Quantum Markovian master equation for a system of identical particles interacting with a heat reservoir

Physica A 176 (1991) 366-386 North-Holland

QUANTUM MARKOVIAN MASTER EQUATION FOR A SYSTEM OF IDENTICAL PARTICLES INTERACTING WITH A HEAT RESERVOIR B.V. BONDAREV Moscow Aviation Institute, Volokolamskoye Shosse 4, 125871 Moscow, USSR

Received 30 July 1990 Revised manuscript received 14 January 1991

The author's earlier quantum kinetic theory of systems in thermal contact with a heat reservoir is now applied to systems of identical particles. A statistical description of such systems is provided by a hierarchy of many-particle density matrices for which a system of "linked" equations is obtained. The quantum Markovian master equation is derived, which makes it possible to determine the single-particle density matrix. Consideration is given to three examples of the application area of this equation, viz. the latter is described for three cases in which particles experience (1) random walks in a crystal lattice, (2) migration in a homogeneous isotropic continuum, (3) thermoactivated transitions between quantum states. In each of these cases the density matrix is found to describe the stationary state of a system of identical particles.

I. Introduction A great variety of applications of the physics of solids, chemical physics, nuclear physics, as well as other fields of science are often faced with the problem of motion of two or more particles (quasi-particles) in a stochastic condensed medium with allowance for their interaction with the environment and with each other. This problem is closely related to the general theory of irreversible processes and with the kinetic theory of systems which are in thermal contact with heat reservoirs [1-4]. The description of the motion of particles in a stochastic medium in terms of the classical (nonquantum) statistical theory is based as a rule on Smoluchowski equation, the Fokker-Planck equation, ztnd on the Liouville equation. These equations take no account of the wave properties of particles and, therefore, in a number of cases they are inadequate to describe the phenomena under study. Such cases necessitate o3btaining and using quantum analogues of the above equations. For example, quantum equations describing the motion of one 0378-4371/91/$03.50 (~) 1991- Elsevier Science Publishers B.V. (North-Holland)

B.V. Bondarev / Quantum Markovian master equation

367

particle in a stochastic medium are reported in refs. [5-11], and quantum master equations for systems of identical particles are developed in refs. [12-14]. Many-particle systems, where cooperative phenomena occur, are often in a stationary or even in a nonequilibrium state rather than in a thermodynamic equilibrium state. The investigation of such systems can be provided by master equations which take into account both the interaction between the partide~, themselves and their interaction with the environment. The development of th,z quantum statistical kinetics of such many-particle systems is stimulated by keen interest in different cooperative phenomena, such as superconductivity, superfluidity, strong magnetism, the condensation of electrons and holes in semiconductors, etc. [15, 16]. To describe the motion of the exiton among the sites of a crystal lattice, Haken and Strobl used the yon Neumann equation to derive the master equation on the basis of a stochastic model [5]. In ref. [17] this model is generalized to an arbitrary quantum system and the master equation is obtained for describing the system evolution of a Markovian process. This equation is employed here to describe the migration of a quasi-particle in a stochastic medium. In quantum mechanics, a statistical description of many-particle systems is made by means of a hierarchy of many-particle density matrices which must satisfy some system of "linked" equations [18]. The major purpose of this work is to obtain such equations for a system of identical particles whose evolution may be viewed as a Markovian process. A starting point will be the quantum Markovian master equation reported in ref. [17]. This equation and its properties will be briefly described in section 2 of the present article. Sections 3 and 4 will be concerned with the quantum Markovian master equations for one particle moving in a homogeneous isotropic continuum or performing random walks in a crystal lattice. In section 5, we shall obtain the quantum Markovian master equation for a system of identical particles and derive from it the master equation for the single-particle density matrix. Some applications of this equation will be considered in the subsequent sections. The master equation for a system of identical particles which perform random walks among the crystal lattice sites will be derived in section 6. in next seciion 7, we shaii employ this equation to obtain the master equation for a system of identical particles in a homogeneous isotropic continuum. Finally, section 8 will deal with the master equation for a system of identical particles for the particular case in which random external actions on the system cause thermoactivated transitions of particles between quantum states. In all of the considered examples, we shall determine the single-particle density matrices which describe the stationary states of the system and show that in the enexgy

B.V. Bondarev / Quantum Markovian master equation

368

representation the diagonal elements of these matrices are the Fermi-Dirac functions in the case of fermions or the Bose-Einstein functions in the case of bosons.

2. Quantum Markovian master equation Consider a certain nonequilibrium system which interacts with a "large" system being in the state of thermodynamic equilibrium. The latter system plays the role of a heat reservoir. Under some general conditions we may assume that a Markovian process occurs in the system under consideration. In this case the system evolution is described by the following master equation [3]: ihlJll, = ~

Blz,z,l,PZ2, ,

(2.1)

2~2 p

where Pir =- Ps,s; = p(t, Si, s~)

(2.2)

is the density matrix, i = 1, 2; s is a set of quantum numbers determining the system state; B,z,z, ,, = B ( s , , s2; s~, s'l)

(2.3)

are the matrix elements of the so-called binary operator, which are the coefficients in the linear differential equation (2.1). The summation on the right side of (2.1) is with respect to s 2 at.~d s~. The density matrix possesses the following properties: P~,,=P,,,

(2.4)

and W, = 1,

(2.5)

I

where W, - W ( s , ) = p,,

(2.6)

is the probability that the system is in state s~. These properties place certain limitations on the matrix elements of the binary operator, namely, quantities (2.3) must satisfy the conditions

B . V Bondarev / Quantum Markovian master equation

369

B * I 2.2'1' = - B l ' 2 ' , 21 ,

(2.7)

2

(2.8)

B!2,2,1 = 0. I

These conditions will be satisfied automatically if the matrix elements of the binary operator are defined as [3, 17] 8,,,,,,

=

+

(2.9)

where

F~2.2.,.

= -F1.2..2, ,

(2.10)

/'2.2 = ~'~ F12.2., ,

(2.11)

I

612 = ~ ( $ 1 - - S 2 )

are

generalized Kronecker symbols.

the

There is another the property completely positive. In order that definiteness of the density matrix operator rl2.2,10 has the form [19, /'12.2. I, =

A

12t~2.1 • -- t~i2A 2. I, +

of the density matrix, namely it must be eq. (2.1) secures a conservation of positive it is necessary and sufficient that the binary 20] (212'1 ---'---,

ih ~ ," " j & "tn"tk) " 1 2 " 1 " 2 '~ ' j.k

where A lz is a self-adjoint operator, Ax2* = A2~, and cj~ is any positive matrix. In particular case it may be such that cjk = Cj6ik, where cj are positive coefficients. Using formulae (2.9) and (2.12) we can rewrite eq. (2.1) in the form ihj6,,. = ~ (A,2P2 ,. - p,2A2,,) 2 ~tal 2 to22,a2,1,

j.k

_ / [ .u ,., . . .u ~. .., / . t , , i ,

+ P12a22 ' U 2 " l ' I f

2.2'

•

(2.13)

In the operator notation this equation is as follows [19, 20]: ih/~ = [m,/5] + ih ~ cjk (a(J;/$a (k~* - ½[a(k)'a (j', ~]+ },

(2.14)

j,k

where [A, t3] is the commutator, [&/~]+ is the anticommutator between d and b, the sign t denotes Hermitian conjugation.

370

B.V. Bondarev I Quantum Markovian master equation

3. Particle in a homogeneous isotropic continuum

Let/5 be the statistical operator describing the motion of a particle of mass m in a stochastic continuous medium. If the medium is homogeneous and isotropic, master equation (2.14) for the single particle is 2iD it~b=[A,/sl + ~ {'i/5'it-½['£'i,/s]+} 2iy ~ . + --~ {b/sb* - ½[l~*b,/5]+ } ,

(3.1)

where

A=B+

,[ /3 p, nP+me

= ~

+'

1

,62 + 0

(3.2)

is the particle Hamiltonian; ~, `6 and 0 are the operators of the coordinates, momentum and potential energy, respectively, F = - V U is the external force acting on the particle; /3 = (kBT) -~ is the reverse temperature; D is the diffusion coefficient; a =/33' is the friction coefficient; ci =`6 + lih/3/~,

ih/3 /~ = ~ + 4--mm/~"

(3.3)

Substituting the operators (3.2) and (3.3) into eq. (3.1) and performing a series of simple transformations, we obtain the following equation [17]: iD it~ = [H, b l - --g- {[~,[~, bl] + ½ih/3[,& [/~', hl+l + (¼h/3)2[/', IF,/sll} iy {[~, [,,/5]] + ih/3~

h

(h/3'~z [/~, [i0, /5]]} .

[e. [`6./51+1 + \4m/

(3.4)

This equation is a quantum analogue of the Fokker-Planck equation for one particle. If we assume that/3 = 0 in (3.4), we shall get the equation encounted in the quantum theory of continual measurements [7-11]. In the coordinate representation eq. (3.4) takes on the form ih

Op

h2

+iaD((V +V')-{V+V'-

h

)"

½/3(F + F ' ) } - 1/32(F-F')2)p

+ 2---ram

\4m

(V + V')2J p '

(3.5)

B.V. Bondarev I Quantum Markovian master equation

371

where p = O ( t , r, r'). Eqs. (3.4) and (3.5) have the property that p is self-adjoint and completely positive for all t. It is not difficult to make sure that the function

p(r, r') = n exp

rn (r-r') 2+im ~-~ F -

2/362

(r - r ' )

)

(3.6)

represents the stationary solution of eq. (3.5), which describes a system of noninteracting particles in a constant homogeneous force field: U = - F . r , F = const.; n = const, is the particle concentration. One may consider the form of the right-hand side of eq. (3.4) as a decomposition into a Hamiltonian part and a dissipative part:

ih~=[H,/~1 + Dr3. The construction of the dissipative part has no direct physical interpretation. To understand its meaning better let us derive an equation for the Wigner function

w(t,r,p)=(2~rh)

r )exp (i) -~

p(t,r+~r,r-½

p'r'

dr',

(3.7)

which is a quantum analogue of the classical distribution function. Using eq. (3.5) we find that at F = const, the desired equation is at follows:

Ow - P "Vw - F "Vpw + D r " ( V - ~ F ) w 0-[m

(3.8) This equation is similar to the Fokker-Planck equation for a Brownian particle. However, eq. (3.8) contains additional terms of a quantum mechanical nature. Substituting expression (3.6) into formula (3.7) yields the equilibrium solution of eq. (3.8),

{

(

( t3 V'" /3 w(p) = n k ~ ] exp -2-ram p

m F a

)2} "

(3.9)

This is none other than the Maxwell distribution function. There can be no doubt that eq. (3.4) correctly describes the motion of the single particle in a stochastic medium. However, this equation becomes unsuitable for describing a system of identical particles even if they do not interact with each other. The correct equation for a system of noninteracting

372

B.V. Bondarev / Quantum Markovian master equation

identical particles should lead to the Fermi-Dirac or Bose-Einstein quantum distribution functions but not to the Maxwell distribution function.

4. Discrete model of r a n d o m walks

Suppose that some particle performs random walks in a cubic lattice. This motion can be conveniently described by the density matrix (4.1)

O,,, -- OR,RI = p(t, R t , R[)

in the site representation. Using eq. (2.13) we may construct the following equation [17]:

iht~,t, = ~ (H,2p2,,- P12H2,,) + iti ~, {.tr, z;2_,+,,.,,p2.2_t. ,, - ½(P2, + P 2 , ' ) P , , ' } 2

iT( ~t

,2

(R,-Rt)p,,,

+ ½/3(R!-R[)'E(Jt2P2,,

2

+ P12J21 ,)

. . "J2,1, ) ) , ",1 ~2 E (112 " J22'P2'I ' - 2JI 2" 022,J2, 1' + Pl,J,2, 2,2'

(4.2)

where H12 = H(R~, R2) are the matrix elements of the Hamiltonian, which we regard as phenomenological quantities, HI2 = - I 1 _ 2 + Ul~12 ~ - I g +

g = R t - R z,

l_g = I~,

U(R1)3¢, la= o = 0 ,

lim Ig = 0 ,

(4.3)

U 1 = U(Rt) is the potential energy of the particle occupying site R~, "/rl2.2, l, -~" "/'J'(R 1, R2; R~, R[) = VP12Pt:2: ,

(4.4)

P2, = P(R2, R l ) =

(4.5)

P21-(°)e x p { _ ½ f l ( U 2

_ U1)}

is the probability of a tbermoactivated transition of a particle from site R~ to . C0) _ _(o) site R2, be21 --//12 =pg0) iimg_,~ ~g'(°)= 0, P2.2-1*1' == p ( t , R2, R 2 - R! + R [ ) ,

(4.6) J12 = (Ri - R2)H12 •

B.V. Bondarev / Quantum Markovian master equation

373

Functions Ig and p~) are such that

E gl~ = o

E g/~g __~o, = 0 ,

,

g

g

h 8~,~,

(4.7)

g g"gflg= m ~"

g,g,,p~O) = 2D6~,, , g

where g~, are the Cartesian coordinates of the vector g, g, v -- 1, 2, 3. In another notation we can write eq. (4.2) as

ihlSR~" = --~'~ Ig(PR+g.R'- PR.R'+~) + (UR -- UR')PRR' g

+ ih ~ {(PR.R+gP~,R,+g)a/2pR+~.R,+~ -- t~(pn.g.g + PR'+g.R')Pn.R'} g

h

(s - R')~R,,, + ½t3(R - R'). F_, g1,,(p,,+,.,,.- p,.,.+,:) g

~ 2 Eg Eg " g. g'td~, (p.+~+r.., + 2p.+~. +~ + p . . , . , ~ ) . (4.8) If function (4.1) varies slowly in going from site to site, it ,,maybe replaced by a continuous function which coincides with it at the lattice sites up to a constant:

p(t,R,R')=op(t,r,r'),

(4.9)

where v is the volume per lattice site. Let us substitute function (4.9) into eq. (4.7). If functions (4.9) and U(r) have R + g or R' + g as their arguments, we expand them into series in powers of g,. Leaving only the terz,s containing the first and second derivatives of p and U, and using formulae (4.7) we arrive at eq. ~.3.5). '

5. Hierarchy of density matrices for a system of identical particles. Master equatie~i Consider a system of N identical particles. A statistical description of this system is provided by the many-particle density matrix [18]

374

B.V. p(N)

Bondarev

-~ P_ l( .N. .)N , I ' . . . N '

/ Quantum

p(N)

-~" -

Markovian

master equation

' ;V) ( t , O l l , " " " ' OIN~ Oll , " " " , OZ '

(5.1)

where a i is the set of quantum numbers determining the state of one particle, i = 1 , . . . , N. Density matrix (5.1) satisfies the normalization condition

E

• " "

I

E P_(N) l...N,I...N

= N!

(5.2)

N

the summation being performed with respect to variables a I . . . . , a N. The density matrix pU,) of a complex of n particles is defined as follows:

pO,) =

P- (l .'.). n , l ' . . . n '

1 --

(N-

p(.)

--

Z

, ..,ct,;a~,...,a

:,)

Z _(N) •

n)l

( t , a 1,

•

,,+t

Pi...N,

"

I'...n'

n+I...N

"

(5.3)

N

It is not difficult to show with the aid of equality (5.2) that n-particle density matrix (5.3) satisfies the condition ~' _(n) 1

• ..

n

_

ul..n.t..,

N!

(5.4)

v'/n'-n

(i)

Of the greatest practical interest is the one-particle density matrix Ply,, whose diagonal element =

=

(1)

(5.5)

is the occupation probability for state a~, i.e. the probability that state a t is occupied by one of the particles of the system. If n = 1, condition (5.4) gives ~'~ Wt = N.

(5.6)

I

A remarkable property of many-particle density matrices is their symmetry (or antisymmetry). For example, density matrix (5.1) describing a system of fermions reverses its sign if any two of the arguments og . . . . . a N (or a ' l , . . . , a~) are transposed, i.e. it is antisymmetric. As regards the density matrix for a system of bosons, there is no sign reversal under such transpositions• Let us assume that the evolution of a system of identical particles represents a Markovian process. In this case the system can be described by eq. (2.1). Taking into account that the state of the system is determined by the set of quantum numbers {a 1. . . . , a N } = S we may write eq. (2.1) in the form

375

B.V. Bondarev / Quantum Markovian master equation

ihptIN.!.N, ,'...N'

-- X

10

• .

.

X . .X.

Xa,

N" 1"

N"

. .

,,.,,... .N. 0 ~ N . ) . .N..,-...,,. . . ,,,,,,,...,,..,, . .

.

(5.7)

The symmetry of the density matrix lays sufficiently stringent requirements on the form of the coefficients on the right side of eq. (5.7). To have an idea of its form, let us first consider a system of noninteracting identical particles. In the simplest case when the system contains only one particle the master equation is as follows: "" P"l (I ,' ) = ID,

5'. no12,2,1,P22, -(')

2,2'

(5.8)

,

where the binary operator elements satisfy the conditions B * 12.2'1' =

--B1"2'.21 ,

(5.9)

(5.10)

E B12.2'1 = 0 . 1

The master equation for a two-particle system (N = 2) in the general case is of the form

i.L-(:, t ~ p 12,1 ' 2 '

=

~'~ E -

(2>

(5.11)

/~12,34;3'4',! '2"1034,3'4 ' '

3,4 3',4'

The coefficients in this equation must obey conditions (2.7) and (2.8): B 12.34;3.4..1.2. = - BI.2..3.4.;34A2 ,

(5.12)

E B12,34;3,4,12 = 0 . 1.2

(5.13)

If the two particles do not interact with each other, the coefficients in eq. (5.11) must be expressed in terms of the coefficients B l:.z,x, in eq. (5.8) and the Kronecker symbols. Taking into account that both sides of eq. (5.11) must be symmetric (or antisymmetric), and the binary operator form must be determined by formulae (2.9) and (2.12), we arrive at the equation [21] (2) (2) n (2) (2) • ,~ -(2) l/'~p 12.1 'z' = E { B I 3 , 4 1 , P 3 2 , 4 2 ' + B23,41'PI3,42' + / ~ 1 3 , 4 2 ' P 3 2 . 1 ' 4 + B 2 3 , 4 2 ' P 1 3 A ' 4 3,4

, (2) I (2) B 31,,42' + -- 1(B13,24 + ,, D24,13)P34.1,2,~Pl2,34(

B42,.31,)} .

(5.14)

B.V. Bondarev / Quantum Markovian master equation

376

The two-particle density matrix is such that

p(2)*

n(2) 1 2 , 1 ' 2 ' = F" 1 ' 2 ' , 1 2

_(2) = 2

P 12,12

(5.15) at N = 9--

(5.16)

•

1,2

In addition, it is antisymmetric for fermions: (2) 12,1'2' =

(2) --P21,1'2'

~

(2) ~PI2,2't'

.

=

,a(2) /~'21,2'1'

(5.17)

and symmetric for bosons: _(21 P12,1'2'

( 2 ~~ ~--" P 2 1 , 1 ' 2 '

_(21 (2) . / ' # 1 2 , 2 ' 1 ' ~- P 2 1 , 2 ' 1 '

=

"

(5.18)

It is easy to make sure that equalities (5.15) and (5.16) are integrals of eq. (5.14) if the elements of the one-particle binary operator B12,2,t, satisfy conditions (5.9) and (5.10). Properties (5.17) and (5.18) can easily be shown to agree with eq. (5.14). It should be recalled that eq. (5.14) describes two noninteracting identical particles. It may seem that the absence of the force interaction between the particles must guarantee their statistical independence, i.e. the probability that one particle is in state a l and the other in state a 2 must be equal to the product of the prooabilities,

W12 (= W(1)W(l) 2' " 1 ) " 2

"

(5.19)

This is however not the case. Doubts about the statistical independence of two noninteracting particles arise when we make a unitary transformation of the density matrix /_(2~ in passing from one representation to another. We can /12,1'2' see that relation (5.19) is not invariant under unitary transformations. The physical reason for "he statistical interdepen6ence of two noninteracting particles is their indistinguishability in virtue of which the density matrix must be either symmetric or non-symmetric. When ~t,l,~'-~ up eq. ~,-,.x-~l, t~ 1,~ we concentrated our attention on its symmetry and agreement with equalities (5.17) and (5.18). As a result, it so happened that the equation has acquired a property which may be called inseparability. This property consists in the fact that the two-particle density matrix p~2)l,2, satisfying eq. (5.14)cannot be expressed in any way in terms of the single-particle density matrices ptl!, which satisfy eq. (5.8). This property is also a consequence of the antagonism between the indistinguishability of the particles and their statistical independence.

B.V. Bondarev /

Quantum

377

Markovian master equation

Now using eq. (5.14) as the pattern we set up a similar equation for density matrix (5.1), which describes a system of N noninteracting particles. The equation of interest is as t011ows: :,!=.'.." I I I ' P(N) l...N,I'

N -E E [1"~. ^(N) • ..N' ~ k"*i0P 1 . . i=1

--

.i-1

0i+1

...

N, 1'...N"

0

^(N)

Pl...N,1'...i-l'Oi+l'..N'GOi')

+ E 0.0'

i N

N.,,

1

1 ~

,-,.o,,+,

.

(N)

- (Dio,jo'Pt..,i-toi+l...i-lo'i+l...N,r...iv' 2 i./=i

"F P- l(N) ...N,l'...i-I'Oi+l',..j-I'O'j+l'.

.. N ' D o i ' .

x [l'j' ) )

(5.20) where Gio = G(c~,, Oto),

c , 2 = n,,_ ± E

D,0.0 T = D ( a , , c~,,; a o, %), 1

+

3

(5.21) ~,

(1)

- D34.12)P43 ,

E

3.4

H12 = H(a~, a:) is the one-particle Hamiltonian,

D,2.2,1, =

(5.22)

B12,2,1, - HI2~2,1, + 812H2,1,

is the dissipative part of the binary operator, the plus corresponds to a system of bosonos and the minus corresponds to a system of fermions. The construction of right side of eq. (5.20) is similar to that of eq. (2.13). Therefore eq. (5.20) conserves the positive definitene~.s of the density matrix and other properties. It should be noted that we had to deviate somewhat from the pattern and add the terms containing quantities (5.21) into the right side of eq. (5.20). The grounds for this deviation will be presented below. It is not difficult to derive from eq. (5.20) a system of linked equations for the hierarchy of density matrices (5.3) of systems of particles. We omit the somewhat lengthy and not interesting calculations and write the final equation for the one-particle density matrix Inpll,

l~\r21'

2

+

3

~ D,~,~.z',' p(Z) 22' + 2,2'

3 p42 J ~1 E {(Dt3.24 _ D -.4 . 13)( P34.1'2-(2)- P ~ " )_(Z)X. 2,3,4

(~),,J~,/J42.31',-, + t_(2) ~,Pl2.34 -- 0~1'~ 3 11924 D31',42)}

(5.23)

B.V. Bondarev ,' Quantum Markovian master equation

378

The presence of terms with the product of the one-particle density matrix in eq. (5.23) is caused by the presence of the terms with quantities (5.21) on the right side of eq. (5.20). The necessity of these terms can be explained in the following manner. It follows from definition (5.3) that _(2) = ( N - 1)p~lt!

(5.24)

P12,1'2 2

If we replace the two-particle density matrix expression (2)

vt2,t,2,-(2)in

this equation by the

^(I) (I)

(5.25)

1012,1'z' - - P 1 1 ' 1 0 2 2 ' ,

we obtain the relation (2) (t)] ~'~(101z,,'z - - P_(I)_ ll'P22 t 2

=

A(I) --PII' "

(5.26/

The two-particle density matrix enters into eq. (5.23) as combination (5.25). As follows from relation (5.26), this fact guarantees the absence of terms containing N on the right side of eq. (5.23) after its convolution. The presence of such terms would make eq. (5.23) incorrect. It should be pointed out that all the equations for the n-particle density matrices, starting from (5.20) and finishing with (5.23), are invariant under arbitrary unitary transformations of the density matrices. To obtain some useful practical results from the foregoing equations, we have to break off some of the first equations from the system of linked equations. This can be most simply done with the first equation. To this end, we must accept certain approximate relations between the two-particle and one-particle density matrices. The simplest (and most likely the only) relations of this kind are as follows: 0(21 ~(1),,.~(1) + (1) (I) 12.1'2' ~" P I I ' / J 2 2 ' - - 1 0 1 2 ' P 2 1 ' ,

(5.27)

where the plus corresponds to bosons and the minus to fermions. Substituting expressions (5.22) and (5.27) into eq. (5.23) yields [21] ihhll,

= Z B12.2,1,1022, 2,2'

+_1 Z -- 22,3,4

{(B13,24 - B 2 4 , 1 3 ) P 3 2 P 4 1 ,

+ PI4P23(B42,31,

-- B31,42)}

,

(5.28) where

B,V. Bondarev / Quantum Markovian master equation

379

_(1) PlI' ~ Pll' '

B I 2 , 2 , 1 , --- H 1 2 8 2 , 1 , -

8 1 2 H 2 , I. + D 1 2 , 2 , 1 , .

It is not difficult to verify that a solution of eq. (5.28) is self-adjoint, normalized and completely positive for all t > to, if it is such at t = t o. Eq. (5.28) may be employed for describing the evolution of the system of interacting particles. For this purpose the Hamiltonian of the particle in the mean-field approximation should be taken as follows: H,2

H ~ ) + 2 ~. /"/13,24 ,,(2) P43 3.4

(5.29)

"

Here ** u ( l12) is the one-particle Hamiltonian (4.3), H 12) 13.24 is the interaction Hamiltonian of two particles. Using formulae (2.9) and (2.12) we can rewrite eq. (5.28) in the operator notation, = [A, b(1 ± b)l

+ ½ ih ~ cj,{[&~J~A6a~*' ~, 1 -+ t~]+ - [&'*~ '~(1 -+ t~)6 ~j', t~]+} •

(5.30)

l,k

6. Lattice gas The form of the right side of eq. (5.28) makes it possible to write the master equation for a system of identical particles if the binary operator B12.2 u. describing the motion of the single particle is known. Thus, for example, instead of eq. (4.2) we shall have the master equation for a system of identical particles performing random walks among lattice sites [21],

3

/

3

+ ih -~--~{ql'12;2-1*l'.l'P2,2-1+l'

-

1 -

+ P..,.)P,,.}

2

+- ~ ih ~ {(~',2;3.,-2÷3 2,3

~r3.,-2+s:,2)P23P,-2+3.,'

-Fp12P3.t'-2+3(~23:I'-2+3,1' - ~ 1 ' - 2 + 3 . 1 ' : 2 3 ) }

B.V. Bondarev / Quantum Markovian master equation

380

( k , - R;)2p,,, + ½~(R,- s ; ) . ~ : (J,~o~,, + o,d~,,) 2

, '

-- 1"$~ E (JI2"J23P31' "-2.I12" 2,3

ioL

-Y-~

P23J31 , --I- P12J23.J31,

)

}

~ {(R z + R 2 - 2R3). J,2P:,3P2i' + 2(R, - R;)" Pt2J23P3,' 2,3

+ o,~o,_d3,. (2R.,- R , - R'l)),

(6.1)

where Oil' = p(t,R~,R~). Let us determine the stationary solution of eq. (6.1) for the case of a homogeneous constant external force field

UR=-F'R,

(6.2)

where F = const. In this case the density matrix must be invariant under translation, i.e. it must of the form

p.R, = p(R- R').

(6.3)

Now apply the Fourier transform to function (6.3):

1 O.s, = ~LL E~ wk exp{ik. (R - R')},

(6.4)

where N L is the number of sites in the lattice. Which form has the function w~,? The answer is predetermined. This must be the Bose-Einstein function for a system of bosons or the Fermi-Dirac function for a system of fermions. If in spite of being complicated eq. (6.1) will give us the correct answer in our particular case, we may hope that it may be useful in other cases as well. Substituting function (6.4) into eq. (6.1) and taking into account the formula (5.29) we get the equation

Vk"

Wk+/3Wk(1-----wk)Vk ek - - F ' k o~

=0,

(6.5)

where e k = e~" + Z Vkk,Wk, k'

(6.6)

B.V. Bondarev I Quantum Markovian master equation

381

is the energy of particle with a wave vector k, ~kc,) =

-Z

g

1~ exp(ik, g)

(6.7)

is the kinetic energy, 2 V~k, ~ ~ Z ~'~ Z ~t~t t4t2) RR' ; R +g, R' +g' exp{i(k, g + k ' - g ' ) } NL

•

R'

g

(6.8)

•'

is the energy of interaction of two particles with the wave vectors k and k'. The solution of eq. (6.5) is the known function W k -~

exp

I~k

ff

F k

/z

~- 1

,

(6.9)

where/z is the chemical potential. Eq. (6.9) implicitly specifies the function w k for the interacting particles so as the energy (6.6) depends on w k.

7. Master equation for a system of identical particles moving in a h o m o g e n o u s isotropic c o n t i n u u m

Taking into consideration that eq. (5.30) corresponds to eq. (2.14), let us replace eq. (3.1) by the equation iD iht~ = [ A , P(1 +- 15)] + y

~ ^ ^,

{[apa, 1 +

,6]+ - [ti +(1 -+ 15)d, 15] + }

iv + -~- {[l~t~b*, 1 + t)]+ - [b*(1 + p)b, 15]+}.

(7.1)

Substitution of the operators (3.2) and (3.3) into eq. (7.1) yields

ih~ = [/:t, t~(l - b)] io {[

/h~

2

)

_-'- ~/3D{'~[[p,^ P]+, 1521- [P, b0t~l + [,0, 15Pbl} T-d [~'[~' 151.1 + \ 4 m l +

Ot

- 2--~ { ½[[/~' f]+' 1521 - [/~' 150] + [f, 15/~151} •

(7.2)

B.V. Bondarev / Quantum Markovian master equation

382

In the coordinate representation eq. (7.2) takes on the form

-V'2)+U-U')(p+-fp.,,pr.,dr")

ap ( h 2 .--2 ih-~- =\-~--mml,V

+ ihO{(17 + W ) e - 113(17 + 17').(F + F ' ) -

~[3Z(F

-

F')2}p

-T- [ ih~D J dr" {½(17.F +17'.F')p,,,,p,,,,, + Fp,,,,,.(17 +17")p,,,, + F ' . p,,.(17' + 17")m, + F " . (17 + 17')p.,pr,,} iy ~[(r-

r,)Z

(n2.]

h:/3 ( r - r ' ) . ( 1 7 - V ' ) (17 + + ~ \T-m'm/

17,)5}o

iha ~" "~m f dr" {3Prr"Pr" + ( r - r") "Vp,,,,p¢,,, + (r' -- r") "V'p,r,,pr,,r, + ( r - r'). P,r"17"Pr",'},

(7.3)

where P =

~rr'-~

p(t, r, r'),

U = U(r) : Uo(r ) + f ~ ( r - r') p,.,. dr',

(7.4) (7.5)

Uo(r ) is the potential energy of particle in the external force field, ~ ( r - r') is the energy of two interacting particle~, F = -VU. The stationary state of the system of particles in a homogeneous constant external field (U 0 = - F o . r , F 0 = const.) is described by density matrix (7.4) of the form P,r' = (2703

Wk

exp{ik. (r - r')} d k .

(7.6)

Here the function w k satisfies the normalization condition (2~r)3

w~dk=n,

n being the concentration of particles. Substitute function (7.6) into eq. (7.3). For the system of noninteracting particles a number of manipulations will lead us to the equation V~" Vkw k +/3w~(1 ± Wk)Vk ~ - - ce

B.V. Bondarev / Quantum Markovian master equation

383

in which

~k = ~ 1 (t~k)2The solution of eq. (7.7) is function (6,9). Putting r' = r in eq. (7.3) we arrive at the continuity equation

0n Ot

-V.j

(7.8)

where n = p(t, r, r) is the concentration of particles, and j is the particle flux density vector defined as

j= _{ ih

~m(V-V')(p+-f

p,:p,..,, dr") +

D(V+V'-[3F)p}.=,

+-~ f (13D(F+F")+ a---(r-r"))p,,.p:rdr ". -

(7.9)

m

If the external force field is homogeneous, and the density matrix is of the form of (7.6), it is not difficult to derive from formula (7.9) that J = (2~) ~

)

m hk + ~DF o

wk(1 +-- w k ) d k .

8. The principle of detailed balance Consider a system of identical particles whose evolution is described by the density matrix p,,.

=

p(t, ot,, Or',)

(8.1)

in the energy representation where the Hamiltonian has the form Hll, --- E 18,,!, .

(8.2)

Here E 1 = E(a~) is the particle energy in state a I. In this case, to the equilibrium state of the system there must correspond the diagonal density matrix

011 = WtStl' , where

(8.3)

384

B.V. Bondarev

I Quantum

' ¥ = ( e x p { / 3 ( E , , - ~ ) } -v-1) -t to[

Markovian

master equation

(8.4)

"

Let us assume that the positive binary operator ~12,2'1' has the form /"12,2' ! ' = eti,812t~2'l ' + ih'h'i2.2'l' E 8 1 - 2 + 3 8 1 ' - 2 ' + 3 3

(8.5)

,

where ~1-2+3 - ~ ( a t - ~z + a 3 ) , e ~ , = A t - A ~ , + ih ~-'~ t.jkt* - _(/)_~k), 1 U 1, j,k

A t = A ( a t ) is a real function, ' ~ i.~

h(m)h(n) "r

'~'12,2°1 ' = L ' ~ " " m n V l 2

'-'1'2' ~'

in. r!

C/k and C.,. are positive matrices.

Using expressions (2.9) and (8.5) we transform eq. (5.28) to

i/~thl,

=

{~,,,-

}(Ell + et'l')}Plt'

± } E ( g t 2 - - E21 + ~ 2 t ' 2

+ i h ~ ' ~ {1rt2-z-t+t.l' 'P.._-1~ +t' 2

~' (~r2,,12

+

rr2t..t

El'2)Pt2P2t'

,2)p,,,}

+ } ~h Z ((~,...,.,-,_+3- ~, t 2+,:,~)p~,pt-~+3, 2,3

-I" P12P3,1,_2+3(q]'23;l,_2+3,1 , -- 71"1,_2+3,1,;23)} •

(8.6)

Substitution of the density matrix (8.3) into eq. (8.6) leads to the equality ~'~ { p ~ z ( 1 ± Wt)Wz-p2~(1 ± W2)Wt} = 0 , 2

(8.7)

where Pl2 = qrt2.21 is the transition probability of a particle from state o~2 to state a~. Let the transition probability is determined by the formula PiE =/'t2-(°)exp{_ ½f l ( E l _ ez)}

(8.8)

where /'t2 _(o) = P21. _(o) Function (8.4) satisfies the equation pt2(1 - Wt ) W 2 = p2t(1 - W 2 ) W 1 , which expresses the principle of detailed balance.

(8.9)

B.V. Bondarev / Quantum Markovian master equation

385

9. Conclusion

Summarizing, we recollect that the main purpose of the present article is to obtain the quantum master equation for a system of identical particles in contact with a heat reservoir. This equation (5.28) is obtained on the condition that the system evolution represents a Markovian process. The interaction of the particles with one another can be taken into account in the mean-field approximation, which is quite acceptable for a system of particles in a condensed medium where the major contribution comes from the collective interactions of the particles but not to their pairwise collisions. Sections 6 to 8 are concerned with the applications of the master equation to particular physical models. The application area of eq. (5.28) is likely far wider than the examples considered. At least, the equations obtained allow some generalizations. The density matrix PsR,, the equation for which is presented in section 6, describes, in a one-band approximation, a system of particles performing random walks among lattice sites. In a more general case, it is necessary to use _ts,~,), which depends on variables s and s', where s is a set the density matrix PRR' of quantum numbers determining the state of the particle on a lattice site. The (s.s' master equation for the density matrix PRR' can also be set up on the basis of eq. (5.28). Eq. (7.3) describing the evolution of a system of identical particles in a stochastic continuum may be generalized to the case where particles represent carriers of a charge q and experience the action of a magnetic field. In this case the operator V should be substituted by V-iqA/hc, where A is a vector potential. No principal difficulties can be encountered in developing an analogous quantum kinetic theory for systems of particles of different sorts. The equations obtained in this article and their generalizations may serve as a helpful tool in kinetical investigations of cooperative phenomena in condensed media.

References [1] J.A. McLennan, Phys. Rev. 115 (1959) 1405; Phys. Fluids 4 (1961) 1319; Adv. Chem. Phys. 5 (1963) 261. [2] C.R. Willis and P.G. Bergmann, Phys. Rev. 128 (1962) 391. [3] K. Blum, Density Matrix Theory and Applications (Plenum, New York, London, 1981). [4] V.G. Bariakhtar and E.G. Petrov, Kinetic Phenomena in Solids (Naukova Dumka. Kiev, 1989).

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B,V. Bondarev / Quantum Markovian master equation

[5] H. Haken and G. Strobl, in: The Triplet State, A.B. Zahlan ed. (Cambridge Univ. Press, London, I967) p. 311; Z. Phys. 262 (1973) 135. [6] M, Grover and Ri Silbey, J. Chem. Phys. 54 (1971) 4843. [7] A. Barchieili, L. Lanz and C.M. Prosperi, Nuovo Cimento B 72 (1982) 79. [8] A. BarchieUi, Nuovo Cimento B 74 (1983) 113; Phys. Rev. A 34 (1986) 1642. [9] E. Joos and H.D. Zeh, Z. Phys. B 59 (1985) 223. [10] G.C. Ghirardi, A. Rimini and T. Weber, Phys. Rev. D 34 (1986) 470. [11] C.M. Caves and G.J. Milburn, Phys. Rev. A 36 (1987) 5543. [12] R. Balian and M. V6n6roni, Ann. Phys. (NY) 135 (1981) 270. [13] A,1. Onipko, Chem. Phys. 89 (1984) 371. [14] J.L. Nero, Phys. Lett. B 138 (1984) 241; Ann. Phys. (NY) 173 (1987) 443. [15] V.G, Valeev, G.F. Zharkov and Yu.A. Kukharenko, The kinetic theory of nonequilibrium processes in superconductors, in: Nonequilibrium Superconduction, V.L. Ginzburg, ed., Transactions of the Institute for Physics of the USSR Academy of Sciences, vol. 174 (Nauka, Moscow, 1986) pp. 155-214. [16] C.D. Jeffries and L.V. Keldysh, eds., Electron-Hole Drops in Semiconductors (Nauka, Moscow, 1988), [17] B.V. Bondarev, The master equation describing Markovian processes in a quantum system, and its applications, VINITI paper No. 7692-V89 (Moscow, 1989); Phys. Lett. A 153 (1991) 326. [18] N,N. Bogolubov and N.N. Bogolubov Jr, Introduction into Quantum Statistical Mechanics (Nauka, Moscow, 1984). [19] G. Lindblad, Commun. Math. Phys. 48 (1976) 119. [20] V. Gorini, A. Frigerio, N. Verri, A. Kossakowski and E.C.G. Sudarshan, Rep. Math. Phys. 13 (1978) 149. [21] B,V. Bondarev, On the use of the density matrix formalism in the quantum theory of solids, VINITI paper No. 3710-V90 (Moscow, 1990).