Kinetic equations for open quantum system in the occupation number representation
Master equation for density matrix of an open many–particle system is derived in the occupation number representation. The Born approximation with respect to system–bath interaction is utilized and the fast relaxation within the system is assumed to be fulfiled. The reduction of a linear master equa...
Saved in:
| Published in: | Вопросы атомной науки и техники |
|---|---|
| Date: | 2001 |
| Main Author: | |
| Format: | Article |
| Language: | English |
| Published: |
Національний науковий центр «Харківський фізико-технічний інститут» НАН України
2001
|
| Subjects: | |
| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/80027 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| Journal Title: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Cite this: | Kinetic equations for open quantum system in the occupation number representation / E.G. Petrov // Вопросы атомной науки и техники. — 2001. — № 6. — С. 261-264. — Бібліогр.: 11 назв. — англ. |
Institution
Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1859708695595712512 |
|---|---|
| author | Petrov, E.G. |
| author_facet | Petrov, E.G. |
| citation_txt | Kinetic equations for open quantum system in the occupation number representation / E.G. Petrov // Вопросы атомной науки и техники. — 2001. — № 6. — С. 261-264. — Бібліогр.: 11 назв. — англ. |
| collection | DSpace DC |
| container_title | Вопросы атомной науки и техники |
| description | Master equation for density matrix of an open many–particle system is derived in the occupation number representation. The Born approximation with respect to system–bath interaction is utilized and the fast relaxation within the system is assumed to be fulfiled. The reduction of a linear master equation to a nonlinear set of kinetic equations for one–particle distribution functions is carried out at the condition of strong particle–particle interaction. As an example, the procedure of derivation of kinetic equations for description of electron transfer through specific molecular nanostructures like molecular wires is demonstrated with taking into consideration the strong Coulomb repulsion between the transferred electrons.
|
| first_indexed | 2025-12-01T04:30:42Z |
| format | Article |
| fulltext |
KINETIC EQUATIONS FOR OPEN QUANTUM SYSTEM IN THE
OCCUPATION NUMBER REPRESENTATION
E.G. Petrov
Bogolyubov Institute for Theoretical Physics, National Academy of Sciences, Kiev, Ukraine
e-mail: epetrov@bitp.kiev.ua
Master equation for density matrix of an open many–particle system is derived in the occupation number
representation. The Born approximation with respect to system–bath interaction is utilized and the fast relaxation
within the system is assumed to be fulfiled. The reduction of a linear master equation to a nonlinear set of kinetic
equations for one–particle distribution functions is carried out at the condition of strong particle–particle interaction.
As an example, the procedure of derivation of kinetic equations for description of electron transfer through specific
molecular nanostructures like molecular wires is demonstrated with taking into consideration the strong Coulomb
repulsion between the transferred electrons.
PACS: 02.50.Ey, 02.50.Wp, 05.20.Dd, 05.60.+w
1. INTRODUCTION
Correct and the most complete description of
evolution of an open quantum system (QS) to its
equilibrium state occurs with the nonequilibrium density
matrix method [1,2]. The method brings to a
Generalized Master Equation (GME) for density matrix
of the QS, σ ( )t . In Born approximation with respect to
interaction H int between the QS and the bath, the GME
reads (cf. e.g. Refs. [3–5]
[ ] ( ) ( ) , ( )σ σ τ τt i H t d e i H H
t
B= − − − +∫
0 2
0
1 0
( )[ ][ ] ( )× −− +H H t eB
i H HB
int int, , .τ τρ σ τ 0 (1)
Here
( )[ ]A = ≡ − +Tr A A i H HB Bρ ττ, exp 0
( )[ ] ,exp 0 τ+× BHHiA (2)
where H0 is the QS Hamiltonian, and =ρ B
( ) ( )[ ]TkHTrTkH BBBBB −−= expexp is the equilibri-
um density matrix of heat bath. In GME (1), one takes
H =int 0 . If it is not the case, then the substitutions
H H H0 → +0 int and H H Hint int int→ − have to be
performed.
Integro–differential operator equation (1) represents
a non–Markovian form of GME. Let evτ be the
characteristic time of evolution process which exceeds
strongly the characteristic time τ d that specifies the
kernel’s decrease in the integral-containing part of
Eq. (1). Therefore, in line with general principles of the
reduction of non-Markovian equation to the Markovian one
[1,4] one can extend the upper integration limit in Eq. (1) up
to ∞ , putting simultaneously σ τ σ( - ) ( )t t≅ . Such a
substitution reduces an integro–differential GME (1) to a pure
differential GME. Below we shall utilize a tetradic form of
differential GME (the Redfield’s equation [3–5]),
( )∑ σ−σ−σω=σ
r
nrrmrmnrnmnmnm tVtVitit )()()()(
− ′ ′
′ ′
′ ′
∑ Γ nm
n m
n m
n m
tσ ( ) , (3)
which is valid for elements σ σnm t n t m( ) ( )= of the
QS density matrix. Here, the manifold { }n relates to a
complete set of QS basic functions so that
( ) ( )H E n n n V n n
n
nn
nn
nn0 1= + − ′∑ ∑ ′
′
′δ , (4)
H F n nnn
nn
int
.= ′′
′
∑ (5)
In Eq. (4), E n( ) is the QS energy in state n whereas
V V n H nnn n n′ ′= = ′*
0 is the transition matrix element
between different states. Operator == ′′
*ˆˆ
nnnn FF
nHn ′= int characterizes the coupling between the QS
and the heat bath. Quantities Vnn ′ , and Fnn ′ specify the
dynamic and relaxation transitions in the QS.
Eq. (3) is a basic one for derivation of kinetic
equations describing an evolution process in different
type of open QS. Two first terms in the right part of
Eq. (3) describe a pure dynamic behavior of the QS (
[ ]ω nm E n E m≡ −( ) ( ) is the transition frequency)
while the third term is responsible for a nonequilibrium
process which is characterized by a relaxation
supermatrix
{∑ ∫
∞
τωτ
′′
′′ δτ=Γ
r
i
nrnrmm
mn
nm
mreFFd
0
2
ˆˆ1
τωτ
′′δ+ rni
rmrmnn eFF ˆˆ
} .ˆˆˆˆ τω
′
τ
′
τωτ
′′
′′ −− nmnm i
nnmm
i
nnmm eFFeFF (6)
(Strongly, the form (6) is valid at small value of nnV ′
and if )()( nEnE ′≈ [5,6]).
2. NONLINEAR FORM OF KINETIC
EQUATION
The further specification of kinetic equations is
dictated by the relation between the pure dynamic and
relaxation processes in the QS. We restrict ourselves by
consideration of transfer processes in a condensed
matter where the bath vibrational levels imitate a quasi
continuous spectrum. In this case, the transition
PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY. 2001, № 6, p. 261-264. 5
probabilities coincide with those found in the framework
of Fermi Golden Rule (i.e. at small couplings nnV ′ and if
)()( nEnE ′≈ [6]). As a result, the basic set of
equations (3) is transformed to a separate set of
equations for diagonal elements σ nn t( ) only. The set
reads as
( ) ( ) ( ) ( )σ σ σnn nn
n
nn n nt R t t= − −′
′
′ ′∑ (7)
where quantity
( )R R V E n E nnn n n nn′ ′ ′= = − ′
2 2π
δ
( ) ( ) (8)
specifies the transition probability between the QS states
n and ′n . It is very important that each QS state has
a complex structure with a manifold of substates, { }α .
Just a relaxation within such a manifold (caused by
interaction with a heat bath) is responsible for
appearance of the irreversibility of a common evolution
process in open QS. Let, for instance, manifold { }α
relate to vibrational substates of electronic term j so that
α= jn and thus α= jEnE )( is the energy of the α -
th substate of site j while α ′′α′ = jjnn VV is the coupling
between the vibrational substates of terms j and ′j .
Fast relaxation between the vibrational substates of each
term brings (for a short characteristic time relτ ) to a
Boltzmann distribution within the vibrational manifold.
As far as the population of each state is defined by
diagonal element )()( tt jjnn αασ=σ the ratio
( )[ ]σ σα α α α α αj j j j j j Bt t E E k T( ) ( ) exp′ ′ ′= − − (9)
is satisfied at any time relt τ> > . [In Eq. (9), kB and T
are the Boltzmann constant and the temperature,
respectively.] Therefore, if inter–term transitions occur
with the characteristic time τ τtr rel> > , one can utilize a
property (9) without any limitations. In particular, the
property (9) allows us to transform Eq. (7) to Pauli–like
equation
[ ]P t P t P tj j j j j j j
j
( ) ( ) ( )= − −→ ′ ′ → ′
′
∑ κ κ (10)
for integral term populations P t tj j j( ) ( )= ∑ σ α αα .
Quantities
( ) ( )κ π δα α
α α
α α αj j j j j j jV W E E E→ ′ ′ ′
′
′ ′= −∑2 2
(11)
with
( ) ( ) ( )∑
α ′
α ′αα −−= TkETkEEW BjBjj expexp (12)
being the distribution function for the j -th term, are the
inter–term transfer rates.
Rate equations (11) describe a linear one–particle
transfer. However, if particle–particle correlations are
important these linear equations do not reflect an actual
situation. One of important examples is a distant
electron transfer (ET) in molecular nanostructures like
molecular wires [5,7-9]. Just in such mesoscopic
systems, a Coulombic interaction between the
transferred electrons is shown to form the specific
nonlinear intramolecular currents [10,11].
The main goal of the present work is to derive
kinetic equations for description of ET in the condition
of strong electron–electron correlation. We restrict
ourselves by the systems where ET occurs on the
background of fast relaxation within the set of sublevels
belonging each site of electron localization. It means
that one can start from a general set of linear equations
(7) where now the n specifies many–particle
electronic states of the whole QS. Below we shall utilize
the occupation number representation where
{ }n N N= = ∏ λ
λ
(13)
and thus manifold { }N fixes electron occupation
numbers N λ = 0 1, for each single–electron state λ .
The structure of many–particle state allows us to
introduce the following form for diagonal elements of
density matrix,
{ } { }σ σ
λ
λ
nn N N Nt t P t( ) ( ) ( ).= = ∏ (14)
Here, quantity P t tN N Nλ λ λ
σ( ) ( )= satisfies the
normalization condition,
P tN
N
λ
λ
( ) ,=∑ 1 (15)
and determines the occupancy of the λ -th QS state,
P tλ ( ) , via relations
( )P t N P t N P tN
N N
Nλ λ λλ
λ λ
λ
( ) ( ) ( )= = −∑ ∑ −1 1 (16)
and
( ) .)()(1)(1 1∑∑
λ
λ
λ
λ −λλλ =−=−
N
N
N
N tPNtPNtP (17)
The structure of many–particle density matrix,
Eq. (14) supposes the derivation of self–consistent set of
equations for one–particle functions )(tPN λ or that is
the same, distribution functions == λλ )()( 1 tPtP
)(1 0 tP λ−= . To derive the corresponding set of equati-
ons we substitute form (14) in Eq. (7) which now reads
as
{ } { }
{ }
d
dt
P t V P tN N N
N
Nλ λ
π
λ λ
( ) ( )= −
∏ ∑ ∏′
′
2 2
{ }( ) { }( )( )−
′ −′∏ P t E N E NN λ
λ
δ( ) (18)
Here,
{ }( ) { }( )E N E N U N= +∑ λ λ
λ
(19)
is the QS energy with taking into consideration an
interaction between the particles (term { }( )U N ).
Quantity { } { } { } { }V N V NN N tr′ = ′ is the matrix
element between many–particle QS states with Vtr
being the transfer operator. Linear form (18) generates
automatically nonlinear kinetic equations for one–
particle distribution functions. Actually, let one multiply
6
Eq. (18) by the N ξ and then sum over all occupation
numbers N λ . With utilization of Eqs. (15)–(17) one
derives ( )
)()()( tPtPtPN
N N
NN ξ
ξ≠λ ξ≠λ
ξ =∑ ∑ ∏
λ ξ
λξ and thus
∑ ∑ ∏
′ λ
ξ′ξ
π−=
λ
}{ }{
2
}}{{ )(2)(
N N
NNN tPNVtP
{ }( ) { }( )( )NENEtPN −′δ
− ∏
λ
′λ
)( (20)
This nonlinear equation written for one–particle
occupancies P tξ ( ) is the main result of the present
work. Note that form (24) works in conditions of fast
relaxation in QS. The further specification of Eq. (24) is
dictated by concrete transfer process.
3. KINETIC EQUATION FOR SHORT
MOLECULAR WIRE
As an example we consider the ET through a short
molecular wire where interaction between the
transferred electrons is so large that no more than one
transferred electron be captured by the wire in the
course of ET across the wire [11]. Let mµE and 0
mµE be
the energies of the -th vibrational state when the
transferred electron does or does not occupy the m -th
wire unit while ksE is the electron energy when an
electron (with the wave vector k ) occupies the
conduction band of the left (=L) or the right (=R)
microelectrodes. Linear wire of units is assumed to
contact with the corresponding electrodes via their
terminal units 1=m and Mm = . No any magnetic
interactions are supposed to be in the system “electrode
L–Wire–electrode R” (LWR–system). With omitting the
spin identification we operate with two type of single–
electron states, ks=λ and mµ=λ . Thus, just the
occupation numbers 1,0=ksN and 1,0== ∑ µ mµm NN
along with the electronic occupancies )(tPsk and
∑ µ µ= )()( tPtP mm exhibit as the main quantum and
statistical characteristics of an electron in the LWR–
system. Electron energy (19) of this system has a form
( )[ ]∑∑ −++= µ ′µ
m
mmmm NENENENE 1})({ 0
LL
k
kk
.})({RR NUNE ++ ∑
q
qq (21)
This value depends strongly on the number of electrons
captured by the wire units (via the numbers 1,0=mN ),
and a Coulomb repulsion between the transferring
electrons (via term })({NU ).
Single–electron transitions are caused by the
transition operator
∑
λ ′λ
λ ′
+
λλ ′λ=
,
aaTVtr (22)
where +
λa and λa are electron creation and electron
annihilation operators with respect to a single electron
state λ while λ ′λT is the corresponding transition matrix
element. Let one derive kinetic equation for the
occupancy )(L tP k . In this case, only single–electron
states 11,L, µ=λ ′λ k participate in the ET process. E.g.
the ET occurs between the -th vibrational level of site
1=m and the k -th band state. The corresponding
coupling reads 11 µµVT L ′=λ ′λ k . [We employ a Condon
approximation, when matrix elements mn mnV µν are
factorized to the form where is the pure electronic
coupling while is the overlap integral between the
vibrational functions]. Therefore,
( ) ( )[ ] 111L
*
L1LL}}{{ 11 µµ ′′−′+−=′ NNVNNVV NN kkkk
kk LL11 1,1, NNNN −′−′ δδ×
∏∏∏ ′
≠
′
≠′
′ δδδ×
′′
qkk
qqkk RRLL ,
1
,, NN
j
NNNN jj (23)
and thus
∑ ∑ ∑ ∑ ∑
µ µ ′µ
µ ′µπ−=
k q
kk
L R 1 11
2
11
2
LL
2)(
N N N Nmm
VtP
( ) ( )})({})({1 1L NENENN ′−δ−× k
[ ])()()()(
11L11L 11 tPtPtPtP NNNN µ−−µ −×
kk
.)()(
R
1
∏∏
≠
µ×
q
q
tPtP N
m
N mm (24)
Now we take into consideration the fact of fast
relaxation within electronic terms. It has been already
noted that such a relaxation brings to important ratio (9)
between the partial site occupancies. Similar ratio exists
for the occupancies )(tP
mN µ . It reads =µ ′µ )()( tPtP
mm NN
( )[ ( ) ( ) ]TkNEETkNEE jmmmmm B
00
B 1exp
00
−−−−−= µ ′µµ ′µ
and thus expresses the vibration occupancies )(tP
mN µ
via the integral occupancies )(tP
mN as
( ) ( )[ ]0,
0
1,)()(
mmmm NmNmNN EWEWtPtP δ+δ= µµµ (25)
where Gibbs distribution functions for electronic terms
are defined by Eq. (12). Now, bearing in mind a weak
dependence of Coulombic repulsion on vibrational states µ
we can sum a right part of Eq. (24) over all vibrational states
mµ (except 1µ and 1µ′ ). With normalization conditions
1)( =∑ mµ mµEW and 1)( 0 =∑ mµ mµEW it yields
∑ ∑ ∑ ∑ ∑
µ ′µ
µ ′µπ−=
k q
kk
L R 1 11
2
11
2
LL
2)(
N N N Nm
VtP
( ) ( ) ( )[ ]0,
0
11,11L 1111
1 NN EWEWNN δ+δ−× µ ′µk
( ))()()()(
1L1L 11 tPtPtPtP NNNN −−−×
kk
( ) .})({})({)()(
R
1
NENEtPtP N
m
Nm
′−δ× ∏∏
≠ q
q (26)
Note now that for a short molecular wire under
consideration, the repulsion of transferring electrons is
assumed to be too strong to allow the appearance of
more then one transferring electron within the wire.
Physically, this fact indicates that energy conservation
law, })({})({ NENE ′= , containing in δ –function of
Eq. (26), is assumed to be fulfilled if only a single
hopping electron is captured by the wire in the course of
ET, i.e. at conditions 11 ≤∑ =
N
m mN and 11 ≤′∑ =
N
m mN .
Just these conditions one has to take into consideration
when the summation over occupation numbers is carried
out. To specify this circumstance, we introduce the
7
"repulsion multiplier" ( )∏ = −≡ N
m mNQ 10 1 which is
equal 1 for empty wire (i.e. at 01 =∑ =
N
m mN ) and 0 if
even one transferring electron occupies the wire (i.e. at
01 >∑ =
N
m mN ). The introduction of enables one to omit
the Coulombic contributions })({NU and })({NU ′ in
the energies involved in δ –function of Eq. (26). With
substitution the ( )})({})({ NENE ′−δ for the
( )µµ ′ −+δ 1
0
10 EEEQ ak and utilization of Eqs. (15-17) we
reduce Eq. (26) to the following compact form
( )∑
µ ′µ
µµ ′ −+δµ ′µπ−= 1
0
1
22
LL
2)( EEEVtP akkk
( )( ) ( ) ( )[ ])()(1)(1)( 11L1
0
1L tPEWtPtPEWtP µµ ′ −−−× kk
( )∏
≠
−×
N
m
m tP
1
)(1 (27)
This equation defines automatically the variation in time
of the number of electrons which are capable to be
transferred through a molecular wire from electrode L to
electrode R, i.e. the quantity )(L tN = =∑ k k )(L tP
=
)(R tN− . As far as )(tNL is the bulk characteristic, it
varies slightly in the course of the ET through a
molecular wire. It means that the )(L tP k has a minor
distinction from equilibrium Fermi distribution function
( )[ ]{ } 1
BFLL 1exp)( −
− += TkEEEn kk (28)
where FE is the Fermi energy of electrode. This let one
put ( )kk LL )( EntP ≅ in right part of Eq. (27) and thus
( )∏
=
−χ−=
N
m
m tPtN
1
LL )(1)(
( ) ( ) .)(11)(
2
11L ∏
=
− −δ−χ+
N
m
mm tPtP (29)
Rate constants Lχ and L-χ follow from Eq. (27) at
( )kk LL )( EntP ≅ and read as
( )∑
µ ′µ
µ ′µπ=χ
k
L
22
LL
2
kk EnV
( ) ( )[ ] ,L
0
11
0
1 kEEEEW −−δ× µ ′µµ ′ (30)
and
( )[ ]∑
µ ′µ
− −µ ′µπ=χ
k
L
22
LL 12
kk EnV
( ) ( )[ ] .L
0
111 kEEEEW −−δ× µ ′µµ (31)
Analogously, one can derive a complete set of equations
for all site occupancies )(tPm . For instance,
( ) ( )∏
=
− −+χ−=
N
m
mL tPtPgtP
2
111 )(1)()(
( ) ( ) .)(1)()(1
2
22
1
∏∏
≠=
−+−χ+
N
m
m
N
m
mL tPtPrtP (32)
It is clearly seen from Eqs. (27) and (32) that just
nonlinear factors ( )∏ −m m tP )(1 caused by extremely
strong Coulomb repulsion between the transferred electrons
are responsible for possible blocking the ET process.
Similar situation appears e.g. at resonant electron
tunneling in semiconductor heterostructures [12].
In conclusion, we note that the approach proposed in
the present paper gives possibility to derive different
form of nonlinear kinetic equations for one–particle
distribution functions if only the characteristic time of
transfer process exceeds strongly the characteristic times of
relaxation processes within electronic terms of the QS.
The work is supported in part by the Russian–Ukrainian
Programme on Nanophysics and Nanoelectronics.
REFERENCES
1. A.I. Akhiezer, S.V. Peletminskii. Methods of
Statistical Physics. London: “Pergamon Press”, 1981,
384 p.
2. K. Lindenberg, B.J. West. The
Nonequilibrium Statistical Mechanics of Open and
Closed systems. New York: “VCH Publishers”,
1990, 322 p.
3. W.T. Pollard, A.K. Felts, R.A. Friesner. The
Redfield equation in condensed-phase quantum
dynamics // Adv. Chem. Phys. 1996, v. XCIII, p. 77-
135.
4. E.G. Petrov. Averaged master equation for a
quantum system coupled to a heat bath with
fluctuating energy levels// Phys. Rev. E. 1998, v.
57, №1, p. 94-99.
5. D. Segal, A. Nitzan, W.B. Davis, M.R. Wasi-
liewski, M.A. Ratner. Electron transfer rates in
bridged molecular systems 2. A steady-state
analysis of coherent tunneling and thermal
transitions // J. Phys. Chem. B. 2000, v. 104, p.
3817-3829.
E.G. Petrov, Ye.V. Shevchenko, V.I. Teslenko,
V. May. Nonadiabatic donor–acceptor electron transfer
mediated by a molecular bridge: A unified description
of the superexchange and hopping mechanism // J.
Chem. Phys. 2001, v. 115, №15, p. 7107-7122.
6. A. Aviram, M. Ratner, Eds. Molecular
Electronics. New York: “Annals of the New York
Academy of Sciences”, 1998, 426 p.
7. E.G. Petrov, V.I. Teslenko. Tunnel currents
through a molecular wire. I. Quasielastic inter-electrode
current // Ukr. J. Phys. 1999, v. 44, № 10, p. 1298-
1305.
8. E.G. Petrov, V.I. Teslenko. Tunnel currents
through a molecular wire. II. Control over inter-
electrode current by the thermal wire units // Ukr.
J. Phys. 1999, v. 44, № 12, p. 1500-1507.
9. J. Chen, M.A. Reed, A.M. Rawlett,
J.M. Tour. Large on-off ratios and negative
differential resistance in a molecular electronic
device // Science 1999, v. 286, p. 1550-1552.
10. E.G. Petrov, P. Hänggi. Nonlinear electron
current through a short molecular wire // Phys. Rev.
Lett. 2001, v. 86, № 13, p. 2862-2865.
11. S.A. Gurvitz, H.J. Lipkin, Ya.S. Prager.
Interference effects in resonant tunneling and the
Pauli principle // Phys. Lett. A. 1996, v. 212, p. 91-
96.
8
1. INTRODUCTION
2. NONLINEAR FORM OF KINETIC EQUATION
3. KINETIC EQUATION FOR SHORT MOLECULAR WIRE
REFERENCES
|
| id | nasplib_isofts_kiev_ua-123456789-80027 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1562-6016 |
| language | English |
| last_indexed | 2025-12-01T04:30:42Z |
| publishDate | 2001 |
| publisher | Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| record_format | dspace |
| spelling | Petrov, E.G. 2015-04-09T15:57:27Z 2015-04-09T15:57:27Z 2001 Kinetic equations for open quantum system in the occupation number representation / E.G. Petrov // Вопросы атомной науки и техники. — 2001. — № 6. — С. 261-264. — Бібліогр.: 11 назв. — англ. 1562-6016 PACS: 02.50.Ey, 02.50.Wp, 05.20.Dd, 05.60.+w https://nasplib.isofts.kiev.ua/handle/123456789/80027 Master equation for density matrix of an open many–particle system is derived in the occupation number representation. The Born approximation with respect to system–bath interaction is utilized and the fast relaxation within the system is assumed to be fulfiled. The reduction of a linear master equation to a nonlinear set of kinetic equations for one–particle distribution functions is carried out at the condition of strong particle–particle interaction. As an example, the procedure of derivation of kinetic equations for description of electron transfer through specific molecular nanostructures like molecular wires is demonstrated with taking into consideration the strong Coulomb repulsion between the transferred electrons. The work is supported in part by the Russian–Ukrainian Programme on Nanophysics and Nanoelectronics. en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Вопросы атомной науки и техники Kinetic theory Kinetic equations for open quantum system in the occupation number representation Кинетические уравнения для открытой квантовой системы в представлении чисел заполнения Article published earlier |
| spellingShingle | Kinetic equations for open quantum system in the occupation number representation Petrov, E.G. Kinetic theory |
| title | Kinetic equations for open quantum system in the occupation number representation |
| title_alt | Кинетические уравнения для открытой квантовой системы в представлении чисел заполнения |
| title_full | Kinetic equations for open quantum system in the occupation number representation |
| title_fullStr | Kinetic equations for open quantum system in the occupation number representation |
| title_full_unstemmed | Kinetic equations for open quantum system in the occupation number representation |
| title_short | Kinetic equations for open quantum system in the occupation number representation |
| title_sort | kinetic equations for open quantum system in the occupation number representation |
| topic | Kinetic theory |
| topic_facet | Kinetic theory |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/80027 |
| work_keys_str_mv | AT petroveg kineticequationsforopenquantumsystemintheoccupationnumberrepresentation AT petroveg kinetičeskieuravneniâdlâotkrytoikvantovoisistemyvpredstavleniičiselzapolneniâ |