Non-equilibrium reversible dynamics of work production in four-spin system in a magnetic field
A closed system of the equations for the local Bloch vectors and spin correlation functions is obtained by decomplexification of the Liouville-von Neumann equation for 4 magnetic particles with the exchange interaction that takes place in an arbitrary time-dependent external magnetic field. The anal...
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irk-123456789-1199792017-06-11T03:03:30Z Non-equilibrium reversible dynamics of work production in four-spin system in a magnetic field Ivanchenko, E.A. A closed system of the equations for the local Bloch vectors and spin correlation functions is obtained by decomplexification of the Liouville-von Neumann equation for 4 magnetic particles with the exchange interaction that takes place in an arbitrary time-dependent external magnetic field. The analytical and numerical analysis of the quantum thermodynamic variables is carried out depending on separable mixed initial state and the magnetic field modulation. Under unitary evolution, non-equilibrium reversible dynamics of power production in the finite environment is investigated. За допомогою декомплексифiкацiї рiвняння Лiувiля-Неймана одержана замкнута система рiвняньлокальних векторiв Блоха та спiнових кореляцiйних функцiй для системи 4 магнiтних частинок з обмiнною взаємодiєю, якi знаходяться в довiльному залежному вiд часу магнiтному полi. Виконано аналiтичний та чисельний аналiз динамiки квантових термодинамiчних параметрiв в залежностi вiд сепарабельного початкового стану системи та модуляцiї магнiтного поля.В умовах унiтарної еволюцiї дослiджено нерiвноважну оборотну генерацiю роботи у найближчому оточеннi. 2011 Article Non-equilibrium reversible dynamics of work production in four-spin system in a magnetic field / E.A. Ivanchenko // Condensed Matter Physics. — 2011. — Т. 14, № 2. — С. 23005:1-12. — Бібліогр.: 33 назв. — англ. 1607-324X PACS: 05.30.-d, 03.65.Aa, 44.05.+e, 05.70.Ln DOI:10.5488/CMP.14.23005 arXiv:0901.3637 http://dspace.nbuv.gov.ua/handle/123456789/119979 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
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A closed system of the equations for the local Bloch vectors and spin correlation functions is obtained by decomplexification of the Liouville-von Neumann equation for 4 magnetic particles with the exchange interaction that takes place in an arbitrary time-dependent external magnetic field. The analytical and numerical analysis of the quantum thermodynamic variables is carried out depending on separable mixed initial state and the magnetic field modulation. Under unitary evolution, non-equilibrium reversible dynamics of power production in the finite environment is investigated. |
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Ivanchenko, E.A. |
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Ivanchenko, E.A. Non-equilibrium reversible dynamics of work production in four-spin system in a magnetic field Condensed Matter Physics |
| author_facet |
Ivanchenko, E.A. |
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Ivanchenko, E.A. |
| title |
Non-equilibrium reversible dynamics of work production in four-spin system in a magnetic field |
| title_short |
Non-equilibrium reversible dynamics of work production in four-spin system in a magnetic field |
| title_full |
Non-equilibrium reversible dynamics of work production in four-spin system in a magnetic field |
| title_fullStr |
Non-equilibrium reversible dynamics of work production in four-spin system in a magnetic field |
| title_full_unstemmed |
Non-equilibrium reversible dynamics of work production in four-spin system in a magnetic field |
| title_sort |
non-equilibrium reversible dynamics of work production in four-spin system in a magnetic field |
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Інститут фізики конденсованих систем НАН України |
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2011 |
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http://dspace.nbuv.gov.ua/handle/123456789/119979 |
| citation_txt |
Non-equilibrium reversible dynamics of work production in four-spin system in a magnetic field / E.A. Ivanchenko // Condensed Matter Physics. — 2011. — Т. 14, № 2. — С. 23005:1-12. — Бібліогр.: 33 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
AT ivanchenkoea nonequilibriumreversibledynamicsofworkproductioninfourspinsysteminamagneticfield |
| first_indexed |
2025-07-08T17:00:53Z |
| last_indexed |
2025-07-08T17:00:53Z |
| _version_ |
1837098919000014848 |
| fulltext |
Condensed Matter Physics, 2011, Vol. 14, No 2, 23005: 1–12
DOI: 10.5488/CMP.14.23005
http://www.icmp.lviv.ua/journal
Non-equilibrium reversible dynamics of work
production in four-spin system in a magnetic field
E.A. Ivanchenko∗
Institute for Theoretical Physics, National Science Center “Kharkov Institute of Physics and Technology”,
1 Akademichna Str., 61108 Kharkiv, Ukraine
Received January 5, 2011, in final form May 25, 2011
A closed system of the equations for the local Bloch vectors and spin correlation functions is obtained by de-
complexification of the Liouville-von Neumann equation for 4 magnetic particles with the exchange interaction
that takes place in an arbitrary time-dependent external magnetic field. The analytical and numerical analysis
of the quantum thermodynamic variables is carried out depending on separable mixed initial state and the
magnetic field modulation. Under unitary evolution, non-equilibrium reversible dynamics of power production
in the finite environment is investigated.
Key words: reversible dynamics, work production, spin system, magnetic field
PACS: 05.30.-d, 03.65.Aa, 44.05.+e, 05.70.Ln
1. Introduction
The classical thermodynamic heat engine converts heat energy into mechanical work with the
help of a classical mechanical system in which some gas expands and pushes a piston in a cylinder.
Such heat engine receives energy from a high-temperature reservoir. Part of energy from this reser-
voir is converted to mechanical work, and part of it is transferred to a low-temperature reservoir.
The classical heat engine reaches its peak efficiency when it is reversible. Due to the impossibility
of constructing an ideally reversible heat engine, in 1824 Carnot [1] offered the mathematical model
for an ideal heat engine which is not only reversible, but is also cyclic. In the last decades great
efforts have been made to investigate the quantum properties of the working substance, the search
and practical implementation of the quantum analogue of the Carnot cycle in microsystems.
The operation of quantum heat engines that employ multi-level systems as working agents,
for example harmonic oscillators, free particles in a box, three-level atoms or electrons subjected
to magnetic fields, were introduced in [2–4]. The two-level quantum systems similar to particles
with spin 1/2 are essential ingredients for quantum computation, but the coupled spin systems
can also be used as quantum thermodynamic engines [5–10]. The quantum analogue of the Carnot
cycle requires a dynamical description of the working medium, the power output and the heat
transport mechanism. The spin system “working gas” in an external field has its own physical
properties. These properties at weak interaction with the environment (heat baths) can, as a rule,
be deformed slightly. The purpose of this paper is to predict these properties and find out what
can be expected concerning the derivation of an equation for the reduced matrix of a system in
contact with an environment, usually considered in the Markovian approximation.
Based on quantum mechanics, we investigate the limiting case when interactions of the working
gas, consisting of 4 particles with spin l/2 in a variable magnetic field, with heat baths are to equal
to zero. In the other words, a unitary time evolution is studied, and the state of the system during
this evolution is interpreted using thermodynamic concepts.
The paper is organized as follows. In section 2 we introduce the model Hamiltonian. In section 3
we use the Bloch representation, in terms of the local Bloch vectors and spin correlation functions,
to write down the Liouville-von Neumann equation for a density matrix for four particles with spin
∗E-mail: yevgeny@kipt.kharkov.ua
c© E.A. Ivanchenko, 2011 23005-1
http://dx.doi.org/10.5488/CMP.14.23005
http://www.icmp.lviv.ua/journal
E.A. Ivanchenko
1/2 with an exchange interaction in a variable magnetic field. We describe the conservation laws
which effectively supervise the numerical calculations. In section 4 we describe the local quantum
thermodynamic parameters of the spin subsystems. Our numerical results are detailed in section 5.
For the separable mixed initial state we numerically investigate the quantum thermodynamics of
a particle in the environment of three others depending on the modulation of an operating field
and initial states. Our results are summarized and discussed in section 6. It will be numerically
found that under unitary dynamics, the work production of one part of a system is compensated by
absorption of work produced by the other part. Work production by a subsystem is accompanied
by entropy growth and vice-versa. The entropy decreases when work is absorbed by the subsystem.
Some necessary additional details for numerical results are presented in the appendix.
2. Model Hamiltonian
The Hamiltonian of four coupled particles e, p, n, u with spin 1/2 in the external ac magnetic
field h = (h1 , h2 , h3) looks like
Ĥ = he
is
e
i + hp
i s
p
i + hn
i s
n
i + hu
i s
u
i + 2(Jepseis
p
i + Jenseis
n
i + Jeuseis
u
i
+Jpnspi s
n
i + Jpuspi s
u
i + Jnusni s
u
i), (2.1)
where he
i , hp
i , hn
i , hu
i are the Cartesian components of the external magnetic field in the energy
units, operating on the corresponding particle ( we set the Bohr magneton µB equal to 1); sei =
1
2
σi ⊗ σ0 ⊗ σ0 ⊗ σ0, s
p
i = 1
2
σ0 ⊗ σi ⊗ σ0 ⊗ σ0, s
n
i = 1
2
σ0 ⊗ σ0 ⊗ σi ⊗ σ0, s
u
i = 1
2
σ0 ⊗ σ0 ⊗ σ0 ⊗ σi are
the matrix representation of spin operators (2.1); the Pauli matrices are
σ0 =
(
1 0
0 1
)
, σ1 =
(
0 1
1 0
)
, σ2 =
(
0 −i
i 0
)
, σ3 =
(
1 0
0 −1
)
;
⊗ is the symbol of direct product [11]; Jep, Jen, Jeu, Jpn, Jpu, Jnu are the constants of isotropic
exchange interaction between spins; the summation over e, p, n, u is absent.
3. Decomplexification of the Liouville-von Neumann equatio n
The Liouville-von Neumann equation for the density matrix ρ, describing the dynamics of
four-spin system, looks like
i∂tρ = [Ĥ, ρ], ρ(t = 0) = ρ0 . (3.1)
Let us present the solution of the equation (3.1) as
ρ =
1
16
Rαβγδσα ⊗ σβ ⊗ σγ ⊗ σδ , ρ+ = ρ, Trρ = 1, R0000 = 1. (3.2)
Hereinafter summation is taken over the repeating Greek indices from zero up to three, and over
Latin indices from one up to three. The four coherence vectors (the Bloch vectors) widely used in
the magnetic resonance theory, are written as
Ri000 = Tr ρ σi ⊗ σ0 ⊗ σ0 ⊗ σ0 , (3.3a)
R0i00 = Tr ρ σ0 ⊗ σi ⊗ σ0 ⊗ σ0 , (3.3b)
R00i0 = Tr ρ σ0 ⊗ σ0 ⊗ σi ⊗ σ0 , (3.3c)
R000i = Tr ρ σ0 ⊗ σ0 ⊗ σ0 ⊗ σi . (3.3d)
These vectors characterize the local properties of individual spins, whereas the other tensors de-
scribe the spin correlations. All correlation functions are in the limits
− 1 6 Rαβγδ 6 1. (3.4)
23005-2
Non-equilibrium reversible dynamics of work production...
As i∂tρ
n = [Ĥ, ρn], (n = 1, 2, 3, . . . ) at unitary evolution there is a enumerable number of
conservation laws Tr ρ = C1 = 1, Tr ρ2 = C2 , . . . , where Cn are the constants of motion, from
which only the first C2 , C3 , . . . , C16 are algebraically independent [12]. From the conservation
of purity, for which (ρ2)ik
def
≡ (ρ)ik , the polynomial (square-law) invariants are obtained. The
square polynomials also control the signs Rαβγδ . The length of the generalized Bloch vector bepnu
is conserved under unitary evolution:
bepnu =
√
R2
αβγδ − 1 . (3.5)
Having inserted the equation (3.2) into (3.1), multiply the equation (3.1) by all elements of the
basis σα ⊗ σβ ⊗ σγ ⊗ σδ in turn and take the trace for each equation to get the time derivatives of
the correlation functions as Rαβγδ
i∂tRαβγδ = Tr [Ĥ, ρ]σα ⊗ σβ ⊗ σγ ⊗ σδ ; α = (0, 1, 2, 3), . . . , δ = (0, 1, 2, 3). (3.6)
A detailed form of the system equation (3.6) is in the appendix. The derivation algorithm of the
system equation (A1)–(A15) is presented in [13]. The Liouville-von Neumann equation accepts a
real form in terms of the functions Rαβγδ as a closed system of 255 differential equations for the
local Bloch vectors and spin correlation functions.
In case of equivalent particles at h
e = h
p = h
n = h
u = h, Jep = Jen = Jeu = Jpn = Jpu =
Jnu = J from the equations (A1)–(A4) and (A5)–(A10) it follows that the square length of the
total magnetization (Rq000 + R0q00 + R00q0 + R000q)
2 and the forms Rii00 , Ri0i0 , . . . , R00ii are
conserved.
In the system (A1)–(A15), assuming, for example, Jeu = Jpu = Jnu = 0, we get a closed system
of equations for the description of three-qubit dynamics [15].
The set of equations (A1)–(A15) with the given initial conditions has wide applications, since
the magnetic field enters the form of arbitrary functions. First of all, it allows to make numerical
calculations for continuous (paramagnetic resonance in a continuous mode) as well as for pulse
modes (nuclear magnetic resonance). Secondly, by means of this system it is possible to investigate
the entanglement dynamics of qubits in a magnetic field [14] since the entanglement measures
are expressed in terms of the reduced density matrices or of populations. Thirdly, an important
application of the system (A1)–(A15) is quantum approach to the Carnot cycle [1–7],where the
working body is a finite spin chain.
4. Quantum thermodynamic variables
In the external field, the energy of a four-spin system is defined by the formula
E = Tr Ĥ(t)ρ =
1
2
(he
iRi000 + hp
iR0i00 + hn
iR00i0 + hu
iR000i)
+ 2(JepRii00 + JenRi0i0 + JpnR0ii0 + JeuRi00i + JpuR0i0i + JnuR00ii). (4.1)
The change of the total energy expectation value is equal to ∂tE = ∂tTr ˆH(t)ρ = Tr ∂t ˆH(t)ρ +
Tr ˆH(t)∂tρ. Due to the equation of motion (3.1) we have Tr ˆH(t)∂tρ = −iTr ˆH(t)[ ˆH(t), ρ] = 0. For
an external static field, the Hamiltonian is independent of time, hence ∂tE = 0, that is the energy
of the system is constant in time.
The change of work W can be associated with a term where only the spectrum changes ∂tW =
Tr ∂t ˆH(t)ρ. The change of heat Q is then ∂tQ = Tr ˆH(t)∂tρ. We have,
∫ t
0
dt∂tW =
∫ t
0
dtTr ∂t ˆH(t)ρ
= (integration by parts)= Tr ˆH(t)ρ(t)−Tr ˆH(0)ρ(0)−
∫ t
0
dtTr [ ˆH(t), ∂tρ] = ∆Wsys . The last integral
is equal to zero due to the equation of motion. Therefore, the work production in the system is
given by an expression of the form:
∆Wsys = Tr Ĥ(t)ρ(t) − Tr Ĥ(0)ρ0
=
t
∫
0
1
2
[(∂th
e
i)Ri000 + (∂th
p
i )R0i00 + (∂th
n
i )R00i0 + (∂th
u
i )R000i] dt. (4.2)
23005-3
E.A. Ivanchenko
The change in finite time t of the full system energy which, in our problem, consists of the performed
work and heat energy is equal to the work production since in the closed system heat energy is not
produced.
The reduced density matrices describe the dynamics of subsystems (4 matrices of individual
particles, 6 matrices of two particles, 4 matrices of three particles) and, for example, for a particle
e, and the coupled particles ep and epn, it can be written as
ρe(t) = Tr pnuρ =
1
2
(
1 +R3000 R1000 − iR2000
R1000 + iR2000(t) 1−R3000
)
, (4.3a)
ρpn =
1
4
R0βγ0σβ ⊗ σγ , ρepn =
1
8
Rαβγ0σα ⊗ σβ ⊗ σγ . (4.3b)
The matrices (4.3a)–(4.3b) are determined by the system solution (A1)–(A15), since the equations
for the reduced matrices are not closed. From the system (A1)–(A15) it follows that the spin flip
probabilities of p, n from their initial state are equal to
P p =
1−R0300
2
, P n =
1−R0030
2
. (4.4)
In the longitudinal field h = (0, 0, h3) with equal coupling constants J on the equations (A1)–
(A4) it follows that
∂tmq = ε3sqh3ms , (4.5)
where mq = Rq000 + R0q00 + R00q0 + R000q and m3 is the invariant of motion. The system of
equations for m1 , m2 has zero solutions m1 = 0, m2 = 0 for the initial condition (A16). We
believe that transverse coherences Rq000 = R0q00 = R00q0 = R000q = 0 for q = 1, 2 because the
numerical solution confirms that each term in m1 , m2 is equal to zero during unitary evolution,
that is the reduced matrix
ρe =
1
2
(
1 +R3000 0
0 1−R3000
)
is diagonal. In this case it is possible to correctly define the local dynamic temperature. The local
or dynamic temperature for a two-level system can be defined according to the basic meaning of a
thermal state [9, 16]:
T e(t) = −
Ω(t)
kB ln(p1/p0)
, (4.6)
where Ω(t) is the transition frequency in the e two-level system, equal to h3 . We assume that the
interaction strength J between the particles is small in comparison to the energy level spacing. In
this case, the interaction will not significantly alter the instantaneous energy eigenvalues of the
system and hence we can meaningfully define the temperature of individual particles, since each
will remain in the instantaneous thermal form (4.6), with the energy level spacing being the same
as in the absence of interaction. Further we set the Bolzmann constant kB equal to 1. Therefore,
all temperatures are in energy units. It is necessary to note a generalized approach to temperature,
work and heat without weak coupling approximation between the particles [17].
The e-spin entropy is equal to
Se(t) = −Tr ρe ln ρe = −p0 ln p0 − p1 ln p1 , (4.7)
where the local populations are equal to
p0 =
1−R3000
2
, p1 =
1 +R3000
2
. (4.8)
As the system is isolated, its entropy as a whole is constant. Hence, the system is reversible. In the
reversible system, the subsystems should be reversible, that is, there exist entropy fluxes between
23005-4
Non-equilibrium reversible dynamics of work production...
subsystems if the initial state is non-equilibrium. The entropy rate ∂tS
e of the e-spin due to the
p, n, u environment is defined by the formula:
∂tS
e =
∂tR3000
2
ln
1−R3000
1 +R3000
. (4.9)
The stated (4.6), (4.7) and (4.9) concern each particle.
The calculation of the work carried out by subsystems is done in the papers [9] by means
of ST diagrams. It is shown numerically that the entropy S and temperature T are dependent
thermodynamic variables. For a closed trajectory in the ST plane, the change of the total energy
∆Wsys is equal to zero and consequently the area captured by the closed curve in the ST plane
determines the work during a reversible cycle
∆W = −
∮
TdS = −
tc
∫
0
T (t)∂tSdt, (4.10)
where tc is the duration of a cycle, and the sign is defined according to the rule saying that if, at a
path tracing clockwise, the area is situated on the right it obtains a minus sign (heat pump). The
spin system is isolated and consequently, the Carnot cycles can only refer to sub-systems of the
4-spin system. We use (4.10) for e, p, n, u particles.
The energy of the coupled particle e in a magnetic field h3 in an environment of three others
is equal to 1
2
h3R3000 . We use the formula ∂t
(
1
2
h3R3000
)
= 1
2
(∂th3)R3000 +
1
2
h3∂tR3000 . The work
production (i.e., the heat production) by spin e during a cycle is equal to
∫ tc
0
1
2
(∂th3)R3000dt
(
∫ tc
0
1
2
h3∂tR3000dt
)
. During that cycle the energy change of the particle e
∫ tc
0
∂t(
1
2
h3R3000)dt is
equal to zero, hence
tc
∫
0
1
2
(∂th3)R3000dt = −
tc
∫
0
1
2
h3∂tR3000dt.
Having inserted the equations (4.6), (4.9) into (4.10) we conclude that
∆W e = −
∮
T edSe =
tc
∫
0
1
2
(∂th3)R3000dt.
Thus, the definitions of temperature (4.6), entropy (4.7) and work (4.10) are coordinated with the
work/heat production for parameters for which the ST plots are closed. It also concerns p, n, u
particles.
The Klein-von Neumann inequality looks like
− Tr ρ ln ρ 6 −
m
∑
i=1
ρii ln ρii 6 lnm, (4.11)
where −
∑m
i=1 ρii ln ρii is the diagonal entropy, d, m is the number of system states. For the initial
diagonal state, the diagonal entropy possesses the property d(0) 6 d(t) = −
∑m
i=1 ρii ln ρii [18].
The dynamics of a purity measure P = Tr ρ2 is connected with the dynamics of entropy
S = −Trρ ln ρ as follows. If the entropy is equal to zero, the system is in a pure state. At the
maximum entropy, the system is in the maximum mixed state. The purity P has the maximum
value 1 for a pure state and the minimum value in the mixed state, equal to 1/m, where m is the
number of system states. The subsystem purity is expressed in terms of the square length of the
local or generalized Bloch vector (3.5): the purity for the subsystems pn and epnu is equal to
P np =
1
4
(
1 + bnp2
)
, P epnu =
1
16
(
1 + bepnu2
)
, (4.12)
23005-5
E.A. Ivanchenko
accordingly, where bnp =
√
∑3
i,j=1 R
2
0ij0 is the length of the generalized Bloch vector of the pn
system.
Let us define the entanglement measure p and n for spins according to [19] on system solutions
having entered the two-particle entanglement tensor:
m0ij0 = R0ij0 −R0i00R00j0 . (4.13)
The tensor m0ij0 is equal to zero when the two-particle correlation function R0ij0 is factorized in
terms of the local Bloch vectors (3.3a)–(3.3d) and thus the matrix will be separable, i.e., ρpn =
ρp ⊗ ρn. By means of this tensor we shall define a measure of the two-particle entanglement in the
pn subsystem
mpn
SM =
√
√
√
√
1
3
3
∑
i,j=1
m2
0ij0 . (4.14)
This measure is equal to zero for a separable state and it is equal to 1 for the Greenberger-Horne-
Zeilinger maximum entangled state. This measure is applicable both to the pure and mixed states
(in all 6 two-particle measures).
5. Numerical results
The quantum thermodynamic devices are subdivided into heat pumps and heat engines de-
pending on functional purpose. In our model, the work absorbed or done by the system arises due
to the displacement of the power levels by a magnetic field [20–22], as well as it also depends on
the initial state of the system.
We will consider the effect of a variable magnetic field
h = (h1 , h2 , h3) (5.1)
on the dynamics of a system in the case when the fields operating on the spins e, p, n, u are equal
to h, where h1 = 0, h2 = 0, h3 = c+ 1.5 sinωt (c is the static part of the magnetic field), with an
external field frequency ω = 0.04 and with all the exchange constants equal to 0.01. The values of
magnetic field frequency are given in the energy units.
We study the behaviour of a four-spin system in terms of local thermodynamic concepts: tem-
perature (4.6), entropy (4.7) and work (4.10) depending on the parameters of a possible non-
equilibrium initial state:
(i) e, p, u particles have the same temperature T e(0) = T p(0) = T u(0) = 0.2, and the
temperature of the n particle is equal to T n(0) = 0.6;
(ii) T e(0) = T p(0) = 0.2, T n(0) = T u(0) = 0.25;
(iii) T e(0) = 0.21, T p(0) = 0.225, T n(0) = 0.264, T u(0) = 0.28.
For the parameters (i) of the initial state (A16), let us choose the following values for our
numerical results: he
3(0) = hp
3(0) = hn
3(0) = hu
3(0) = c. In figure 1 (i), c=1.5 the parametric
dependences on entropy and temperature for a cycle are shown, i.e., ST cycles with negative
work (bold line) which are identical for e, p, u spins. The spin n produces some work (thin
line) from the (e, p, u) – environment, while each of the spins e, p, u (bold line) absorbs work.
The work performed by n spin, is equal to the area limited by a closed curve according to the
formula (4.10) and is equal to 0.1331514. The e, p, u spins perform an amount of work equal to
3×(−0.0443838) = −0.1331514. Thereby, the work production of the four-spin system ∆Wsys (4.2)
is equal to zero in full compliance with the general results for the isolated system [23–27]. It
confirms the use of temperature (4.6) and entropy (4.7) as effective thermodynamic characteristics.
A feature of this initial state and modulation magnetic field is that these cycles are repeated without
any deformation. (It is known that the system equations (A1)–(A15) with periodic coefficients,
according to the Floquet theory, have periodic or quasi-periodic solutions i.e. the Floquet theory
does not exclude the periodic solutions as well. This depends on the set of coefficients. We have
presented this set.) In other words, each cycle comes to an end returning to the same initial state.
23005-6
Non-equilibrium reversible dynamics of work production...
0.4 0.8 1.2
0.00
0.09
0.18
0.27 c=1.5
S
T
-n
+e,p,u
(i)
0.4 0.8 1.2
0.00
0.09
0.18
0.27
S
T
+e,p,u
-n
c=1.8(i)
0.0 0.2 0.4
0.005
0.010
0.015
S
T
c=1.5(ii)
+e,p
-n,u
0.0 0.2 0.4 0.6
0.009
0.018
0.027
S
+e
+p
-u
-n
T
c=1.8(iii)
Figure 1. ST diagrams for the non-equilibrium initial state with the different values of the
static field c = he
3(0) = hp
3(0) = hn
3(0) = hu
3(0) and initial temperatures: (i) T e(0) = T p(0) =
T u(0) = 0.2, T n(0) = 0.6, (ii) T e(0) = T p(0) = 0.2, T n(0) = T u(0) = 0.25, (iii) T e(0) =
0.21, T p(0) = 0.225, T n(0) = 0.264, T u(0) = 0.28. All spins are in an identical field h and the
exchange constants are equal to 0.01. It is seen that the field h = (0, 0, c + 1.5 sinωt) causes
periodic cycles with parameters of the initial state ω = 0.04. The duration of a cycle is equal to
tc = 2π/ω = 157.08. The arrows indicate the direction of circulation. The circulation direction
becomes opposite for all particles after the replacement of a frequency sign ω. In addition to
the direction of circulation, the signs +,− specify the negative or the positive work of particles.
The vertical arrows in the plots specify the return points.
The closure of the ST plots does not depend on the amplitudes of the driving field (only the
form and the area may change), but it critically depends on the frequency ω [9] and the module
of the exchange constants, it does not depend on ferromagnetic or antiferromagnetic character of
the working gas [28]. After replacement of a frequency sign ω, the circulation direction becomes
opposite for all particles. If the value of the temperature parameter is T n(0) > 0.2, then the particle
n does work, and the spins e, p, u absorb work. The purity of the whole system decreases with an
increase of T n(0), but cycle-after-cycle and periodicity remain as it is described, and the areas
characterizing work increase with preservation of the algebraic sum which is equal to zero. For
T n(0) < 0.2 the spins e, p, u produce work, and the spin n absorbs it.
If T n(0) approaches 0.2, that is, at the equilibrium initial state T e(0) = T p(0) = T n(0) =
T u(0) = 0.2 the entropy of each subsystem is constant and the work production of each subsystem
is equal to zero, since at the initial moment there is no temperature gradient in the system (passive
or immovable state [29]).
For the initial state (i), c=1.8 the work performed by the n spin is equal to 0.1329435. The
e, p, u spins perform an amount of work equal to 3× (−0.0443145) = −0.1329435.
The numerical calculations also show that for the initial state (ii), c=1.5 particles e, p absorb
23005-7
E.A. Ivanchenko
work equal to 2(−0.002265), and particles n, u produce the same work 2(+0.002265) [30].
In the case of initial state (iii) for c=1.8 particles e, p absorb work which is equal to (−0.0041)+
(−0.0029), and particles n, u produce the same work (0.0020 + 0.0050).
We have the results of the ST diagrams in figure 1 at different static parts of the magnetic
field c for the initial non-equilibrium states. The numerical analysis shows that the balance of work
production for (i), (ii), (iii) in fact does not depend on the static magnetic field, only the local
temperatures and the form of the ST diagrams change.
In figure 1 it is seen that in the middle of the cycle, the non-equilibrium reversible system is
converted to the quasi-equilibrium one; the entropy rates ∂tS
i change signs, the temperature rates
∂tT
i are minimum (i = e, p, n, u).
We would like to indicate that in the vicinity of t ≈ 117.8, the local temperature of all particles
goes to zero as the frequencies Ωi(t) for all particles go to zero. It is necessary to notice that for
the minimum local temperatures, the eigenvalues of the Hamiltonian (2.1) come closer to zero, and
with the growth of temperatures the eigenvalues become bigger. The transition probability of each
particle (4.4) is close to 1 and makes one oscillation per cycle. The population ρ16 16 is approximately
equal to 0.925 during a cycle. For the opposite sign he
3(0) = hp
3(0) = hn
3(0) = hu
3(0) = −1.5, ρ1 1 ≈
0.925 the thermodynamic characteristics do not change. In figure 1 it is clearly seen that if the
initial state of a subsystem is more disordered, a subsystem absorbs/produces more (in absolute
magnitude) work [29].
The cycle of each particle is determined by the direct coupling of the given particle with the
others and does not depend on the coupling constants between other particles. The calculations
confirm that the dependence between the energy of particles p, n Epn = (hp
iR0i00 + hn
iR00i0)/2 +
2JpnR0ii0 and the entropy Spn is also cyclic. This property is carried out by any particle pair.
0 50 100 150
0.3
0.6
0.9
Se+Su
Seu
Peu
Pepnu
Sp+Sn
t
Ppn
Spn
0 45 90 135
0.01
0.02
0.03
0.04
eu
mSM
t
pn
Figure 2. Time dependence on purity of all sys-
tem P epnu and on characteristics eu and pn sub-
systems for one cycle with the parameters, cor-
responding to the ST diagrams in figure 1 (i),
c=1.5.
Figure 3. Time dependence on the local en-
tanglement mSM for subsystems pn (thin line)
and eu (bold line) for one cycle. The parame-
ters correspond to the ST diagrams in figure 1
(i), c=1.5.
The results for additional quantum thermodynamic characteristics of eu and pn subsystems
are presented in figures 2 and 3. Under unitary evolution, the global purity P epnu does not depend
on time. The purity P pn (4.12) has a maximum at an entropy minimum. The work production of
the eu subsystem is accompanied by entropy increase and purity reduction and the inverse process
occurs at work absorption. It is seen that the local entanglement mpn (4.14) has a maximum for a
minimum entropy Spn = −Trρpn ln ρpn. The entanglement (4.14) between the particles is noticeable
in the middle of the cycle and it is bigger between particles with opposite signs in work, and, as
the calculations confirm, grows with the disorder increase in the system. This entropy is always
less or equal to the sum entropies of individual spins in the pn subsystem. Also, the inequality
23005-8
Non-equilibrium reversible dynamics of work production...
of Klein-von Neumann is carried out (4.11), i. e., the diagonal entropy is bigger or equal to the
pn subsystem entropy. For the case presented in figure 1 (i), c=1.5 the diagonal entropy dpn (deu)
coincides with the summed entropies of the individual spins Sp + Sn (Se + Su).
The control of calculations was carried out with the help of the invariants of the motion,
described in sections 3 and 4, and all the correlation functions have been in the limits −1 6
Rαβγδ 6 1.
6. Summary
A closed system of equations is derived for the local Bloch vectors and spin correlation functions
of four two-level systems with exchange interaction, being in a time-dependent external magnetic
field. The invariants of motion necessary for the control of computing have been found. Analytical
and numerical analysis of thermodynamic behaviour in a four-spin weak coupling system depending
on the parameters characterizing the initial non-equilibrium state and modulation of the driving
field was performed. It was numerically found that under unitary dynamics, the work production of
one part of the system is compensated by absorption of work produced by the other part. The work
production by a subsystem is accompanied by entropy growth and vice-versa, while the entropy
decreases with work absorption by the subsystem.
It is shown that in the middle of a cycle, the non-equilibrium reversible system is converted to
a quasi-equilibrium one. Thermalization of non-equilibrium system of 4 spins in finite time under
different initial conditions is numerically shown (figure 1). In other words, the finite non-equilibrium
system in finite time generates an “attractor”, i.e., an intermediate temperature is established for
all particles. It was revealed that the ST cycle of each particle is determined by the direct coupling
of the given particle with the others and weakly depends on the coupling constants between the
other spins.
It was analytically and numerically shown that the formulas for temperature (4.6) and en-
tropy (4.7) are the effective thermodynamic characteristics for work calculation in a weakly coupled
spin system by means of ST diagrams in the course of unitary evolution.
The study of the “spin gas” properties is necessary for the implementation of quantum thermo-
dynamic cycles in spin systems [31–33]. Taking into account the environment, zero balance will be
broken and the system will become a heat engine or a heat pump depending on the temperatures
of the heat baths.
Acknowledgements
The author is grateful to Zippa A.A. for constant invaluable support and to the referees for
their objective comments that improved the text in many points.
23005-9
E.A. Ivanchenko
Appendix
The detailed form of the system equation (3.6) is as follows:
∂tRq000 = εisqh
e
iRs000 + εstq(J
epRts00 + JenRt0s0 + JeuRt00s), (A1)
∂tR0q00 = εisqh
p
iR0s00 + εstq(J
epRst00 + JpnR0ts0 + JpuR0t0s), (A2)
∂tR00q0 = εisqh
n
iR00s0 + εstq(J
enRs0t0 + JpnR0st0 + JnuR00ts), (A3)
∂tR000q = εisqh
u
iR000s + εstq(J
euRs00t + JpuR0s0t + JnuR00st), (A4)
∂tRqk00 = εisqh
e
iRsk00 + εiskh
p
iRqs00 + Jepεksq(Rs000 −R0s00)
+ εtsq(J
enRskt0 + JeuRsk0t) + εtsk(J
pnRqst0 + JpuRqs0t), (A5)
∂tRq0k0 = εisqh
e
iRs0k0 + εiskh
n
iRq0s0 + Jenεksq(Rs000 −R00s0)
+ εtsq(J
epRstk0 + JeuRs0kt) + εstk(J
pnRqst0 + JnuRq0ts), (A6)
∂tRq00k = εisqh
e
iRs00k + εiskh
u
iRq00s + Jeuεksq(Rs000 −R000s)
+ εtsq(J
epRst0k + JenRs0tk) + εtsk(J
puRqt0s + JnuRq0ts), (A7)
∂tR0qk0 = εisqh
p
iR0sk0 + εiskh
n
iR0qs0 + Jpnεksq(R0s00 −R00s0)
+ εtsq(J
epRtsk0 + JpuR0skt) + εstk(J
enRsqt0 + JnuR0qts), (A8)
∂tR0q0k = εisqh
p
iR0s0k + εiskh
u
iR0q0s + Jpuεksq(R0s00 −R000s)
+ εstq(J
epRst0k + JpnR0tsk) + εstk(J
euRsq0t + JnuR0qst), (A9)
∂tR00qk = εisqh
n
iR00sk + εiskh
u
iR00qs + Jnuεksq(R00s0 −R000s)
+ εstq(J
enRs0tk + JpnR0stk) + εstk(J
euRs0qt + JpuR0sqt), (A10)
∂tRqkl0 = εisqh
e
iRskl0 + εiskh
p
iRqsl0 + εislh
n
iRqks0
+ Jepεksq(Rs0l0 −R0sl0) + Jpnεlsk(Rqs00 −Rq0s0)
+ Jenεlsq(Rsk00 −R0ks0) + JeuεstqRtkls + JpuεtskRqslt + JnuεtslRqkst , (A11)
∂tRqk0l = εisqh
e
iRsk0l + εiskh
p
iRqs0l + εislh
u
iRqk0s
+ Jepεksq(Rs00l −R0s0l) + Jpuεlsk(Rqs00 −Rq00s)
+ Jeuεlsq(Rsk00 −R0k0s) + JenεstqRsktl + JpuεtskRqstl + JnuεstlRqkst , (A12)
∂tRq0kl = εisqh
e
iRs0kl + εiskh
n
iRq0sl + εislh
u
iRq0ks
+ Jenεksq(Rs00l −R00sl) + Jeuεlsq(Rsok0 −R00ks)
+ Jnuεlsk(Rq0s0 −Rq00s) + JepεtsqRstkl + JpnεstkRqstl + JpuεstlRqskt , (A13)
∂tR0qkl = εisqh
p
iR0skl + εiskh
n
iR0qsl + εislh
u
iR0qks
+ Jpnεksq(R0s0l −R00sl) + Jpuεlsq(R0sk0 −R00ks)
+ Jnuεlsk(R0qs0 −R0q0s) + JepεtsqRtskl + JenεtskRtqsl + JeuεstlRsqkt , (A14)
∂tRqklm = εisqh
e
iRsklm + εiskh
p
iRqslm + εislh
n
iRqksm + εismhu
iRqkls
+ Jepεksq(Rs0lm −R0slm) + Jenεlsq(Rsk0m −R0ksm)
+ Jpnεlsk(Rqs0m −Rq0sm) + Jeuεmsq(Rskl0 −R0kls)
+ Jpuεmsk(Rqsl0 −Rq0ls) + Jnuεmsl(Rqks0 −Rqk0s). (A15)
Concrete calculations were carried out for the separable mixed initial state Sep-Mix :
ρSep−Mix(0) = ρe ⊗ ρp ⊗ ρn ⊗ ρu, (A16)
23005-10
Non-equilibrium reversible dynamics of work production...
where
ρi =
1
2 coshhi
3(0)/2T
i(0)
exp
(
−
hi
3(0)
2T i(0)
σ3
)
, (A17)
is the mixed state of the i particle, T i(0) is the initial temperature, hi
3(0) is the initial field, i =
(e, p, n, u).
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E.A. Ivanchenko
Нерiвноважна оборотна динамiка роботи в 4-спiновiй
системi у магнiтному полi
Є.О. Iванченко
Iнститут теоретичної фiзики, Нацiональний науковий центр “Харкiвський фiзико-технiчний iнститут”,
Харкiв, Україна
За допомогою декомплексифiкацiї рiвняння Лiувiля-Неймана одержана замкнута система рiвнянь
локальних векторiв Блоха та спiнових кореляцiйних функцiй для системи 4 магнiтних частинок з
обмiнною взаємодiєю, якi знаходяться в довiльному залежному вiд часу магнiтному полi. Викона-
но аналiтичний та чисельний аналiз динамiки квантових термодинамiчних параметрiв в залежностi
вiд сепарабельного початкового стану системи та модуляцiї магнiтного поля. В умовах унiтарної
еволюцiї дослiджено нерiвноважну оборотну генерацiю роботи у найближчому оточеннi.
Ключовi слова: оборотна динамiка, генерацiя роботи, спiнова система, магнiтне поле
23005-12
Introduction
Model Hamiltonian
Decomplexification of the Liouville-von Neumann equation
Quantum thermodynamic variables
Numerical results
Summary
|