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— 1 — Quantum master equation for a system of identical particles and the anisotropic distribution of the interacting electrons Boris V. Bondarev Moscow Aviation Institude, Volokolamskoye Shosse 4, 125871 Moscow, Russia 3 1 0 A R T I C L E I N F O A B S T R A C T 2 n a Article history: J We consider an open quantum many-particle system in which 1 there are dissipative processes. The evolution of this system is 2 described by a kinetic equation for the density matrix. From the Keywords: equation describing a random Markov process in this system, we ] h Density matrix for N obtain an equation for the single-particle statistical operator. This c particles equationdescribesthe evolutionofasystemofidenticalparticlesin e Equation with dissipative amean-fieldapproximation.Theequationforinteractingparticlesin m terms thermodynamicequilibriumwasobtained.Thedistributionfunction - Hierarchy of density matrices of a system of interacting electrons in metals has multivalence t a Equation for single-particle in a certain region of wave vectors. Among many solutions one t s density matrix is isotropic. Other solutions have the anisotropy of the electron . t Mean-field approximation distributionoverthe wavevectors.The anisotropyarisesasa result a Anisotropy of distribution of repulsion and attraction between electrons. m Interacting electrons - d 2012 Elsevier B.V. All rights reserved n o c [ 1. Introduction 1 v Themostgeneralstatisticaldescriptionofanysysteminquantummechanicsisperformedbyusingthe 2 1 statistical operator [1]. If the system state changes with time, then the statistical operator is a function 7 of time: 4 ̺ˆ=̺ˆ(t). (1.1) . 1 WhenthesystemtakesarandomMarkovprocess,thisfunctioncanbefoundfromthekineticequation 0 of the form [2 – 14] 3 1 i¯h̺ˆ˙= H,̺ˆ +i¯hdˆ̺ˆ, (1.2) : v which is called the Liouville - von Neumann e(cid:2)quatio(cid:3)n. In this equation, H is the Hamiltonian of the Xi system;dîstheso-calleddissipativeoperatordescrbibingdissipativeprocessesinthesystem.Thestatistical operator should be normalized, self-adjoint and positive. Any solution obf (1.2) that has a physical r a significance and has these properties at the initial moment of time t should have them for all t>t . So 0 0 the equation (1.2) should have the form i¯h̺ˆ˙= H,̺ˆ +i¯h C aˆ ̺ˆ, aˆ+ + aˆ , ̺âˆ+ . (1.3) kl k l k l (cid:2) (cid:3) Xk,l (cid:16)(cid:2) (cid:3) (cid:2) (cid:3)(cid:17) b This equationis calledthe Lindblad equation[6].Here a is anoperator,k =1,2,... – the number ofthe k operator, C - constants. kl Let us write the equation (1.3) for the quantum harmonic oscillator: 1 1 i¯h̺ˆ˙= H,̺ˆ + i¯hA aˆ̺ˆ, aˆ+ + aˆ, ̺âˆ+ + i¯hB aˆ+̺ˆ, aˆ + aˆ+, ̺âˆ , (1.4) 2 2 (cid:2) (cid:3) (cid:16)(cid:2) (cid:3) (cid:2) (cid:3)(cid:17) (cid:16)(cid:2) (cid:3) (cid:2) (cid:3)(cid:17) b — 2 — where 1 H =h¯ω aˆ+aˆ+ , (1.5) 2 ! A and B are constants. The operator aˆ habs the form 1 ipˆ aˆ= +√κxˆ . (1.6) √2¯hω √m ! Equation(1.4)surprisinglyaccuratelydescribesthechangeofthestateofthequantumharmonicoscillator with time, provided that A =eβh¯ω , (1.7) B where β =1/kT is the inverse temperature. 2. The hierarchy of statistical operators Considerthe systemconsistingof N identicalparticles.The statisticaloperator̺ˆdescribingthe state of this system can be written as follows: ̺ˆ=̺ˆ(1, 2,...,N). (2.1) Here the numbers in parentheses denote the indices of the variables which are affected by this operator. The statistical operator like any other operator describing the state of the system of identical particles must be symmetric: ̺ˆ(..., i,..., j,...)=̺ˆ(..., j,..., i,...). (2.2) Let us assume the following normalization condition for the statistical operator: Tr ̺ˆ(1,...,N)=N!. (2.3) 1...N If the system state changesover time, the statisticaloperator is a function of time: t: ̺ˆ=̺ˆ(t). When we can be sure that the system takes a random Markov process, this function can be found from the equation i¯h̺ˆ˙= ,̺ˆ +i¯hdˆ̺ˆ. (2.4) H The Hamiltonian operator in this equation can(cid:2)be wr(cid:3)itten as the sum b N N N 1 (1,...,N)= H(i)+ H(i,j), (2.5) H 2 i=1 i=1 j=1 X XX j6=i b b b whereH(i)isasingle-particleHamiltonian,i.e.theenergyoperatorofasingleparticlewithoutconsidering its interaction with other particles; H(i,j) is the operator of the interaction of two particles. A single- particlebHamiltonian may contain dissipative terms. A two-particle Hamiltonian should be symmetric: b H(i,j)=H(j,i). (2.6) Under this condition, the Hamiltonian (2.5)bwill also bbe symmetric. Supposethattheactionofthedissipativeoperatordônthestatisticaloperator̺ˆleadstotheexpression 1 dˆ̺ˆ= aˆ̺ˆ, aˆ+ + aˆ, ̺âˆ+ . (2.7) 2 (cid:16)(cid:2) (cid:3) (cid:2) (cid:3)(cid:17) Here the indices k and l near the operator aˆ are omitted . — 3 — Supposethatthedissipativeprocessesoccurringinthesystemarecausedbythestochasticinteraction ofparticleswithaheatreservoir.Moreover,eachparticleinteractswithitindependentlyofotherparticles. In this case the many-particle operator aˆ in the expression (2.7) can be written as the sum N aˆ(1,...,N)= aˆ(i), (2.8) i=1 X where the operator aˆ(i) characterizes the effect of the heat reservoir on one of the particles. Substitution of the sum (2.8) in the formula (2.7) gives N N 1 dˆ̺ˆ= aˆ(i)̺ˆ, aˆ+(j) hh + aˆ(i), ̺âˆ+(j) . (2.9) 2 Xi=1 Xj=1(cid:16)(cid:2) (cid:3) (cid:2) (cid:3)(cid:17) Itisimpossibletosolvetheequation(2.4)withsuchacomplexright-handside.Thereforethepractical interestiscarriedbythe single-particlekineticequationforthestatisticaloperator̺ˆ(1),whichisdefined by the relation 1 ̺ˆ(1)= Tr ̺ˆ(1, 2,...,N). (2.10) (N 1)! 2...N − This definition and condition (2.3) implies the normalization condition Tr ̺ˆ(1)=N . (2.11) 1 We define the two-particle statistical operator ̺ˆ(1,2) as 1 ̺ˆ(1,2)= Tr ̺ˆ(1, 2,...,N). (2.12) (N 2)! 3...N − This operator satisfies the normalization condition Tr ̺ˆ(1,2)=N (N 1). (2.13) 12 − 3. The quantum kinetic equation for single-particle density matrix To obtain the equation for the single-particle operator ̺ˆ(1), let us apply the operation of coagulation Tr to both sides of the equation (2.4). As 2...N Tr H(i), ̺ˆ(1,...,N) 0 i ≡ by definition (2.10), we obtain (cid:2) (cid:3) b N Tr H(i), ̺ˆ(..., i,...) =(N 1)! H(1), ̺ˆ(1) . 2...N − i=1 X(cid:2) (cid:3) (cid:2) (cid:3) b b Due to the identity Tr H(i,j), ̺ˆ(1,...,N) 0, ij ≡ taking into account the property (2.6) an(cid:2)d the definition (2.12(cid:3)) we have b N N 1 Tr H(i,j), ̺ˆ(1,...,N) = 2...N 2 i=1 j=1 XXj6=i(cid:2) (cid:3) b N N N 1 = Tr H(1,j), ̺ˆ(1,...,N) + H(i,j), ̺ˆ(1,...,N) = 2...N 2 ! j=2 i=2 j=2 X(cid:2) (cid:3) XXj6=i(cid:2) (cid:3) b b — 4 — = (N 1)! Tr H(1,2), ̺ˆ(1,2) . − 2 Similarly (cid:2) (cid:3) b 1 Tr dˆ̺ˆ = Tr aˆ(1)̺ˆ, aˆ+(1) + aˆ(1), ̺âˆ+(1) + 2...N 2 2...N (cid:2) (cid:3) (cid:2) (cid:3) N + aˆ(i)̺ˆ, aˆ+(1) + aˆ(i), ̺âˆ+(1) + aˆ(1)̺ˆ, aˆ+(i) + aˆ(1), ̺âˆ+(i) + Xi=2(cid:16)(cid:2) (cid:3) (cid:2) (cid:3) (cid:2) (cid:3) (cid:2) (cid:3)(cid:17) N N + aˆ(i)̺ˆ, aˆ+(j) + aˆ(i), ̺âˆ+(j) = ! Xi=2 Xj=2(cid:16)(cid:2) (cid:3) (cid:2) (cid:3)(cid:17) 1 = (N 1)! aˆ(1)̺ˆ(1), aˆ+(1) + aˆ(1), ̺ˆ(1)aˆ+(1) + Tr aˆ(2)̺ˆ(1,2), aˆ+(1) + 2 − 2 (cid:16)(cid:2) (cid:3) (cid:2) (cid:3) (cid:2) (cid:3) + aˆ(1), Tr ̺ˆ(1,2)aˆ+(2) , 2 as (cid:2) (cid:3)(cid:17) Tr aˆ(i), ̺âˆ+(j) 0, Tr aˆ(i)̺ˆ, aˆ+(j) 0. i ≡ j ≡ Bringing together these ex(cid:2)pressions, we o(cid:3)btain the equation (cid:2) (cid:3) i¯h̺ˆ˙(1)= H(1),̺ˆ(1) +Tr H(1,2), ̺ˆ(1,2) + 2 1 (cid:2) (cid:3) (cid:2) (cid:3) + i¯h aˆ(1)̺ˆ(1), aˆ+(1)b+ aˆ(1), ̺ˆ(1)aˆ+(1b) + Tr aˆ(2)̺ˆ(1,2), aˆ+(1) + 2 2 (cid:16)(cid:2) (cid:3) (cid:2) (cid:3) (cid:2) (cid:3) + aˆ(1), Tr ̺ˆ(1,2)aˆ+(2) . (3.1) 2 (cid:2) (cid:3)(cid:17) Wedenotethematrixelements̺ˆ(1),̺ˆ(1,2),H(1),H(1,2)andaˆ(1)inanα-representationasfollows: ̺α1α′1 =̺11′ , b ̺αb1α2,α′1α′2 =̺12,1′2′, Hα1α′1 =H11′, Hα1α2,α′1α′2 =H12,1′2′ , aα1α′1 =a11′, and write the equation (3.1) in matrix form as: i¯h̺˙11′ = H12̺21′ ̺12H21′ + H12,34̺34,1′2 ̺12,34H34,1′2 + − − Xα2 (cid:0) (cid:1) Xα2 Xα3 Xα4 (cid:0) (cid:1) 1 + i¯h 2a ̺ a+ a+ a ̺ ̺ a+ a + 2 12 23 31′ − 12 23 31′ − 12 23 31′ Xα2 Xα3 (cid:0) (cid:1) 1 + i¯h ̺ a a+ a+ a + a a+ a+ a ̺ . (3.2) 2 12,34 42 31′ − 42 31′ 12 34− 12 34 42,31′ Xα2 Xα3 Xα4 (cid:16) (cid:0) (cid:1) (cid:0) (cid:1) (cid:17) The two-particle density matrix describing the fermion system state must be antisymmetric. This matrix can be approximately expressed by means of the single-particle density matrix as follows: ̺ =̺ ̺ ̺ ̺ . (3.3) 12,1′2′ 11′ 22′ − 12′ 21′ Substituting thisexpressionintotheequation(3.2),weobtaintheequationforthesingle-particledensity matrix 1 i¯h̺˙11′ = H12̺21′ −̺12H21′ + 2 i¯h 2a12̺23a+31′ −a+12a23̺31′ − ̺12a+23a31′ + Xα2 (cid:16) (cid:17) Xα2 Xα3 (cid:0) (cid:1) 1 + i¯h ̺ ̺ a a+ a+ a a a+ +a+ a + 2 12 34 43 21′ − 43 21′ − 23 41′ 23 41′ Xα2 Xα3 Xα4 (cid:16) (cid:0) (cid:1) — 5 — + a a+ a+ a a a+ +a+ a ̺ ̺ , (3.4) 12 34− 12 34 − 14 32 14 32 43 21′ where (cid:0) (cid:1) (cid:17) H =H +2 H ̺ . (3.5) 12 12 13,24 43 Xα3 Xα4 Let us write the equation (3.4) in operator form 1 i¯h̺ˆ˙= H,̺ˆ + i¯h aˆ̺ˆ, aˆ+ + aˆ, ̺âˆ+ + 2 (cid:2) (cid:3) (cid:16)(cid:2) (cid:3) (cid:2) (cid:3)(cid:17) 1 b + i¯h Tr (aˆ+̺ˆ)aˆ Tr (aˆ̺ˆ)aˆ+, ̺ˆ + ̺âˆ+, ̺âˆ + aˆ+̺ˆ, aˆ̺ˆ , (3.6) 2 2 2 − where ̺ˆ=̺ˆ(1), (cid:16)(cid:2) (cid:3) (cid:2) (cid:3) (cid:2) (cid:3)(cid:17) H =H(1)+2 Tr H(1,2), ̺ˆ(2) (3.7) 2 is anaveragedsingle-particleHamiltonian.The equatio(cid:2)n(3.6) canbe(cid:3)givenmore compactformif we use b b b the notation ′ 1 H =H + i¯h Tr (aˆ+̺ˆ)aˆ Tr (aˆ̺ˆ)aˆ+ . (3.8) 2 2 2 − (cid:16) (cid:17) In this case we have b b ′ 1 i¯h̺ˆ˙= H,̺ˆ + i¯h aˆ̺ˆ, aˆ+(1 ̺ˆ) + (1 ̺ˆ)aˆ, ̺âˆ+ . (3.9) 2 − − (cid:2) (cid:3) (cid:16)(cid:2) (cid:3) (cid:2) (cid:3)(cid:17) Ingeneralthe rightsidesbof equations(3.8)and(3.9)may containseveralterms describingdissipative effects which depend ondifferent operatorsaˆ . Taking this into accountwe generalizethe equation (3.8) k and write it as follows: ′ i¯h̺ˆ˙= H,̺ˆ +i¯h C aˆ ̺ˆ, aˆ+(1 ̺ˆ) + (1 ̺ˆ)aˆ , ̺âˆ+ , (3.10) kl k l − − k l (cid:2) (cid:3) Xk,l (cid:16)(cid:2) (cid:3) (cid:2) (cid:3)(cid:17) b where ′ H =H + i¯h C Tr (aˆ+̺ˆ)aˆ Tr (aˆ ̺ˆ)aˆ+ . (3.11) kl 2 l k− 2 k l Xk,l (cid:16) (cid:17) b b We write the equations (3.10) and (3.11) in matrix form: ′ ′ 1 i¯h̺˙11′ = H12̺21′ −̺12H21′ + 2 i¯h 2Γ12,31′ ̺23 −Γ23,12̺31′ − Γ31′,23̺12 + Xα2 (cid:16) (cid:17) Xα2 Xα3 (cid:0) (cid:1) 1 + i¯h ̺ ̺ Γ Γ Γ +Γ + 2 12 34 43,21′ − 21′,43 − 23,41′ 41′,23 Xα2 Xα3 Xα4 (cid:16) (cid:0) (cid:1) + Γ Γ Γ +Γ ̺ ̺ , (3.12) 12,34− 34,12 − 14,32 32,14 43 21′ where (cid:0) (cid:1) (cid:17) ′ 1 H =H + 2H + i¯h Γ Γ ̺ , (3.13) 12 12 13,24 2 12,43− 43,12 43 Xα3 Xα4 (cid:16) (cid:0) (cid:1)(cid:17) Γ =2 C a a+ . (3.14) 12,31′ kl 12,k 31′,l k,l X 4. The principle of detailed balance Suppose that at some point in time the density matrix becomes diagonal: ̺11′ =W1δ11′ , (4.1) — 6 — where W W is the probability of the state α being filled by a particle. The substitution of matrix 1 ≡ α1 1 (4.1) into (3.12) leads to the equation W˙ = P (1 W )W P (1 W )W , (4.2) 1 12 1 2 21 2 1 − − − Xα2 (cid:16) (cid:17) where P =Γ (4.3) 12 12,21 istheprobabilityofthetransitionofaparticlefromstateα tostateα duringatimeunit.Thetransition 2 1 probability P can always be represented as 12 P12 =P1(o2)e−21β(ε1−ε2), (4.4) where (o) (o) P =P , 12 21 ε – the average energy of a particle in the state α , which is an eigenvalue of operator (3.13): i i ε =ε +2 H W , (4.5) 1 1 12,12 2 Xα2 where ε =H . 1 11 When a fermion system comes to the state of thermodynamic equilibrium, the particle distribution over the states must be subjected to the principle of detailed balance. We show that this distribution follows from the equation (4.2). Since the probability W no longer depends on time, the right side of 1 this equation is equal to zero. Moreover,according to the principle of detailed balance each term should be equal to zero. Thus the principle of detailed balance in this case is expressed by P (1 W )W =P (1 W )W . (4.6) 12 1 2 21 2 1 − − Wesubstitutetheexpression(4.4)intothisequation.Aftersimpletransformationsweobtaintheequality 1 W 1 W − 1 e−βε1 = − 2 e−βε2, W W 1 2 in which the left side depends on α , and the right one - on α . This is only possible if both sides are 1 2 equalto the same constantvalue. Denote this value e−βµ, where µ is the chemicalpotential. We get the equation 1 W − =eβ(ε−µ). (4.7) W Using the equality (4.5) we could show that this equation has an anisotropic solution [15]. Fromtheequation(4.7)wefindthattheequilibriumdistributionofnoninteractingparticlesoverstates is described by the Fermi - Dirac function 1 W = . (4.8) 1+eβ(ε−µ) 5. The electrons in a metal Consider electrons in metal. Equation (4.7) is transformed to 1 wk ln − =β(εk µ). (5.1) wk − Here, the function wk describes the electron distribution over wave vectors k, εk =εk+ εkk′wk′ (5.2) k′ X — 7 — is the average energy of an electron, εk is the energy of an electron without taking into account its interaction with other electrons, εkk′ is the interaction energy of electrons with wave vectors k and k′. Assume the following approximate formula for the interaction energy εkk′ =Iδk+k′ −Jδk−k′, (5.3) where I is the repulsion energy of electrons with vectors k and k′, J is the attraction energy of the electrons with vectors k and k′. The substitution of (5.3) into (5.2−) gives εk =εk+Iw−k J wk, (5.4) − and the equation (5.1) becomes 1 wk ln −wk =β εk+Iw−k−J wk−µ . (5.5) (cid:0) (cid:1) Replace in this equation the vector k by the vector k: − ln 1−w−wk−k =β εk+Iwk−J w−k−µ , (5.6) (cid:0) (cid:1) where ε−k =εk. The equations (5.5) and (5.6) form a system with two unknowns wk and w−k. Thissystemhasthesolutionsoftwotypes.Oneofthemdescribestheisotropicdistributionofelectrons over wave vectors and the other describes the anisotropic distribution. If w−k = wk, then each of the equations (5.5) and (5.6) turns into the equation 1 wk ln − =β εk+(I J)wk µ . (5.7) wk − − (cid:16) (cid:17) In the equations (5.5) and (5.6) the unknown functions wk and w−k are complex functions, in which the roleofanintermediatevariableis playedbythe kinetic energyεk ofanelectron:w−k =w1(εk) and wk =w2(εk). The functions w1 =w1(ε) and w2 =w2(ε) are the solutions of the equations 1 w 2 1 ln − = 2ǫ+(1 f)w (1+f)w , (5.8) 2 1 w τ − − 1 (cid:16) (cid:17) 1 w 2 2 ln − = 2ǫ+(1 f)w (1+f)w , (5.9) 1 2 w τ − − 2 (cid:16) (cid:17) where ε µ 4θ ǫ= − , τ = , J +I J +I the ratio of the energy I and J is determined by the parameter J I f = − . J +I Without loss of generality, we may accept the condition w (ǫ) w (ǫ). (5.10) 1 2 ≤ Let us introduce new variables d and s by means of relations w w =d, w +w =1+s. (5.11) 2 1 1 2 − — 8 — Due to (5.10) the quantity d is non-negative: d 0. The maximal value of d is equal to one: d [0,1]. ≥ ∈ The quantity s takes on values from 1 to 1: s [ 1,1]. Let us solve equations (5.11) with respect to − ∈ − w and w : 1 2 1 1 w = (1+s d), w = (1+s+d). (5.12) 1 2 2 − 2 We transformequations(5.8)and (5.9)using (5.12)to a formconvenientfor their numericalsolution. To do this we first subtract one equation from another, then we add these equations. As a result, we obtain the following system: (1+d)2 s2 − =e4d/τ , (1 d)2 s2 − −  (5.13) ǫ= τ ln (1−s)2−d2 + 1 (1+s)f .  8 (1+s)2 d2 2 − These equations allow us to represent the energy ǫ and the probabilitiesw and w as functions of the 1 2 parameterd. Using these functions it is not difficult to constructthe graphs of the functions w =w (ǫ) 1 1 and w =w (ǫ) for different values of temperature. These graphs are shown in Fig. 1 for the case when 2 2 the interaction energy of electrons I and J is such that J =3I. At the same time f =1/2. The solution of equation (5.7) is also a function of electron kinetic energy εk: wk = w0(εk). The function w =w (ǫ) can be found from the equation 0 0 1 w 4 0 ln − = ǫ f w , (5.14) 0 w τ − 0 (cid:0) (cid:1) whichisacorollaryofequation(5.7).Thesolutionw =w (ǫ)ofequation(5.14)isamongthesolutions TT0 0 of equations (5.8) and (5.9). Indeed, if we put in these equations w = w , then each of them takes the 1 2 form (5.14). Fig. 1 shows the graphs of all three functions w = w (ǫ), w = w (ǫ) and w = w (ǫ). From this 0 0 1 1 2 2 figure it is clear that at sufficiently low temperatures the distribution function for some values of the energyǫ cantake notone but severalvalues.The multivaluedness of functions w =w(ǫ) suggeststhat at the same temperature there can be different equilibrium macrostatesofa systemof conduction electrons in metal. These macrostates differ from each other by the distributions of electrons in Bloch states. In fact, only the macrostate of electrons, in which their energy is minimal, is realized, provided that this state is stable and cannot be destroyed by any external effects. The graphs in Fig. 1 give an idea of the changes in the distribution of electrons in states that occur when the metal temperature changes. When the temperature τ 1, only the isotropic distribution of ≥ electronsoverwavevectorsdescribedbythefunctionwk =w0(εk)ispossible.Thegraphofthefunction w =w (ε)forallvaluesofthetemperaturepassesthoughthepointΩwithcoordinatesε=µ+1(J I) 0 0 2 − and w = 1. As the temperature decreases, the slope of the curve at this point increases. When the 2 temperature is sufficiently low,the curve ofw =w (ε) bends so that it becomes similar to the letter Z. 0 0 Forτ =1 onthe curvew =w (ε)atthe pointΩ aclosedcurveZ originates.Its sizeincreasesasthe 0 0 temperature decreases. The shape of the curve also changes. The value τ =1 corresponds to the critical temperature I +J T = . (5.15) c 4k B Atτ (τ′,1),whereτ′ issomecriticalvalue,theverticallineintersectsthecurveZ nomorethanintwo ∈ points(thecurve2inFig.1c).Atτ <τ′ thecurveZ bendssothattheverticallineintersectsitfourtimes in some places (Fig. 1b). In the limit, τ 0, the curve Z transforms into a polygon AB C OC B A, 1 1 2 2 → that looks like the letter Z.This polygonis shownin Fig. 1а. The curve w =w (ε) divides the curve Z 0 0 into two parts. The lower part corresponds to w =w (ε) and the upper - to w =w (ε). 1 1 2 2 Let us give more detailed consideration to the distribution of electrons over states at T = 0. In the — 9 — limit, τ 0, the solution of (5.14) takes on the form → 1 if ε µ+J I, ≤ − ε µ w (ε)= − if µ ε µ+J I, (5.16) 0  J −I ≤ ≤ − 0 if ε µ. ≥ The graph of this function is shown inFig. 1a as the dotted broken line B AOC . 2 1 Inthelimit,τ 0,theequations(5.8)and(5.9)givethefollowingdependenceoftheprobabilityw on → the electron kinetic energy ε, which describes the anisotropic distribution of electrons over wave vectors: 1 if ε µ I, ≤ − w(ε)= wi(ε) if µ I ε µ+J , (5.17)  0 if ε −µ+≤J≤, ≥ where i=1 or 2. The values of the functions w =w (ε) and w =w (ε) form pairs, such that 1 1 2 2 w (ε)=0 and w (ε)=1 (5.18) 1 2 at µ I ε µ+J, or − ≤ ≤ 1 w (ε)= (ε µ+I) and w (ε)=1 (5.19) 1 2 J − at µ I ε µ+J I, or − ≤ ≤ − 1 w (ε)=0 and w (ε)= (ε µ) (5.20) 1 2 J − at 0 ε µ+J. The brokenline AB C O in Fig. 1а correspondsto w =w (ε), and the line AB C O 1 1 1 1 2 2 ≤ ≤ – to w =w (ε). 2 2 6. Conclusions ThequantumMarkovequationforN-particledensitymatrixdescribestheevolutionofastrictlyopen system with allowance for the dissipative operator. From this equation, we obtained the approximate kinetic equationfor a single-particledensity matrix. This equation describes the evolutionof a system of identicalparticles.Weobtainedtheintegralequationforthefunctionofdistributionofparticlesinstates where a many-particlesystemis in thermodynamic equilibrium. We showedthat for equilibriumsystems of interacting particles the distribution over wave vectors is anisotropic. — 10 — References [1] J. von Neumann, Mathematical Foundations of Quantum Mechanics (Nauka, Moscow, 1964). [2] Y.R.Shen, Phys. Rev. 155 (1967) 921. [3] M.Grover, R.Silbey, Chem. Phys. 52 (1970) 2099, 54 (1971) 4843. [4] A.Kossakowski,Rep. Math. Phys. 3 (1972) 247. [5] V.Gorini, A.Kossakowskiand E.C.G.Sudarshan, J. Math. Phys. 17 (1976) 821. [6] G.Lindblad, Commun. Math. Phys. 48 (1976) 119. 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