Negative magnetic susceptibility and nonequivalent ensembles for the mean-field φ^{6} spin model

0.26 MB·English

by  S. A. Alavi

Negative magnetic susceptibility and
nonequivalent ensembles for the mean-field Φ6
spin model
1
1
0 S.A.Alavi†,S.Sarvari
2

n Department of Physics, Sabzevar Tarbiat Moallem university , P. O. Box
a 397, Sabzevar, Iran.
J

3
†Email:[email protected],[email protected]
]
h
c Keywords: Statistical mechanics, Mean-field φ6 spin models, Ensemble
e inequivalence, Nonconcave entropies.
m
- Abstract
t
a
t We derive the thermodynamic entropy of the maen-field φ6 spin model in
s
. theframeworkofmicrocanonicalensembleasafunctionoftheenergyandmag-
t
a netization. Using the theory of large deviations and Rugh’s microcanonical
m formalism we obtain the entropy and its derivatives and study the thermody-
- namic properties of φ6 spin model. The interesting point we found is that like
d φ4 model the entropy is a concave function of the energy for all values of the
n
magnetization, but is nonconcave as a function of the magnetization for some
o
c values of the energy. This means that the magnetic susceptibility of the model
[ can be negative for fixed values of the energy and magnetization in the mi-

crocanonical formalism. This leads to the inequivalence of the microcanonical
2
v and canonical ensembles. It is also shown that this mean-field model,displays
8 a first-order phase transition driven by the magnetic field. Finally we compare
8 the results of the mean-field φ6 and φ4 spin models.
8
5
. 1 Introduction
2
1
0 The microcanonical and canonical ensembles are the two main probability dis-
1
tributions with respect to which the equilibrium properties of statistical me-
:
v chanical models are calculated. Despite the two ensembles model two different
i physical situations, it is generally believed that the ensembles give equivalent
X
results in the thermodynamic limit; i.e., in the limit in which the volume of the
r
a system tends to infinity. The equivalence of the microcanonical and canonical
ensembles is most usually explained by saying that although the canonical en-
sembleisnotafixed-mean-energyensemblelikethemicrocanonicalensemble,it
must ’converge’ to a fixed-mean-energy ensemble in the thermodynamic limit,
and so must become or must realize a microcanonical ensemble in that limit.
This convergence can be proved to hold for noninteracting systems such as the
1
perfectgasandforavarietyofweaklyinteractingsystems. Forgeneralsystems,
however, neither is this convergence valid nor is the conclusion true concerning
ensemble equivalence which this convergence is intended to motivate. In fact,
inthepastthreeandahalfdecades,numerousstatisticalmodelshavebeendis-
coveredhavingmicrocanonicalequilibriumpropertiesthatcannotbeaccounted
for within the framework of the canonical ensemble. For systems with short-
range interactions, the choice of the statistical ensemble is typically of minor
importance and could be considered a finitesize effect: differences between, say,
microcanonical and canonical expectation values are known to vanish in the
thermodynamic limit of large system size, and the various statistical ensembles
become equivalent . In the presence of long-range interactions this is in general
notthecase,andmicrocanonicalandcanonicalapproachescanleadtodifferent
thermodynamic properties even in the infinite-system limit . In the astrophysi-
cal context, nonequivalence of ensembles and the importance of microcanonical
calculations have long been known for gravitational systems. Although equiv-
alence of ensembles had been proven only for short-range interactions, it was
tacitly assumed by most physicists to hold in general. Therefore it came as a
surprise to many that equivalence does not necessarily hold for long-range sys-
tems, in particular in the presence of a discontinuous phase transition. Until
now, the phenomenon of nonequivalent ensembles has been identified and ana-
lyzed almost exclusively by determining regions of the mean energy where the
microcanonical entropy function is anomalously nonconcave or by determining
regions of the mean energy where the heat capacity, calculated microcanoni-
cally, is negative. The existence of such nonconcave anomalies invalidates yet
another tacit principle of statistical mechanics which states that the one should
alwaysbeabletoexpressthemicrocanonicalentropy, thebasicthermodynamic
functionforthemicrocanonicalensemble, astheLegendre-Fencheltransformof
the free energy, the basic thermodynamic function for the canonical ensemble.
Indeed,ifthemicrocanonicalentropyistobeexpressedastheLegendre-Fenchel
transform of the canonical free energy, then the former function must necessar-
ily be concave on its domain of definition. Hence, if the microcanonical entropy
has nonconcave regions, then expressing it as a Legendre-Fenchel transform is
impossible. When this occurs, we say that there is thermodynamic nonequiva-
lence of ensembles. More recently, nonconcave anomalies in the microcanonical
entropyaswellasnegativeheatcapacitieshavebeenobservedinmodelsoffluid
turbulenceandmodelsofplasmas,inadditiontolong-rangeandmean-fieldspin
models, including the mean-field XY model and the mean-field Blume-Emery-
Griffiths (BEG) model.
One of the motivations for the study of Φ6 models is the search for soliton.
Another reason for interest is that Φ6 models are the simplest systems with
continuous variables that exhibit a rich phase diagram, with first- and second-
order phase transitions and a tricritical point. This structure was observed in
the study of the quantum mechanics of a single site, three-well potential when
classical,perturbativeandmean-fieldargumentswereusedandbubblesolutions,
their relation to the phase transitions and the question of their stability, both
relativistically and non-relativistically. In recent years the Φ6 model and its
2
application in different physical systems including the following problems have
been studied extensively: the crossover from a quantum φ6 theory to a renor-
malized two-dimensional classical nonlinear sigma model, alpha matter on a
lattice, first-orderelectroweakphasetransition(EWPT)duetoadimension-six
operator in the effective Higgs potential, first-order phase transitions in con-
fined systems, effective Potential and spontaneous symmetry breaking in the
noncommutativeΦ6 model, bubbledynamicsinquantumphasetransitions, the
canonical transformation and duality in the Φ6 theory, hermitian matrix model
Φ6 for 2D quantum gravity, phase structure of the generalized two dimensional
Yang-Mills theories on sphere, tricritical Ising model near criticality, sponta-
neous symmetry breaking at two Loop in 3D massless scalar electrodynamics,
Isingmodelintheferromagneticphase,statisticalmechanicsofnonlinearcoher-
ent structures, kinks in the Φ6 model, growth kinetics in the Φ6 N-Component
model, stability of Q-balls, the liquid states of pion condensate and hot pion
matter, instantons and conformal holography, first-order phase transitions in
superconducting films, field-theoretic description of ionic crystallization in the
restricted primitive mode.
This increasing interest in Φ6 model and its many application in physics, moti-
vated us to study the statistical mechanics of Φ6 spin models.
2 The mean-field Φ6 model and its thermody-
namic
TheHamiltonianofthemean-fieldΦ6 modelisgivenbythefollowingexpression
(cid:88)N P2 q2 q4 q6 1 (cid:88)N
H = i − i + i + i − q q (1)
2 4 4 6 4N i j
i=1 i,j=1
where q and p are the canonical coordinates of unit mass,moving on a line
i i
(q ,p ). The entropy of the system in terms of its mean energy and magnetiza-
i i
tion is defined as1
1 (cid:90)
S((cid:15),m)= lim ln δ((cid:15)(x)−(cid:15))δ(m(x)−m)dx (2)
N→∞N
As is sated in [1], if S((cid:15),m) were concave,then one would calculate this
function from the point of view of the canonical ensemble using the following
steps
• Calculate the partition function :
(cid:73)
Z(β,η)= exp[−βH(x)−ηM(x)]dx (3)
where M =Nm(x)
1InthispaperwechoosekB =1
3
• Calculate the thermodynamic potential defined by
1
ϕ(β,η)=− lim lnZ(β,η) (4)
N→∞N
where ϕ(β,η)=βF(β,η), where F is the free energy of the system.
• Obtain S((cid:15),m) by taking the legendre transform of free energy function
ϕ(β,η);
S((cid:15),m)=β(cid:15)+ηm−ϕ(β,η) (5)
with β and η determined by the equations
∂ ∂
ϕ(β,η)=(cid:15) , ϕ(β,η)=m (6)
∂β ∂η
Asmentionedearlierthesestepsarevalidonlywhentheentropyisconcave. For
nonconcave entropies we use two methods i.e., a) method of large deviation. b)
Rugh’s method, and compare the results.
3 Calculation of the entropy using large devia-
tion method
We are looking for a set of macro variables or mean-fields such that the energy
per particle (cid:15)(x) and the mean magnetization m(x) can be considered as a
function of these variables:
(cid:15)(x)=(cid:15)(µ(x)) , m(x)=m(µ(x)) (7)
(cid:101) (cid:101)
Here we have used µ(x) to denote the macro variables collectively. Corre-
spondingly the entropy can be expressed as a function of these macro variables
or mean fields:
1 (cid:90)
S(cid:101)(µ)=lim ln δ(µ(x)−µ)dx (8)
N
In this method the entropy S((cid:15),m) can be obtained by solving the following
constrained maximization problem:
S((cid:15),m)= sup S(cid:101)(µ) (9)
µ:(cid:101)(cid:15)(µ)=(cid:15),m(cid:101)(µ)=m
If S(cid:101)(µ) is a concave function of the given macrostate M, then S(cid:101)(µ) is the
legendre transform of free energy function defined by :
(cid:90)
Z(cid:101)(λ)= exp[−Nλ·µ(x)]dx (10)
Γ
If S(cid:101)(µ) is concave, one can find it using the so called macrostate generalization
of the legendre transform presented in Eqs. (5) and (6)
S(cid:101)(µ)=inf{λµ−ϕ(cid:101)(λ)} (11)
λ
4
Now, let us return to the mean-field φ6 model. An appropriate choice of
macrostate is the vector M(m,k,ν), where m, k and ν are the mean magneti-
zation, the mean kinetic energy and the mean potential energy, respectively:
1 (cid:88)
k = (P2) (12)
2N i
1 (cid:88) q6 q4−q2
ν = ( i + i i ) (13)
N 6 4
So we can write (cid:15)(µ) as follows:
(cid:101)
m2
(cid:15)(µ)=k+ν− (14)
(cid:101) 4
To calculate the entropy S(cid:101)(µ), we use the legendre transformation. It is shown
in [2] that the S((cid:15),m) is given by the following expression:
(cid:26)1 m2 1 (cid:27)
S((cid:15),m)=sup ln[(cid:15)−ν+ ]+ ln4πe+S(cid:101)(m,ν) (15)
2 4 2
ν
4 Rugh’s formalism for the mean-field φ6 model
Rugh’s microcanonical formalism has been discussed in[1,2]. We assume the
HamiltonianH(x;M)inwhichMisthetotalmagnetization. Themicrocanonical
entropy of the system is given by:
(cid:90)
S(E,M)=ln δ(H(x;M)−E)dx (16)
Γ
whereEisthetotalenergyofthesystem. InRugh’smethodthethermodynamic
quantities are defined through derivatives of the entropy [1,2], and calculated
by introducing a vector Y in Γ in such a way that, Y ·∇H =1, so we have:
∂ 1
S(E,M)=(cid:104)∇.Y(cid:105) = (17)
∂E E,M T(E,M)
and
(cid:28) (cid:29)
∂ ∂H
S(E,M)=− ∇.( Y) (18)
∂M ∂M
E,M
In general for an arbitrary observable A(x), we have [1]:
∂ 1
(cid:104)A(cid:105) =(cid:104)∇.(AY)(cid:105) − (cid:104)A(cid:105) (19)
∂E E,M E,M T(E,M) E,M
and
(cid:28) (cid:29) (cid:28) (cid:29) (cid:28) (cid:29)
∂ ∂H ∂H ∂A
(cid:104)A(cid:105) =− ∇.( AY) + ∇.( Y) (cid:104)A(cid:105) +
∂M E,M ∂M ∂M E,M ∂M
E,M E,M E,M
(20)
5
where (cid:104)A(cid:105) denotes the average of the abserable A. By incorporating the mag-
netization constraint into the Hamiltonian H(x) one can obtain the Hamil-
tonian H(x;M). This can be done by eliminating one of the q , for example
i
q =M −(cid:80)N−1q . For the mean-field φ6 model we have :
N i=1 i
N(cid:88)−1p2 1 1,(cid:88)N−1 N(cid:88)−1 q2−q4 1
H(x;M) = i − p p − ( i i − q6)
2 2N i j 4 6 i
i=1 i,j i=1
N−1 N−1 N−1
1 (cid:88) 1 (cid:88) 1 (cid:88)
− (M − q )2+ (M − q )4− (M − q )6
4 i 4 i 6 i
i=1 i=1 i=1
M2
− (21)
4N
(22)
Following Rugh[1], we choose the vector Y as: Y = 1 (p ,...,p ,0,...,0)
2kc 1 N−1
where K is the kinetic part of H(x;M). From Eq.(17) we have:
c
(cid:28) (cid:29)
1 N −3
= (23)
T(E,M) 2k
c E,M
and from Eq.(18) we have:
(cid:28) (cid:29)
∂ m N −3
S(E,M)= 1 − (m −m ) (24)
∂M T(E,M) 3 5 2k
c E,M
Here m = 1 (cid:80)N q3 and m = 1 (cid:80)N q5. We define the effective mean-field
3 N i=1 i 5 N i=1 i
h(E,M) as follows:
∂S(E,M)
h(E,M)=−T(E,M) (25)
∂M
using Eq.(24), we have:
(cid:28) (cid:29)
N −3
h(E,M)=−m+T(E,M) (m −m ) (26)
3 5 2k
c E,M
magneticsusceptibilityandtheheatcapacitycanbecalculatedbydifferentiating
of h(E,M) and T(E,M), see Eqs.(19) and (20).
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Figure1: Entropyasafunctionofthemeanmagnetizationmfordifferentvalues
of the mean energy (cid:15)=0.16, 0.08, 0.04, 0.00 and -0.04 (from top to bottom).
5 Results of the large deviation method for mean-
field φ6 model
The entropy density S((cid:15),m) of the mean-field φ6 model has been shown as a
function of mean energy (cid:15) and mean magnetization m for five different values
of (cid:15) in fig.1. One of them is above the critical value (cid:15) ∼=0.102 and the rest are
c
below. The effective magnetic field h(E,M) has been plotted as a function of
m for different values of (cid:15), in fig.2. It is worth mentioning that the data in fig.1
and fig.2 are obtained using the large deviation method i.e., Eq.(15). In fig.3
the magnetic susceptibility has been plotted as a function of magnetization m
for different values of (cid:15).
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Figure 2: Effective magnetic field as a function of magnetization m for different
values of (cid:15)=0.16, 0.08, 0.04, 0.00 and -0.04 (from top to bottom).
6 Ensemble inequivalence for mean-field φ6 model
Now, we study the inequivalence of microcanonical and canonical ensemble, for
the case of φ6 model. This is in close relation with nonconcavity of the entropy
S((cid:15),m). The effective magnetic field h(E,M) is shown in fig.2. We observe
that h(E,M) can be negative for positive values of m when (cid:15) < (cid:15) . In the
c
canonical ensemble, the magnetization m(β,h) and the magnetic field h have
thesamesign. Thismeansthat,inthecanonicalensemble,themagnetizationof
the mean-field φ6 model is always in the direction of the magnetic field. But in
the microcanonical ensemble the situation is different. h(E,M) and m can have
opposite signs, so the canonical and microcanonical ensembles are inequivalent.
Formoreunderstandingofthethisprobleminmicrocanonicalensemble andits
relationwithnonconcavityoftheentropyS((cid:15),m),wehaveplottedthemagnetic
susceptibility and m derivative of the entropy in figs.(3) and (4) respectively.
SinceT(E,M)isdirectlyproportionaltothekineticenergy,itisalwayspositive
andthismeansthatthesignofh(E,M)isalwaysoppositetothesignof ∂S((cid:15),m).
∂m
Thenh(E,M)isnegativeform>0. Bycomparingfigs.(3)and(4),wefindthat
this happens when the entropy S((cid:15),m) is a concave function of the magnetiza-
tion e.g., the case for which (cid:15)=0.16, then ∂S((cid:15),m) is necessarily negative when
∂m
m > 0(fig.4), which implies that h(E,M) is necessarily positive (for m > 0).
This is in agreement with the results of canonical ensemble. Let us now study
the magnetic susceptibility in the microcanonical and canonical ensemble. In
thecanonicalensemble,themagneticsusceptibilityattheconstanttemperature
8
Figure 3: Magnetic susceptibility as a function of magnetization m calculated
using the large deviation methode for differennt values of (cid:15)=0.16, 0.08, 0.04,
0.00 and -0.04 (from top to bottom).
Figure 4: ∂S((cid:15),m) for diffrent values of (cid:15)=0.16, 0.08, 0.04, 0.00 and -0.04 (from
∂m
top to bottom), culculated using the large divation method.
9
is defined by the following expression:
∂m(β,h)
χT(β,h)= (27)
∂h
It is easy to show that χT(β,h) is always positive. In the microcanonical en-
semble χT is a function of (cid:15) and m, and given by:
(cid:20)∂h((cid:15),m) (cid:21)−1
χT(E,M)= | (28)
∂m T((cid:15),m)
where h((cid:15),m) is given by Eq.(25). Using Eq.(25) and (28) one can show that:
∂2S (cid:20)∂2S ∂2S ∂2S (cid:21)−1
χT((cid:15),m)=−T−1((cid:15),m) −( )2 (29)
∂(cid:15)2 ∂m2 ∂(cid:15)2 ∂m∂(cid:15)
For the mean-field φ6 model, ∂2S is always negative, so if for some values
∂(cid:15)2
of m (when (cid:15) < (cid:15) , see fig.3), ∂2S is positive, then χT will be negative as can
c ∂m2
be checked by Eq.(29). But we already shown that in the canonical ensmble
χT is always positive, this again means that microcanonical and canonical en-
sembles are nonequivalent. It is worth to mention that this is a consequence of
nonconcavity of the entropy S((cid:15),m) .
7 conclusion
The entropy of the mean-field φ6 model is concave as a function of the energy,
but is nonconcave as a function of the energy and magnetization. This leads
to a very important difference between the thermodynamic properties of this
model in the microcanonical and canonical ensembles. We shown that the ef-
fective magnetic field in the microcanonical ensemble can have a sign opposite
to that of the magnetization m, which is in contrast to the case of canonical
ensembleforwhichthemagnetizationmisalwaysinthedirectionoftheapplied
magnetic field. The magnetic susceptibility which in microcanonical ensemble
is a function of the energy and magnetization can be negative but it is always
positiveinthecanonicalensemble. Thesearetwoimportantdifferencesbetween
microcanonical and canonical ensemble which make them nonequivalent. The
mean-field φ6 model like φ4 model displays a first-order phase transition driven
by the magnetic field in the canonical ensemble which is a consequence of the
nonconcavity of the entropy, as a function of magnetization (for certain values
oftheenergy). Inadditiontodisplayafirst-orderphasetrasitiondrivenbythe
magnetic field the mean-field φ6 model displays a second order phase transition
driven by the temperature or the energy.
10