Non-extensive thermodynamics of 1D systems with long-range interaction
Non-extensive thermodynamics of 1D systems with long-range interaction
S. S. Apostolov1,2, Z. A. Mayzelis1,2, O. V. Usatenko1 ∗, V. A. Yampol’skii1
A. Ya. Usikov Institute for Radiophysics and Electronics
Ukrainian Academy of Science, 12 Proskura Street, 61085 Kharkov, Ukraine
V. N. Karazin Kharkov National University, 4 Svoboda Sq., 61077 Kharkov, Ukraine
A new approach to non-extensive thermodynamical systems with non-additive energy and entropy
is proposed. The main idea of the paper is based on the statistical matching of the thermodynamical
systems with the additive multi-step Markov chains. This general approach is applied to the Ising
spin chain with long-range interaction between its elements. The asymptotical expressions for the
energy and entropy of the system are derived for the limiting case of weak interaction. These
thermodynamical quantities are found to be non-proportional to the length of the system (number
of its particle).
PACS numbers: 05.40.-a, 02.50.Ga, 05.50.+q
One of the basic postulates of the classical statistical
physics is an assumption about the particle’s interaction
range considered to be small as compared with the sys-
tem size. If this condition does not hold the internal and
free energies, entropy, etc. are no more additive physical
quantities. Due to this fact the definitions of the tem-
perature, entropy, etc. are not evident. The distribution
function of the non-extensive system is not the Gibbs
function, the Boltzman relationship between the entropy
and the statistical weight is not any longer valid.
The non-extensive systems are common in physics [1].
They are gravitational forces [2], Coulomb forces in glob-
ally charged systems [3], wave-particle interactions, mag-
nets with dipolar interactions [4], etc. Long-range corre-
lated systems are intensively studied not only in physics,
but also in the theory of dynamical systems and the the-
ory of probability. The numerous non-Gibbs distribu-
tions are found in various sciences, e.g., Zipf distribution
in linguistic [5, 6], the distribution of nucleotides in DNA
sequences and computer codes [7, 8, 9], the distributions
in financial markets [10, 11, 12], in sociology, physiology,
seismology, and many other sciences [13, 14].
The important model of the non-extensive systems in
physics is the Ising spin chain with elements interacting
at great distances [15, 16]. Unfortunately, the general-
ization of the standard thermodynamic methods for the
case of arbitrary number of interacting particles is im-
possible. There exist solutions for some particular cases
of particles interaction only [16, 17]. The other results
obtained for the chains with long-range interaction are
either the general theorems about the existence of the
phase transitions in the system or the calculations of the
critical indexes [18, 19].
Several algorithms for calculating the thermodynamic
quantities of the long-range correlated systems have been
proposed. Unfortunately, they are not well grounded and
∗usatenko@ire.kharkov.ua
need additional justifications. One of them is the Tsalis
thermodynamics (see [20, 21]) based on axiomatically in-
troduced entropy: S(q)(W ) = (W 1−q − 1)/(1− q). Here
W is the statistical weight and q is the parameter reflect-
ing the non-extensiveness of the system. However, this
expression does not correspond to the entropy of Ising
chain with a long but finite range of interaction. Indeed,
it does not contain the size of the system and remains
non-additive when the length of the system tends to in-
finity. Meanwhile, the entropy has to be additive if the
system is much greater than the characteristic range of
interaction. Thus, the problem of finding the thermody-
namic quantities for the non-extensive systems is still of
great importance.
Recently a new vision on the long-range correlated sys-
tems, based on the association of the latter with the
Markov chains, has been proposed [22]. The binary
Markov chains are the sequences of symbols taking on
two values, for example, ±1. These sequences are de-
termined by the conditional probability of some symbol
to occur after the definite previous N symbols and this
conditional probability does not depend on the values of
symbols at a distance, greater than N , P (si = s|T
i,∞) =
P (si = s|T
). Here T±
are the sets of L sequen-
tial symbols (si±1, si±2, . . . , si±L). The unbiased addi-
tive Markov chain is defined by the additive conditional
probability function,
P (si = 1 | T
i,∞) =
F (r)si−r . (1)
The function F (r) is referred to as the memory func-
tion [23]. It describes the direct interaction between the
elements of the chain contrary to the binary correlation
function K(r) = sisi+r that also takes into account the
indirect interactions. Here . . . means statistical aver-
aging over the chain. As was shown in [24], the corre-
lation function can be found from the recurrence rela-
tion K(r) =
r′=1 2F (r
′)K(r − r′), that establishes the
one-to-one correspondence between these two functions.
http://arxiv.org/abs/0704.0887v1
In the limiting case of small F (r), this equation yields
K(r) ≈ δr,0 + 2F (r).
We consider the chain of classical spins with hamilto-
H = −
j−i<N
ε(j − i)sisj , (2)
where si is the spin variable taking on two values, −1
and 1, and ε(r) > 0 is the exchange integral of the ferro-
magnetic coupling. The range N of spin interaction may
be arbitrary but finite.
We find thermodynamical quantities, specifically, en-
ergy and entropy, of such a chain being in thermody-
namical equilibrium with a Gibbs thermostat of temper-
ature T . The hamiltonian (2) does not include the terms,
corresponding to the interaction of the system with the
thermostat. Nevertheless, this interaction leads to the
relaxation of the temperature in different parts of the
chain, and the thermostat fixes its temperature of the
whole spin chain. A great many numerical procedures,
for example, the Metropolis algorithm, were elaborated
for achieving the equilibrium state. The algorithm con-
sists in the sequential trials to change the value of a ran-
domly chosen spin. The probability of spin-flip is deter-
mined by the values of spins on both sides of the chain
within the interaction range. Thus, the Ising spin chain
in the equilibrium state can be characterized by the con-
ditional probability, P (si = s | T
i,∞, T
i,∞), of some spin
si to be equal to s under the condition of definite val-
ues of all other spins on both sides from si. In Ref. [25],
a chain with the two-sided conditional probability func-
tion, which is independent of symbols at a distance of
more than N , was considered. These chains were shown
to be equivalent to the N -step Markov chain. So, to
calculate the statistical properties of the physical object
with long-range interaction, it is sufficient to find the
corresponding Markov chain.
In this work, we demonstrate that the Ising spin chains
with long-range interaction being in the equilibrium state
are statistically equivalent to the Markov chains with
some conditional probability function. Then, using the
known statistical properties of the Markov chains, we
calculate the thermodynamical quantities of the corre-
sponding spin chains.
First, we present a method of defining a thermodynam-
ical quantity Q (e.g., energy, entropy) for a subsystem of
arbitrary length L (a set of L sequential particles) of any
non-extensive system. This subsystem, denoted by S,
interacts with the thermostat and with the rest of the
system. The interaction with the latter makes it impos-
sible to use the standard methods of statistical physics.
The internal energy of the entire system is not equal to
the sum of the internal energies of S and the rest of the
system.
In order to find the condition of the system equilibrium
we divide the ensemble of the system into the subensem-
bles with fixed 2N closest particles from both sides of S.
We denote these two ”border” subsystems of length N
by letter B. All the other particles, except for S and B,
are denoted by R:
. . . si−N
︸ ︷︷ ︸
si−N+1 . . . si
︸ ︷︷ ︸
si+1 . . . si+L
︸ ︷︷ ︸
si+L+1 . . . si+L+N
︸ ︷︷ ︸
si+L+N+1 . . .
︸ ︷︷ ︸
Within every subensemble, the subsystem B is fixed and
plays the role of a partition wall conducting the energy
and keeps its internal energy unchanged. The energy of
system S + R equals to the sum of energies of S and
R and their energy of interaction with B. Thus, within
every subensemble we can use the equilibrium condition
between S and R, analogous to that for extensive ther-
modynamics:
∂ lnWS(ES |B)
∂ lnWR(ER|B)
, (3)
where WS(ES |B) and WR(ER|B) are the statistical
weights of systems, in which the energy of S is ES , those
of R is ER, and B is fixed. We refer to these statistical
weights as the conditional statistical weights.
In a similar way, we introduce any conditional ther-
modynamical quantity Q(·|B). The real quantity Q(·) is
the conditional one, averaged over the subensembles with
different environments B:
Q(·) =
Q(·|B)
Using this way of calculating the thermodynamical quan-
tities, one does not need to find the distribution function
of the system S. Nevertheless, it is quite appropriate
for the non-extensive systems and yields the thermody-
namical values that could be measured experimentally.
If the system is in the thermal contact with the Gibbs
thermostat, within every subensemble with a fixed B,
the equilibrium condition between thermostat and sub-
system S yields the temperature of S equal to T . Thus,
the averaged temperature of the subsystem S is also T .
The conditional entropy can be introduced as the log-
arithm of the conditional statistical weight: S(ES |B) =
lnW (ES , S|B). Equality dS(ES |B) = dES/T is fulfilled
for the conditional quantities S and E. Meanwhile, such
a relation is not valid for the averaged entropy and en-
ergy.
Note that the presented algorithm for calculating the
thermodynamical quantities is rather general and can be
applied to other quantities, e.g., to the probability for
some spin to take on the definite value under the condi-
tion of the particular environment.
Now we proceed to the application of this algorithm
for the analysis of Ising spin chain with long-range inter-
action. The theory of Markov chains is built around the
expression for the conditional probability function. So, in
an effort to find a Markov chain that corresponds to a
given Ising system, we have to find its conditional prob-
ability function. To this end, we consider one spin si
as a system S and the conditional probability P as a
quantity Q in the above-mentioned algorithm. In the ex-
tensive thermodynamics, this probability is determined
by the Gibbs distribution function. According to our al-
gorithm, one can find that the probability of event si = 1
under the condition of the fixed spins from B is given as
p(si = 1|B) =
1 + exp
2ε(r)
(si−r + si+r)
It should be pointed out that this expression is sim-
ilar to the Glauber formula [26] written in some other
terms. Our result has been that the conditional proba-
bility of the si occurring is determined by two subsys-
tems of length N on both sides of si only and does not
depend on the remoter spins. As mentioned above, this
is a property of the N -step Markov sequences.
To arrive at an explicit expressions for the energy and
entropy of the subsystem S we consider the case of weak
interaction,
ε(r) ≪ T . In this case, Eq. (5) corre-
sponds approximately to the additive Markov chain with
the memory function F (r) ≈ ε(r)/2T and, thus, its bi-
nary correlation function is
K(r) ≈ δr,0 + ε(r)/T. (6)
Figure 1 shows that the Ising spin chain being in equi-
librium state is statistically equivalent to the Markov
chain. The solid circles describe the binary correlation
function of the additive Markov chain with the memory
function F (r) = ε(r)/2T . The open circles correspond
to the correlation function of the spin chain being equi-
librated by the Metropolis scheme.
It can be shown that the Metropolis scheme also yields
the same expression (5) for the conditional probability.
Due to this fact one does not need to use the Metropolis
algorithm to find an equilibrium state of the spin chain
but can generate a Markov chain with a corresponding
conditional probability function.
0 20 40 60 80 100
0.000
0.001
0.002
0.003
0 20 40 60 80
0.000
0.002
FIG. 1: (Color online) The correlation functions of the ad-
ditive Markov chain determined by memory function F (r) =
ε(r)/2T (solid circles) and the spin chain being equilibrated
by the Metropolis scheme (open circles). The inset represents
the function ε(r)/T .
Now we examine binary spin chain determined by the
Hamiltonian (2) with si = ±1 and find the non-extensive
energy and entropy of subsystem S containing L par-
ticles. Since the explicit expressions for the correlation
function were derived for the additive Markov chains with
the small memory function only, we suppose the energy of
interaction to be small as compared to the temperature,
ε(r) ≪ T .
The proposed algorithm of finding thermodynamic
quantities is considerably simpler while finding the en-
ergy of the system S of length L. The energy is the
averaged sum of products of pairs of spin values,
E(T ) =
ε(|k − j|)sksj −
j,k=i
ε(|k − j|)sksj .
Formally, this expression depends on the index i of the
first spin in the system S. However, we study the ho-
mogeneous systems, so function E(T ) in Eq. (7) does
not contain i as an argument. Energy in Eq. (7) can be
calculated via the binary correlation function only with
the arguments, less than N , without finding conditional
energies. Using Eq. (6), we arrive at
E(T ) = −
ǫ2NL+
ε2(i)min{i, L}
/T, (8)
with ǫ2N =
ε2(i). If the system length is much greater
than memory length N this expression yields extensive
energy. Indeed, in this case subsystems S and R interact
nearly extensively. In the opposite limiting case we get:
E(T ) ≈ −(4ǫ2NL− ε
2(1)L2)/2T, L ≪ N. (9)
If one regards the whole chain of the length M , forming
the circle, as the system S in the above-mentioned sense,
the additive energy is
E(T ) = −ǫ2NM/T. (10)
This result is very natural because the extensive thermo-
dynamics is valid.
It is seen, that the non-extensive energy is expressed
in terms of binary correlation functions only. This is not
correct for other thermodynamical quantities, e.g. the
entropy of the system S. Formally, to find the entropy,
we have to calculate all conditional entropies by integra-
tion of dS(E|B) = dE(S|B)/T , and then to average the
result over all realizations of the borders. However, at
high temperatures, to a first approximation in the small
parameter
ε(r)/T we can change the order of these
operations and calculate the averaged entropy by inte-
grating this formula taken with the averaged energy. A
constant of integration is such that the chain is com-
pletely randomized at T → ∞, and its entropy is equal
to ln 2L. Thus, we obtain
S(T ) = L ln 2 + E(T )/2T. (11)
Here E(T ) is determined by one of Eqs. (8)-(10). This
expression describes the non-extensive entropy of the sys-
tem S. However, while the energy is non-extensive in the
main approximation, the entropy is non-extensive as a
first approximation in the parameter
ε(r)/T . The
dependences of the non-extensive energy and entropy on
size L of the system S are given in Fig. 2 for step-wise
interaction ε(r). The solid line corresponds to additive
quantities, it is the asymptotic of these quantities for the
large system length.
It should be emphasized, that knowing the energy and
entropy we can find some other thermodynamic quanti-
ties. For example, at high temperatures the heat capac-
ity can be determined in a similar way as for the entropy
calculating. One can use the classical formula CV (T ) =
TdS(T )/dT with averaged entropy from Eq. (11) and
obtain the heat capacity value as CV = −E(T )/T .
This simple procedure of its finding is valid for the
case of high temperatures as the main approximation
only. In the general case, CV (T ) is not determined by
CV (T ) = TdS(T )/dT . This relation holds for the condi-
tional quantities only.
At high temperatures, we calculated the averaged non-
additive energy and entropy without using the condi-
tional ones. In the opposite limiting case of low temper-
atures, the calculation of conditional quantities proves to
be necessary.
Thus, we have suggested the algorithm of evaluat-
ing thermodynamic characteristics of non-extensive sys-
tems. The value of certain thermodynamical quantity
0 500 1000 1500
N=500
N=100
FIG. 2: (Color online) The specific non-extensive energy
E × 103/L and entropy (S/L − ln 2) × 105 vs. the size L
of system for step-wise interaction ε(r). The memory length
N is indicated near the curves. The constant limiting value
of energy is −4ǫ2N/2T ≈ −5× 10
can be obtained by averaging the corresponding condi-
tional quantity. This method is applied to the Ising spin
chain. The explicit expressions for the non-additive en-
ergy and entropy are deduced in the limiting case of high
temperatures as compared to the energy of spins interac-
tion. At high temperatures, the equilibrium Ising chain
of spin turns out to be equivalent to the additive multi-
step Markov chain.
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