Nonequilibrium thermodynamics with binary quantum correlations

Nonequilibrium thermodynamics with binary quantum correlations

K. Morawetz Münster University of Applied Sciences, Stegerwaldstrasse 39, 48565 Steinfurt, Germany International Institute of Physics- UFRN, Campus Universitário Lagoa nova, CEP: 59078-970 / C.P. 1613, Natal, Brazil Max-Planck-Institute for the Physics of Complex Systems, 01187 Dresden, Germany

The balance equations for thermodynamic quantities are derived from the nonlocal quantum kinetic equation. The nonlocal collisions lead to molecular contributions to the observables and currents. The corresponding correlated parts of the observables are found to be given by the rate to form a molecule multiplied with its lifetime which can be considered as collision duration. Explicit expressions of these molecular contributions are given in terms of the scattering phase shifts. The two-particle form of the entropy is derived extending the Landau quasiparticle picture by two-particle molecular contributions. There is a continuous exchange of correlation and kinetic energies condensing into the rate of correlated variables for energy and momentum. For the entropy, an explicit gain remains and Boltzmann’s H-theorem is proved including the molecular parts of the entropy.

05.60.Gg. 05.70.Ln, 47.70.Nd,51.10.+y,

I Introduction

Highly non-equilibrium Fermi systems occur in various fields of physics, e.g. electrons driven by fast lasers, nucleons in nuclear reactions or atoms in ultra-cold gases. The dynamics of such systems is often too complex to be treated by exact quantum statistical approaches. This is caused by the strong interaction. A feasible microscopic picture is provided by quasi-classical simulations of single-particle trajectories in self-consistent force fields and randomly selected binary collisions. Although these simulations solve in principle a kinetic equation offering the complete single-particle distribution in phase space, the main results are hydrodynamical quantities like the particle flow and the corresponding density profile, because of their clear interpretation.

The relation between the single-particle distribution and the particle density is trivial as long as binary collisions are so fast that they can be treated as instantaneous. If the finite duration of collisions becomes important, a part of particles is hidden in collision states found here as molecular states and a more sophisticated evaluation is necessary. In this paper we evaluate the hydrodynamic and thermodynamic quantities as functionals of the single-particle nonequilibrium distribution including nonlocal collisions of finite duration. We derive balance equations for densities of particles, momentum, energy and the entropy. As will be seen, the finite duration of collisions leads to molecular contributions in all balance equations.

History of nonlocal collisions

The very basic idea of the Boltzmann equation from 1872 Boltzmann (1872), to balance the drift of particles with dissipation, is used in all mentioned fields allowing for a number of improvements that make it possible to describe phenomena far beyond the range of the validity of the original Boltzmann equation. In these improvements the theory of gases differs from the theory of condensed systems.

In the theory of gases, the focus was on the so called virial corrections that take into account a finite volume of molecules and an effective pressure caused by their interaction. The original Boltzmann equation cannot describe virial corrections because the instant and local approximation of scattering events implies an ideal gas equation of state. To extend the validity of the Boltzmann equation to moderately dense gases, Clausius and Boltzmann included the space nonlocality of binary collisions Chapman and Cowling (1990). For the model of hard spheres, Enskog Enskog (1972) has further extended the nonlocal collision integrals by statistical correlations. It was later modified to the nowadays used revised Enskog theory van Beijeren and Ernst (1973). An effort to describe the virial corrections for real particles, in particular when their de Broglie wave lengths are comparable with the potential range, has resulted in various generalizations of Enskog’s equation Waldmann (1957, 1958, 1960); Snider (1960, 1964); Bärwinkel (1969a); Thomas and Snider (1970); Snider and Sanctuary (1971); Rainwater and Snider (1976); Balescu (1975); McLennan (1989); Laloë (1989); Tastevin, Nacher, and Laloë (1989); Nacher, Tastevin, and Laloë (1989); Loos (1990a, b); de Haan (1990a, b, 1991); Laloë and Mullin (1990); Snider (1990, 1991); Nacher, Tastevin, and Laloë (1991a, b); Snider (1995); Snider, Mullin, and Laloë (1995). By closer inspection one finds that all tractable quantum theories deal exclusively with non-local corrections. The statistical correlations in quantum systems would require an adequate solution of three-particle collisions from Fadeev equations Beyer, Röpke, and Sedrakian (1996); Papp, Krassnigg, and Plessas (2000); Papp and Plessas (1996). A systematic incorporation of the latter one into the kinetic equation, however, is not yet fully understood, therefore we discuss only binary processes.

In the theory of condensed systems, a historical headway was the Landau concept of quasiparticlesBaym and Pethick (1991a) with three major modifications of the Boltzmann equation: the Pauli blocking of scattering channels, the underlying quantum mechanical dynamics of collisions, and the single-particle-like excitations (quasiparticles) instead of real particles. Unlike in the theory of gases, the scattering integrals of the Boltzmann equation remain local in space and time and the Landau theory does not include a quantum analog to (non-local) virial corrections.

Although nuclear matter is a dense Fermi liquid and very much benefits from the Landau concept, it was felt that nonlocal contributions are missing. Attempts started from numerical experiments Halbert (1981) within the cascade model and nonlocal corrections of Enskog type Malfliet (1983), incorporated into the Monte-Carlo codes for the Boltzmann equation Kortemeyer, Daffin, and Bauer (1996) using a method developed within the classical molecular dynamics Alexander, Garcia, and Alder (1995). These implementations of Enskog’s corrections to nuclear reactions did not improve the agreement with experimental data Bonasera, Gulminelli, and Molitoris (1994). One of the discussed reasons for this disagreement were the statistical correlations studied in detail for classical hard-sphere modelEnskog (1972); Chapman and Cowling (1990); Hirschfelder, Curtiss, and Bird (1964); Schram (1991); Cohen (1962); Weinstock (1963a, b, 1965); Kawasaki and Oppenheim (1965); Dorfman and Cohen (1967); Goldman and Frieman (1967); van Beijeren and Ernst (1979).

It turned out that the original Enskog corrections are not suited for nuclear matter, because the dominant correction is the finite duration of the nucleon-nucleon collision, not its nonlocality Danielewicz and Pratt (1996). Pioneering simulations were thus heading in the wrong direction, because the hard spheres lead to the excluded volume and thus to a compressibility lower than the one of an ideal gas, while the finite duration increases the compressibility because a fraction of particles is bounded in short-living molecules.

Collisions with nonlocal corrections obtained from realistic nucleon-nucleon scattering phase shifts Morawetz et al. (1999a) were first implemented in simulationsMorawetz et al. (1999b) of nuclear reactions. As a result, the hydrodynamic properties changes and a hot neck between two reacting nuclei shows a longer lifetime which increases the production of hot protons and neutrons reducing the discrepancy between experimental and simulated data Morawetz et al. (2001). It was encouraging that the nonlocal corrections have no effect on the run time of simulations. This is in contrast with quasiparticle contributions, because the back-flow (within the Landau local approach) is a hard numerical problem. Within the nonlocal approach, one can view non-dissipative interactions among particles as zero-angle ‘collisions’. Replacing rejected Pauli-blocked collisions by nonlocal zero-angle collisions, modified velocities of quasiparticles and back-flows are simulated on no numerical cost Lipavský, Morawetz, and Špička (2001); Morawetz et al. (2001).

The simulated corrections Lipavský, Morawetz, and Špička (2001); Morawetz et al. (2001) mimic the collision delay by seeding particles after the local collision into positions from which they would have arrived asymptotically when colliding with the true delay. Quantum studies of gases based on Waldmann’s equationWaldmann (1957) generalized by SniderSnider (1960, 1964) and further developed by Laloë, Mullin, Nacher and TastevinLaloë (1989); Tastevin, Nacher, and Laloë (1989); Nacher, Tastevin, and Laloë (1989); Laloë and Mullin (1990); Nacher, Tastevin, and Laloë (1991a, b) directly result in instantaneous collisions with nonlocal corrections modeling the collision delay. The invisibility of the collision delay in Snider’s approach follows from the absent off-shell motion during collisions. The Wigner collision delay is the energy derivative of the scattering phase, but Snider’s approach provides the scattering phase only for the ‘on-shell’ energy equal to the sum of final single-particle energies. The same limitation applies to the approach of de Haande Haan (1990a, b, 1991) who confirmed the results of Laloë, Nacher and Tastevin using Balescu’s formal derivation of kinetic equations Balescu (1975).

It should be noted that the approach of Laloë, Tastevin and Nacher is limited to non-degenerate systems, therefore their quasiparticle features are given by other than exchange processes.

Collision delay

Unlike nonlocal corrections in space, the true collision delay is still a problem for implementations. In spite of the lack of an effective simulation scheme, here we want to discuss properties of the kinetic equation with collision delay. In particular in dense systems the collision delay has to be treated properly because during the collision particles contribute to the background as part of the Pauli blocking of states.

Let us outline the quantum collision delay and its interpretation. In the weak-coupling limit the scattering rate of two particles is given by the matrix element of the interaction potential between the plain waves of initial and final states. For strong potentials, the wave function cannot completely penetrate the core but can be enhanced at moderately short distances. In the strong-coupling case one thus has to take into account the reconstruction of the wave functions by the interaction potential. It reflects the finite duration of the collision usually called internal dynamics. The build-up of the wave function means that the particles can be found at a given distance with an increased probability. Within the ergodic interpretation of the probability it means that they have to spend a longer time at this distance than with an uncorrelated motion corresponding to the concept of the dwell time Hauge and Støvneng (1989). A recent review Kolomeitsev and Voskresensky (2013) shows numerous definitions of the collision delay and discusses their relevance to the quantum kinetic theory. The gradient expansion supports the Wigner delay time as energy derivative of the scattering phase shift.

Within many-body Green’s functions, the space and time nonlocal corrections are treated on equal footage and the internal dynamics of collisions is described by the two-particle T-matrix which has been introduced to describe the reconstruction of the wave function in the collision process. The physical content of the T-matrix is, however, easily wasted when one derives the kinetic equation. Most of the Green’s function studies result in Landau’s kinetic equation, where the scattering integral is instant and local. This contradiction follows from the second ’well established’ approximation which is the neglect of gradient corrections to the scattering integral. The headway for a Green’s function treatment of non-instant and nonlocal corrections to the scattering integral was done by Bärwinkel Bärwinkel (1969a) who also discussed the thermodynamical consequences of these correctionsBärwinkel (1969b). The present approach is based on non-equilibrium Green’s functions Lipavský, Morawetz, and Špička (2001); Spička, Lipavský, and Morawetz (1997) known as the generalized Kadanoff-Baym (GKB) formalism taking into account consistently all first-order gradient corrections Morawetz, Lipavský, and Špička (2001).


The entropy as a measure of complexity, or inversely as the loss of information, plays a central role in processes like nuclear or cluster reactions, where the kinetic and correlation energy of projectile and target particles transform into heat. In nuclear matter, mainly the single-particle entropy Ivanov, Knoll, and Voskresensky (2003); Peshier (2004); Alberico et al. (2008); Moustakidis et al. (2010); Suraud and Reinhard (2014) is discussed as it is in ultra-cold atoms Baur and Mueller (2010). The equilibrium entropy has been given in a form of cluster expansion where the two-particle part is represented by the two-particle correlation function Kirkwood (1942) which has been calculated numerically for different systemsLaird and Haymet (1992); Nayar and Chakravarty (2013). The genuine two-particle part of the quantum entropy is still an open question as well as its nonequilibrium expression.

Equilibrium values are presented in terms of either general Green’s functions Vanderheyden and Baym (1998); Miyake and Tsuruta (2015) or in expansion of coupling parameters Blaizot, Iancu, and Rebhan (2001). General expressions of the entropy in terms of -derivable functionals Carneiro and Pethick (1975) would require a tremendous reduction in order to understand the contribution of correlated parts and single-parts explicitly in an applicable form. We proceed another way and employ the nonlocal kinetic equation which contains single and two-particle correlated parts to extract the correlated entropy from balance equations. Therefore we are using the infinite ladder summation condensed in the T-matrix since the correlated entropy is a result essential beyond the one-loop approximation Vanderheyden and Baym (1998). The second advantage of our approach is that we present the nonequilibrium expression of the entropy where no extremal principle for -derivable functionals can be given.

Very often the entanglement entropy is also investigated if one set of variables is traced off from the density operator which provides the information exchanged between the two subsystems Peschel and Eisler (2009). The extraction of two particles out of a many-body state leads to a different entanglement entropy Samuelsson, Neder, and Büttiker (2009) than the one of the reduced density matrix. Similarly the calculation of either spatial-dependent or momentum-dependent one- and two-particle entropies yields different results Laguna et al. (2016). The majority of approaches calculate the classical entropy in various approximations Puoskari (1999); Hernando and Blum (2000). Here we will obtain the quantum one- and two-particle entropies explicitly in terms of phase shifts of the scattering T-matrix.

Outline of the paper

First we give the nonlocal kinetic equation derived first in Špička, Lipavský, and Morawetz (1998); Lipavský, Morawetz, and Špička (2001) and present important symmetries of the collision integral in chapter III. Then we derive the thermodynamic quantities as balance equations for density, momentum, energy, and entropy from this kinetic equation in chapter IV. We show that besides the usual balance equations for quasiparticles, where the integrals over the collision integral vanishes, the nonlocality of the collision process induces explicit molecular contributions. In Chapter V we summarize the forms of balance equations discussing their statistical interpretation and prove Boltzmann’s H-theorem. The summary is in chapter VI.

Ii Nonlocal kinetic theory

ii.1 Nonlocal kinetic equation

The nonlocal kinetic equation reads


with the scattering-in


and the scattering-out


Thorough the paper all distribution functions and observables have the arguments


and a bar indicates the reversed sign of s. If not otherwise noted all derivatives are explicit ones, i.e off-shell derivatives keeping the energy argument of as independent variable.

In the scattering-out (scattering-in is analogous) one can see the distributions of quasiparticles describing the probability of a given initial state for the binary collision. The hole distributions describing the probability that the requested final states are empty and the particle distribution of stimulated collisions combine together in the final state occupation factors like . The scattering rate covers the energy-conserving -function, and the differential cross section is given by the modulus of the T-matrix reduced by the wave-function renormalizations Köhler (1995). We consider here the linear expansion in small scattering rates, therefore the wave-function renormalization in the collision integral is of higher order.

All ’s are derivatives of the scattering phase shift ,


according to the following list


The quantum kinetic equation (1) unifies the achievements of transport in dense gases with the quantum transport of dense Fermi systems and was derived starting with the impurity problem Špička, Lipavský, and Morawetz (1997); Spička, Lipavský, and Morawetz (1997) and then for arbitrary Fermi systems Špička, Lipavský, and Morawetz (1998); Lipavský, Morawetz, and Špička (2001). The quasiparticle drift of Landau’s equation is connected with a dissipation governed by a nonlocal and non-instant scattering integral in the spirit of Enskog corrections. These corrections are expressed in terms of shifts in space and time that characterize the non-locality of the scattering process Morawetz et al. (1999a). In this way quantum transport was possible to recast into a quasi-classical picture suited for simulations. The balance equations for the density, momentum and energy include quasiparticle contributions and the correlated two-particle contributions beyond the Landau theory as we will demonstrate.

Figure 1: Displacements in the effective collision of real particles. In the scattering-out process, the momenta and correspond to the initial states of particles and , the momenta and to the final states. In the scattering-in process, the picture is reversed.

As special limits, this kinetic theory includes the Landau theory as well as the Beth-Uhlenbeck equation of state Schmidt, Röpke, and Schulz (1990); Morawetz and Roepke (1995) which means correlated pairs. The medium effects on binary collisions are shown to mediate the latent heat which is the energy conversion between correlation and thermal energy Lipavský, Špička, and Morawetz (1999); Lipavský, Morawetz, and Špička (2001). In this respect the seemingly contradiction between particle-hole symmetry and time reversal symmetry in the collision integral was solved Špička, Morawetz, and Lipavský (2001). Compared to the Boltzmann-equation, the presented form of virial corrections only slightly increases the numerical demands in implementations Morawetz et al. (1999b); Morawetz (2000); Morawetz, Ploszajczak, and Toneev (2000); Morawetz et al. (2001) since large cancellations in the off-shell motion appear which are hidden usually in non-Markovian behaviors. Details how to implement the nonlocal kinetic equation into existing Boltzmann codes can be found in Morawetz et al. (2001).

Let us summarize the properties of the nonlocal kinetic equation (1). The drift is governed by the quasiparticle energy obtained from the single-particle excitation spectrum. The scattering integral is non-local and non-instant, including corrections to the conservation of energy and momentum as it is illustrated in figure 1. Neglecting the nonlocal shifts, the standard quasiparticle Boltzmann equation results with Pauli-blocking.

Iii Symmetries of collisions

Integrating the kinetic equation it will be helpful to perform two transformations, once to interchange incoming and outgoing particles and once to exchange the collision partners and .

iii.1 Transformation A

The integrated kinetic equation (1) is invariant if we interchange particles and or labels and . This is realized by the substitution


The local T-matrix obeys this symmetry


However, this substitution changes the derivatives of the phase of the T-matrix (5) as


leading to the relation between the displacements (6)


Relations (10) merely show that the reference point has been moved to the partner particle and shifts were correspondingly renamed, see Fig. 2.

Figure 2: Transformation A as interchange of particle and leading to (10).

The and remain unchanged as well as the invariant combination


which is the distance between final and initial geometrical centers of the colliding pair. It can be interpreted as the distance on which particles travel during .

The quasiparticle energies (4) shift according to


where we denote only changes of the arguments of (4) explicitly.

iii.2 Transformation B

The interchange of initial and final states, and , is accomplished by the substitution


The general symmetry of the T-matrix with respect to the interchange of the initial and final states,


implies that the differential cross section, the collision delay and the energy gain do not change their forms by this substitution. All gradient corrections are explicitly in the form of -corrections. Under this substitution, the space displacements effectively behave as if we invert the collision,


while the other ’s keeps their values and the combination (11) is invariant again. This is illustrated in figure 3.

Figure 3: Transformation B as interchange of incoming and outgoing particles leading to (15).

The distributions of the in-scattering term can be translated into the ones of the out-scattering term if we interchange out- and in-going collisions which means to apply transformation B. Consequently, the arguments of the quasiparticle energies (4) transform as


where the denotes the shift of momentum arguments by , the spatial arguments by , and the time arguments by . If we transform the scattering-in (2) with this transformation we obtain the order of distributions as the scattering-out (3) however with these shifts in the different functions. Therefore we will abbreviate in the following


where we denote from (3) with


In case where this term will appear as prefactor to s we could ignore the shifts inside since our theory is linear in s. But consistently we will keep the shift as being the one of the out-scattering.

iii.3 Symmetrization of collision term

In (1), the differential cross section and the energy argument of the scattering phase shift is based on initial states . The transformation B interchanges initial and final states and this sum energy becomes . For a convenient implementation, we thus introduce the sum energy at the center of the collision


From the energy-conserving -functions follows that on the energy shell the centered energies equal to the arguments of the T-matrix,


The centered energies (19) are the physically natural choice and we favor them against resulting from the quasiparticle approximation of Green’s functions. The centered energy argument, however, gives a non-trivial contribution to the factor of the energy-conserving functions. This comes from the fact that any energy argument in the scattering-in term is given in terms of while the scattering-out has the argument . Changing to the centered energies (19) means mathematically


which one gets by comparing


in linear order of s. Using (21) with and after substitution (19) the -function of the scattering-in reads


and the scattering-out is given by inverse signs of the s at . Please remember that the quasiparticle energies and distributions have the shifts according to (4) and a bar indicates the reversed sign.

Iv Nonequilibrium thermodynamic properties

iv.1 Local conservation laws

Now we ask about the consequences of the kinetic equation to the thermodynamic properties. Far from equilibrium, the traditional thermodynamic quantities as the temperature and chemical potential do not capture the time-dependent properties of the system. Accordingly, we want to express the thermodynamic observables as functionals of the time-dependent quasiparticle distribution. Therefore we will multiply the kinetic equation (1) with a variable and integrate over momentum. It results in the equation of continuity, the Navier-Stokes equation, the energy balance and the evolution of the entropy, respectively. All these conservation laws or balance equations for the mean thermodynamic observables


will have the form


The additional gain on the right side might be due to an energy or force feed from the outside or the entropy production by collisions. The external potential is absorbed here in the quasiparticle energy and we can concentrate on the internal contributions due to correlations. Then we will obtain a gain only for the entropy while for density, momentum and energy the time change of the density is exclusively caused by the divergence of the current .

We will show from the balances of the kinetic equation that the particle density, momentum flux (pressure), energy and entropy density consist of a quasiparticle part and a correlated contribution , respectively. The latter one takes the form of a molecular contribution as if two particles form a molecule. Proving the conservation laws (24) and showing that also the currents consists of will be the ultimate goal to convince us about the consistency of the nonlocal kinetic equation.

iv.2 Drift contributions to balance equations

The balance of quantity requires to evaluate the -weighted momentum integrals of the kinetic equation (1),


The left-hand side includes terms which have been treated many times within the Boltzmann theory and later extended to the Landau theory of Fermi liquids Smith and Jensen (1989); Baym and Pethick (1991b). Let us consider them first.

iv.2.1 Density balance from drift

For the density , the left-hand side of the kinetic equation (25) gives


In Landau’s theory the integration over the local collision integral of the Boltzmann equation is zero and one finds that the divergence of the quasiparticle current


and the time derivative of the quasiparticle density


sum to zero in the form of (24). The quasiparticle current includes the quasiparticle back flow Lipavský and Špička (1994) which appears due to a non-symmetry of the quasiparticle energy resulting from a non-symmetry of the quasiparticle distributions.

iv.2.2 Energy

We integrate now the kinetic equation (1) multiplied with the energy . The drift side


results in the divergence of the quasiparticle energy current


and the first term of (29)


When ’s tend to zero the collision integral vanishes after integration over and the energy balance is (29). Obviously (31) has to be rearranged into the time derivative. In the absence of non-local collisions which corresponds to Landau’s concept of quasiparticles, the quasiparticle energy equals the functional derivative of the energy density,


With the help of (32) the drift term (29) attains the desired form,


Landau’s functional relation (32) is consistent with the Boltzmann equation and is particularly useful for phenomenological quasiparticle energies Landau (1957); Baym and Pethick (1991b). The variational energy (32) makes the conservation laws very convenient and results in correct collective motion. Here we use the quasiparticle energy identified as the pole of the Green’s function. Except for simple approximations, these two definitions lead to different values of quasiparticle energies. In the theory of liquid He, the difference between these two definitions is know as the rearrangement energy Glyde and Hernadi (1983). A relation between these quasiparticle energies and the rearrangement energy has been discussed in Lipavský, Špička, and Morawetz (1999).

The simplicity of Landau’s variational approach makes his concept of quasiparticles very attractive. On the other hand, the Green’s function pole represents the true dispersion law of single-particle excitation, therefore the pole definition leads to a better description of the local distribution of particles. Of course, it is on cost of more complex balance equations. The quasiparticle contributions for all thermodynamical quantities we discuss are complete if we evaluate the collision contributions.

iv.2.3 Balance of forces

For the momentum balance one multiplies the kinetic equation (1) with the -th component of momentum , i.e. , and integrates over momentum . The results in the time derivative of the momentum density of quasiparticles


The other parts of the drift side can be rearranged by integration by parts as


Eq. (32) allows to write the last term of (35) as the gradient of the energy density,


In such a way (35) becomes the quasiparticle stress tensor Smith and Jensen (1989); Baym and Pethick (1991b),


The quasiparticle momentum-force balance from the drift becomes therefore


as the one for the local Boltzmann equation or Landau’s theory since without shifts the collision integral vanishes due to momentum conservation. We will obtain additional contributions from the nonlocal collision integral.

iv.2.4 Entropy

Finally, the single-particle entropy density distribution is given by Landau (1957)


which is the generalization of the classical expression towards quantum effects including the Pauli-blocking. The first sum in (39) is the entropy of particles but with the quantum quasiparticle distribution. The second sum one can consider as the entropy of holes as if they are just a second sort of particles.

Since any derivative of (39) leads to the derivative of the distribution it is advisable to multiply the kinetic equation (1) with and to integrate over . The drift side becomes


It results into the divergence of the quasiparticle entropy current


and the time derivative of the quasiparticle entropy


as integral over (39). The arguments of , etc. follow the notation of (4).

For the entropy balance, the collision integral does not vanish even neglecting shifts providing an explicit entropy gain. The interesting question is how the molecular part of entropy will look like, what balance we get and whether we can prove Boltzmann’s H-theorem, i.e. the second law of thermodynamics. If we manage to derive the expressions including shifts and to prove the H-theorem, this includes, of course, then also the simpler case for local Boltzmann equation neglecting shifts.

iv.3 Molecular contributions to observables from collision integral

Besides the known quasiparticle contributions to the observables of the last chapter, there appear explicit binary correlations due to the nonlocal collision integral. The remaining parts of this chapter presents a new systematic way to derive these correlated observables.

iv.3.1 Expansion properties

Now we search for the terms arising from the nonlocal collision integral (1). Multiplying the latter one with , integrating and applying the B-transform to the in-scattering part, we obtain from (22) the structure


where we abbreviated (18) and adopt the notation (4) of the arguments for the observable . One has (16) when B-transforming and in this notation . The factors (22) transform into


where the unchanged out-scattering one is just (22) with reversed signs and the in-scattering one appears since we have applied transformation B to (22). Since our theory is linear in s we ignore the shifts inside , i.e. we can use when they appear as factors with s.

To start with the treatment of all the following expansions it is very helpful to observe that we can consider the arguments of s before expansion either being or alternatively up to first order in s due to the energy conservation in . To see this we expanding the equality up to first order for any


with the corresponding derivative . From this we now subtract the equality due to the -function, to get the relation


which we will use later.

iv.3.2 Correlated observables

In order to make the different parts transparent we concentrate successively on specific terms and collect them together in the end.

For the time derivative and terms in (43) and (44), we employ transformation A, add with the original expression and divide by 2 resulting into the terms under integration


The terms in the brackets form the total (on-shell) derivative as follows. From the definition (6) we have the identity and therefore for any argument


This means we can write for (47)