distribution function [136, 144, 116]. Concerning optical x-ray edge experiments P(W) can be associated with absorption and emission spectra, see Secs. 3.3 and 3.5.
Work distribution functions are related to dynamical correlation functions via Fourier transformation [167] P(W) = Z dt eiW tG(t), G(t) =eiH(λi)te−iHH(λf)t, h. . .i= Tr e−βH(λi) Zi . . . , (3.8)
where HH(λf) = U†(tf)H(λf)U(tf) with U the full time evolution operator obeying the operator equation analogue to the Schr¨odinger equation, see Eq. (3.5). The simplest case and the most important one for what follows is that of a quench where a parameter of the Hamiltonian is changed suddenly. For such a particular protocol we have that ti =tf = 0 yielding a trivial time evolution operator U = 1. The dynamical correlation function G(t) then reduces to the expectation value of two counter propagating time evolution operators
G(t) = eiH(λi)te−iH(λf)t. (3.9)
Considering the contribution in the Hamiltonian that is suddenly switched on as a pertur- bation G(t) may be interpreted as a measure for the “dynamical“ strength of the pertur- bation and its influence on the system’s time evolution - G(t) compares the dynamics of the unperturbed system with the perturbed one. This becomes even more transparent in the zero temperature limit when the system is initially prepared in the ground state |ψ0i
of the Hamiltonian H(λi)
G(t) =hψ0|eiH(λi)te−iH(λf)t|ψ0i. (3.10)
where the dynamical correlation function G(t) reduces to the overlap of two states time- evolved with different Hamiltonians. In this way it quantifies the stability of quantum motion [131]. The modulusL(t) =|G(t)|2ofG(t) is termed the Loschmidt echo [78, 89, 162]
and may be associated with a return probability in the spirit of Loschmidt’s gedanken
experiment about microreversibility in all physical laws. For more details see Sec. 4.
3.2
Work fluctuation theorems
If an external force acting on the system under study is sufficiently weak for a perturbative treatment, the system’s properties are completely determined by linear response theory where the influence of the external perturbation is fully encoded in the equilibrium fluc- tuations. This manifests in the famous fluctuation-dissipation theorem [24]. For external forces that cannot be considered as weak the fluctuation dissipation theorem is violated and the system is driven out of equilibrium. It has been discovered recently that there ex- ists a more general class of fluctuation theorems that are valid also for systems arbitrarily far away from equilibrium [25]. Among these are the so-called work fluctuation theorems such as the Jarzynski equality [80], the Crooks relation [32], and the Bochkov-Kuzovlev theorem [19, 27].
W
av∆F
W
P(W)
Figure 3.3: Schematic plot of a work distribution function P(W). The average work
performed Wav ≥∆F is bounded from below by the free energy difference ∆F due to the
second law of thermodynamics. There are individual but unlikely processes withW <∆F where work can be extracted from the system. Note, however, that this does not correspond to a ”violation“ of the second law of thermodynamics, see text.
3.2.1
Jarzynski equality
Preparing the system of interest in a canonical state at a temperature T the second law of thermodynamics gives a lower bound on the average work performed
Wav =
Z
dW W P(W)≥∆F. (3.11)
set by ∆F where ∆F =F(λf, T)−F(λi, T) is determined by the difference in free energies
of the system at the same temperature T but for different parameters λf and λi. The
lower bound is reached for reversible processes. In Fig. 3.3 a schematic plot of a work
distribution function is shown. Although the average work Wav is larger than ∆F there
are individual processes withW <∆F and a nonzero probability to extract work from the system. Note, however, that this does not correspond to a ”violation“ of the second law of thermodynamics as the average work still obeys Eq. (3.11).
There is, however, a much stronger result than the lower bound given by the second law of thermodynamics that is called the Jarzynski equality [80]
he−βWiP =e−β∆F, h. . .iP = Z
dW . . . P(W). (3.12)
It relates the work distribution function to free energy differences not by a lower bound but via an equality. First shown for classical systems [80], the Jarzynski equality has later been proven also for quantum systems [123]. It connects an inherently nonequilibrium quantity - the work distribution function P(W) - with the purely equilibrium quantities temperature
T = β−1 and free energies ∆F. Weighting the work distribution with an exponential
exp[−βW] washes out all the nonequilibrium details. This is even more remarkable because Eq. (3.12) is valid for arbitrary nonequilibrium protocols. The probability distribution P(W) depends on the precise path in parameter space whereas its weighted integral only
3.2 Work fluctuation theorems 53
λ
0λ
f F B∆F
W
P
B(−W)
P
F(W)
W
avF−W
avBFigure 3.4: Within the protocol (F) of the forward process the system parameterλis varied fromλ0 to a value λf. For the backward process (B) the protocol is executed in the time- reversed way. The right-hand side shows a schematic plot of the forward work distribution function PF(W) and the work distribution function PB(−W) for the backward process at minus the work performed. As a consequence of the Crooks relation both distributions intersect each other exactly at one point whereW = ∆F irrespective of the nonequilibrium protocol. For more details see text.
about the initial preparation and a hypothetical equilibrium system at temperatureT and valueλf of the external parameter.
The Jarzynski equality allows to deduce equilibrium quantities, ∆F, from nonequilib- rium measurements. The equilibrium information is hidden in the work distributionP(W) and can be revealed by the appropriate average. Consequently, the Jarzynski equality has been used to determine free energy differences of classical systems in experiments [36, 110]. Concerning quantum systems, however, there has been no experimental verification due to difficulties to measure the corresponding work distribution function. Recently, there has been a proposal using cold ions in a harmonic traps [73]. This proposal has, however, not been realized yet. In Sec. 3.3 below it will be shown that work distributions of quantum impurity models have been measured for decades in terms of x-ray edge spectra which in principle allows for the study of the Jarzynski equality in experiments.
The second law of thermodynamics can be inferred from the Jarzynski equality using Jensen’s inequality [168]. It should be noted, however, that this does not constitute a proof of the second law as the derivation of Eq. (3.12) already relies on basic thermodynamic principles through the initial canonical state. In contrast to the second law of thermody- namics the Jarzynski equality also holds for microscopic systems provided the initial state before the start of the nonequilibrium protocol can be described by a canonical ensemble. Therefore, Eq. (3.12) constitutes an extension of the second law to microscopic systems.
3.2.2
The Crooks relation
Consider now both the forward and backward nonequilibrium protocols. Within the back- ward process the system is initially prepared in the canonical state of the Hamiltonian H(λf) at the same temperature T. At time t = 0 the nonequilibrium protocol is started
where the parameter of the backward processλB(t) changes according toλB(t) = λF(tf−t) where λF(t) is the parameter dependence for the forward protocol. In the following the work distribution functions for the forward and backward protocols will be denoted by
PF(W) and PB(W), respectively. From the second law of thermodynamics one can again
deduce for the average workWB
av of the backward process
WavB = Z
dW W PB(W)≥ −∆F ⇒
Z
dW W PB(−W)≤∆F (3.13)
where the free energy difference ∆F = F(λf, T)−F(λi, T) is defined as in Eq. (3.11). Based on this observation the mean of PB(−W) is located at a value smaller than ∆F as is schematically indicated in Fig. 3.4. As a consequence PF(W) and PB(−W) have to intersect each other at least in one point in case where their support is not too small as is typically the case for nonzero temperatures. As it turns out, however, see below, there cannot be more than one intersection point.
Independent of the details of the nonequilibrium protocol and the considered system the forward at backward work distributions are connected via a universal relation - the Crooks relation [32]
PF(W) PB(−W)
=eβ(W−∆F). (3.14)
First proven for classical systems [32] its validity has later been extended to closed [168, 166] and open quantum systems [26]. For open quantum systems however, the meaning of the free energy difference ∆F changes and contributions from the coupling to the environment have to be included [26].
As is the case for the Jarzynski equality, see Eq. (3.12), the Crooks relation establishes a connection between nonequilibrium (PF(W),PB(W)) and equilibrium quantities (T =β−1, ∆F). The Crooks relation is more general than the Jarzynski equality because the former can be deduced from the latter as its integral version [32]:
PF(W)e−βW =PB(−W)e−β∆F ⇒ Z
dW e−βWPF(W) = e−β∆F. (3.15) Despite of being more general, an additional advantage of the Crooks relation is that it is local in W in contrast to the Jarzynski equality for which P(W) has to be known for all relevant values ofW. This is especially important for experiments as due to the exponential weighte−βW in Eq. (3.12)P(W) is amplified exponentially for negative values ofW. Thus,
the small values of P(W) for negative W have to be resolved with a high precision in
measurements. On the other hand, the Crooks relation requires the measurement of two different work distributions. For certain cases, however, as will be shown below, see Sec. 3.3, the Crooks relation can also manifest in a single work distribution.
For the value W = ∆F the two work distribution functions PF(∆F) = PB(−∆F)
have to intersect each other. As eβW is a monotonously increasing function there can
only be this one intersection point. In case of zero temperature initial states the Crooks relation implies that PF(W) = 0 for W < ∆E and PB(−W) = 0 for W > ∆E with