 respectively. Eachdiagram must have a χ3 term in the expansion. We firstshow the combinatorial method to obtain the prefactor:Diagram (a) has a factor of 1/3! from the number of permutations of the χ variables, canceling the 3! fromthe χ derivatives. Diagram (b) has a factor of 1/2! fromthe number of permutations of the χ variables, a factorof 1/2! from the Taylor series of the exponential, a factorof 2 from the binomial expansion of V 2, and the 3! fromthe χ derivatives, leaving a factor of 3. Diagram (c) has

a factor of 1/3! from the Taylor series of the exponential,a factor of 3 from the binomial expansion of V 3, a factorof 1/2! from the number of permutations of the δQ vari-ables, a factor of 2 from the functional derivatives actingon W , and the 3! from the χ derivatives, leaving a factorof 3. The NIP is simpler to derive: We divide the numberof permutations of the external legs, m!, by the numberof equivalent permutation of the elements of the diagramthat leave it unchanged. The number of equivalent per-mutations of diagrams (a,b,c) are 3!, 2!, 2!, leaving theoverall factors 1, 3, 3.

(a) (c)(b)

FIG. 6: Three examples of diagrams contributing to thefourth cumulant.

The computation of these diagrammatic contributionsis best understood by a little practice on some examples.Consider three of the diagrams that contribute to thefourth cumulant drawn in Fig. 6. The diagrams symbol-ically represents the combinations:

(a) = ∂

∂QI 2m

∂ 2

∂Q2QmI Q

2m , (61a)

(b) = ∂ 3∂Q3

I mI Q3m , (61b)

(c) = Q2m

∂ 2

∂Q2I m

2I Q2m . (61c)

To figure out the numerical prefactors, we divide 4! (4 isthe number of external legs) by the symmetry factor of the diagram. We first consider the symmetry factor of (a): The upper two legs may be flipped, and the lowertwo legs may be independently flipped where the dottedarrows join without altering the topology of the diagram.Therefore, the symmetry factor is 2 × 2 = 4, and theNIP is 4!/4 = 6. Moving on to diagram (b), the three

lower legs may be permuted amongst themselves to givea symmetry factor 3!, and therefore the NIP is 4!/3! = 4.Finally, diagram (c) may be flipped about its center fora symmetry factor of 2, giving a NIP of 4!/2 = 12.

C. Operator approach.

In the stationary limit, t ≫ τ C , the action takes theform S = tH (Q,λ,χ) so that the evaluation of the cumu-lant generating function reduces to finding the station-ary point of the Hamiltonian H as a function of vari-ables λ and Q. This can be done by solving equations

∂ QH = 0 and ∂ λH = 0. The generating function is thenobtained by substituting the solutions {Q̄, λ̄} into theHamiltonian. In the previous section we have shown thatthis problem can be solved using path integral methods,and the solution can be represented diagrammatically.In the next section we will exploit the full strength of the path integral formalism in order to generalize the di-agrammatics to an arbitrary network, and for the caseof time-dependent charges. However, in the stationarylimit, the conceptual simplicity of the problem of find-ing the stationary point of the function H indicates thatthere should exist a simple iterative procedure for evalu-ating the cumulants up to a given order. In this section

we use classical mechanics methods to prove that this isindeed the case.We first make the variable transformation iλ → λ, and

iχ → χ, so that the Hamiltonian becomes a real func-tion. For χ = 0 the saddle point is located at {Q0, λ0}.For non-zero χ the saddle point moves to a new posi-tion {Q̄, λ̄}, which depends on χ, and the HamiltonianH ( Q̄, λ̄, χ) becomes the generator of cumulants of thecurrent,

I n = dnH ( Q̄, λ̄, χ)/dχn|χ=0. (62)


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15

By expressing the total χ derivative in terms of partialderivatives, the average current can be written as

I = (∂ χ + Q′∂ Q + λ

′∂ λ)H (Q,λ,χ)|{χ=0,Q0,λ0}, (63)

where Q′ = dQ/dχ, λ′ = dλ/dχ are χ dependent. Wewish to eliminate the functions Q′ and λ′ and to expressthe cumulant in terms of the partial derivatives of H .This is done by applying a total derivative to the equa-

tions of motion: [∂ QH ]′ = [∂ λH ]′ = 0 and leads to twoequations for Q′ and λ′ which may be solved,

Q′ = {∂ λH, ∂ χH }

{∂ QH, ∂ λH }, λ′ = −

{∂ QH, ∂ χH }

{∂ QH, ∂ λH }, (64)

where {A, B} is the Poisson bracket, defined as {A, B} =∂ λA ∂ QB − ∂ QA ∂ λB. The solutions have to be insertedinto the Eq. (63).

The advantage of this representation is clear: Now theright hand side of the Eq. (63) (before taking the χ = 0saddle point) depends only on variables λ, Q, and χ.Therefore, we can apply the procedure again in orderto express the high-order cumulant in terms of partial

derivatives. This procedure solves the problem by givinga single operator,

D = ∂ χ + {∂ λH, ∂ χH }∂ Q − {∂ QH, ∂ χH }∂ λ

{∂ QH, ∂ λH } , (65)

which, being applied n times to a given Hamiltonian H and evaluating the resulting expression at the χ = 0 sad-dle point, gives cumulants of current:

I n = DnH (Q,λ,χ)|{χ=0,Q0,λ0}. (66)

This approach is obviously more simple compared to thediagrammatic method, since in the diagrammatics, af-ter drawing all of the diagrams, they have to be eval-

uated individually by taking many partial derivatives of the Hamiltonian and evaluating them at the χ = 0 saddlepoint. With this new approach, given the HamiltonianH , the operator D may be constructed (65) and with amathematical program, an arbitrary cumulant may beeasily computed (66).

It is easy to see the importance of the minimal correla-tion coordinates in this solution. After applying D severaltimes, the derivative quotient rule generates a large num-ber of denominators, {∂ QH, ∂ λH } = (∂ Q∂ λH )(∂ λ∂ QH )−(∂ Q∂ QH )(∂ λ∂ λH ). At χ = 0, as we argued previously,∂ Q∂ QH = 0, and it is possible to change coordinates sothat ∂ Q∂ λH = −1. As a result, the denominator in (66)is equal to 1, which greatly simplifies the expansion. Fi-nally, we would like to stress that the operator approach,introduced in this section for the one node case, can beeasily generalized to a network.

D. Network Cascade Diagrammatics: CorrelationFunctions.

Consider now a general network. In the Sec. IVB, wesaw that the dominant contribution to Eq. (47) arises

from tree-level diagrams. On time scales t ≫ τ C , thetime dependence drops out, and the current cumulantsare static. We now generalize the diagrammatic rulespresented in the section IV B to investigate time- andnode-dependent correlation functions of conserved andabsorbed charges, Eq. (19). To define the network, wemust arbitrarily label the current flow, yielding a directednetwork. By doing so we fix the signs of the elements

H αβ = −H βα of the Hamiltonian. In particular, theelements of the generalized conductance matrix Ĝ,

Gαβ = ∂ 2H

∂ (iλα)∂Qβ, (67)

(evaluated at Q = Q0,Λ = 0) are negative or positive de-pending on the chosen direction. If we segregate absorb-ing (a) and conserving (c) nodes, the conductance matrix

Ĝ may be put in block form. Two of them, the blocksĜcc (real symmetric) and Ĝac will be relevant. This givesus the necessary tool to define the generalized minimalcorrelation coordinates. We consider the frequency de-pendent response by letting the evolution time extent to

infinity, and introduce the time Fourier transform of thevariables {Qc,Λc, χc, χa}, where the vector {χc, χa} is atime-dependent source term introduced to produce cor-relation functions of the conserved and absorbed currents[see Eq. (19)].

Following the steps of section IVB, we again split theaction into two parts, S = S 0 +

dt V , where

S 0 = i

dt[−ΛcQ̇c + Λc ĜccQ

c + (χc Ĝcc + χa Ĝac)Q

c]

= i

dωdω′

2π [Λc(iω′ + Ĝcc)Q

c

+(χc Ĝcc + χa Ĝac)Q

c] δ (ω + ω′), (68)

and where we have dropped the δ in front of the variablesfor simplicity. As in Sec. IVB, the generalized minimalcorrelation coordinates are defined by shifting and rescal-ing the Λc variables in order to eliminate the χ variablesin Eq. (68). However, because χ is now a vector, the pro-portionality factor must be a frequency dependent ma-trix,

Λc(ω) → D̂†(ω)[Λc(ω) + Ĝ†ccχc(ω) + Ĝ†caχ

a(ω)]. (69)

Here D̂(ω) is the matrix network propagator,

D̂(ω) = −(iω Ê + Ĝcc)−1, (70)

and Ê is the identity matrix. It is straightforward to ver-ify that after the shift, the functional

dt V becomes the

generator of cumulants of minimal correlation currents,i.e. of the currents which are solutions of the Langevinequations:

I cα = −iωQcα = −iω

βγ

Dαβ(ω)Ĩ βγ (71a)

I aα =βγα′

Gαα′Dα′β(ω)Ĩ βγ +γ

Ĩ αγ , (71b)


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16

where Ĩ αβ are the bare noise sources as defined in Eq.(5). We finally rescale χc(ω) → χc(ω)/(iω) in order toreplace conserved currents with charges, Ic → Qc.

The total action now acquires the following form

S = (2πi)−1

dω Λc(−ω)Qc(ω)

+ dtV Qc, D̂†(Λc + χc) + D̂† Ĝ†ca

χa, χa , (72)where the simplified form of the Λ argument of V fol-lows after composing the various transformations. Fol-lowing the plan of the previous section, we replace thecharge and counting variables {Q(ω),Λ(ω)} by func-tional derivatives with respect to the charge and counting

sources {J(ω),K(ω)}, and take the V term outside of thefunctional integral. The functional integrals may now beperformed to obtain

W (J,K) = exp

dωdω′

2π J(ω′)K(ω)δ (ω + ω′)

.

(73)The perturbation V must now be expanded in a Tay-lor series with respect to all variables. The time depen-dence only appears through the variables themselves, sothe expansion coefficients will be time independent, withthe exception of the propagator Dαβ(ω) multiplying thecounting variables.

V =∞

{iα,jα,kα,lα}=0

δ j1+···+jn

δ (Qc1)j1 · · · δ (Qcn)

jn(I a1 )

l1 · · · (I ar )lr(Qc1)

i1 · · · (Qcq)iq(Qc1)

k1 · · · (Qc p)kpm

×(χa1)

l1

l1! · · ·

(χar )lr

lr! ×

(χc1)i1

i1! · · ·

(χcr)iq

iq! ×

λk11k1!

· · ·λkp pk p!

× (Qc1)

j1

j1! · · ·

(Qcn)jn

jn! . (74)

As in the one node case, the vertices δ QcαI aβ vanish.

We note again that the notation chosen for the expan-sion coefficients in Eq. (74) connects the formalism de-scribed here with the Langevin equation point of view.The minimal correlation cumulant . . .m may be cal-culated either by the expansion procedure described byEqs. (72,74), or by expressing the physical currents andcharges in terms of the current source cumulants by solv-

ing the Langevin equations for currents and charges,given by Eq. (71).

The nth order irreducible correlatorI a1 (ω1) · · · Q

cn(ωn) may be expressed as a tree-level

diagram with n external lines representing absorbedcurrents I aα and conserved charges Q

cα. Every vertex

is local in time, so if there are p legs at a vertex,each is assigned an independent frequency, while thetime integral imposes overall frequency conservation,δ (

i ωi). The cascade rules are generalized as follows:

1. Every vertex represents the object

δ Qc

1(ω1) · · · δ Qc

l (ωl)I

a

l+1(ωl+1) · · · Q

c

n(ωn)m ,which is multiplied by a δ -function conserving over-all frequency, δ (

ni=1 ωi).

2. The minimal correlation cumulantsI al+1(ωl+1) · · · Q

cn(ωn)m may be evaluated by

expressing them in terms of cumulants of sourcesĨ nαβ via the solutions (71) of the Langevin

equations, or by Eq.(74) if the Hamiltonian isknown.

3. The internal dashed arrow goes from Qcα(ω) toδ Qcα(ω). It conserves the node index α and the fre-

quency ω .46

4. External lines for absorbed currents and conservedcharges originate from I aα(ω) or Q

cα(ω) of the ver-

texes. They conserve the node index and the fre-quency.

5. Sum over all internal node indices, and integrateover all internal frequencies to remove all but oneof the frequency delta functions.

6. The result has to be multiplied by the total numberof inequivalent permutations.

The cascade rules are easily extended to the field the-ory (see Sec. III). The functional analog to the inverseconductance matrix is the operator

ˆG

−1

(r − r′

) ≡

δ 2h

δλ(r) δρ(r′) = −δ (r − r′

)∇ˆ

D∇ . (75)

The diffusion propagator (iω + Ĝ−1)−1 can be used tosolve the Langevin equations (42) for the density ρ(ω, r)and current I (ω) in order to evaluate minimal correlationcumulants. We would like to stress that these cumulantsare limited to second order only, because in the diffusionlimit the noise sources are Gaussian. The summationover node indices is replaced with an integration over thecoordinate r.


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17

V. APPLICATIONS

The formalism presented above is intentionally ab-stract and general. This is to facilitate maximum appli-cability and

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