Jun 2, 2017 - tached at infinity to a heat reservoir Ri with (inverse) temperature βi .... which provides a bridge between microscopic quantum physics and the ...
Quantum Fluctuations of Entropy Production for Fermionic Systems in Landauer-B¨ uttiker State Mihail Mintchev Istituto Nazionale di Fisica Nucleare and Dipartimento di Fisica dell’Universit` a di Pisa, Largo Pontecorvo 3, 56127 Pisa, Italy
arXiv:1706.00561v1 [cond-mat.stat-mech] 2 Jun 2017
Luca Santoni Institute for Theoretical Physics, Princetonplein 5, 3584 CC Utrecht, Netherlands
Paul Sorba LAPTh, Laboratoire d’Annecy-le-Vieux de Physique Th´eorique, CNRS, Universit´e de Savoie, BP 110, 74941 Annecy-le-Vieux Cedex, France The quantum fluctuations of the entropy production for fermionic systems in the LandauerB¨ uttiker non-equilibrium steady state are investigated. The probability distribution, governing these fluctuations, is explicitly derived by means of quantum field theory methods and analysed in the zero frequency limit. It turns out that microscopic processes with positive, vanishing and negative entropy production occur in the system with non-vanishing probability. In spite of this fact, we show that all odd moments (in particular, the mean value of the entropy production) of the above distribution are non-negative. This result extends the second principle of thermodynamics to the quantum fluctuations of the entropy production in the Landauer-B¨ uttiker state. The impact of the time reversal is also discussed. I.
INTRODUCTION
The entropy production is a measure for irreversibility and represents an essential characteristic feature of nonequilibrium systems. In the quantum context the entropy production is fundamental for understanding the deep interplay between microscopic and macroscopic physics and in particular, the second principle of thermodynamics. For this reason the study of the entropy production is receiving a constant attention [1]-[6]. A variety of offequilibrium states [7]-[10] and different physical systems [11]-[15] have been already analysed. In addition, the fluctuation relations which have been established [16][21], provide universal information about the nature of the entropy production and the related time reversal breaking. ✤✜
β1 , µ1
... ✲
L1 ✣✢
✓✏ ✛ ... S ✒✑ L2
✤✜
β2 , µ2
✣✢
FIG. 1: (Color online) Two-terminal junction.
In this article we investigate the entropy production in quantum systems which are schematically shown in Fig. 1. Each of the two semi-infinite leads Li is attached at infinity to a heat reservoir Ri with (inverse) temperature βi ≥ 0 and chemical potential µi ∈ R. The capacity of the reservoirs is assumed to be large enough, so that the processes of emission and absorption of particles do not change the parameters of Ri . A point-like defect is localised at x = 0 and is described by a unitary scattering matrix S. A non-vanishing transmission probability |S12 |2 drives the system away from equilibrium, provided that the temperatures and/or chemical
potentials are different. The departure from equilibrium is characterised by the presence of incoming and outgoing matter and energy flows from the reservoirs Ri . The study of these flows started with the pioneering work of Landauer [22] and B¨ uttiker [23], who developed an exact scattering approach, going far beyond the linear response approximation. The Landauer-B¨ uttiker (LB) framework is the basis of modern quantum transport theory and has been successfully generalised [24]-[26] and applied to the computation of the noise power [27]-[32] and the full counting statistics [33]-[38]. In what follows we apply the LB approach to the study of the entropy production. We concentrate on fermionic systems, discussing the bosonic case elsewhere [39]. The basic ingredients of our investigation are: ˙ x), which de(i) a suitably defined field operator S(t, scribes the entropy production; (ii) a non-equilibrium steady state ΩLB , which captures the physical properties of the system shown in Fig. 1. With this input, all the information about the entropy production is codified in the sequence of n-point correlation functions (n = 1, 2, ...) ˙ 1 , x1 , ..., tn , xn ) = hS(t ˙ 1 , x1 ) · · · S(t ˙ n , xn )iLB , wn [S](t (1) the expectation value h· · ·iLB being computed in the LB state ΩLB . Previous research in the quantum context has been ˙ which describes the mean value mainly focussed on w1 [S], of the entropy production. Adopting quantum field theory methods, we address in this paper the problem of the quantum fluctuations, which are fully characterised ˙ deby (1) with n ≥ 2. The correlation functions wn [S] pend on 2n space-time variables, which complicate the
2 analysis for large n. In order to simplify the problem, we follow the standard approach [33]-[38] to full counting ˙ statistics and investigate the zero frequency limit Wn [S] ˙ integrating the quantum fluctuations over long of wn [S], ˙ take period of time. We show that in this limit Wn [S] the form Z ∞ dω ˙ = Wn [S] Mn (ω) , (2) 2π 0 where ω is the energy and Mn are the moments of a probability distribution ̺. The derivation of ̺ represents a key point of our investigation. In fact, we extract from ̺ the basic information about the entropy production at the microscopic level. The fundamental quantum process, which takes place in our system, is the emission of a particle from the reservoir Ri and the subsequent absorption by Rj . We derive from ̺ the probability pij for this event at any energy ω and determine the corresponding entropy production σij = [(βi − βj )ω − (βi µi − βj µj )] |S12 | .
(3)
In the absence of transmission (S12 = 0) one has σij = 0 in agreement with the fact that the two heat reservoirs are disconnected and the system is in equilibrium. The antisymmetry of σij implies furthermore that the entropy production for emission and absorption of a particle by the same reservoir vanishes, as expected on general grounds. Moreover, σ12 and σ21 have opposite sign which, combined with the fact that p12 6= 0 and p21 6= 0, leads to the conclusion that both processes with positive and negative entropy production are necessarily present at the microscopic level. Nevertheless, we demonstrate below that the process with positive entropy production dominates in the state ΩLB , implying that all moments {Mn (ω) : ω ≥ 0, n = 1, 2, ...} of ̺ obey Mn (ω) ≥ 0 ,
T . As expected [41], ΩLB is not T -invariant and generates another non-trivial off-equilibrium state ΩTLB = T ΩLB with different physical properties because the inversion t 7→ −t converts the process of emission (absorption) in the state ΩLB to absorption (emission) in ΩTLB . We derive and analyse the probability distribution ̺T , correspond˙ ing to the S-fluctuations in ΩTLB . It turns out that the T associated entropy production rates σij differ from (3) T and that the odd moments M2k−1 respect (4) only on a subset of the heat bath parameters. The paper is organised as follows. In the next section we describe the basic physical properties of the system. We also introduce the entropy production operator S˙ and the LB representation incorporating the non-equilibrium properties of the system in Fig. 1. The n-point correlation functions of S˙ in the LB state ΩLB are derived in section III. In section IV we reconstruct the probability distribution ̺ associated with the entropy production, solving the corresponding moment problem. The physical properties of ̺ are also discussed and the role of time reversal is elucidated. Section V is devoted to our conclusions. Finally, the appendices collect some technical details.
II.
PRELIMINARIES
In this section we summarise the basic non-equilibrium features of systems of the type shown in Fig. 1. Throughout the paper we adopt the coordinates {(x, i) , : x ≤ 0, i = 1, 2}, where |x| denotes the distance from the defect and i labels the lead.
A.
Conserved currents and entropy production
(4)
for any value of the temperatures and chemical potentials of Ri . In addition, Mn (ω) vanishes for any ω only at the equilibrium β1 = β2 and µ1 = µ2 . For even n the inequality (4) follows directly from the fact that ̺ is a true probability distribution, whereas for odd n it is a consequence of the specific values (3) of entropy production. It generalises to the quantum fluctuations the result of Nenciu [7] Z ∞ dω ˙ hS(t, x)iLB = M1 (ω) ≥ 0 (5) 2π 0 about the mean value of the entropy production in ΩLB , which provides a bridge between microscopic quantum physics and the second law of thermodynamics. In this respect, the bound (4) represents an extension of the second principle to the quantum fluctuations of the entropy production. The result (4) is an intrinsic characteristic feature of the LB state. In order to illustrate this fact, we investigate the response of the system in Fig. 1 to time reversal
Let us start by fixing the symmetry content. We consider in this paper physical systems in which both the particle number and the total energy are conserved. Accordingly, the correlation functions are invariant under global U (1) transformations and time translations. These symmetries imply the existence of a conserved particle and energy currents (jt , jx ) and (θtt , θxt ). Local conservation implies ∂t jt (t, x, i) − ∂x jx (t, x, i) = 0 , ∂t θtt (t, x, i) − ∂x θxt (t, x, i) = 0 .
(6) (7)
In order to generate global conserved charges from jt and θtt , which define the particle number and total energy respectively, one must impose the Kirchhoff’s rules 2 X i=1
jx (t, 0, i) =
2 X
θxt (t, 0, i) = 0 ,
i=1
which are assumed in what follows.
(8)
3 The total energy of our system has two components: heat energy and chemical energy. Since the chemical energy density is given by µi jt (t, x, i), for the heat density one has [40] qt (t, x, i) = θtt (t, x, i) − µi jt (t, x, i) .
(9)
Accordingly, the heat current reads qx (t, x, i) = θxt (t, x, i) − µi jx (t, x, i) .
(10)
Following [40], we introduce at this point the entropy production operator [3, 7] ˙ x) = − S(t,
2 X
βi qx (t, x, i) .
(11)
i=1
The definition (11) involves the non-equilibrium heat currents flowing in the leads Li and the equilibrium temperatures βi of the heat reservoirs. The operator (11) will be the main subject of our investigation below. It is instructive at this stage to describe the basic physical process taking place in the system in Fig. 1 and generating the entropy production. The conservation laws (6,7) obviously imply the local heat current conservation ∂t qt (t, x, i) − ∂x qx (t, x, i) = 0 .
(12)
However, if µ1 6= µ2 the heat current violates the Kirchhoff rule. One has in fact 2 X i=1
qx (t, 0, i) = (µ1 − µ2 )jx (t, 0, 1) .
(13)
Since the total energy is conserved, both the heat and chemical energies are in general not separately conserved. Therefore, for µ1 6= µ2 the junction in Fig. 1 operates as energy converter without dissipation [41]. The two possible regimes are controlled by the expectation value of the operator Q˙ = −
2 X
qx (t, x, i)
(14)
i=1
˙ < 0 the in the underlying non-equilibrium state. If hQi junction transforms heat to chemical energy. The op˙ > 0. A detailed posite process takes place if instead hQi study of this phenomenon of energy transmutation in the LB sate ΩLB has been performed in [41]. The above general considerations apply to the system in Fig. 1 with any dynamics preserving the particle number and total energy. In this sense they are universal. For concretely evaluating the quantum fluctuations as˙ one should fix the dynamics and the sociated with S, non-equilibrium state. This is done in the next subsection.
B.
Dynamics and the LB state - the Schr¨ odinger junction
Non-equilibrium systems of the type in Fig. 1 behave in a complicated way and the linear response or other approximations are usually not enough for fully describing their complexity. For this reason the existence of models, which incorporate the main non-equilibrium features, while being sufficiently simple to be analysed exactly, is conceptually very important. One such example is provided by particles, which are freely moving along the leads and interact only in the junction x = 0. This hypothesis accounts remarkably well [42] for the experimental results [43] about the noise in mesoscopic conductors and has been recently confirmed [44] even in the case of fractional charge transport in quantum Hall samples. At the theoretical side, our previous analysis in [41], [31], [38] and [32] shows that this setup represents an exceptional testing ground for exploring general ideas about quantum transport. One concrete realisation of the above scenario is the Schr¨odinger junction, where the dynamics along the leads is fixed by (the natural units ~ = c = kB = 1 are adopted throughout the paper) 1 2 (15) ∂x ψ(t, x, i) = 0 , i∂t + 2m supplemented by the standard equal-time canonical anticommutation relations. The junction represents physically a point-like defect localised at x = 0. The associated interaction determines the scattering matrix S, which is fixed by requiring that the bulk Hamiltonian −∂x2 /2m admits a self-adjoint extension in x = 0. All such extensions are defined [45]-[46] by the boundary condition lim−
x→0
2 X j=1
[λ(I − U)ij + i(I + U)ij ∂x ] ψ(t, x, j) = 0 , (16)
where I is the identity matrix, U is a generic 2×2 unitary matrix and λ > 0 is a parameter with dimension of mass. Eq. (16) guaranties unitary time evolution and implies [45]-[46] the scattering matrix S(k) = −
[λ(I − U) − k(I + U)] , [λ(I − U) + k(I + U)]
U ∈ U (2) ,
(17)
k being the particle momentum. Equation (17) defines a meromorphic function in the complex k-plane. The family (17) involves scale invariant and therefore k-independent elements. These, so called critical points, are given by S = U Sd U ∗ ,
U ∈ U (2) ,
Sd = diag(1, −1) ,
(18)
where ∗ stands for Hermitian conjugation. In the additional two isolated points S = ±I the transmission probability vanishes and the system is in equilibrium. It has
4 been argued in [48] that scale invariance preserves the universal features of one-dimensional quantum transport, leading at the same time to relevant simplifications. In ˙ x)i fact, we derive in section III.A the mean value hS(t, LB at criticality in explicit form. The scattering states associated to (17) read [49] χ(k; x) = e−ikx I + eikx S∗ (k) , k ≥ 0, (19)
Postponing the discussion of the general case, let us assume for the moment that S(k) has no bound states. Then, the solution of the quantum boundary value problem (15,16) is given by ψ(t, x, i) =
2 Z X j=1
0
∞
dk −iω(k)t e χij (k; x)aj (k) , 2π
Both (15) and (16) are invariant under global U (1) phase transformations and time translations. The relative conserved particle and energy currents have the well known form jx (t, x, i) =
1 [(∂t ψ ∗ ) (∂x ψ) + (∂x ψ ∗ ) (∂t ψ) 4m − (∂t ∂x ψ ∗ ) ψ − ψ ∗ (∂t ∂x ψ)](t, x, i) ,
θxt (t, x, i) =
(22)
respectively. Plugging the solution (20) in (21,22), one can express the heat current (10) and therefore the entropy production field operator in terms of the generators of A+ . The result is
Z ∞ Z i dk ∞ dp it[ω(k)−ω(p)] ˙ S(t, x) = e 4m 0 2π 0 2π 2 2 n o X X × βi [2µi − ω(k) − ω(p)] χ∗li (k; x) [∂x χij ] (p; x) − [∂x χ∗li ] (k; x)χij (p; x) aj (p) . a∗l (k) l,j=1
(21)
and
(20)
where ω(k) = k 2 /2m is the dispersion relation and the operators {ai (k), a∗i (k) : k ≥ 0, i = 1, 2} generate a standard anti-commutation relation algebra A+ .
i [ψ ∗ (∂x ψ) − (∂x ψ ∗ )ψ] (t, x, i) , 2m
(23)
i=1
˙ x) as a quadratic element of the This equation defines S(t, algebra A+ . In order to extract the physical information we are interested in, one must fix a representation of A+ . The physical setup in Fig. 1 suggests to adopt the LB representation of A+ , which generalises the equilibrium Gibbs representation to the case of systems driven away from equilibrium by a particle and energy exchange with more then one heat reservoir. A field theoretical construction of the Hilbert space {HLB , (· , ·)} of this representation is given in [49]. For deriving the expectation values of (23) one can concentrate on the 2n-point function (ΩLB , a∗l1 (k1 )am1 (p1 ) · · · a∗ln (kn )amn (pn )ΩLB ) ha∗l1 (k1 )am1 (p1 ) · · · a∗ln (kn )amn (pn )iLB ,
≡
III.
ENTROPY PRODUCTION CORRELATION FUNCTIONS
A.
The one-point function
It is natural to start with the one point function ˙ x)iLB , which gives the mean value of the entropy hS(t, production in the LB state ΩLB . Using (23) and (A3) for n = 1, one easily obtains the integral representation (5) with M1 (ω) = τ (ω) [γ2 (ω) − γ1 (ω)][d1 (ω) − d2 (ω)] .
(24)
which can be represented as a kind of Slater determinant, whose explicit form (A3) is given in appendix A. Using (A3) we derive in what follows the correlation functions of the operator S˙ in the LB representation HLB and discuss the physical implications.
(25)
Here √ τ (ω) = |S12 ( 2mω)|2
(26)
is the transmission probability, γi (ω) = βi (ω − µi ) ,
i = 1, 2
(27)
and di (ω) is the Fermi distribution di (ω) =
1 1 + eγi (ω)
(28)
5 of the reservoir Ri . One can easily check now that both square brackets [· · · ] of (25) have always the same sign or vanish simultaneously. Therefore, M1 (ω) ≥ 0 ,
(29)
which proves (4) for n = 1. In addition, M1 (ω) = 0 for any ω implies the equilibrium regime β1 = β2 and µ1 = µ2 . ˙ x)i , given by It is worth mentioning that hS(t, LB (5, 25), is both time and position independent. The
˙ x)i = (λ2 − λ1 ) hS(t, LB
t-independence follows from the energy conservation, whereas the x-independence is a consequence of the heat current conservation (12). Clearly, these are peculiar ˙ The study properties of the one-point function w1 [S]. ˙ of {wn [S] , : n ≥ 2} in the next subsection reveals a more complicated behaviour. Let us explore in conclusion the scale invariant regime. At criticality the transmission probability τ is constant and plugging (25) in (5) one can perform the ωintegration explicitly. The result is
τ τ 1 1 1 1 λ1 λ2 , (30) − −e −e ln 1 + eλ2 − ln 1 + eλ1 + (β1 − β2 ) Li Li 2 2 2π β2 β1 2π β12 β22
where λi ≡ βi µi are dimensionless parameters and Li2 is the dilogarithm function.
on µi . One has in fact ˙ x)iβ1 =β2 = lim hS(t, LB
β→∞
0 ,
−µ1 ln 2 2π
∞ ,
,
for µ1 < µ2 < 0 , for µ1 < µ2 = 0 , for 0 < µ1 < µ2 , (31)
as shown in Fig. 2. If µ1 = µ2 = µ the behaviour of the entropy production is illustrated in Fig. 3. In this case the origin of the entropy increase is the difference between the temperatures of the two heat reservoirs. One finds ˙ x)iµ1 =µ2 = 0 , lim hS(t, LB
µ→−∞
FIG. 2: (Color online) Entropy production (in units of 1/β) for β1 = β2 = β and different values of µi .
˙ x)iµ1 =µ2 = lim hS(t, LB
µ→∞
π(β1 − β2 )2 (β1 + β2 )τ , 12β12 β22
(32)
as displayed in Fig. 3. Finally, for τ = 1 the defect at x = 0 is absent and one obtains from (30) the mean entropy production of two heat reservoirs connected with a homogeneous infinite lead.
B.
The n-point function
First of all we observe that the correlation function ˙ depends on the time differences wn [S] tˆk ≡ tk − tk+1 , FIG. 3: (Color online) Entropy production (in units of 1/β) for µ1 = µ2 = µ and different values of βi .
Some particular cases are of interest. For β1 = β2 = β the mean entropy production (30) is entirely driven by the chemical potentials and vanishes at high temperatures β → 0. The behaviour at low temperatures depends
k = 1, ..., n − 1 ,
(33)
which is a consequence of the time translation invariance of ΩLB . Since the defect at x = 0 violates translation ˙ depends on all the coordinates invariance in space, wn [S] {xl : l = 1, ..., n} separately. In order to simplify the analysis and avoid those variables, which are marginal for the entropy production, we perform the zero frequency limit adopted already in the classical studies [27]-[31] of
6 quantum noise. In this limit the quantum fluctuations are integrated over long period of time, which both sim˙ and preserves the basic charplifies the structure of wn [S]
˙ 1 , ..., xn ; ν) = Wn [S](x
Z
∞
−∞
dtˆ1 · · ·
Z
∞
acteristic features of the entropy production. Concretely, we consider for n ≥ 2 the Fourier transforms
ˆ ˆ ˙ 1 , x1 , ..., tn , xn ) , dtˆn−1 e−iν(t1 +···tn−1 ) wn [S](t
−∞
and perform the zero-frequency limit ˙ = lim Wn [S](x ˙ 1 , ..., xn ; ν) . Wn [S] ν→0+
(35)
Using the definition (23) of S˙ and the correlation function (A3), one finds Z ∞ dω ˙ Wn [S] = [γ2 (ω) − γ1 (ω)]n Dn (ω) . (36) 2π 0 The explicit form of the factor Dn (ω) in the integrand is given by equation (B1) in appendix B. It is a sum of determinants, which depend on the scattering matrix (17) and the Fermi distribution (28), but is xi -independent. Therefore the position dependence drops out in the limit ˙ is a function of S and (βi , µi ). It has (35) and Wn [S] been shown in [32] that the bound states of S, if they exists, do not contribute in the limit (35) as well. Despite of these significant simplifications, at the first sight the integrand of (36) for generic n might look still complicated. As shown in appendix B however, this is not the case and the final expression reads Z ∞ dω ˙ = Wn [S] Mn (ω) , (37) 2π 0
n ≥ 2,
(34)
Our goal in the next section will be to show that the integrands (38,39) represent indeed the moments of a probability distribution and to reconstruct this distribution.
IV. PROBABILITY DISTRIBUTION GOVERNING THE ENTROPY PRODUCTION
The fluctuations of a quantum observable give rise in general to a quasi-probability distribution. Familiar examples are the Wigner function [50], some distributions stemming from coherent states in quantum optics [51, 52] and more recent examples associated with timeintegrated observables [53, 54] in the context of full quantum statistics [33]-[36]. In this section we show that S˙ generates in the LB state ΩLB a true probability distribution ̺. The idea is to reconstruct ̺ from the moments (38,39), solving the underlying moment problem.
A.
Solution of the moment problem
with M2k−1 = τ k (γ2 − γ1 )2k−1 c1 , M2k = τ k (γ2 − γ1 )2k c2 .
(38) (39)
Here k = 1, 2, ..., the ω-dependence of all factors has been suppressed for conciseness and the following combinations c1 ≡ d1 − d2 ,
c1 ≡ d1 + d2 − 2d1 d2 ,
(40)
have been introduced for convenience. The explicit form (38,39) of the integrands Mn represents a key point of our analysis of the fluctuations of the entropy production. First of all, from (38,39) one infers the result (4) announced in the introduction, namely that all Mn are nonnegative. In fact, the argument about the positivity of M1 applies actually for all odd values of n. The inequality (39) for even values of n follows instead from c2 =
eγ1 + eγ2 ≥ 0. (1 + eγ1 ) (1 + eγ2 )
(41)
We are looking for a function ̺ with domain D such that Z Mn = dσ σ n ̺(σ) , n = 0, 1, ... , (42) D
where Mn are given for n ≥ 1 by (38,39) and M0 = 1
(43)
provides a normalisation condition. The parameter σ describes the entropy production. There exist [55] three possible choices for the domain D of σ: the whole line D = R, the half line D = R+ and a compact interval D = [a, b]. In order to determine D we have to investigate the Hankel determinants M0 M1 · · · Mn M1 M2 · · · Mn+1 Hn ≡ . (44) . .. .. .. .. . . . Mn Mn+1 · · · M2n
7 A necessary and sufficient condition for the existence of ̺ on R is [55] Hn ≥ 0 ,
∀ n = 1, 2, ...
(45)
Using (38,39,43) one gets 2
τ c21 ) ,
H0 = 1 , H1 = τ (γ1 − γ2 ) (c2 − 3 H2 = τ (γ1 − γ2 )6 (1 − c2 )(c22 − τ c21 ) ,
Hn≥3
(46) = 0 . (47)
Combining the inequalities 0 ≤ c2 ≤ 1 ,
Summarising, the entropy production σ in the LB state ΩLB gives rise to the so called Hamburger moment problem D = R. Moreover, since Hn≥3 = 0 the general theory [55] implies that ̺ is fully localised at three different values of σ. Once the domain D has been determined, the explicit form of the distribution ̺ can be recovered [55] by performing the Fourier transform
̺(σ) = c22 − c21 ≥ 0 ,
(48)
which follow directly from the explicit form (40) of ci and using 0 ≤ τ ≤ 1, one gets that both H1 and H2 are non-negative. Since in addition, M M2 = τ 2 (γ1 − γ2 )4 (c21 τ − c22 ) ≤ 0 , (49) H′2 ≡ 1 M2 M3
the domains R+ and [a, b] are excluded [55].
Z
∞
−∞
dλ −iλσ e ϕ(λ) 2π
of the generating function
ϕ(λ) =
∞ X (iλ)n Mn . n! n=0
̺(σ) =
√ √ √ √ 1 1 (c2 − c1 τ )δ σ − (γ1 − γ2 )] τ + (1 − c2 )δ(σ) + (c2 + c1 τ )δ σ − (γ2 − γ1 )] τ . 2 2
Equation (53) confirms that the entropy production is indeed localised in three points on the σ-line. It is convenient to adopt at this stage the variables σij defined by (3), which read √ σij = (γi − γj ) τ (54) in terms of γi . Then ̺ can be rewritten the form ̺(σ) = p12 δ(σ − σ12 ) + p δ(σ) + p21 δ(σ − σ21 )
(55)
with p12 =
√ 1 (c2 − c1 τ ) , 2
p21 =
√ 1 (c2 + c1 τ ) , 2
p = 1 − c2 .
(56) Here pij is the probability of emission of a particle by the reservoir Ri and absorption by Rj , whereas p is the probability for emission and absorption by the same reservoir R1 or R2 . One can easily show in fact that p12 + p + p21 = 1 ,
pij ∈ [0, 1] ,
p ∈ [0, 1] ,
(57)
implying that ̺ is a true probability distribution. As anticipated in the introduction, we have shown that both processes with positive and negative entropy production appear at the quantum level. It is quite intuitive
(51)
Employing (38,39,43) one finds
√ √ √ ϕ(λ) = 1 + i c1 τ sin λ(γ2 − γ1 ) τ + c2 cos λ(γ2 − γ1 ) τ − 1 ,
and
(50)
(52)
(53)
that if the transport of a particle from the red to the blue reservoir in the isolated system in Fig. 1 increases the entropy, the opposite process is decreasing it. The crucial point is that according to (56) both these events have a non-vanishing probability and are present without invoking any time reversal operation. Since ̺ is not smooth but a generalised function, in order to illustrate graphically its behaviour it is convenient to introduce the δ-sequence 2 2 ν δν (σ) = √ e−ν σ , ν > 0, (58) π and consider the smeared distribution ̺ν (σ) = p12 δν (σ − σ12 ) + p δν (σ) + p21 δν (σ − σ21 ) . (59) As well known, for ν → ∞ one has ̺ν → ̺ in the sense of generalised functions. The plots of ̺ν for finite values of ν nicely illustrate the physics behind the distribution ̺. One example is reported in Fig. 4. We see that the events with positive entropy production in the LB state dominate those with negative one. This feature, which persists for any ν > 0, holds also in the limit ν → ∞ and provides an intuitive explanation of the bound Mn ≥ 0. It is instructive in this point to derive the ratio P+ /P− where P± is the probability to have positive/negative en-
8 where |ηT | = 1 and T is an anti-unitary operator in HLB with T 2 = I. Using (21,22) one easily gets
FIG. 4: (Color online) The smeared distribution ̺ν with ν = 4/6, γ1 = 21, γ2 = 1 and τ = 1/4.
tropy production. Without loss of generality one can assume for this purpose that σ12 > 0. Then √ p12 c2 − c1 τ P+ √ = = P− p21 c2 + c1 τ √ (60) √ √ (1 − τ ) + (1 + τ ) eσ12 / τ √ . = √ √ (1 + τ ) + (1 − τ ) eσ12 / τ Equation (60) generalises the fluctuation relation, discussed in [16]-[21], to the case of a quantum point-like defect characterised by the transmission probability τ . In the limit τ → 1 the defect disappears and one recovers from (60) the result of Crooks [16] lim
τ →1
P+ = eσ12 , P−
(61)
originally obtained in the context of stochastic dynamics. Summarising, the probability distribution (53) fully describes the entropy production zero-frequency fluctuations in the LB state ΩLB . It is natural to expect that the behaviour of ̺ depends on the choice of this state. This expectation is confirmed in the next subsection, where ˙ the S-fluctuations in the state generated from ΩLB by time reversal are explored.
B.
The impact of time reversal
T jx (t, x, i) T −1 = jx (−t, x, i) , (63) T θtx (t, x, i) T −1 = −θtx (−t, x, i) , (64) Since hθtx (t, x, i)iLB 6= 0, the overall minus sign in the right hand side of (64) implies that T ΩLB 6= ΩLB . Therefore, T generates another state ΩTLB = T ΩLB ∈ HLB of the system in which the processes of emission and absorption are inverted with respect to those in ΩLB . This crucial physical difference between ΩLB and ΩTLB has a ˙ deep impact on the S-fluctuations, which are codified in the correlation functions ˙ 1 , x1 , ..., tn , xn ) = hS(t ˙ 1 , x1 ) · · · S(t ˙ n , xn )iT wnT [S](t LB ˙ 1 , x1 ) · · · S(t ˙ n , xn ) T Ω ) , ≡ (T Ω , S(t LB
(65)
where (· , ·) is the scalar product in HLB . One can naively think that S˙ simply changes sign under time reversal, but this is not the case because according to (64) the particle component of the heat current (10) preserves its sign. The study of (65) closely follows our previous analysis and is not reported for conciseness. For the probability distribution one finds T T ̺T (σ) = p12 δ(σ − σ12 ) + p δ(σ) + p21 δ(σ − σ21 ) , (66)
where T σij = − [(βi − βj )ω + (βi µi − βj µj )] |S12 | √ = σij − 2(βi − βj )ω τ .
T ψ(t, x, i)T −1 = ηT ψ(−t, x, i) ,
MT2k−1
(62)
(67)
Comparing (67) with (54), one concludes that the entropy production, associated with the elementary emission-absorption processes in ΩTLB and ΩLB , is different. This fact induces a modification of the moments according to MT2k−1 = τ k [(γ2 − γ1 ) − 2(β2 − β1 )ω]2k−1 c1 ,
(68)
= τ [(γ2 − γ1 ) − 2(β2 − β1 )ω]
(69)
MT2k
As before, we consider the field ψ defined by (20) in the LB representation {HLB , (· , ·)} of the algebra A+ . The time reversal operation acts as usual
LB
k
2k
c2 ,
with k = 1, 2, .... As before, MT2k are non-negative in the whole range of heat bath parameters, but this is no longer true for the odd moments MT2k−1 . One finds for instance
1 1 τ k [(β1 − β2 )ω]2k−1 ≤ 0, − β ω β ω 1+e 2 1+e 1 = 2k−1 1 1 k τ [β(µ1 − µ2 )] ≥ 0, − 1+eβ(ω−µ1 ) 1+eβ(ω−µ2 )
for µ1 = µ2 = 0 , for β1 = β2 = β ,
(70)
9 which simply reflects the fact that the energy current changes sign (64) under time reversal reversal, whereas the particle current is invariant (63). From (70) it follows in particular that ˙ x)iT ≤ 0 , hS(t, LB ˙ x)iT ≥ 0 , hS(t, LB
for µ1 = µ2 = 0 ,
(71)
for β1 = β2 = β .
(72)
Therefore, the state ΩTLB respects the second principle of thermodynamics only on a subset of the heat bath parameters. This result confirms the distinguished role of the LB state ΩLB in non-equilibrium quantum thermodynamics. V.
OUTLOOK AND CONCLUSIONS
The present paper pursues further the quantum field theory analysis of the physical properties of the LB nonequilibrium steady state. It focuses on the quantum fluctuations of the entropy production in the fermionic system shown in Fig. 1. The junction acts as a nondissipative converter of heat to chemical potential energy and vice versa. During the energy transmutation, particles are emitted and absorbed by the heat reservoirs, which induces a non-trivial entropy production. Processes with positive, vanishing and negative entropy production occur at the quantum level. In order to characterise the relative impact of these events, we investigate the correlation functions of the entropy production operator in the LB state. The one-point function describes the mean entropy production, whereas the n-point functions with n ≥ 2 capture the relative fluctuations. We discover that in the zero frequency limit these fluctuations generate a true probability distribution, whose moments are all positive. Since the first moment describes
ha∗l1 (k1 )am1 (p1 ) · · · a∗ln (kn )amn (pn )iβ,µ =
the mean entropy production, this remarkable property can be interpreted as a kind of extension of the second principle of thermodynamics to the non-equilibrium quantum fluctuations in the LB state. The response of the system to time reversal shows that the positivity of the moments in the whole domain of heat bath parameters is a characteristic feature of the LB state. The results of this paper can be generalised in several directions. Along the above lines one can study multi terminal systems as well as the Tomonaga-Luttinger liquid away from equilibrium [56, 57]. The impact of the quantum statistics on the entropy production represents also a challenging problem from both physical and mathematical point of view. We are currently investigating [39] this impact in the bosonic version of the fermion system studied above.
Acknowledgments
The work of LS is supported by the Netherlands Organisation for Scientific Research (NWO). Appendix A: Correlation functions in the LB representation
In their original work [22, 23] Landauer and B¨ uttiker derived the two- and four-point correlation functions of {ai (k), a∗i (k) : k ≥ 0, i = 1, 2} in the LB representation {HLB , (· , ·)} using quantum mechanical tools. If one is interested in generic n-point functions, it is more convenient to adopt the formalism of second quantisation developed in [49]. The correlation function (24), needed for the derivation of the entropy production fluctuations, is defined in this formalism by
1 −K ∗ Tr e al1 (k1 )am1 (p1 ) · · · a∗ln (kn )amn (pn ) , Z
ki > 0, pi > 0 ,
(A1)
where K=
Z
0
∞
2 dk X γi [ω(k)]a∗i (k)ai (k) , 2π i=1
Z = Tr e−K .
Referring for the details to [38, 49], we report the final result ∆l1 m1 (k1 , p1 ) ∆l1 m2 (k1 , p2 ) · · · e l2 m1 (k2 , p1 ) ∆l2 m2 (k2 , p2 ) · · · −∆ ha∗l1 (k1 )am1 (p1 ) · · · a∗ln (kn )amn (pn )iLB = .. .. .. . . . −∆ e ln m2 (kn , p2 ) · · · e ln m1 (kn , p1 ) −∆ Here
(A2)
. ∆ln mn (kn , pn ) ∆l1 mn (k1 , pn ) ∆l2 mn (k2 , pn ) .. .
(A3)
where dl (ω) is the Fermi distribution (28) and ∆lm (k, p) ≡ 2πδ(k − p)δlm dl [ω(k)] , e lm (k, p) ≡ 2πδ(k − p)δlm del [ω(k)] , ∆
(A4) (A5)
del (ω) = 1 − dl (ω) =
eγl (ω) . 1 + eγl (ω)
(A6)
10 Appendix B: Derivation of Dn
The key observation now is that Dn can be represented in the form
In this appendix we omit the ω-dependence for conciseness. After some algebra, using (A3) one finds for the factor Dn in the integrand of (36) the following sum of determinants Ti1 i1 di1 Ti2 i1 di2 · · · Tin i1 din 2 −Ti1 i2 dei1 Ti2 i2 di2 · · · Tin i2 din X Dn = . .. .. .. .. . . . . i1 ,...,in =1 −T de −T de · · · T d
Tr e−L J n , Dn = Tr (e−L )
i1 in i1
i2 in i2
in in in
(B6)
which can be verified by explicit computation using (B4,B5). One has at this point that ∞ X Tr e−L eiηJ (iη)n Dn = . n! Tr (e−L ) n=0
(B7)
(B1)
Here di and dei are given by (28) and (A6) and the matrix T is defined in terms of S by T11 = −T22 = |S12 |2 ≡ τ , T12 = T21 = −S11 S21 ,
(B2) (B3)
the bar indicating complex conjugation. In order to compute Dn we introduce an auxiliary algebra of fermionic oscillators generated by {ai , a∗i : i = 1, 2}, which satisfy [ai , a∗j ]+ = δij ,
[ai , aj ]+ = [a∗i , a∗j ]+ = 0 .
(B4)
Let us introduce also the quadratic operators L=
2 X i=1
γi a∗i ai ,
J=
2 X
a∗i Tij aj .
(B5)
The right hand side of (B7) has been previously computed [38] for the full counting statistics of the particle current (21). Using the result of [38], one finds ∞ X √ √ √ (iη)n Dn = 1 + ic1 τ sin(η τ ) + c2 cos(η τ ) − 1 , n! n=0 (B8) were ci are defined by (40). From (B8) it follows that
1, Dn = τ k c 1 τ k c , 2
n = 0, n = 2k − 1 , n = 2k ,
k = 1, 2, ... , k = 1, 2, ...
(B9)
implying the result (38,39).
i,j=1
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