Journal of Statistical Mechanics: Theory and Experiment (2006) P01011 (23 pages)
Fluctuation theorem for transport in mesoscopic systems
David Andrieux and Pierre Gaspard
Center for Nonlinear Phenomena and Complex Systems,
Université Libre de Bruxelles, Code Postal 231, Campus Plaine, B-1050 Brussels, Belgium
The fluctuation theorem for the currents is applied to several mesoscopic systems on the basis
of Schnakenberg’s network theory, which allows one to verify its conditions of validity. A graph
is associated with the master equation ruling the random process and its cycles can be used to
obtain the thermodynamic forces or affinities corresponding to the nonequilibrium constraints. This
provides a method to define the independent currents crossing the system in nonequilibrium steady
states and to formulate the fluctuation theorem for the currents. This result is applied to out-ofequilibrium diffusion in a chain, to a biophysical model of ion channels in a membrane, as well as to
electronic transport in mesoscopic circuits made of several tunnel junctions. In this later, we show
that the generalizations of Onsager’s reciprocity relations to the nonlinear response coefficients also
hold.
Keywords: Nonequilibrium steady state, fluctuating current, affinities, thermodynamic forces,
linear and nonlinear responses, reciprocity relations, fluctuation theorem, diffusion, membrane ion
channel, mesoscopic conductor, tunnel junction.
I.
INTRODUCTION
The discreteness of the composition of matter into particles manifests itself in the form of time-dependent fluctuations of the physical properties on mesoscopic scales from the micrometer down to the nanometer. This is the
case for the electric and particle currents crossing nonequilibrium mesoscopic systems. These fluctuating transport
phenomena can be described in terms of stochastic processes based on a master equation ruling the time evolution
of the probability to observe the system in a certain state. The random jumps from state to state are described by
transition rates which enter the master equation. Although these rates are determined by nonequilibrium kinetics
and can thus have various dependences on the parameters of the system, the assumption of local thermodynamic
equilibrium allows one to find general relationships between the rates of the forward and backward transitions and the
nonequilibrium constraints imposed on the system. This has been formulated in a very general way by Schnakenberg
who developed a network theory for Markovian stochastic processes [1]. In this theory, a graph is associated with the
master equation and its cycles, i.e., the cyclic paths of the graph, can be used to relate the transition rates to the
thermodynamic forces or affinities characterizing the nonequilibrium constraints on the system.
Recently, we have been able to derive from Schnakenberg’s network theory a fluctuation theorem for the currents
crossing a nonequilibrium mesoscopic system [2, 3]. This fluctuation theorem is valid far from equilibrium and applies
even if the currents are driven by general state-dependent nonequilibrium constraints. Moreover, we have shown
that, thanks to this fluctuation theorem, the Onsager reciprocity relations can be generalized to higher orders in the
nonequilibrium response of the system. The fluctuation theorem for the currents [2, 3] extends previous results about
fluctuating currents and is related to fluctuation theorems for entropy production and dissipated work as obtained in
other contexts [4–17].
These fluctuation relations can be derived in either the thermostatted-system approach [4–8] or the stochastic
framework [9–13]. There exist stationary [4, 6, 8–13] and transient [5, 14] versions of the fluctuation theorem. The
differences between the transient and stationary versions are discussed in Refs. [14, 15]. More recently, results have
also been obtained for externally driven systems [16, 17].
The purpose of the present paper is to show how Schnakenberg’s network theory can be used to apply the fluctuation
theorem for the currents to typical transport phenomena in mesoscopic systems.
The first system we consider is out-of-equilibrium diffusion in a chain of cells containing fluctuating numbers of
particles. This model of diffusion is often used to derive the fluctuating diffusion equation and its consequences [18].
Here, we apply the fluctuation theorem to this model and, moreover, we obtain analytic expressions for the generating
function of the fluctuating currents in the chain.
The second system is a biophysical model introduced by Schnakenberg for the transport of ions across membranes
[1]. This system is shown to satisfy the conditions of validity of the fluctuation theorem for the currents, which can
thus be applied under general conditions.
The same result applies to the statistics of electrons transmitted across mesoscopic conductors composed of several
2
junctions [19, 20]. Here, we use a stochastic description based on a master equation for the probability that the
system transiently contains a certain number of electrons. In these systems, the electric currents obey the fluctuation
theorem from which higher-order generalizations of Onsager’s reciprocity relations can be derived, as we shall explain.
The plan of the paper is the following. We summarize the results on the fluctuation theorems for stochastic processes
in Sec. II. The case of out-of-equilibrium diffusion is considered in Sec. III. A model of ion channel in membranes is
studied in Sec. IV and, finally, the case of transport in mesoscopic conductors with several tunnel junctions in Sec.
V. The conclusions and perspectives are drawn in Sec. VI.
II.
A.
THE FLUCTUATION THEOREM
Master equation and Schnakenberg’s associated graph
At the mesoscopic scale, the presence of fluctuations requires a stochastic description for the evolution of the
probability P (ω, t) to observe the system in some coarse-grained state ω at the time t. For Markovian random
processes, its time evolution is typically ruled by a master equation of the form
dP (ω, t) X =
W+ρ (ω 0 |ω)P (ω 0 , t) − W−ρ (ω|ω 0 )P (ω, t)
dt
0
(1)
ρ,ω
ρ
where Wρ (ω 0 |ω) is the rate of the transition ω 0 →ω induced by the elementary process ρ ∈ {±1, ±2, ..., ±r} [23, 24].
It is assumed that, for any transition with a non-vanishing rate, the rate of the reversed transition does not vanish
either. In some simple systems, such processes can be exactly derived from the underlying deterministic dynamics [21]
by introducing an appropriate partition of the phase space while in other systems, such processes can be rigorously
derived from the underlying Hamiltonian classical or quantum dynamics in some scaling limit [22]. In general, such
Markovian equations are derived from the assumption of a good separation of time scales between the microscopic
and mesoscopic levels [23].
The master equation (1) admits a stationary solution dP/dt = 0, which describes either a nonequilibrium steady
state or the equilibrium state, whether nonequilibrium constraints are imposed to the system or not.
Schnakenberg has shown that the fundamental properties of a nonequilibrium random process can be studied by
associating a graph with the master equation [1]. This graph is defined as follows: Each state ω of the system
+ρ
corresponds to a vertex while the edges represent the different transitions ω ω 0 allowed between the states. In
−ρ
this respect, two states can be connected by several edges if several elementary processes ρ allow transitions between
ρ
them. A conventional orientation is given to the graph G, for instance, by using the direction of the transitions ω 0 →ω
corresponding to the elementary processes with ρ > 0.
In order to identify all the cycles of a graph, Schnakenberg has proposed a method based on the concept of maximal
tree. In general a given graph G has several maximal trees T (G). Every maximal tree T (G) of the graph G should
satisfy the following properties:
(1) T (G) is a covering subgraph of G, i.e., T (G) contains all the vertices of G and all the edges of T (G) are edges
of G;
(2) T (G) is connected;
(3) T (G) contains no circuit (i.e., no cyclic sequence of edges).
The edges sl of G which do not belong to T (G) are called the chords of T (G). If we add to T (G) one of its chords
~ l , which is obtained from T (G) + sl by removing all
sl , the resulting subgraph T (G) + sl contains exactly one cycle C
the edges which are not part of the cycle. A maximal tree T (G) together with its associated fundamental set of cycles
~1, C
~ 2 , ..., C
~ l , ...} provides a decomposition of the graph G. We notice that all the maximal trees of a graph can be
{C
obtained by linear combinations of a given maximal tree T (G) with its associated cycles as described in Ref. [1].
The transition rates Wρ (ω|ω 0 ) of the master equation (1) depend on the thermodynamic forces or affinities due
to the nonequilibrium constraints imposed on a system because of its exchanges with energy or particle reservoirs
at different temperatures or chemical potentials. Schnakenberg made the observation that the ratio of the products
~ l of the graph is independent of the states
of the transition rates along the two possible directions of any cycle C
composing the cycle and only depends on the affinities of the nonequilibrium constraints imposed to the system [1].
~ l ) according to
This property is assumed here also and is the basis to introduce the affinities A(C
Y W+ρ (ω|ω 0 )
~
= eA(Cl )
0
W−ρ (ω |ω)
~l
ρ∈C
(2)
3
In the equilibrium state, the affinities vanish and we recover the conditions of detailed balance between every forward
and backward transition.
~ l we can associate a corresponding current jl which is the current crossing the chord sl corresponding
To each cycle C
~ l hence more currents jl than independent currents γ
to this cycle. We notice that there can exist more cycles C
between the reservoirs. The reason is that the graph contains all the possible coarse-grained states and transitions of
the mesoscopic level of description while the currents γ between the reservoirs are typically macroscopic and fewer
than the coarse-grained states. Consequently, several currents jl may contribute to a macroscopic current γ and
~ l may have the same affinity corresponding to the same global current γ from reservoir to reservoir:
several cycles C
~
~ l ∈ γ.
A(Cl ) = Aγ for all C
B.
The fluctuation theorem for the currents
~
PThe instantaneous current associated with a cycle Cl is defined as the current passing on its chord sl as jl (t) ≡
δ(t
−
t
)
where
t
are
the
times
when
random
jumps
occur on the chord sl . We use the convention that jl is
i
i
i∈Z i
oriented as the graph G, i.e. i is equal to (−)1 if the transition is (anti)parallel to sl . The generating function of
these fluctuating currents is defined as
Rt 0
P
0
1
Q({λl }; {Al }) ≡ lim − lnhe− l λl 0 dt jl (t ) i
t→∞
t
(3)
where h·i denotes a statistical average with respect to the stationary state. The fluctuation theorem for the currents
states that the generating function (3) obeys the symmetry
Q({λl }; {Al }) = Q({Al − λl }; {Al })
(4)
~ l ). This theorem has been proved in Refs. [2, 3].
where Al ≡ A(C
The different mesoscopic currents jl (t) associated with the same global current γ can be regrouped as
X
jγ (t) =
jl (t)
(5)
l∈γ
Accordingly, the generating function of these macroscopic currents
Rt 0
P
0
1
Q({λγ }; {Aγ }) ≡ lim − lnhe− γ λγ 0 dt jγ (t ) i
t→∞
t
(6)
~ l ∈ γ. As a consequence of Eq. (4), the global
can be obtained from Eq. (3) by setting λl = λγ for all the cycles C
currents (5) obey the fluctuation theorem
Q({λγ }; {Aγ }) = Q({Aγ − λγ }; {Aγ })
(7)
The Legendre transform of the generating function
"
H({αγ }) = Max{λγ } Q({λγ }) −
#
X
λγ αγ
(8)
γ
is the decay rate of the probability that the currents averaged over some time interval t take given values {αγ }:
Z t
1
Prob
dt0 jγ (t0 ) ' αγ
∼ exp [−H({αγ })t]
(9)
t 0
The fluctuation theorem (7) implies that the ratio of the probabilities to observe opposite values for the currents is
simply related to these values and the corresponding affinities according to
hn R
oi
t
Prob 1t 0 jγ (t0 )dt0 ' αγ
X
hn R
oi ' exp
Aγ αγ t
t
Prob 1t 0 jγ (t0 )dt0 ' −αγ
γ
(t → ∞)
(10)
4
This relation is very general and holds for any nonequilibrium process as long as Schnakenberg’s conditions (2) are
satisfied.
The mean value of the current jα (t) is given by
Jα =
X
1X
∂
1 X
=
Lαβ Aβ +
Q({λ }; {A })
Mαβγ Aβ Aγ +
Nαβγδ Aβ Aγ Aδ + · · ·
∂λα
2
6
{λ =0}
β
β,γ
(11)
β,γ,δ
The expansion in powers of the affinities gives Onsager’s linear-response coefficients as well as higher-order coefficients
characterizing the nonlinear response of the system with respect to the nonequilibrium constraints {A }. As shown
elsewhere [2, 3], the fluctuation theorem for the currents (7) implies not only Onsager’s reciprocity relations Lαβ = Lβα
[25], but also further relations for these higher-order coefficients. The third-order response coefficients are given by
Mαβγ =
1
(Rαβ,γ + Rαγ,β )
2
(12)
with
Rαβ,γ
∂
≡
∂Aγ
Z
+∞
−∞
h[jα (t) − hjα i] [jβ (0) − hjβ i]ist dt
{A =0}
(13)
The fourth-order response coefficients should form the totally symmetric tensor
2Nαβγδ − Tαβ,γδ − Tαγ,βδ − Tαδ,βγ
(14)
together with the coefficients
Tαβ,γδ
∂
∂
≡
∂Aγ ∂Aδ
Z
+∞
−∞
h[jα (t) − hjα i] [jβ (0) − hjβ i]ist dt
{A =0}
(15)
The higher-order response coefficients can thus be expressed in terms of microscopic quantites and present more
complicated symmetries than the Onsager-type symmetry at the level of the linear response. We notice that the
generalized fluctuation theorem and the preceding higher-order reciprocity relations can be applied to the bulk or
boundary driven lattice gases and other examples considered in Ref. [10], for which Schnakenberg’s conditions (2) are
verified.
C.
The fluctuation theorem for the entropy production
On the other hand, the fluctuation theorem for the entropy production corresponding to a trajectory ω(t) =
ρ1
ρ2
ρ3
ρn
ω0 −→ ω1 −→ ω2 −→ · · · −→ ωn over the time interval t concerns the quantity:
Z(t) ≡ ln
Wρ1 (ω0 |ω1 )Wρ2 (ω1 |ω2 ) · · · Wρn (ωn−1 |ωn )
W−ρ1 (ω1 |ω0 )W−ρ2 (ω2 |ω1 ) · · · W−ρn (ωn |ωn−1 )
(16)
introduced by Lebowitz and Spohn [10] and which measures the lack of detailed balance. Its generating function is
defined by
1
q(η) ≡ lim − lnhe−ηZ(t) i
t→∞
t
(17)
q(η) = q(1 − η)
(18)
and obeys the symmetry
of the fluctuation theorem.
The generating function (17) can be obtained in terms of the leading eigenvalue of the equation
dG(ω, t) X =
W+ρ (ω 0 |ω)η W−ρ (ω|ω 0 )1−η G(ω 0 , t) − W−ρ (ω|ω 0 )G(ω, t)
dt
0
ρ,ω
as shown in Ref. [10].
(19)
5
In a nonequilibrium steady state, the entropy production is given by
X
1
di S Aγ Jγ
= lim hZ(t)i =
dt st t→∞ t
γ
(20)
as shown in Refs. [1, 10]. In particular cases where a single current crosses the system, we notice that the fluctuation
theorems for the current and the entropy production are related to each other by setting λ = ηA since q(η) = Q(ηA).
This will be illustrated in the following sections.
III.
OUT OF EQUILIBRIUM DIFFUSION
The model studied here is a discrete version of a diffusive process maintained out of equilibrium by imposing different
concentrations at the ends of a chain composed of L cells. The different cells are denoted by Xi with i = 1, 2, ..., L.
The ends of the chain are in contact with particle reservoirs A and B. The particles are transported from cell to cell
according to the possible jumps:
A X1 X2 · · · XL B
(21)
The coarse-grained states of the model are the numbers of particles in each cell:
ω = (N1 , N2 , ..., NL ) ∈ NL
(22)
which form a lattice of non-negative integers. This process is ruled by the master equation (1) with the transition
rates
W+i (· · · , Ni + 1, Ni+1 − 1, · · · | · · · , Ni , Ni+1 , · · · ) = k+i (Ni + 1)
W−i (· · · , Ni − 1, Ni+1 + 1, · · · | · · · , Ni , Ni+1 , · · · ) = k−i (Ni+1 + 1)
(23)
(24)
modeling the chaotic mixing in the cells. We suppose that the chain is homogeneous so that the rate constants between
all the cells are the same
k+i = k+
k−i = k−
(25)
At the ends of the chain, the concentrations of particles are constant in time because we suppose the reservoirs are
arbitrarily large so that we impose the nonequilibrium boundary conditions N0 = hAi and NL+1 = hBi which are the
mean numbers of particles in the reservoirs A and B. Figure 1 depicts the graph of the random process in a chain
containing L = 2 cells. In this example, the six transitions ±0, ±1, and ±2 are possible.
In order to convince oneself that the above random process defines a model of diffusion, we can introduce the
particle density n(x) in each cell as
n(iδ) =
hNi i
V
(26)
where V is the volume and δ the length of each cell. By expanding in terms of the spacing δ between the sites, we
find that
∂n(iδ)
δ
δ
= k+ iδ −
n[δ(i − 1)] + k− iδ +
n[δ(i + 1)]
∂t
2
2
δ
δ
− k− iδ −
n(δi) − k+ iδ −
n(δi) + O(δ 3 )
2
2
h δ2
i
= −∇[δ(k+ − k− )n] + ∇
(k+ + k− )∇n + O(δ 3 )
2
which becomes, in the continuous limit δ → 0, the diffusion equation for the particle density:
∂n(x, t)
= −v ∇n(x, t) + D ∇2 n(x, t)
∂t
(27)
with the drift velocity and the diffusion coefficient respectively defined by
v = δ(k+ − k− )
D = δ2
(k+ + k− )
2
(28)
6
N2
4 •
•
•
•
•
3•
•
•
•
•
•
•
•
•
•
•
+1
+2
2•
•
1•
•
0 •
0
•
•
•
•
1
2
3
4
+0
N1
FIG. 1: Graph associated with the random process of the diffusive transport (21) over a chain of length L = 2.
The system sustains a single independent current between the two particle reservoirs. The affinity associated with
~ in which a particle is transported from the left-hand reservoir
this current can be obtained by considering a cycle C
A to the right-hand one B:
+0
+1
(N1 , N2 , ..., NL )→(N1 + 1, N2 , ..., NL )→ · · ·
+(L−1)
+L
→ (N1 , N2 , ..., NL + 1) →(N1 , N2 , ..., NL )
(29)
The affinity calculated along such cycles takes the value
~ = ln
A(C)
hAi
k+
+ (L + 1) ln
hBi
k−
(30)
The fluctuation theorem (10) here applies and predicts that the fluctuations of the current transmitted in any section
of the chain satisfies the symmetry (10) with the affinity given by Eq. (30).
If the drift velocity arises from a constant external field F , a possible choice for the transition rates k± is the
following:
k± = k e±φ
with φ =
zeF `
2(L + 1)kB T
(31)
where ` = (L + 1)δ is the total length of the chain, ze the electric charge of the ions, and T the temperature. In the
continuous limit, the drift velocity and the diffusion coefficient (28) are now given by
zeF
+ O(δ 4 )
kB T
D = kδ 2 + O(δ 4 )
v = D
(32)
(33)
and we recover Einstein’s relation for the mobility µ = D/kB T of the particles. In this case, the macroscopic affinity
(30) becomes
~ = ln
A(C)
hAi zeF `
+
hBi
kB T
(34)
in terms of the concentrations imposed at the boundaries and the total potential energy due to the external field in
units of the thermal energy kB T . The current fluctuation theorem for this system can thus be expressed in terms of
7
the fluctuating current crossing a given section of the chain and it satisfies the symmetry (10) with the macroscopic
affinity (34):
i
h R
t
h
i
Prob 1t 0 ji (t0 )dt0 ' α
~
h R
i ' exp A(C)αt
(t → ∞)
(35)
t
Prob 1t 0 ji (t0 )dt0 ' −α
Let us now derive the generating functions for the currents (6) and the entropy production (17) in the purely diffusive
case v = 0 or equivalently k+ = k− . In the nonequilibrium steady state, the stationary probability distribution of the
master equation (1) is given by the multi-Poissonian
Pst (N1 , ..., NL ) =
L
Y
e−hNi i
i=1
1
hNi iNi
Ni !
(36)
with the mean values
hNi i = hAi +
hBi − hAi
i
N +1
(37)
corresponding to a linear profile of concentration between both reservoirs with a gradient of concentration
∇n =
hBi − hAi
V (L + 1)δ
(38)
Therefore, the mean flux of particles moving between the cells i and i + 1 takes the value
J = k (hNi i − hNi+1 i) = −k
hBi − hAi
L+1
(39)
which is nothing else than Fick’s law, j = −D∇n, for the current density j = J δ/V .
Let us first consider the fluctuation theorem for the entropy production. The solution to the eigenvalue problem
M̂η Gη = −q(η)Gη , where M̂η is the operator of Eq. (19), can be obtained by searching the eigenvector which gives
back the stationary solution (36) in the case η = 1 when the master equation (1) is recovered. This solution is given
by
Gη (N1 , ..., NL ) =
L
Y
i=1
e−hNi i
1 η
αi (η)Ni
Ni !
(40)
where
(L + 1 − i)hAiη + ihBiη
L+1
(41)
k hAi + hBi − hAi1−η hBiη − hAiη hBi1−η
L+1
(42)
αi (η) =
The corresponding eigenvalue is
q(η) =
It vanishes q(η) = 0 at thermodynamic equilibrium hAi = hBi and the symmetry q(η) = q(1 − η) of the fluctuation
theorem (18) is clearly satisfied.
The mean entropy production can be obtained by differentiating Eq. (42)
dq
hAi − hBi hAi
(0) = k
ln
dη
L+1
hBi
which corresponds to the entropy production of such process [23]:
Z
Z
di S
(∇n)2
dq
= dr σ(r) = dr D
=
(0)
dt
n
dη
with the local entropy production σ(r).
(43)
(44)
8
In the continuous limit, the number L of sites must go to infinity while the distance δ between them vanishes and
the diffusion coefficient (27) is kept constant. Introducing the area a ≡ V /δ of the cells forming the conductor, we
thus obtain the generating function (42)
D η
η 1−η
q(η) = a
nA + nB − n1−η
(45)
A nB − nA nB
`
in terms of the diffusion coefficient, the total chain length and area, and the concentrations nA = hAi/V and nB =
hBi/V imposed at the boundaries.
Let us now look at the fluctuations in the particles currents. We know from the current fluctuation theorem that
the fluctuation relation will be valid at every section of the chain thanks to the arbitrary choice of the maximal tree
used to generate the current fluctuation theorem. Nevertheless, we can use another approach to obtain this result.
We calculate the generating function of the joint current distribution and see that the current fluctuation theorem is
recovered if we look at a given section of the chain, but which is arbitrary. With this aim, we first solve the eigenvalue
problem where the different elementary processes have been separated by associating a different parameter λi to each
pair of forward and backward transitions. The eigenvalue problem is based on the equation
d
F (N1 , ..., NL ) =
dt
k A e−λ0 F (N1 − 1, N2 , ...) − k N1 F (N1 , N2 , ...)
+k (N1 + 1) e+λ0 F (N1 + 1, N2 ...) −
..
.
−λi
+k (Ni + 1) e
F (..., Ni + 1, Ni+1 − 1, ...) −
+λi
F (..., Ni − 1, Ni+1 + 1, ...) −
+k (Ni+1 + 1) e
..
.
−λL
+k (NL + 1) e
F (..., NL−1 , NL + 1) −
+k B e+λL F (..., NL−1 , NL − 1) −
k A F (N1 , N2 , ...)
k Ni+1 F (..., Ni , Ni+1 , ...)
k Ni F (..., Ni , Ni+1 , ...)
k B F (..., NL−1 , NL )
k NL F (..., NL−1 , NL )
(46)
from which the master equation (1) is recovered if λi = 0 for all i = 0, 1, 2, ..., L. The eigensolution of Eq. (46) has
the form
F (N1 , ..., NL , t) = e
−Q̃t
L
Y
C Ni
i
i=1
(47)
Ni !
with the leading eigenvalue
Q̃(λ1 , ..., λL ) =
L
L
X
X
i
k h
hAi + hBi − hAi exp −
λi − hBi exp
λi
L+1
i=0
i=0
(48)
Since the different currents are not independent, we notice that
P the function (48) does not satisfy any well defined
symmetry because every transformation λi → Bi − λi with i Bi = lnhAi/hBi leaves this function invariant. Nevertheless, if we look at one particular site i of the chain by setting all the others λj equal to zero, we recover the
symmetry of the current fluctuation theorem at any site i. This corresponds to the different possibilities of choosing
the maximal tree and to the elimination of the dependent contributions:
Q(λ) = Q̃(λ, 0, ..., 0) = Q(A − λ)
(49)
~ given by Eq. (34) at zero field F = 0. In this regard, the current and its fluctuations
with the affinity A = A(C)
have a global behavior in the sense that the current is conserved and that the symmetry of the current fluctuation
theorem is expressed in terms of the global affinity, not the local ones.
The current generating function can be redefined by introducing the parameters ηi ≡ λi /Ai , with the local affinities
Ai ≡ ln
hNi i
hNi+1 i
(50)
The generating function becomes
k hAi + hBi
L+1
− hAi1−η0 hN1 iη0 −η1 hN2 iη1 −η2 ...hNL iηL−1 −ηL hBiηL
Q(η1 , ..., ηL ) =
− hAiη0 hN1 iη1 −η0 hN2 iη2 −η1 ...hNL iηL −ηL−1 hBi1−ηL
(51)
9
and the corresponding fluctuation theorem is satisfied [3]:
Q(η1 , ..., ηN ) = Q(1 − η1 , ..., 1 − ηN )
(52)
We recover the generating function (42) of the fluctuating quantity (16) if all the parameters {ηi } take the same value
η:
q(η) = Q(η, ..., η)
(53)
so that the fluctuation theorem (18) is a consequence of the more general relation (52).
As before, we can take the continuous limit and the generating function of the different reactions (52) becomes
Q(η1 , ..., ηL ) = a
L
L
X
X
ni+1 ni+1 i
D h
nA + nB − nA exp
− nB exp −
ηi ln
ηi ln
`
ni
ni
i=0
i=0
(54)
2
with n0 = nA and nL+1 = nB . In the continuous limit, ln( nni+1
) ' δ ∇n
n + O(δ ) and the sum becomes an integral:
i
Z `
hZ `
h
∇n i
∇n io
D n
nA + nB − nA exp
dx η(x)
− nB exp −
dx η(x)
Q[η(x)] = a
`
n
n
0
0
(55)
µ
so that the generating function has now become a functional of η(x). The quantity − ∇n
n = −∇( T ) is nothing else
than the thermodynamic force. The mean local entropy production is given by the functional derivative
D
∇n
δQ =
(nB − nA )
≡ a σ(r)
(56)
δη(x) η(x)=0
`
n
The total mean entropy production is obtained by integration over the whole length
Z
Z `
δQ D
nB
di S
≡ dr σ(r) =
dx
=a
(nB − nA ) ln
dt
δη(x)
`
nA
η(x)=0
0
(57)
which is the entropy production of nonequilibrium thermodynamics. On the other hand, the generating functional of
the currents becomes
h Z `
i
h Z `
io
D n
Q̃[λ(x)] = a
nA + nB − nA exp −
dx λ(x) − nB exp
dx λ(x)
(58)
`
0
0
The flux of the current density, i.e., the intensity of the current, is given by the functional derivative
Z
δ Q̃ Σ
= a j = j · dΣ
δλ(x) λ(x)=0
(59)
As before, the current fluctuation theorem is obtained from Eq. (58) if we consider the current crossing a given section
of the chain by taking λ(x) = λδ(x). The symmetry is valid for the global macroscopic affinity.
IV.
ION CHANNEL IN A MEMBRANE
Here, we want to illustrate the fluctuation theorem for the currents on a biophysical model introduced by Schnakenberg [1]. In typical biological situations, cell membranes separate electrolyte solutions with very different chemical
compositions maintained out of equilibrium by the metabolism. This requires an active transport across the membrane. The active transport is performed by membrane proteins of different kinds which are powered for instance by
ATP hydrolysis. Besides active transport, some membrane proteins are responsible for a passive transport without
expenditure of energy. Since cell membranes are permeable very selectively to particular type of ions, some of these
proteins should be very narrow membrane pores specialized to the transport of particular kinds of ions. This is the
case for open ion channels which selectively transport small ions such as Na+ , K+ , Cl− , or Ca2+ [26].
Following Schnakenberg [1], we can model an ion channel pore by a one-dimensional array of L − 1 stable sites for
one particular type of ions. Assuming the electrostatic repulsion prevents situations with more than one ion occupying
the channel, the model is described by L states: i = 1, ..., L − 1 for the occupation of the L − 1 sites, and the state
i = L for the empty channel. This model is represented by the simple graph of Fig. 2. The transport process is
10
random and described by the master equation (1) with the states ω = i. The transition probabilities per unit time
including an applied electric field are given by
W (i|i + 1)
W (i + 1|i)
W (L|1)
W (L|L − 1)
=
=
=
=
ki eφ , i = 1, 2, ..., L − 1
ki e−φ , i = L, 1, ..., L − 2
c kL eφ
c0 kL e−φ
(60)
where φ = zeF ∆/2LkB T in terms of the membrane thickness ∆, the ionic valency z and the applied electric field F . c
and c0 correspond to the ionic concentrations on the left and right sides of the membrane containing the ion channel.
The quantities k±i are internal rate constants and we used a cyclic ordering: L + 1 ≡ 1.
.
..
..
.
2
L−1
1
L
FIG. 2: Graph of the model of a membrane ion channel.
Calculating the quantity (2) along the unique cycle, we find
W (L|1)W (1|2) · · · W (L − 1|L)
c
~
= 0 e2Lφ = exp A(C)
W (L|L − 1) · · · W (2|1)W (1|L)
c
(61)
~ ≡ ln(c/c0 ) + zeF ∆/kB T corresponds to the affinity given by the overall difference of the electrochemical
where A(C)
potentials of the involved ion across the membrane. It does not depend on the internal rate constants or on the
number of sites L. We notice that the affinity vanishes at equilibrium where c0 = c exp(2Lφ). The Schnakenberg
assumption is verified and the current fluctuation theorem (10) is valid. We thus know on a general ground that
h R
i
t
h
i
Prob 1t 0 ji (t0 )dt0 = α
~
h R
i ' exp αA(C)t
(t → ∞)
(62)
t
Prob 1t 0 ji (t0 )dt0 = −α
where i = 1, ..., L can take any value as any edge can be chosen as a chord (this is a consequence of current conservation
along the cycle). Here, the integrated current corresponds to the signed cumulated number of ions that have crossed
the channel.
We illustrate this result by an analytical calculation in a specific situation. Let us consider the case where the rate
constants take the same value ki = k for i = 1, ..., L, as well as the concentrations c = c0 = 1. It is easy to see that,
in this case, the stationary probability distribution is uniform: Pst (i) = 1/L for all i = 1, ..., L. The motion is then
identical to a biased random walk in the sense that the probability distribution µ(m, t) to make a total displacement
of m states during a time t obeys the equation:
dµ(m, t)
= k eφ µ(m − 1, t) + k e−φ µ(m + 1, t) − 2k cosh φ µ(m, t)
dt
(63)
This equation is solved by introducing the function
F (s, t) ≡
+∞
X
sm µ(m, t)
(64)
m=−∞
which obeys the equation
e−φ
∂t F (s, t) = k e s +
− 2k cosh φ F (s, t)
s
φ
(65)
11
The solution for the initial condition µ(m, t = 0) = δ0m is given by
µ(m, t) = e−2kt cosh φ Im (2kt) emφ
(66)
where we use the fact that the Bessel functions of integer order Im (z) are generated by [27]:
+∞
X
1
exp kt s +
=
sm Im (2kt)
s
m=−∞
(67)
The two first moments of the probability distribution of m(t) are given by
hm(t)i =
+∞
X
m µ(m, t) = 2kt sinh φ
(68)
m=−∞
and
2
h[m(t) − hm(t)i] i = 2kt cosh φ
(69)
so that the mean current across the membrane and the corresponding diffusion coefficient are given by
2k
1
hm(t)i =
sinh φ
Lt
L
(70)
k
1
2
h[m(t) − hm(t)i] i = 2 cosh φ
2L2 t
L
(71)
J = lim
t→∞
and
D = lim
t→∞
The fluctuation theorem (62) can be directly derived by calculating the probability to have a cumulated number n
of ions crossing an edge up to time t. This probability is given by the sum over every site i of the stationary probability
Pst (i) to be on site i multiplied by the probability µ(m, t) to make the number of steps from site i necessary to perform
n passages across the selected edge. As every site is equivalent, a counting gives the aforementionned probability
Z t
1n
0
0
L µ(nL, t)
Prob
ji (t )dt = n =
L
0
+ (L − 1) [µ(nL + 1, t) + µ(nL − 1, t)]
+ (L − 2) [µ(nL + 2, t) + µ(nL − 2, t)]
+ ···
o
+ 1 [µ(nL + L − 1, t) + µ(nL + 1 − L, t)]
(72)
Now, using the following properties of the Bessel functions
Im (z) = I−m (z),
lim Im (z)/Im0 (z) = 1
z→∞
we can calculate the ratio (62) which is given, in the long-time limit t → ∞, by
hR
i
t
h
i
Prob 0 ji (t0 )dt0 = +n
~ n
hR
i ' exp (2Lφ n) = exp A(C)
t
Prob 0 ji (t0 )dt0 = −n
(73)
(74)
as it should. By using again Eq. (72), the generating function of the current is given by
Rt
0
0
1
Q(λ) = lim − lnhe−λ 0 ji (t )dt i
t→∞
t
λ
= 2k cosh φ − cosh φ −
L
(75)
where the limit t → ∞ is required. The current generating function (75) has the symmetry of the fluctuation theorem
Q(λ) = Q(2Lφ − λ)
(76)
12
as it should. The mean current (70) and the diffusion coefficient (71) are given by
1 ∂ 2 Q ∂Q and
D=−
J =
∂λ λ=0
2 ∂λ2 λ=0
The decay rate of the probabilities appearing in the current fluctuation theorem (62)
Z t
1
Prob
ji (t0 )dt0 = α ∼ exp [−H(α)t]
t 0
can here be analytically calculated by taking the Legendre transform of the generating function (75) to get


s
s
2
2
Lα
Lα
Lα

− Lφα + Lα ln 
H(α) = 2k cosh φ − 2k 1 +
+ 1+
2k
2k
2k
(77)
(78)
(79)
which vanishes if α takes the value of the mean current (70) as expected. It satisfies the symmetry property
~ α
H(−α) − H(α) = 2Lφ α = A(C)
(80)
which is equivalent to the fluctuation theorem for the currents according to Eqs. (9)-(10). In this particular case
where ki = k and c = c0 = 1, the current fluctuation theorem (62) can be explicitly verified.
Here, there is only one affinity and one corresponding flux. The Onsager reciprocity relations are thus trivially
satisfied but we can illustrate the obtention of the response coefficients in terms of the generating function of the
current (75). The mean current can be expended in term of the affinity so that
J =
2k
1
1
sinh φ = LA + MA2 + N A3 + · · ·
L
2
6
(81)
where L = k/L2 , M = 0 and N = k/(4L4 ). The linear response coefficient L can be obtained by calculating the
second derivative of the generating function with respect to λ at equilibrium [2, 3, 10]
1 ∂2Q
k
(0; 0) = 2
(82)
2 ∂λ2
L
The third-order response coefficient M can be obtained from the knowledge of the power spectrum in the vicinity of
equilibrium according to Eqs. (12)-(13). In our case, the power spectrum at zero frequency is given by 2D, which
depends quadratically on the affinity near equilibrium:
L=−
2D =
2k
k 2
2k
cosh φ = 2 +
A + ···
L2
L
4L4
(83)
so that the third-order coefficient M vanishes according to Eqs. (12)-(13). Explicit expressions for the higher-order
response coefficients can be derived in a systematic way [2, 3].
Following the reasoning of Subsec. II C, we can also find the generating function for the fluctuating quantity (16).
Defining η ≡ λ/(2Lφ), we recover the generating function q(η) from Eq. (75) as
q(η) = Q(2Lφη) = 2k {cosh φ − cosh [φ(1 − 2η)]}
(84)
One can verify that this function applies even at finite time t, contrary to the function of the current fluctuation
theorem.
V.
FULL COUNTING STATISTICS IN MESOSCOPIC CONDUCTORS
The nature of the current flow at low temperatures through mesoscopic structures has received a lot of attention
during recent years. After initial focus on the conductance, which measures the average number of electrons transmitted in time, there has been an increasing interest for the noise power, a measure for the variance of the transmitted
charge [20]. The next logical step is to study the full distribution function of the charge transmitted through a
mesoscopic conductor. After the pioneering work of Ref. [28], several method have been developed in order to obtain
the full counting statistics (FCS) in mesoscopic conductors. One of them is based on the Keldysh Green’s functions
formalism [29]. In the semiclassical limit, one can also describe the full counting statistics of the currents in terms of a
stochastic path integral [30] or a cascade of Boltzmann-Langevin equations describing the fluctuations of the currents
[31]. The purpose of this section is to apply the current fluctuation theorem in this recent context, based on a master
equation description.
13
A.
Stochastic models
We consider two mesoscopic tunnel junctions coupled in series as described in Ref. [19]. A schematic representation
of the system is given in Fig. 3. When the charging energy Ec = e2 /2C, where C = CL + CR , is larger than the
thermal energy kB T , electron tunneling events across the junction become correlated and give rise to a variety of
phenomena such as Coulomb blockade, leading to steps in the current-voltage characteristics [19]. The voltage in the
central region between the two junctions VM fluctuates depending on the number N of excess electrons in this region.
The voltage drop across the left junction, VL − VM (N ), and the right junction, VM (N ) − VR , are found using classical
electrodynamics to be
CR
Ne
V +
+ Vp
C
C
Ne
CL
V −
− Vp
VM (N ) − VR =
C
C
VL − VM (N ) =
(85)
(86)
The additional voltage Vp has been included to account for any misalignement of the Fermi level in the middle region
with respect to the Fermi levels of the left and right leads when V and N are zero [19]. In this semiclassical description,
the state of the system is determined by the probability P (N, t) to have a number of excess electrons N in the middle
region, which obeys the master equation:
X X
dP (N, t)
=
Wρ (N ± 1|N )P (N ± 1, t) − Wρ (N |N ± 1)P (N, t)
dt
±
(87)
ρ=L,R
The system is controlled by four tunneling rates: the rate for electrons to tunnel into the central region from the
left WL (N |N + 1) and right WR (N |N + 1), and the rate for electrons to tunnel out of the central region to the left
WL (N |N − 1) and right WR (N |N − 1). These rates are computed via Fermi’s golden rule and take the form
Wρ (N |N ± 1) =
1
±e[VM (N ) − Vρ ] − Ec
e2 Rρ 1 − eβ{∓e[VM (N )−Vρ ]+Ec }
(88)
with ρ = L, R and the inverse temperature β = 1/(kB T ). This model successfully reproduces the experimental data
on the complicated structure of the I-V characteristics [19].
The graph of the system is depicted in Fig. 4. Calculating the quantity (2) along the cycle shown in Fig. 4, one
finds
WR (N |N + 1)WL (N + 1|N )
eV
= exp
WL (N |N + 1)WR (N + 1|N )
kB T
(89)
One can thus extract the potential difference applied to the double junction by calculating the quantity (2) along the
cycles of the graph. The current fluctuation theorem can be applied and the distribution of the charge transmitted
to the double junction therefore satisfies
R
t
Prob 1t 0 jρ (t0 )dt0 = α
eV
R
' exp α
t
(t → ∞)
(90)
t
kB T
Prob 1t 0 jρ (t0 )dt0 = −α
where ρ = L, R are the chords which can be chosen to correspond to the left or right junction.
The fluctuation theorem for the currents and its consequences presented in Sec. II can be extended to multi-terminal
systems, i.e. systems with three or more junctions such as the one of Fig. 5. In such cases, the system is crossed by
several independent currents. The master equation ruling the stochastic process is similar to Eq. (87) with the sum
extending to ρ = L, R, G, where G corresponds to the third junction often called the gate (see Fig. 5). The graph
of the circuit of Fig. 5 is depicted in Fig. 6, which shows that a possible maximal tree is composed of the edges G
~ 1 via the edge L
joining all the vertices. Therefore, two cycles can be attached to each one of these edges: the cycle C
with the affinity
~ 1 ) = eVG
A1 = A(C
kB T
(91)
~ 2 via the edge R with the affinity
and the cycle C
~ 2 ) = e(V + VG )
A2 = A(C
kB T
(92)
14
The currents Jα with α = 1, 2 can be expanded in power series of the affinities Aα according to Eq. (11). Since
the conditions of application of the current fluctuation theorem holds, we infer that the reponse coefficients obey the
Onsager reciprocity relations as well as their higher-order generalizations (12)-(15).
The breaking of the Onsager symmetry have been investigated in several theoretical [32, 33] and experimental
works [34, 35]. The main conclusion of these works is that the Onsager symmetry does not apply when considering
the third-order coefficient Mαβγ in presence of a magnetic field. This is not in contradiction with our work as we
predict for this coefficient an expression in terms of the derivatives of the power spectrum (12) but no particular
symmetry. The symmetries we predict concern the even response coefficients Lαβ , Nαβγδ , . . . and present a more
complicated form than the usual Onsager symmetry as they involve some combinations of different quantities such
as the derivatives of the power spectrum. In this regard, we notice that the zero-frequency current noise Sαβ can be
computed from the generating function (7) by taking second derivatives
Z +∞
∂2Q
h[jα (t) − hjα i] [jβ (0) − hjβ i]ist dt
(93)
Sαβ = −
(0, {Aγ }) =
∂λα ∂λβ
−∞
This quantity is of central interest to characterize the fluctuations of the currents in mesoscopic conductors [20].
Moreover, we saw in Sec. II that the knowledge of the power spectrum in the vicinity of equilibrium gives access to
the nonlinear response coefficients and their nonlinear symmetries.
CL
I
M
.
RL
CR
RR
V
FIG. 3: Schematic representations of the two junctions which are denoted L for left and R for right. They have the resistances
RL and RR , and capacitances CL and CR respectively. The two junctions are driven by an ideal constant voltage source V .
I = J is the current intensity.
(1)
...
L
N−1
.
N
L
.
N+1
...
.
...
...
R
R
FIG. 4: Graph associated with the random process of the conductor of Fig. 3.
B.
Resonant-level model
We consider in this subsection a special case of the previous model. We will assume that there is a single resonant
level at energy Ei which can be placed between the chemical potentials µρ in the leads ρ = L, R by changing the
gate voltage [36]. In a strongly interacting case the Coulomb repulsion excludes the double occupancy of the resonant
level. The system can thus only be found in two microscopic states corresponding to the presence or the absence of
an electron. The transition rates take the form [36]
Wρ (0|1) = 2Γρ fρ (Ei )
Wρ (1|0) = Γρ [1 − fρ (Ei )]
(94)
(95)
15
CL
CR
M
•
RL
I1
RR
CG
RG
I2
VG
V
FIG. 5: Schematic representations of the three junctions which are denoted L for left , R for right, and G for gate. They
are characterized by their respective resistances RL , RR , and RG , and capacitances CL , CR , and CG . The three junctions are
driven by two ideal constant voltage source V and VG . I1 = J1 and I2 = J2 are the current intensities in the two main loops
of the circuit.
N−1
...
...
N
L
N+1
N+2
...
.
.
...
1
.
...
.
G
2
...
...
...
R
FIG. 6: Graph associated with the random process of the conductor of Fig. 5.
where fρ (E) ≡ {1 + exp[(E − µρ )/kB T ]}−1 is the Fermi distribution function and the factor 2 comes from the
two possible orientations of the spin of the tunneling electrons. Γρ are the corresponding tunneling rates. Here,
Schnakenberg’s property reads
WL (0|1)WR (1|0)
∆µ
= exp
WR (0|1)WL (1|0)
kB T
(96)
with ∆µ = µL − µR = eV so that the current fluctuation theorem can be applied. For this two-state process, Bagrets
and Nazarov found the following generating function [36]
1
Q(λ) = ΓL + ΓR − (ΓL + ΓR )2 + 4ΓL ΓR (e−λ − 1)f− (Ei ) + (eλ − 1)f+ (Ei ) 2
(97)
where f+ (Ei ) = fR (Ei )[1 − fL (Ei )] and f− (Ei ) = fL (Ei )[1 − fR (Ei )]. One can immediately check that the symmetry
∆µ
Q(λ) = Q
−λ
(98)
kB T
is verified and consequently that the current satisfies Eq. (90). This result illustrates the fluctuation theorem for the
currents in the special case where the Coulomb repulsion is strong enough to prevent more than one electron in the
well.
VI.
CONCLUSIONS AND PERSPECTIVES
In this paper, we have shown how the fluctuation theorem for the currents [2, 3] can be applied to transport in
mesoscopic systems. In such systems, the currents crossing the system over some time interval are affected by fluctuations and are thus statistically distributed according to some probability distributions describing the nonequilibrium
steady state of the system. The probability distributions depend on the specificities of the system and the nonequilibrium constraints imposed on it. Nevertheless, the ratio of the probability for a given value of the currents over the
16
probability for the corresponding negative value turns out to be an exponential function of the time with a coefficient
related to the thermodynamic forces or affinities. The conditions of applications of this general result can be deduced
from Schnakenberg’s network theory [1] which is based on the assumption (2). As a consequence, the fluctuation
theorem for the currents applies to broad classes of mesoscopic systems, as we have here shown.
In the model of diffusion in the chain of L cells, the graph associated with the random process is defined on a lattice
NL , the nodes of which are labeled by L non-negative integers giving the numbers of particles in the L cells of the
chain. The product of the ratios of forward and backward transition rates along the cycles of this graph only depend
on the thermodynamic force or affinity of the nonequilibrium constraints. In this system, there is a single independent
current to which the fluctuation theorem applies. For this model, we have derived an analytical expression for the
generating function of the fluctuating currents between the different cells composing the chain in the particular case
of unbiased diffusion and we have discussed the connection between the fluctuation theorems for the current and the
entropy production.
In Schnakenberg’s biophysical model of transport in a membrane ion channel [1], the graph of the random process is
a simple loop connecting all the states. The conditions (2) are satisfied for the application of the fluctuation theorem,
which allows us to obtain a general property about the probability distribution of the fluctuating currents in spite of
the absence of knowledge on their analytical expression which is only derived in a particular case.
We have also studied stochastic processes describing the fluctuations in the electronic currents across mesoscopic
conductors [19, 20]. Here also, the fluctuation theorem applies to the currents identified by Schnakenberg’s graph
analysis based on Eqs. (2). This provides very general results amenable to experimental verification. In the case
of a system made of three tunnel junctions sustaining two independent currents, we have shown that the nonlinear
response coefficients obey generalizations of Onsager’s reciprocity relations [2, 3]. Our study shows the possibility to
test experimentally these higher-order generalizations of Onsager reciprocity relations in such mesoscopic conductors.
Acknowledgments. The authors thank Professor G. Nicolis for support and encouragement in this research.
D. Andrieux is Research Fellow at the F.N.R.S. Belgium. This research is financially supported by the “Communauté
française de Belgique” (contract “Actions de Recherche Concertées” No. 04/09-312) and the National Fund for
Scientific Research (F. N. R. S. Belgium, contract F. R. F. C. No. 2.4577.04).
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