Nonlinear generalized master equations: quantum case

A system of N≫1 interacting spinless quantum particles, described by a statistical operator F(t), is considered. A time-dependent projection operator formalism for a family of projectors, which select a statistical operator FS(t) for a group of S < N relevant particles by integration of the variables of the irrelevant N − S ‘environment’ particles, is presented. This formalism results in a nonlinear version of the inhomogeneous Nakajima–Zwanzig generalized master equation (GME) for the relevant part of the statistical operator F(t), which contains an undesirable initial condition term related to the irrelevant part of the statistical operator depending on N variables. By introducing an additional operator identity, the obtained nonlinear inhomogeneous Nakajima–Zwanzig equation is exactly converted into a homogeneous nonlinear equation, accounting for initial correlations in the kernel governing its evolution. Both of these equations are equivalent to the corresponding evolution equations for FS(t) and take into account the dynamics of the environment particles. The equations for FS(t) are specialized in the linear approximation for the particle density n and analyzed for a one-particle statistical operator F1(t) . It turns out that, in addition to the correlations caused by interparticle interaction, the irrelevant initial condition term in the inhomogeneous equation for F1(t) , defined by the two-particle correlations, also contains quantum correlations conditioned by the quantum particle statistics. The latter do not vanish with time and cannot be disregarded in the conventional way by applying e.g. Bogoliubov’s principle of weakening of initial correlations. In order to take initial quantum and other correlations (commonly discarded in an unconvincing way) into consideration, the obtained nonlinear homogeneous GME has been applied. Exact in the first approximation in n, we obtain a new homogeneous equation for F1(t) , which accounts for correlations and is valid for any time moment t. It is shown how this reversible-in-time equation changes at different timescales and converts into the irreversible quantum Boltzmann equation on the macroscopic timescale without the use of the conventional approximations.


Introduction
Rigorous derivation of the effective equations describing the evolution of a nonequilibrium state of a many-particle system remains a principal task in statistical physics.It is naturally expected that these equations should be closed homogeneous equations for marginals, i.e. the distribution functions (classical physics) or statistical operators (quantum case) of a group of S < N particles from a whole system of N ≫ 1 particles.To achieve this goal, several assumptions are usually made.
One of them is related to the initial (at the initial moment of time t 0 ) state of a system.All derivations, e.g. of the Boltzmann equation, are mainly based on either factorized initial condition (random phase approximation (RPA) and propagating from t = t 0 'molecular chaos') corresponding to the uncorrelated initial state for t = t 0 or on Bogoliubov's principle of weakening of initial correlations [1].The first is incorrect J. Stat.Mech.(2024) 013107 in principle [2] and the second allows for obtaining the kinetic equation only for the timescale on which all initial correlations vanish (if this is the case).
On the other hand, the kinetic or transport equations should generally be nonlinear evolution equations.The problem is that the derivation of these equations commonly starts with the linear Liouville or von Neumann equation for the N -particle distribution function or statistical operator.Trying to obtain the equation for the S -particle marginal by integrating over the unnecessary N − S particles' variables, one arrives at the BBGKY (after the names of Bogoliubov, Born, Green, Kirkwood and Yvon) chain of coupled differential equations, in which the evolution equation for an S -particle marginal contains an (S + 1)-particle marginal.Therefore, a decoupling procedure for the BBGKY hierarchy should be applied.For the first equation of the BBGKY chain (S = 1), the two-particle marginal can be presented at any t as the product of the two one-particle ones in the spirit of the 'molecular chaos' approximation, as first proposed by Boltzmann.As a result, one obtains the homogeneous closed nonlinear Boltzmann equation for a one-particle distribution function, i.e. the linear Liouville equation results in a nonlinear evolution equation due to the assumption of the 'propagation of chaos' from t = t 0 onward.However, if molecular chaos is even assumed for the initial state at t = t o , its propagation with time has not been proven yet (see, e.g.[3]).
A similar situation takes place in the second main approach to the derivation of the kinetic equation from the linear Liouville-von Neumann equation: the projection operator method (see, e.g.[4]).This approach leads to so-called generalized master equations (GMEs) for the relevant part of a distribution function (statistical operator), which are linear ones if the time-independent projection operator is used.These GMEs contain irrelevant inhomogeneous terms, which are defined by the multiparticle initial correlations.Again, in order to get rid of the undesirable initial condition terms, the RPA or other approximations are conventionally used.For the derivation of nonlinear GMEs, the time-dependent projection operator should be applied (see [5]).
In this paper, we address the two problems discussed above relating to the proper accounting for the initial state of the multi-particle system in the evolution equations and to the derivation of the nonlinear evolution equations from the basic linear von Neumann equation (quantum case).
We consider a many-body system of spinless quantum particles and introduce a time-dependent projection operator P (t), selecting the relevant part of the N -particle statistical operator F (t).It should be emphasized that, considered here and in [5], the time-dependent projection operator formalism is completely different from the ones developed earlier (see, e.g.[6,7]).These approaches, using the specific forms of projectors, do not result in nonlinear (often simply postulated) GMEs for a density matrix, e.g. in the form of a Lindblad-type master equation whose generator depends parametrically on this density matrix (see, e.g.[4]).
The time-dependent projection formalism presented here generalizes the approach used in [5] on the wide spectrum of the projectors P (t) and on the quantum physics case.The obtained nonlinear version of the time-convolution Nakajima-Zwanzig GME (TC-GME) is equivalent to the nonlinear equation for the S -particle statistical operator F S (t) with a source (irrelevant term) containing all initial at t = t 0 correlations.This equation is considered in the linear approximation in the particle density n and, in particular, for a one-particle statistical operator F 1 (t).
As it turns out, for the quantum system under consideration, it is impossible to treat initial correlations as in the classical physics case, e.g. by ignoring them using either RPA-type approximation or the Bogoliubov principle of weakening of initial correlations with time.In the quantum case, there are the quantum correlations caused by the Bose (in our case) statistics of particles, which are independent of the correlations due to the interparticle interaction, and exist at all times starting from t = t 0 .In order to take the initial correlations (quantum and caused by interaction) into consideration, the obtained nonlinear TC-GME has been exactly converted into the homogeneous (with no irrelevant condition term) nonlinear time-convolution GME (TC-HGME).The obtained equation for the relevant part of the statistical operator and the corresponding equation for the S -particle statistical operator contain the initial correlations in the kernels governing the evolution of these equations and thus is valid at all times, including the initial stage of the evolution when the initial correlations due to interparticle interaction matter.We note that the effective treatment of the initial correlations in the quantum case is made possible due to the obtained nonlinear homogeneous GME.The quantum correlations included in the kernels result in the proper symmetrization of the statistical operators.The equation for F S (t) is then specialized for the first approximation in the particles' density.The new homogeneous equation for F 1 (t), exact in the linear approximation in n and accounting for the initial correlations, is obtained without the use of the conventional approximations.In the space homogeneous case, we show how this equation for F 1 (t) transforms from a reversible into an irreversible nonlinear quantum Boltzmann equation at a large (kinetic) timescale.

General time-dependent projection formalism
We consider a system of N ≫ 1 identical quantum spinless particles.The evolution of such a system can be described by the von Neumann equation: Here, F (t) = F (1, . . ., N ; t) is an N -particle statistical operator (a density matrix), (e.g. in the coordinate representation [F (1, . . ., N ; t)] r,r ′ = F (r 1 , . . ., r N ; r ′ 1 , . . ., r ′ N ; t)), and L(t) is a superoperator acting on any arbitrary operator A(t), particularly on a statistical operator F (t), as where [, ] is a commutator and H (t) is a Hamiltonian of the system.The statistical operator is subject to the normalization condition where Tr N stands for the trace over all N variables.

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The formal solution to the Liouville-von Neumann equation ( 1) can be presented as where T denotes the chronological time-ordering operator, which orders the product of time-dependent operators such that their time arguments increase from right to left, and F (t 0 ) is the statistical operator at some initial moment of time t 0 (initial condition).

Inhomogeneous nonlinear GMEs
Let us introduce the (generally) time-dependent operator P (t) and the operator Q(t) = 1 − P (t), which divide F (t) into relevant and irrelevant parts: Using the von Neumann equation ( 1), the equations of motion for f r (t) and f i (t) (5) can be written as Here, ∂P (t) ∂t denotes the operator obtained by taking the derivative of the operator P (t).The formal solution to the second equation ( 6) is where The latter operator can be presented as the following series: Equation ( 10) is very general and looks like a generalization of the Nakajima-Zwanzig TC-GME (see [8][9][10]) for the relevant part of a statistical operator to the case of any time-dependent operators P (t) and Q(t) = 1 − P (t).Like the conventional TC-GME, equation ( 10) is an exact inhomogeneous integro-differential equation for the relevant part of a statistical operator containing the irrelevant part of a statistical operator at an initial moment of time f i (t 0 ) = F (t 0 ) − f r (t 0 ) as a source.
We note that the action of P (t) on F (t) is no longer a linear operation (in contrast to applying the conventional time-independent projection operator [9]) because P (t) can depend on the time-dependent statistical operator of interest.Therefore, by applying P (t) to the linear Liouville-von Neumann equation ( 1), we generally obtain a nonlinear equation.The operator P (t) does not commute with the derivative ∂ ∂t (the time-independent linear projection operators commute with ∂ ∂t ) and P (t ′ )F N (t) ̸ = f r (t) (for t ′ ̸ = t).Thus, equation (10) is generally a nonlinear equation and, therefore, can be applicable to studying the evolution (transport) properties of many-particle systems, which are described, e.g. by the nonlinear Boltzmann-type equations (in the kinetic regime).
For time-independent P and Q, equation (10) reduces to a conventional linear TC-GME.This type of equation is more suitable for studying, e.g. the evolution of a subsystem interacting with a large system in the thermal equilibrium state (a thermostat).
In what follows, we take into consideration that for the calculation of the expectation values of interest ⟨A S ⟩ t , defined by operators A S = A S (1, . . ., S), normally dependent on a small number S ≪ N of relevant variables, one needs only to know the reduced statistical operator depending on these S relevant variables, i.e.
Here we introduced the S -particle statistical operator F S (t) (see [1]), while T r S = T r 1,2,...,S , T r Σ = T r S+1,...,N (12) stands for the traces over 1, . . ., S relevant and S + 1, . . ., N remaining irrelevant variables ('environment' Σ), correspondingly.For example, for any matrix φ N (1, . . ., N ) defined for an N -particle system, we have in the coordinate representation In view of the existence of correlations between the elements of the considered system, including the initial correlations at the initial time t 0 , the problem is obtaining the closed evolution equation for the S -particle statistical operator F S (t), which depends on a number of variables S < N, which is smaller than N, and is sufficient for the determination of the evolution of the measurable values of interest ⟨A S ⟩ t (11).Thus, we divide an N -particle system under consideration into a subsystem of S particles of interest and an environment Σ of N − S remaining particles, i.e. we have a complex of S particles, which interacts with the environment Σ.
In order to make sense of the separation (5) of F (t) into relevant and irrelevant parts, the operators P (t) and Q(t) must have the properties of 'projection operators'.Let us introduce the operator P (t) of the form where D is a time-independent (super)operator acting on any φ N (1, . . ., N ) and eliminating (integrating) all N − S unnecessary (irrelevant) variables while C (t) is a well-defined time-dependent (operator) function of particle variables (more details on the reasonable specification of P (t) will be given below in section 3).The representation of P (t) ( 14) is suitable for studying many cases of interest, including the cases when C does not depend on time.We also note that splitting a statistical operator into the relevant P (t)F (t) and irrelevant Q(t)F (t) parts makes sense if the operator P (t) satisfies the following condition for projection operators: for any t.Then, if P (t) = C(t)D, equation ( 16) entails that for any time moment t.From the property (17), we have where we assume that the operators D and ∂/∂t commute (D does not depend on time).Equation ( 17) generally represents the normalization condition for some statistical operator C (t) and is, therefore, consistent with the definition that C (t) depends on the irrelevant variables that are removed from F (t) by the operator D (it is not excluded that C (t) can depend also on the relevant variables [11]).The operator C (t) can be chosen in different ways, providing an opportunity to obtain the final equation in the desired convenient form.Thus, f r (t) = P (t)F (t) depends on both relevant and irrelevant variables (a complete set of variables characterizing an N -particle system).Now, the relevant and irrelevant parts of the statistical operator By applying the operator D to equation (10) from the left (see ( 15)), we get the following time-convolution evolution equations for the S -particle statistical operator: where ( 17)-( 19) have been used.We will further consider an isolated system with a time-independent Hamiltonian and the corresponding Liouvillian.Without loss of generality, we can present the Hamiltonian of the system under consideration as where H S , H Σ and H SΣ are related to a subsystem of S selected particles, to an environment Σ of N − S remaining particles, and to their interaction, respectively.The Liouville superoperator L corresponding to (21) is https://doi.org/10.1088/1742-5468/ad0f92

J. Stat. Mech. (2024) 013107
Then, for any matrix φ N (1, . . ., N ), defined for an N -particle system, Using ( 9), ( 18), ( 23) and other relations that follow from (17), we can prove the following equations: where Then, by making use of ( 18), ( 23) and ( 24), equation ( 20) can be finally rewritten as the following TC equation: Equation ( 26) constitutes the main result of this subsection.This is generally (for the time-dependent projection operators) the nonlinear time-convolution equation for F S (t) obtained from the linear von Neumann equation ( 1) for the N -particle statistical operator F (1, 2, . . ., N ; t) by means of the projection operator (14).The first term L S F S (t) is a conventional flow term.The second term DL SΣ C(t)F S (t) represents the self-consistent Vlasov-like field acting on the subsystem of S particles and determined by the environment of N − S particles.
The term C S (t, t 0 ) in equation ( 26) is a collision term with an additional factor accounting for the time-dependence (evolution) of the irrelevant subsystem of N − S particles (the evolution of an isolated complex of N − S particles is defined by the equation The term with the commutator [L S , C(t ′ )] accounts for the case when C (t) depends on the subsystem's S variables and vanishes otherwise.

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The last (irrelevant) term I S (t, t 0 ) in equation ( 26) describes the influence of the initial (at t = t 0 ) correlations on the evolution of the statistical operator F S (t).
There are several options for the selection of the projection operator P (t).As mentioned in the Introduction, one of our goals is to effectively include in our consideration the influence of the initial correlations on the evolution process.In some cases, it can be done by introducing a special time-independent projector (see [11]).But generally, we need an additional equation relating the initial correlation term S(t, t 0 )f i (t 0 ) in equation (10) to the relevant part of the statistical operator f r (t).

Homogeneous GMEs
Our goal is to convert equation ( 10) into a homogeneous equation containing the irrelevant part of the statistical operator f i (t 0 ) = F (t 0 ) − f r (t 0 ) in the modified 'mass' operator acting on the relevant part of the distribution function f r (t) and thus on the S -particle statistical operator F S (t).Such an equation allows treating the initial correlations (contained in f i (t 0 )) on an equal footing with all other correlations (collisions), and is applicable on any timescale, including the initial evolution stage, when the initial correlations, caused by the interaction of particles, are certainly essential.This homogeneous GME also allows avoiding the 'secular terms' growing with time as t − t 0 [1] in the solution to the evolution equation (10), accounting for the initial correlation in the inhomogeneous initial condition term, thereby restricting the applicability of the solution to the small microscopic timescale.
The quantum case is more complicated than the classical one, particularly because of the quantum correlations, which are present at any time even in the absence of interaction between particles, and which do not damp with time.Quantum correlations arise due to symmetry conditions that reflect the statistics of the particles under consideration.For example, in the space representation, the system of N quantum particles is described by the density matrix F (t) = F (r 1 , . . ., r N , r ′ 1 , . . ., r ′ N ; t), which should satisfy the symmetry condition where P ij is the operator of transmutation of any two variables r i and r j , when it stands to the left of F (t), or of r ′ i and r ′ j , when it acts from the right side of F (t) (this rule holds for any operators acting on the matrixes).The plus sign is applicable to bosons and the minus should be used for fermions.
Let us identically represent the irrelevant part of a statistical operator as https://doi.org/10.1088/1742-5468/ad0f92 where U −1 (t, t 0 ) is the backward-in-time evolution operator, T − is the antichronological time-ordering operator arranging the time-dependent operators L(s) in such a way that the time arguments increase from left to right, Hence, an additional identity (28) is obtained by multiplying the irrelevant part by unity F −1 (t 0 )F (t 0 ) and inserting U −1 (t, t 0 )U (t, t 0 ) = 1 and P (t) + Q(t) = 1.In (28), we introduce the parameter of the initial correlations which is a series in f i (t 0 )f −1 r (t 0 ).Thus, we assume that the inverse operator F −1 (t 0 ) exists.As seen from ( 29), the correlation parameter is a series in f i (t 0 )f −1 r (t 0 ) and, therefore, one only needs a formal existence of the operator f −1 r (t 0 ), which is inverse to the relevant statistical operator chosen with the help of the appropriate projection operator P (generally, it can provide some restriction on the class of appropriate projectors).The relevant part is, as a rule, the vacuum (relatively slowly changing) part of a statistical operator, i.e. a part with no correlations (e.g. a product of the one-particle statistical operators).An example of the construction of the appropriate inverse of the relevant part of the statistical operator will be given below in section 3 (equations ( 58) and ( 59)) and section 4 (equations ( 77) and (78)).
We now have two equations, ( 7) and (28), relating f i (t) to f i (t 0 ).Finding f i (t 0 ) from these equations as a function of f r (t) and substituting it in (10), we obtain the equation where the operator R(t, t 0 ) is defined as Applying operator D to equation (30) from the left, making use of ( 14), ( 15), ( 18), (19), ( 23) and (24), it can be rewritten as the following equation for the S -particle statistical operator of interest F S (t) : where Equations ( 30) and (32) represent the central result of this subsection.We have not made any approximation in deriving equation (30), and it is, therefore, an exact integrodifferential equation that takes the initial correlations and their dynamics into account (on an equal footing with collisions) via the modified memory kernel of (10) acting on the relevant part of the statistical operator f r (t).Equation (32) follows from (30) for the Hamiltonian (21) and the projection operator (14).The exact kernel (32) obtained can serve as a starting point for effective perturbation expansions.In many cases, such expansions of homogeneous equations (like (32)) have a much broader applicability domain than those of inhomogeneous equations (like (26)) due to the possibility of avoiding the 'secular' terms growing with time (see, as an example [1], where the transition from the conventional perturbation expansions of the inhomogenous BBGKY equations for the complexes of particles, applicable only on a small timescale, to the expansions of the homogeneous ones results, particularly, in the Boltzmann equation at a large kinetic timescale using the principle of weakening of initial correlations).

Nonlinear equations for S-particle statistical operator in the linear in the particle density approximation
Let us now define the projection operator P (t) as where the definitions ( 11) and ( 12) have been used. https://doi.org/10.1088/1742-5468/ad0f92

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Now, the relevant and irrelevant parts of the statistical operator F (t) are and as earlier (see ( 19)) Then, it is easily seen that the basic equations ( 26) and (32) take the forms and where defined by (33), in which U (t, t 0 ) is replaced by U 1 (t, t 0 ), and we have taken into account that C 1 (t) (34) does not depend on the subsystem's S variables, i.e. [L S , C 1 (t ′ )] = 0.It is important to note that equations (37) and (38) contain the term in the collision integral, which takes into account the influence of the time evolution of the environment of N − S particles on the complex of S particles under consideration.This result is a consequence of using the time-dependent projection operator (34), which allows for going beyond the stationary environment state assumption and makes the obtained evolution equations nonlinear and self-consistent.
We consider here equations (37) and (38) for a system of N spinless quantum particles (these equations for the reduced distribution function F S (t) of a complex of S classical particles were considered in detail in [5]).
Let us consider the case when the Hamiltonian (21) with the two-body interparticle interaction V ij is J. Stat.Mech.(2024) 013107 where K i is the kinetic energy operator of the i th particle with mass m, and V ij is the interparticle interaction.
The Liouville operator L corresponding to (39) in the coordinate representation is where For what follows, it is useful to observe that and therefore the 'Vlasov' term in equations ( 37) and (38) (in the latter, without the contribution of initial correlations) can be rewritten as At N ≫ S, this term is proportional to the particle density n = N /V .We see that the forms of the Liouville operators (40) and integral operator D = T r Σ provide an opportunity for expanding the terms of equations ( 37) and (38) into the environment particle density (N − S)/V series, and that applying the integral operator D to terms containing L SΣ or L Σ and C 1 (t) leads to expressions of at least first order in (N − S)/V .Let us consider equations ( 37) and (38) in the first approximation in the N − S particle density (N − S)/V (in the case of a small subsystem-environment interaction

J. Stat. Mech. (2024) 013107
H SΣ , one can try to expand the kernels of these equations in L SΣ ).The dimensionless small parameter of a perturbation expansion in the density is where r 0 is the effective radius of the interparticle interaction.Condition (44) introduces the time hierarchy where t cor ∼ r 0 /v (v is a particle's mean velocity), t rel ∼ γ −1 t cor .In this approximation, all terms with P 1 (t) can be neglected (in particular, in U 1 (t, t ′ )) because there is already a common prefactor T r Σ L SΣ .In this approximation, the operator U 1 (t, t ′ ) can be reduced to Let us first consider the inhomogeneous equation (37).Then, in the linear approximation in the small density parameter γ (46) we have Equations ( 43) and (47) define the evolution of the S -particle statistical operator F S (1, . . ., S; t) in the linear in (N − S)/V approximation.This equation is closed, but contains the source I 1 S (t, t 0 ), accounting for the evolution in time of initial (at t = t 0 ) correlations.As was already mentioned, the obtained equation is self-consistent in the sense that the collision integral contains the evolution in time of the environment of the complex of the relevant S -particle subsystem of interest.In the linear in n approximation, the environment evolution is described by the term in equation ( 47), which is proportional to n, as it follows from equations ( 43) and (47), and, therefore, can be neglected in the adopted linear approximation in n.
To discuss the obtained results in more detail, let us write down explicitly the equation for a one-particle statistical operator, which is mostly needed for applications.From ( 43) and (47) we have in the linear in n approximation Further simplification is possible in the spatially homogeneous case, when a oneparticle distribution function is F 1 (i; t) = F 1 (r i − r ′ i ; t) and the first (flow) and the second (Vlasov) terms in (48) vanish, as follows from definition (41).Thus, equation (48) for the space homogeneous system acquires a more simple form: It is natural to expect that the nonlinear equation ( 49) can be transformed into a closed nonlinear quantum Boltzmann equation or Vlasov-Landau equation.First of all, we should get rid of the initial correlation term (second term on r.h.s. of (49)).In the quantum case, we can present the initial correlations as where the symmetrization operator P 12 secures the right particles' statistics and g 2 (1, 2; t 0 ) is an irreducible two-particle correlation function arising due to the interparticle interaction V 12 .There are several propositions (although not very convincing) for discarding the initial correlations given by g 2 (1, 2; t 0 ): Bogoliubov's principle of weakening of initial correlations with time, RPA or factorized initial conditions and others allowing for considering the initial correlations g 2 (1, 2; t 0 ) negligible (see, e.g.[12]).However, the quantum correlations given by P 12 F 1 (1; t 0 )F 1 (2; t 0 ) do not damp with time and there is no reason to disregard them.Therefore, it turns out that the homogeneous equation (38), which exactly takes into account the influence of initial correlations on the evolution in time of the S -particle statistical operator, is more convenient.Keeping in mind the derivation of the quantum Boltzmann equation without any artificial suggestions, we will consider equation (38) for a one-particle (S = 1) statistical operator in the linear approximation in the particle density parameter (44).The influence of initial correlations is defined in equation (38) https://doi.org/10.1088/1742-5468/ad0f92J. Stat.Mech.(2024) 013107 by the operator K 1 (t, t 0 ).Given that the action of D results in expressions proportional to at least the first power of the density, and that all terms on the r.h.s. of (38) (except for the first flow term) are already proportional to D, in the first approximation in the density parameter we can neglect all terms with P 1 (t) in (38), including such terms in U (t, t ′ ).Then U (t, t ′ ) = U (t, t ′ ) in this approximation and is given by ( 46).Therefore, we can approximate U −1 (t, t 0 )U (t, t 0 ) in (33) by unity and K 1 0 (t, t 0 ) by K 0 .Hence, in the adopted approximation, the correlation parameter is as it follows from (29) and ( 33).
For what follows, it is convenient and natural to define the projection operator (34) for S = 1 as i.e. we choose the statistical operator for environment particles 2, . . ., N , F N −1 (t) as the product of the one-particle statistical operators (condition (17 is satisfied and therefore (52) is the projector).Then, according to (35), the relevant part of the statistical operator F (t) is a 'vacuum' part with no correlations and the irrelevant part where we have also used the definition (11) for the N -particle statistical operator.Now, f i (t) contains all correlations -quantum and those caused by interparticle interactions.It follows from the fact that F N (t) can be presented in the form of the cluster expansion where we have omitted for shortness the dependence of statistical operators on time (definition (55) is applicable for any t) and N −2 k=1,k̸ =i,j F 1 (k) stands for the product of N − 2 one-particle operators F 1 (k), with k taking values 1, . . ., N but k ̸ = i, j.A twoparticle g 2 (i, j) and a three-particle g 3 (i, j, k) correlation function are defined as (the correlation functions for more particles can be written down in the same way) J. Stat.Mech.(2024) 013107 The terms on the right-hand side of equation ( 56) with one, two, etc permutation operators P ij represent two-, three-particle, etc quantum correlations emerging due to the proper symmetry properties of F N (t).The irreducible two-, three-, etc correlation functions, g 2 (i, j), g 3 (i, j, k), . . .are due to the interaction between particles, and are proportional to an interparticle interaction parameter ε in some power.Note that, in contrast to the classical physics case, each correlation function in (56) contains correlations of all possible orders in ε, e.g.g 2 (i, j) includes two-particle quantum correlations (existing even in the absence of interaction, when ε = 0) and two-particle correlations of the first order in ε (g 2 (i, j)).The same is valid for g 3 (i, j, k), which contains three-particle correlations of the zero, first and second (g 3 (i, j, k)) orders in ε.Now, the correlation parameter (51) can be written as where we defined an operator inverse to the relevant operator (53) as and an inverse one-particle statistical operator is defined by e.g. in the momentum representation.
Let us consider the contribution of initial correlations to the modified Vlasov term of equation ( 38) for S = 1 and projection operator (52): in the linear in n approximation.Proceeding in the same way as in the derivation of equation ( 48), we see that in order to remain in the first order in n approximation, only the two-particle part of the initial correlation parameter (57) Nonlinear generalized master equations: quantum case J. Stat.Mech.(2024) 013107 should be saved.The same is true for other operators in equation (60).Performing the trace over the environment of 2, . . ., N particles, taking into account the normalization condition for F 1 (i; t) (see (11)) and the identity of all particles of the system, we finally get for (60) Here, we have introduced the two-particle initial correlation function at and have taken into consideration that due to the invariance of the Hamiltonian H 2 with regard to renaming the particles 1 ⇄ 2, the symmetrization operator commutes with H 2 , i.e.
Thus, the Vlasov term in equation (38), as it follows from equations ( 48) and (62), acquires the form and is determined only by the two-particle dynamics described by the Hamiltonian H 2 .We also note that (65) is different from the corresponding (modified by initial correlations) Vlasov term obtained in [5] for the classical physics case: as well as the two-particle correlation function (63), equation (65) contains a symmetrization operator P 12 responsible for quantum correlations.The collision term of equation (38) C 1 !(t, t 0 ) for a one-particle statistical operator (S = 1) and projector (52) in the adopted first approximation in n can be obtained in the same way.Accounting for equations (51), (57), ( 61) and (64), and remembering that the term accounting for the evolution of the environment particles 2, . . ., N can be disregarded in the adopted linear in n approximation, it is not difficult to show that the collision integral in (38) reduces to Collecting the results given by equations ( 65) and (66), we obtain the following equation for a one-particle statistical operator (compare with equation (48)): This result is different from that obtained in [5] for the classical physics case by the presence of quantum correlations stipulated by the symmetrization operator P 12 .It is not difficult to show that in the space homogeneous case, equation (67) reduces to Equation ( 67) is the central result of this section.We have obtained a new closed homogeneous nonlinear integro-differential equation for a one-particle density matrix retaining initial correlations, which is valid on any timescale (i.e.we can use equation (67) to consider in detail all stages of the evolution of a one-particle density matrix).No approximation like the Bogoliubov principle of weakening of initial correlations (valid only on a large timescale t ≫ t cor , where t cor is the correlation time caused by the interparticle interaction) has been used for the derivation of equation (67).This equation is exact in the linear approximation in n and, therefore, accounts exactly for the initial correlations and collisions in this approximation.The second (Vlasov-like) term on the right-hand side of (67), linear in the interaction L 12 , is modified by the initial correlations (both quantum and caused by interaction between particles).The quantum correlations do not contribute to this term in the space homogeneous case, as seen from equation (68).The initial correlations, both quantum and due to the interaction, also modify the collision terms of equations ( 67) and (68).

Connection to the quantum Boltzmann equation
Let us consider the space homogeneous case, i.e. equation (68), and put t 0 = 0 hereinafter (the equation is valid for any initial moment t 0 ).Equation (68) is not a kinetic equation in the conventional sense because it is time-reversible (if the correlation function G 2 (1, 2; t) does not change when the particle velocities are reversed).Thus, one of the possibilities to secure fully time-asymmetric behavior is the special choice of an initial condition (e.g. a factorized one, which is not very realistic [2]).
It is instructive to consider the transition from equation (68) to an irreversible kinetic equation describing the evolution into the future (t > 0).As seen from (68), in order to enter the kinetic (irreversible) stage of the evolution, the reversible terms caused by the https://doi.org/10.1088/1742-5468/ad0f92The (operator) δ functions of the Hamiltonian H (entering equation (71) with a twoparticle Hamiltonian H 2 ) can be expressed via the imaginary part of Green's function G(E ): On the other hand, there is the so-called optical theorem relating δ(E − H) (the imaginary part of Green's function G(E ± ) for the full Hamiltonian H = H 0 + H ′ ) and the interaction Hamiltonian H ′ to δ(E − H 0 ) (the imaginary part of Green's function G 0 (E ± ) = (E ± − H 0 ) −1 ) and the T -matrix (see, e.g.[13]): where the T -matrix is defined as where we estimated the mean particle's velocity as 2k B T /m.At rather typical values, e.g.r 0 ∼ 10 −8 cm, ℏ ∼ 10 −27 erg×sec, m ∼ 10 −24 g, k B T ∼ 10 −14 erg, tcor t coh ∼ 1.Although at different parameters this estimation can change, we assume that t coh ∼ t cor .